End-Effect Magnetic Field Analysis of the Halbach Array Permanent Magnet Spherical Motor

The Halbach array permanent magnet (PM) spherical motor (PMSM) consists of a spherical rotor and a spherical-shell stator. The magnetic field distribution surrounding the rotor PMs is closely related to the spherical structure of the Halbach array PMSM. The Halbach array PM configuration characteristics lead to the end leakage magnetic field in the latitudinal direction of the spherical structure, which is end effect. Establishment of an accurate 3-D magnetic field analytical model containing the end leakage magnetic field of the Halbach array PMSM is the foundation of magnetic field distortion research with its influence on eddy-current loss. The magnetic field distribution of the Halbach array PMs in the air gap is calculated using the scalar magnetic potential. Laplace and Poisson equations of the scalar magnetic potential in the spherical coordinate system are derived. To analyze the end leakage magnetic field of the Halbach array PMSM in the latitudinal direction, concept of mechanical pseudo-cycle in the motor latitudinal direction is proposed, and the cycle of Fourier series is redefined. A completed 3-D analytical expression of the Halbach array PMSM magnetic field distribution is derived by the new Fourier series and spatial integral technology, in which the end leakage magnetic field is contained. The results obtained by the proposed analytical method are compared with the numerical results, i.e., the results obtained by the finite-element method, which verify the effectiveness of the analytical method. Magnetic field distribution containing the end leakage magnetic field in different magnetization methods is compared and analyzed, which shows that the Halbach array magnetization has a larger main air-gap magnetic field and a smaller end leakage magnetic field. The effect of motor structure parameters on the end leakage magnetic field is analyzed, and the effect of end leakage magnetic field on eddy-current loss is preliminarily discussed, which shows that eddy-current loss caused by end effect almost accounts for half of the total eddy-current loss. The research on end effect and the effect of motor parameters on end effect are of great significance. In the end, the leakage magnetic field results obtained by the proposed analytical method are verified by the experimental results.