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Abstract-This paper extends the theory of the air-gap magnetic field in permanent-magnet (PM) 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 PM motors where there is a requirement to minimize noise and torque ripple, and maximize efficiency, and a continuing need for improvements in the accuracy and rigor of design calculations. The air-gap 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. This 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 in relation to the accuracy and the speed and practicality of the various methods.Index Terms-Magnetic analysis, permanent-magnet brushless machines.

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