Permanent magnets (PMs) are widely used in magnetic machinery. To better predict the performance of PMs, the magnetic field and force need to be calculated. In general applications, PMs are usually magnetized in a direction parallel to their symmetry axes, and the equivalent magnetic charge method (EMCM) can be used to calculate the corresponding magnetic field and force. However, in some particular cases, such as Halbach arrays, the magnetizing directions or shapes of PMs are irregular and cannot be modeled by the normal EMCM that is limited to the magnets with regular shapes and magnetizing directions. In this paper, an improved EMCM is proposed to calculate PMs with irregular magnetizing directions and shapes. The general rules of magnetic charge distribution are given. The standard forms for calculating magnets with irregular magnetizing directions are obtained through the orthogonal decomposition of the magnetizing direction to simplify the calculation. As an illustration, semi-analytical expressions for magnetic field and force of arc-shaped PMs with irregular magnetizing direction are obtained. Furthermore, the application of this method to calculate the 3-D magnetic field and axial magnetic force of Halbach arrays is discussed and compared with the finite element method (FEM) and experimental measurement. It shows that the improved EMCM has good accuracy and greatly reduces the time cost, which can provide reference for the structure design.
YonnetJP. Passive magnetic bearings with permanent magnets. IEEE Trans Magn1978; 14: 803–805.
2.
YonnetJP. Permanent magnet bearings and couplings. IEEE Trans Magn1981; 17: 1169–1173.
3.
HamlerAGoricanVŠtumbergerB, et al.Passive magnetic bearing. J Magnetism Magn Mater2004; 272–276: 2379–2380.
4.
KarutzPNussbaumerTKolarJW. Magnetically levitated slice motors - an overview. In: 2009 IEEE Energy Conversion Congress and Exposition, San Jose, CA, 20–24 September 2009. IEEE, pp. 1494–1501.
5.
BrennanJR. Applications for magnetic drive rotary positive displacement pumps. World Pumps1999; 1999: 24–28.
6.
TaylorAPIzquierdo ReyesJVelásquez-GarcíaLF. Compact, magnetically actuated, additively manufactured pumps for liquids and gases. J Phys D: Appl Phys2020; 53: 355002.
7.
ZhangHKongFZhuA, et al.Effect of blade outlet angle on radial force of marine magnetic drive pump. Shock and Vibration2020; 2020: 1–18.
8.
SmistrupKStoneHA. A magnetically actuated ball valve applicable for small-scale fluid flows. Phys Fluids2007; 19: 063101.
9.
MorganSCHendricksADSetoML, et al.A magnetically tunable check valve applied to a lab-on-chip nitrite sensor. Sensors2019; 19: 4619.
10.
WoodwardMASittiM. Universal custom complex magnetic spring design methodology. IEEE Trans Magn2018; 54: 1–13.
11.
JanssenJLGPaulidesJJHLomonovaEA, et al.Design study on a magnetic gravity compensator with unequal magnet arrays. Mechatronics2013; 23: 197–203.
12.
WangQYangFYangQ, et al.Study of a permanent magnet spring mechanism and its application to hydraulic valves. Proc Inst Mech Eng C: J Mech Eng Sci2010; 224: 1299–1303.
13.
DongKYuHHuM, et al.Study of axial-flux-type superconducting eddy-current couplings. IEEE Trans Appl Supercond2017; 27: 1–5.
14.
ShinH-JChoiJ-YJangS-M, et al.Design and analysis of axial permanent magnet couplings based on 3D FEM. IEEE Trans Magn2013; 49: 3985–3988.
15.
AcharyaVMBirdJZCalvinM. A flux focusing axial magnetic gear. IEEE Trans Magn2013; 49: 4092–4095.
16.
XuLZhuX. Magnetic planetary gear drive. Proc Inst Mech Eng Part C: J Mech Eng Sci2009; 223: 2167–2181.
17.
ChenWHuQHuY, et al.Magnetic focusing simulator: a 3-D finite-element permanent-magnet focusing system design tool. IEEE Trans Electron Devices2015; 62: 1319–1326.
18.
XuYDunYWangX, et al.Analysis of hybrid magnetic bearing with a permanent magnet in the rotor by FEM. IEEE Trans Magn2006; 42: 1363–1366.
19.
TirMAMirimaniSM. Design of a new structure passive magnetic bearing with radial magnetization using FEM. In: 2014 22nd Iranian Conference on Electrical Engineering (ICEE), Tehran, Iran, 20–22 May 2014. IEEE, pp. 798–802.
20.
AkounGYonnetJP. 3D analytical calculation of the forces exerted between two cuboidal magnets. IEEE Trans Magn1984; 20: 1962–1964.
21.
RoversJMMJansenJWLomonovaEA. Analytical calculation of the force between a rectangular coil and a cuboidal permanent magnet. IEEE Trans Magn2010; 46: 1656–1659.
22.
TanQXinDLiW, et al.Study on transmitting torque and synchronism of magnet couplings. Proc Inst Mech Eng Part C: J Mech Eng Sci1992; 206: 381–384.
23.
LiJTanQZhangY, et al.Study on the calculation of magnetic force based on the equivalent magnetic charge method. Phys Proced2012; 24: 190–197.
24.
RavaudRLemarquandGLemarquandV, et al.Permanent magnet couplings: field and torque three-Dimensional expressions based on the coulombian model. IEEE Trans Magn2009; 45: 1950–1958.
25.
RavaudRLemarquandGLemarquandV, et al.Discussion about the analytical calculation of the magnetic field created by permanent magnets. PIER B2009; 11: 281–297.
26.
RavaudRLemarquandGLemarquandV. Force and stiffness of passive magnetic bearings using permanent magnets. Part 1: axial magnetization. IEEE Trans Magn2009; 45: 2996–3002.
27.
RavaudRLemarquandGLemarquandV. Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: radial magnetization. IEEE Trans Magn2009; 45: 3334–3342.
28.
VučkovićANRaičevićNBIlićSS, et al.Interaction magnetic force calculation of radial passive magnetic bearing using magnetization charges and discretization technique. JAE2013; 43: 311–323.
29.
ChoiH-SLeeS-HParkHII. General formulation of equivalent magnetic charge method for force density distribution on interface of different materials. IEEE Trans Magn2005; 41: 1420–1423.
30.
ChoiH-SHan ParkILeeS-H. Generalized equivalent magnetizing current method for total force calculation of magnetized bodies in contact. IEEE Trans Magn2006; 42: 531–534.
31.
TanQKouZWeiL, et al.A model of magnetic force for thrust bearings of permanent magnets. Proc Inst Mech Eng C: J Mech Eng Sci2005; 219: 979–985.
32.
De MedeirosLHReyneGMeunierG, et al.Distribution of electromagnetic force in permanent magnets. IEEE Trans Magn1998; 34: 3012–3015.
33.
DemenkoALyskawinskiWWojciechowskiRM. Equivalent formulas for global magnetic force calculation from finite element solution. IEEE Trans Magn2012; 48: 195–198.
34.
LubinTMezaniSRezzougA. Simple analytical expressions for the force and torque of axial magnetic couplings. IEEE Trans Energ Convers2012; 27: 536–546.
35.
WangJLinHFangS, et al.A general analytical model of permanent magnet eddy current couplings. IEEE Trans Magn2014; 50: 1–9.
36.
ZhouYLiHMengG, et al.Analytical calculation of magnetic field and cogging torque in surface-mounted permanent magnet machines accounting for any eccentric rotor shape. IEEE Trans Ind Electron2014; 62: 3438–3447.
37.
HalbachK. Design of permanent multipole magnets with oriented rare earth cobalt material. Nucl Instr Methods1980; 169: 1–10.
38.
ZhuZHoweD. Halbach permanent magnet machines and applications: a review. IEE Proc Electr Power Appl2001; 148: 299.
39.
JiangHHuangXHuangX. Calculation and analysis of the 3D magnetic field of Halbach PM array based on equivalent magnetic charge method. In: Proceedings of the 11th International Conference on Electrical Machines and Systems (ICEMS), Wuhan, China, 17‐20 October, 2008, pp. 368–372.
40.
LuoCZhangKZhangW, et al.3D analytical model of permanent magnet and electromagnetic hybrid Halbach array electrodynamic suspension system. J Electr Eng Technol2020; 15: 1713–1721.
41.
SimM-SRoJ-S. Semi-analytical modeling and analysis of Halbach array. Energies2020; 13: 1252.
42.
HuangRFengJ. Magnetic field analysis of permanent magnet array for planar motor based on Equivalent Magnetic Charge method. In: Proceedings of the 10th World Congress on Intelligent Control and Automation, Beijing, China, 6–8 July 2012. IEEE, pp. 3966–3970.
43.
PaulSBobbaDPaudelN, et al.Source field modeling in air using magnetic charge sheets. IEEE Trans Magn2012; 48: 3879–3882.
44.
ZhangXZhangCYuJ, et al.Analytical model of magnetic field of a permanent magnet synchronous motor with a trapezoidal Halbach permanent magnet array. IEEE Trans Magn2019; 55: 1–5.
45.
JinPYuanYLinH, et al.General analytical method for magnetic field analysis of Halbach magnet arrays based on magnetic scalar potential. J Magnetics2013; 18: 95–104.
46.
WangJLinHFangS. Analytical prediction of torque characteristics of eddy current couplings having a quasi-halbach magnet structure. IEEE Trans Magn2016; 52: 1–9.
47.
XiaZZhuZHoweD. Analytical magnetic field analysis of Halbach magnetized permanent-magnet machines. IEEE Trans Magn2004; 40: 1864–1872.
48.
JinPYuanYXuQ, et al.Analysis of axial-flux Halbach permanent-magnet machine. IEEE Trans Magn2015; 51: 1–4.
