Analytic Expressions of Torque and Inductances via Polynomial Approximations of Flux Linkages

Due to slotting effects and non-sinusoidal flux distribution, permanent magnet synchronous motors have relatively large torque ripples. Furthermore, core saturation, along with armature reaction, intensifies the cross coupling between d-axis and q-axis dynamics, and causes changes in the rotor flux linkage. In this paper, it is attempted to develop a closed-form analytical torque equation and inductances, which are known to be uneasy due to the above non-linearities. This approach is based on the flux linkages obtained via finite-element analysis for all current sets in an operation area. First, a flux linkage is obtained for each phase coil as the rotor rotates over one electrical period. At this time, synchronous ac currents are injected to the stator coils. Then, the resulting flux linkages appear as sinusoidal curves with harmonics. They are transformed into a synchronous dq-frame and expanded by Fourier series. Then, the Fourier coefficients, reflecting high-order flux components caused by the permeance change associated with the rotor motion, are dependent on the magnitudes of current. Second, the Fourier coefficients of the same harmonic number are collected over many current samples and interpolated by polynomial functions in dq-axis current. In the interpolation process, the coefficients are obtained through a least square method. Then, an estimate of flux linkage is reconstructed analytically using the resulting interpolations. Analytic expressions for inductances and torque are obtained by dividing or by differentiating the flux linkage estimates. The accuracy is evaluated by comparing the results from finite element method (FEM) and the experimental results.