# Computationally Efficient PM Power Loss Mapping for
## 3 Constant-Torque Operation Region
First, the parameters \(a_{1}\), \(b_{1}\), and \(c_{1}\) in \(f(N)\) are determined. Since the amplitude modulation ratio is set as the reference value and \(g(M)=1\), the fitting formula can be simplified as shown in Equation (4). \[ \begin{split}\left[\begin{array}{ccc}\frac{1}{N_{1}}^{2}& \frac{1}{N_{1}}&1\\ \frac{1}{N_{2}}^{2}&\frac{1}{N_{2}}&1\\ \frac{1}{N_{3}}^{2}&\frac{1}{N_{3}}&1\end{array}\right]\left[\begin{array}{ c}a_{1}\\ b_{1}\\ c_{1}\end{array}\right]=\left[\begin{array}{c}P_{PM-EL}(n_{R},N_{1},M_{1})\\ P_{PM-EL}(n_{R},N_{2},M_{2})\\ P_{PM-EL}(n_{R},N_{3},M_{3})\end{array}\right]\end{split}  \tag{4}\]  
where the amplitude modulation ratio \(M_{1}=M_{2}=M_{3}\), \(g(M_{3})=1\). Both sides are multiplied by the inverse of the coefficient matrix, which is  
\[\left[ \begin{array}{c}a_{1}\\ b_{1}\\ c_{1}\end{array} \right]=\left[ \begin{array}{ccc}\frac{1}{N_{1}}^{2}&\frac{1 }{N_{1}}&1\\ \frac{1}{N_{2}}^{2}&\frac{1}{N_{2}}&1\\ \frac{1}{N_{3}}^{2}&\frac{1}{N_{3}}&1\end{array} \right]^{-1}\left[ \begin{array} []{c}P_{PM-EL}(n_{R},N_{1},M_{1})\\ P_{PM-EL}(n_{R},N_{2},M_{2})\\ P_{PM-EL}(n_{R},N_{3},M_{3})\end{array} \right] \tag{5}\]Second, the parameters \(a_{2}\) and \(b_{2}\) in \(f(N)\) are determined. \[\left[ \begin{array}{cc}\left(M_{4}-M_{0}\right)^{2}&\left(M_{4}-M_{0}\right)\\ \left(M_{5}-M_{0}\right)^{2}&\left(M_{5}-M_{0}\right)\end{array} \right]\left[ \begin{array}{c}a_{2}\\ b_{2}\end{array} \right]=\left[ \begin{array}{c}P_{PM-EL}(n_{R},N_{4},M_{4})\\ P_{PM-EL}(n_{R},N_{5},M_{5})\end{array} \right]/f(N_{3}) \tag{6}\]  
where the carrier ratio \(N_{3}=N_{4}=N_{5}\). On further simplification  
\[\left[ \begin{array}{c}a_{2}\\ b_{2}\end{array} \right]=\left[ \begin{array}{cc}\left(M_{4}-M_{0}\right)^{2}& \left(M_{4}-M_{0}\right)\\ \left(M_{5}-M_{0}\right)^{2}&\left(M_{5}-M_{0}\right)\end{array} \right]^{-1} \left[ \begin{array}{c}P_{PM-EL}(n_{R},N_{4},M_{4})\\ P_{PM-EL}(n_{R},N_{5},M_{5})\end{array} \right]/f(N_{3}) \tag{7}\]  
The reference speed of the prototype is selected as 600 rpm. The prototype works at the rated load, that is, the cross-axis current \(i_{qw}\) is 43.1 A and the d-axis current is 0. The coordination of the motor control parameters, carrier ratio, and amplitude modulation ratio is shown in Table 3. The reference speed is lower than the rated speed because the fundamental frequency corresponding to the speed is 80 Hz, which is relatively low. A large finite element step can be set so that the PM loss corresponding to different carrier ratios can be accurately and quickly calculated to improve the calculation efficiency and accuracy. Through finite element analysis and the calculation method of the above fitting formula, the values of the five constant coefficients in the fitting formula are obtained, as shown in Table 4.  
Figure 2 shows the comparison of additional PM losses under different carrier ratios of the finite element calculation and the fitting calculation. It can be seen that the calculation accuracy of the fitting calculation is high, and it is in good agreement with the finite element results. Under the condition of higher motor speed and higher carrier ratio, the fitting formula will have a little error. The reasons are as follows: First, the error of calculating the sinusoidal PM loss is inherited, which comes from the interaction of core saturation, inductive eddy current reaction, armature reaction of the alternating axis, and slot effect. Second, in the finite element calculation of high speed and a high carrier ratio compared to the error of the PM loss, due to the high-speed case, the fundamental frequency of the phase current is larger. In the case of high reflection, the stator current harmonic frequency is higher and the corresponding frequency is also higher, so a small step needs to be set up for accurate calculation in order to eliminate the meter and higher harmonic current caused by eddy current loss brought by the error.