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This paper deals with the magnet pole shape design of permanent magnet machines for the minimization of torque ripple based on electromagnetic field theory. On the basis of a magnetic vector potential and a two-dimensional (2-D) polar system, analytical solutions for flux density due to permanent magnet (PM) and current are obtained. In particular, the analytical solutions for mathematical expressions of magnets with different circumferential thicknesses can be solved by introducing improved magnetization modeling techniques, resulting in accurate calculations of electromagnetic torque. The analytical results are validated extensively by nonlinear finite element method (FEM). Test results such as back-emf measurements are also given to confirm the analyses. Finally, on the basis of derived analytical solutions, a reduction of torque ripple can be achieved.