Fast calculation of eddy current losses caused by pulse‐width modulation in magnets of surface‐mounted PM machines based on small‐signal time‐harmonic finite element analysis

Abstract

This study presents a novel method for fast calculating the eddy current losses (ECLs) caused by pulse‐width modulation (PWM) harmonic voltages in the permanent magnets (PMs) of surface‐mounted PM synchronous machines (SPMSMs) with distributed windings or concentrated windings. Based on the small‐signal models of the SPMSMs, the functional relationships between the high‐frequency harmonic voltages (HFHVs) in the rotor reference frame and the corresponding ECLs are investigated with the time‐harmonic finite element analysis (THFEA). Instead of using complex analytical PM ECL models, the relationships are directly established with some numerical methods, such as the Gauss quadrature and the piecewise cubic Hermite interpolation method, based on the results from THFEA. With the functional relationships, the total PM ECL caused by all the HFHVs is simply calculated by summing the ECL generated by each harmonic component in the spectra of the PWM voltage in the rotor reference frame. Both two‐ and three‐dimensional (2D and 3D) models of the SPMSMs are calculated. Besides, to further reduce the needed steps of 3D THFEA, an equivalent 2D THFEA method using modified PM conductivities under different‐axis HFHV excitations is proposed. The merits of the proposed method are validated by comparing with the traditional time‐stepping finite element method.

Fast Calculation of Electromagnetic Forces in IPMSMs Under PWM VSI Supply Based on Small-Signal Time-Harmonic Finite Element Method

“…Independent D-or Q-axis HFHV Excitation with THFEA PWM harmonic voltages can be viewed as the combination of a series of high-frequency components whose frequencies are usually near the integer multiples of the carrier frequency. With the FDRTM, the IPMSM model can be linearized and THFEA can be utilized to investigate the relationships between HFHVs and high-frequency flux densities in PMs and SSTs [21]- [23]. It is a natural idea that this method can also be applied to investigate the high-frequency flux densities in the airgap.…”

“…It is a natural idea that this method can also be applied to investigate the high-frequency flux densities in the airgap. According to [21]- [23], the PWM-induced losses in PMs and SSTs can be fast calculated with the THFEA results under independent d-and q-axis HFHV excitations. Hence, in this Sub-section, the same THFEA results are utilized to investigate the relationships between HFHVs and airgap flux density variations without repeating more THFEA.…”

“…The real and imaginary parts of Bsd0 and Bsq0 has been calculated in Section II-B when Vs is 40V and fs is fc+4fo. χsd0 and χsq0 are two constant matrices when neglecting eddy current reaction effects at one fixed working condition, because the flux density is proportional to the voltage amplitude and inversely proportional to the frequency [21]- [23]. Besides, the simple superposition principle can be applied because the model has been linearized with the FDRTM.…”

“…Besides, the superposition principle and other analytical methods [17]- [20] can only be applied under the assumption of neglecting the nonlinearity of SSTs, which is not fit for IPMSMs. To tackle the nonlinearity in IPMSMs, the small-signal time-harmonic FEA (THFEA) based on the frozen differential reluctivity tensor method (FDRTM) has been proposed for fast calculating the PWM-induced permanent magnet (PM) eddy current losses and iron losses [21]- [23]. In this paper, the small-signal THFEA method is further extended for calculating the airgap flux densities and electromagnetic forces caused by HFHVs.…”

“…This paper focuses on presenting a detailed procedure for fast calculating the electromagnetic forces in IPMSMs under PWM VSI supply. Although only the calculation procedure at one working condition is given and the eddy current reaction effect in PMs is neglected, the proposed method is able to consider different working conditions and the eddy current reaction effect, just like the calculation of the PWM-induced losses in [21]- [23], with more THFEA and interpolation methods, which is no longer repeated here. Fig.…”