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Abstract-In this paper, two-dimensional (2-D) analytical magnetic field calculations are used to compute crucial quantities of Brushless permanent-magnet (PM) machines with surface-inset PMs. The analytical magnetic field distribution is based on the subdomain technique in which the slotting and tooth-tip effects have been considered. It includes both different magnetization patterns (e.g., radial, parallel and Halbach) and different spatial distribution of winding (e.g., concentrated and distributed with one or more layers). The saturation effect of the magnetic circuit is neglected. In this investigation, the phase and line induced back electromotive force (EMF) waveforms, electromagnetic/cogging/reluctance torques, self-/mutual-inductances and unbalanced magnetic forces (UMFs) have been analytically calculated. The analytical expressions can be used for Brushless machines having surface-inset PMs with any radius-independent magnetization pattern. In this study, two slotted Brushless PM machines with surface-inset PMs have been selected to evaluate the efficacy of the analytical expressions by comparing the results with those obtained by the finite-element method (FEM). One of the case studies is a 9-slots/8-poles Brushless DC (BLDC) machine with radially magnetized PMs, non-overlapping all teeth wound winding and six-step rectangular armature current waveforms. The other is a 15-slots/4-poles Brushless AC (BLAC) machine with Halbach magnetization PMs, double-layer overlapping winding and sinusoidal armature current waveforms.

This paper presents a two-dimensional analytical model of outer rotor permanent magnet machines equipped with surface inset permanent magnets. To obtain the analytical model, the whole model is divided into the sub-domains, according to the magnetic properties and geometries. Maxwell equations in each sub-domain are expressed and analytically solved. By using the boundary/interface conditions between adjacent sub-regions, integral coefficients in the general solutions are obtained. At the end, the analytically calculated results of the air-gap magnetic flux density, electromagnetic torque, unbalanced magnetic force (UMF), back-electromotive force (EMF) and inductances are verified by comparing them with those obtained from finite element method (FEM). One of the merits of this method in comparison with the numerical model is the capability of rapid calculation with the highest precision, which made it suitable for optimization problems.