Fig. Distribution model of axial magnetic bearing air gap permeance G 6 ~ G 8 G G G ' 7-8 G 3 G 4 G ' 3 G ' 4 G 5 G 6 G 5 G

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1 Electri c Machines and Control Vol. 0 No. 0 Oct DOI /j. emc TM 53 A X The oretical and experimental research on axial and radial magnetic force of axial magnetic bearing ZHANG Yun-peng GAO Shu-ning LIU Shu-qin LI Hong-wei XUE Bo-wen School of Electrical Engineering Shandong University Jinan 5006 China Abstract In order to simplify magnetic levitation system supported by axial magnetic bearings axial and radial magnetic force of axial magnetic bearing is studied in this paper. Based on spatial distribution of magnetic flux in the gap equivalent circuit model was created and the mathematic formula of axial and radial magnetic force in axial magnetic bearing was deduced by virtual work method. Magnetic flux density and field distribution in the gap of axial magnetic bearing were obtained by finite element method. On three-dimensional force measuring experimental platform axial and radial magnetic force of axial magnetic bearing was measured. The change law of magnetic force in terms of exciting current i axial gap g and radial displacement Δr was analyzed and physical mechanism was discussed based on theoretical and simulation results. These results provide reference for research of magnetic suspension system in multi-degree-of-freedom supported by axial magnetic bearings. Keywords magnetic bearing magnetic equivalent circuit axial magnetic force radial magnetic force finiteelement method BS0DX03 05JC

2 Fig. Distribution model of axial magnetic bearing air gap permeance G 6 ~ G 8 G G G ' 7-8 G 3 G 4 G ' 3 G ' 4 G 5 G 6 G 5 G ' 6 ' G 7 G 8 G 7 G ' 8 '

3 0 57 Fig. Equivalent magnetic circuit of axial magnetic bearing. i g 9 Δr F z g Δr i F r g Δr i 3G ' 4 G i g Δr = G G 3G 4 G' = G 3 G 4 G ' 3 G ' 4 μ 0 R πr - 4Δr 8Δr g Δr g 5G ' 6 G o g Δr = G G ' G 5G 6 G' G 5 G 6 G ' 5 G ' 6 G 7 G 8 G' 7G ' 8 = G 7 G 8 G ' 7 G ' 8 F z = W μ 0 R 3 R π R 3 - R - 4Δr g } g F r = W 7 8Δr Δr Δr g ~ G i G o R F z g Δr i F r g Δr i R R 3 g Δr F z g Δr i = W g = - G i F mi g - G o F mo g = μ 0 N i f i R πr - 4Δr G a g Δr = g = G i G o μ 0 R πr - 4Δr 8Δr g Δr g πt - 4Δr μ 0 R R 3 g 8Δr Δr g 3 H m = Ni N i Φ g = H m G a = Ni μ 0 R πr - 4Δr 8Δr g Δr g πt - 4Δr μ 0 R R 3 g 8Δr Δr g 4. F z i g F z i /g W m = F mg / W g = W g W g = F mig in / F mog out / 6 F mi = f i NI F mo = f o NI f i f o 9 6Δr Δr g μ 0 N i f o R 3 R πt - 4Δr g 6Δr Δr g 8

4 58 0 F r g Δr i = - G i F mi e - G o F mo e - μ 0 N i f i R 4 g - 6g Δr g = f o R 3 R 9 3 B z B r r = 0 Δr = 0 B z B r r = 0 mm F z F r Δr 4 a B z 3 4 b i g Δr B r Table Parameters of axial magnetic bearing R /mm 8 R /mm 6 R 3 /mm 0 /mm 0 /mm 0 /mm 45 /mm 5 / 00 g /mm 0 - Δr /mm 0-3 Fig. 3 3 Fig. 4 Finite element simulation model of axial magnetic bearing 4 / / 4 a 4 b ATI Mini40 4 Spatial distribution of magnetic flux density 5 a 5 b

5 0 59 g Δr F z! B A 0 i μ 0 F x F y F z F z B A = 槡 F x F y F r μ 0 8 B z i i 0 Fig. 6 6 F z i Axial force F z dependence on current i 5 Fig. 5 Magnetic force measurement system for axial magnetic bearing 4 F z g Δr i F r g Δr i i g Δr i g Δr Fig. 7 Radial force F r dependence on current i 4. F z F r i 6 F z i g = 0. 5 mm7 F r i F z = F miμ 0 πr F moμ 0 π R 3 - R g F i miμ 0 πr F moμ 0 π R 3 - R F r i 4. F z g 8

6 60 0 F z! /g 4. 3 F r g i =. 4 A Δr = mm F r g 9 g! /g 9 F r! /g F r Fig. 8 8 Magnetic flux density B dependence on current i 0 Fig. 0 Axial force dependence on axial air gap with radial displacement 9 Fig. 9 Axial force dependence on axial air gap without radial displacement 8 /g / Δr g Δr = mm 0 8 F z! /g B z /g Fig. Magnetic flux density B z dependence on /g F z! /g 9 i =. 4 A Δr = mm B z /g 3 B z /gb z! /g F z Δr g F z g

7 0 6 8 F z g Δr i μ 0N i f R f o R R i 3 N /m 8 B z 0 πt - 4Δr g A F z 0 Δr < 0. 5 mm F z = Δr 4 Fig. 4 F z r Axial force F z dependence on radial displacement Δr Fig. F r g Radial force factor F r dependence on axial air-gap g 4. 5 F r Δr 9 i i F r Δr 5 9 F r Δr! = - F miμ 0 R F moμ 0 R 3 R 4 g 6 3 g F r = Fig. 3 Normalized axial forceand radial force Δr Δr = 0 dependence on axial air-gapg N /m Δr K r 4 a

8 6 0 5 Fig. 5 6 Fig. 6 5 F r Δr Radial force F r dependence on radial displacement factor Δr r Radial stiffness dependence on radial displacement Δr NUSSBAUMER T GRUBER W et al. Novel mag- of the CSEE KARUTZ P netically levitated two-level motor J on Mechatronics ASME Transactions on Mechatronics SCHWEITER G BLEULER H TRAXLER A. Active magnetic bearings-basics properties and applications of active magnetic bearings M. ETH Switzerland Hochschlverlag AG CHIBA A FUKAO T ICHIKAWA O et al. Magnetic bearings and bearingless drives M. Tokyo Newnes G E. M J ZHAO Xusheng DENG Zhiquan WANG Xiaolin et al. Research status and development of permanent magnet biased magnetic bearings J. Transactions of China Electrotechnical Society J ZHANG Jian-cheng HUANG Li-pei CHEN Zhi-ye. Research on flywheel energy storage system and its controlling technique J. Proceedings of the CSEE J WANG Xi FANG Jiancheng FAN Yahong et al. Axial force tilting permanent-magnet-biased magnetic bearing with five degrees of freedom and magnetic field decoupling design J. Proceedings. IEEE / ASME Transactions 8 CHEN Shyh-Leh WANG Cheng-Chi. Robust control of a voltagecontrolled three-pole active magnetic bearing system J. IEEE / J ZHU Huangqiu JU Jintao. Development of three-pole magnetic bearings and key technologies J. Proceedings of the CSEE 04

9 JIRO K TADAHIKO S LICHUAN L et al. Miniaturization of a one-axis-controlled magnetic bearing J. Precision Engineering JUNICHI A YUKI H TAKAAKI O et al. A Novel Concept of a Single-Drive Bearingless Motor C / /0 IEEE International E- lectric Machines & Drives Conference IEMDC Canada 0. REN Zhaohui JAHANMIR S HESHMAT H et al. Design a- nalysis and performance assessment of hybrid magnetic bearings for a rotary centrifugal blood pump J. ASAIO Journal RAVAUD R LEMARQUAND G LEMARQUAND V. Force and stiffness of passive magnetic bearings using permanent magnets. part radial magnetization J. IEEE Transactions on Magnetics ZHANG Yunpeng XUE Bowen LIU Shuqin et al. Calculation J of electromagnetic force of axial hybrid magnetic bearing based on TIAN Lulin LI Yan WANG Shanshi et al. Research on magnetism engineering analytical calculation method for bi-barrel fringe effect of magnetic flux J. Electric Machines and Control 59. shaped radial permanent magnetic bearings J. Proceedings of the CSEE J TIAN Lulin LI Yan TIAN Qi et al. Research on the axial magnetic force of axially magnetized bi-annular-shaped permanent magnetic bearings J. Proceedings of the CSEE J ZHANG Yunpeng LIU Shuqin LI Hongwei et al. Calculation of radial electromagnetic force of axial hybrid magnetic bearing based on magnetic circuit analysis J. Transactions of China Electrotechnical Society J M

34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1

34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1 22 2 2018 2 Electri c Machines and Control Vol. 22 No. 2 Feb. 2018 1 2 3 3 1. 214082 2. 214082 3. 150001 DOI 10. 15938 /j. emc. 2018. 02. 005 TM 301. 4 A 1007-449X 2018 02-0033- 08 Research of permanent