Investigation of Bessel Functions in Analytical Modeling of Rotor Eddy Current Losses in PM Machine Rotor With Copper Shield

In high-speed surface-mounted permanent magnet (SPM) brushless machines, time- and space-harmonic excitations can induce significant eddy currents flowing in the rotor conductive parts. An effective measure to suppress the eddy currents is to employ a copper shield between the magnets and the retaining sleeve. In this article, a Fourier-based analytical model is proposed to evaluate the rotor eddy current losses. The model is set up in the cylindrical coordinate system and is able to compute the eddy current fields and losses in all rotor domains, that is, the sleeve, shield, magnets, and shaft. Although it is generally believed that both ordinary Bessel functions (OBFs) and modified Bessel functions (MBFs) are workable for solving the magnetic vector potential, it is argued in this article that only the MBF-based models are numerically stable when the high-conductivity domain (e.g., the copper shield) exists, while the OBF-based models may result in irrational results. Both the OBF- and MBF-based analytical models for a prototype motor are established and comparatively studied, with verification of the finite-element (FE) method. Moreover, the cause of losing analysis accuracy in the OBF-based models is revealed.