This paper extends the theory of the airgap magnetic field in permanent-magnet brushless motors. Scalar and vector potential solutions to the field equations are brought together to unify many of the important practical methods already in use. The theory admits a more general representation of the magnetization vector than has been previously assumed, including both the radial and tangential components, and variation with radius. The work is applied in the design of permanent-magnet motors where there is a requirement to minimize noise and torque ripple, and a continuing need for improvements in the accuracy and rigour of design calculations. The airgap flux-density distribution is at the heart of the design process, and it is desirable to study different magnetization patterns, including imperfections in the magnetization, for a wide range of magnet shapes. The paper shows the application of the analytical solutions in comparison with a new finite-element procedure, with test results on a prototype motor, and with simpler, older methods of calculation based on magnetic equivalent circuits. The comparison brings out many interesting points, not merely in relation to the accuracy but also about the speed and practicality of the various methods.
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