Analytical Determination of the Cogging Torque in Brushless Motors Excited by Permanent Magnets

This paper presents an analytical method how to calculate electromagnetic forces acting in brushless motors excited by permanent magnets. The analytical method is based on the rotating field theory. The flux density, the electric loading - as a substitution of the magnetisation - and the air gap permeance are considered as infinite sums of spatial harmonics. The spatial harmonics of the flux density in the air gap are calculated by use of the conformal mapping of the air gap geometry from the magnetic scalar potential. Each spatial harmonic is characterised by its amplitude, number of pole pairs, frequency and phase angle. The interaction of two different spatial harmonics opens the possibility to recognise the generation of cogging torques systematically and to assess counter measures how to minimise undesirable effects. Thus, time consuming numerical studies by FE method can be avoided, or their number can at least be reduced. The validity of the analytical calculation method was verified by experimental investigations in several test motors. Special attention was paid to the influence of manufacturing tolerances which play the dominant role in the generation of cogging torques in motors with a design that is already optimised in order to achieve small cogging torques. The correlation between the analytical and the numerical results is outstanding, even though the analytical approach reduces the computation time by a factor more then 1000.