Analytical Calculation Of The Cylindrical Stator Coil In Permanent Magnet Spherical Motor

With the development of robots and other advanced technology,many international and domestic scholars pay more attention to permanent magnet spherical motor(PMSM)which can realize three dimensional movement.Among these researches,permanent magnet(PM)eddy current loss is a significant problem for PMSM research and design,which has influence on motor temperature rise.There are two factors caused PM eddy current loss.One factor is the space harmonic caused by stator coils distribution.The other factor is the time harmonic caused by non-sinusoidal stator coils current.Therefore,it is necessary to solve the magnetic field distribution of stator coils.According to the structure characteristics of the cylindrical stator coils in PMSM,this paper adopted two methods to solve the magnetic field produced by the coil.One method was based on potential function,which analyzed the magnetic field in the air gap and the PM produced by the coil using vector magnetic potential and obtained the elliptic equations for each component of vector magnetic potential.The space harmonic magnetic field produced by coil distribution and the time harmonic magnetic field produced by non-sinusoidal current may cause eddy current distribution and magnetization effect of PM.Ignoring the above factors,the magnetic field in the air gap and the permanent magnet can be regarded as a non-rotational field and can be calculated with scalar magnetic potential.This paper combined the double Fourier series decomposition and spherical harmonic functions and obtained the current density analytical expression indicated by spherical harmonic functions.The analytical expression of the stator coil magnetic field distribution was derived using the boundary of current density.The influence of different structure parameters on radial magnetic flux density was also analyzed.The magnetic field of the PMSM in any space produced by one coil at any position was obtained based on Bio-Savart Law by ingeniously establishing the mapping between the global coordinate system of the PMSM and the local coordinate system of stator coil and using Taylor series expansion method,power series method and numerical calculation method respectively.On this basis,the magnetic field of single-layer many-turn coils at any position in any space,multilayer coils at any position on the central axis,multilayer coils at any position in any space were obtained respectively.Based on the established three-dimensional model of the PMSM,the single PM eddy current density was solved using Faraday's law and the distribution of eddy current density was analyzed,which would lay a theoretic foundation for further PMSM eddy current loss research.