# Computationally Efficient PM Power Loss Mapping for
## 3 Constant-Torque Operation Region  
According to [15; 16; 17], the PM loss caused by a harmonic current is proportional to the square of the rotational speed. As the phase current decreases, the amplitude/fundamental current of the harmonic current increases and the eddy current loss caused by the armature reaction changes a little, so the effect of the phase current can be ignored. When the amplitude modulation ratio and speed remain constant, the reciprocal relationship between the additional loss and carrier ratio is a quadratic function of one variable. \begin{table}
\begin{tabular}{c c c c} \hline \hline Rated power & 10.39 kW & Outer diameter of the stator & 234 mm \\ \hline Rated speed & 1039 rpm & Maximum speed & 1400 rpm \\ PM thickness & 8 mm & PM material & NdFeB35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main parameters of a permanent magnet synchronous machine (PMSM). Figure 1: Finite element model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Coefficient** & \(\mathbf{a}\) (\(\text{W/A}^{2}\)) & \(\mathbf{b}\) (\(\text{W/A}^{2}\)) & \(\mathbf{c}\) (\(\text{W/A}\)) & \(\mathbf{d}\) (\(\text{W}\)) \\ \hline Value & 0.0216 & 0.0152 & \(-\)3.836 & 275.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The constant coefficient of the PM loss under sinusoidal current input. When the carrier ratio and speed remain unchanged, the additional loss and the amplitude modulation ratio form a quadratic function of one variable. Here, only the amplitude modulation ratio of less than 1 is considered. Based on the above, the fitting formula of the additional loss under a constant-torque condition can be expressed as  
\[ \begin{split} P_{PM-EL}(n,N,M)=f(N)\cdot g(M)\cdot\left(\frac{n}{n _{R}}\right)^{2}\\ =\left(a_{1}\frac{1}{N^{2}}+b_{1}\frac{1}{N}+c_{1}\right)\left(a_ {2}(M-M_{0})^{2}+b_{2}(M-M_{0})+1\right)\left(\frac{n}{n_{R}}\right)^{2}\end{split}  \tag{2}\]  
where \(P_{PM-EL}\) is the additional PM loss; \(n_{R}\) is the reference speed; \(N\) is the carrier frequency, and its value is \(N=f_{c}/f\), where \(f\) is the fundamental frequency and \(f_{c}\) is the carrier frequency; \(M\) is the amplitude modulation ratio; and \(a_{1}\), \(b_{1}\), \(c_{1}\), \(a_{2}\), \(b_{2}\), and \(c_{2}\) are constant coefficients. In Equation (2), there are five constant coefficients that need to be solved. To solve them accurately and quickly, further simplification is needed. When the carrier ratio and speed remain constant, the PM loss decreases as the amplitude modulation ratio increases. Since the study considers the case where the amplitude modulation ratio is less than 1, \(g(M)\) can be understood as a correction coefficient based on the amplitude modulation ratio of the variable. With the increase in the amplitude modulation ratio, the correction coefficient decreases and the relation with the pulse width modulation is a quadratic function of one variable. When the amplitude modulation ratio is set to the reference amplitude modulation ratio, then \(g(M_{0})=1\). \(f(N)\) is understood as the fitting formula for the additional PM loss based on the variable carrier ratio, and it is related to the carrier ratio as a quadratic function of one variable. On the basis of the relation, the correction factor based on the variable amplitude modulation ratio and the speed was added. Through the above simplification, five special working conditions with different carrier ratios and amplitude modulation ratios can be selected, including three carrier ratios and three amplitude modulation ratios, and the time-step finite element analysis of the PM loss can be carried out to calculate the above five constant coefficients. It should be pointed out that the PM loss analyzed here includes sinusoidal PM loss and additional PM loss. To obtain the additional PM loss, the sinusoidal PM loss should be subtracted first, and the additional PM loss can be obtained through the following equation:  
\[P_{PM-EL}=P_{PM-AL}-P_{PM-SL} \tag{3}\]  
where \(P_{PM-AL}\) is the total PM loss.