Analytical Modeling for the Flux Reversal Permanent Magnet Machine with Halbach Array Magnets in Rotor Slot

Abstract

I. INTRODUCTIONIn recent years, due to the merits of easy thermal management, alleviated torque ripple and robust rotor, flux reversal permanent magnet (FRPM) machines have attracted more and more attention [1]. However, since the large leakage flux leads to the limitation of its output, a series of novel topologies are proposed, such as the consequent-pole FRPM machine [2], the FRPM machine with asymmetric-stator pole configuration [1] and the FRPM machine with Halbach array magnets in rotor slot (HAM-FRPM) [3], etc. This study presents an analytical sub-domain model for the characteristic evaluation of the HAM-FRPM machine. By dividing the HAM-FRPM machine into seven sub-domains and solving the Maxwell equations in polar coordinate for each region according to the boundary conditions, the vector magnetic potential can be obtained. Subsequently, the radial and tangential flux density can be calculated. Finally, the analytical results are compared with the finite element method (FEM) to verify the accuracy.II. ANALYTICAL SUB-DOMAIN MODELOne 12-stator slot/17-rotor tooth HAM-FRPM machine with double-layer winding is investigated in this paper to verify the proposed method. As shown in Fig.1, the permanent magnet (PM) arrangement in the stator of the HAM-FRPM machine is identical to that of the consequent-pole FRPM machine. In addition, the PMs in each rotor slot are the same, which contain a radially magnetized middle PM and two tangentially magnetized side PMs.In this analytical sub-domain modeling, some essential assumptions have been made in advance: (1) The permeability of iron is regarded as infinite. Also, the relative permeability of PMs and coils is assumed to be 1; (2) The end effect and the magnetic field component on the axial direction are neglected; (3) The geometrical structure side of the HAM-FRPM machine is parallel to either r or θ direction in polar coordinate. As shown in Fig. 2, the HAM-FRPM machine is divided into several regions: stator slot (Region i), stator slot opening (Region l and Region u), PM (Region s and Region g), air gap (Region I) and rotor slot (Region j). Maxwell equation is satisfied in each region as equation (1), which can be expressed as a Poisson equation or a Laplace equation and further solved by the separation of variables method. According to the law of magnetic flux continuity and Neumann condition, the unknown coefficients can de derived. Finally, the radial flux density Br and tangential flux density Bθ can be obtained by (2).where J and M are the current density vector and magnetization vector of magnets, respectively. μ0 is the magnetic permeability of vacuum.III. RESULTS AND DISCUSSIONThe air gap flux density comparisons between the analytical prediction and FEM results are exhibited in Fig.3-4, which present a good agreement between the analytical results and FEM solution for both radial and tangential air gap flux density. Hence, the proposed analytical model is verified to be used to predict the performances of the HAM-FRPM machines. Fig.5-6 show the back-EMF and torque waveforms, which validate the correctness of analytical model. More detailed derivation and analysis will be shown in the full paper. **