Analytical solution of air-gap field in motors with parallel magnetized permanent magnets

Analysis of airgap magnetic field is the basis of design and performance analysis of permanent magnet (PM) motors. Based on 2D models in polar coordinates, the analytical formulas of the open-circuit airgap flux density in PM motors equipped with parallel magnetised magnets are derived by solving the Laplace's equation or Poisson's equation in the regions of airgap, PM and shaft core. And based on the rotor with surface-mounted magnets, analytical formulas of rotors with cylindrical or ring-type PMs generally for high-speed PM motors are also obtained by mathematical methods. Considering the slotting effects, the calculation results of magnetic field analysis are modified by 2D relative permeance function. By comparing with finite element analysis, it is proved that this analytical method is correct and reliable because of their good agreement in waveform and magnitude. It provides a powerful tool for optimal design and performance analysis of PM motors.