## 3 Harmonic Loss Model of the PMSM
### Harmonic Iron Loss of PMSM  
As the PMSM is droved by the three-phase PWM inverter, the PWM output voltage of the phase leg in the three-phase inverter will generate the harmonic current in the stator windings of the PMSM, which will cause the harmonic iron loss. The harmonic voltage is calculated with respect to the negative DC bus; hence, the harmonic voltage with respect to the load neutral point can be obtained as  
\[\dot{u}_{an_{mn}}=\dot{u}_{az_{mn}}-\frac{\left(\dot{u}_{az_{mn}}+\dot{u}_{bz_ {mn}}+\dot{u}_{cz_{mn}}\right)}{3} \tag{30}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{\omega_{0}t=\theta_{0}}\) & **A Phase Leg** & **B Phase Leg** & **C Phase Leg** \\ \hline \(0\leq\theta_{0}<\frac{\pi}{3}\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{1}{6}\pi)\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{5}{6}\pi)\) & \(\frac{3}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{2}{3}\pi)\) \\ \hline \(\frac{\pi}{3}\leq\theta_{0}<\frac{2\pi}{3}\) & \(\frac{3}{2}MU_{dc}\text{cos}(\theta_{0}\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{sin}\theta_{0}\) & \(-\frac{\sqrt{3}}{2}MU_{dc}\text{sin}\theta_{0}\) \\ \hline \(\frac{2\pi}{3}\leq\theta_{0}<\pi\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{1}{6}\pi)\) & \(\frac{3}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{2}{3}\pi)\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{5}{6}\pi)\) \\ \hline \(\pi\leq\theta_{0}<\frac{4\pi}{3}\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{1}{6}\pi)\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{5}{6}\pi)\) & \(\frac{3}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{2}{3}\pi)\) \\ \hline \(\frac{4\pi}{3}\leq\theta_{0}<\frac{5\pi}{3}\) & \(\frac{3}{2}MU_{dc}\text{cos}\theta_{0}\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{sin}\theta_{0}\) & \(-\frac{\sqrt{3}}{2}MU_{dc}\text{sin}\theta_{0}\) \\ \hline \(\frac{5\pi}{3}\leq\theta_{0}<2\pi\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{1}{6}\pi)\) & \(\frac{3}{2}MU_{dc}\text{cos}(\theta_{0}-\frac{2}{3}\pi)\) & \(\frac{\sqrt{3}}{2}MU_{dc}\text{cos}(\theta_{0}+\frac{5}{6}\pi)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Phase reference voltage for SVPWM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{i}\) & \(\boldsymbol{y_{1}(\boldsymbol{i})}\) & \(\boldsymbol{y_{2}(\boldsymbol{i})}\) & \(\boldsymbol{x_{1}(\boldsymbol{i})}\) & \(\boldsymbol{x_{2}(\boldsymbol{i})}\) \\ \hline \(1\) & \(\frac{2}{3}\pi\) & \(\pi\) & \(-\frac{\pi}{2}[1+\frac{\sqrt{3}}{2}\text{Mcos}(y+\frac{\pi}{6})]\) & \where \(\dot{u}_{a_{2 nm}}\), \(\dot{u}_{b_{2 nm}}\) and \(\dot{u}_{b_{2 nm}}\) are the harmonic voltages with respect to the negative DC bus in the \(n\)th fundamental signal and the \(m\)th carrier signal. In addition, \(\dot{u}_{an_{nm}}\) is the harmonic voltage with respect to the load neutral point in the \(n\)th fundamental signal and the \(m\)th carrier signal. Based on the voltage equation of the PMSM, the stator harmonic current can be obtained as  
\[I_{s_{mn}}=\left|\frac{\dot{u}_{an_{mn}}}{R_{s}+j\omega_{mn}L_{s}}\right| \tag{31}\]  
where \(I_{s_{mn}}\) is the amplitude of the harmonic current in the \(n\)th fundamental signal and \(m\)th carrier signal. Similar to the calculation of the fundamental iron loss, the harmonic iron loss can be derived from the Bertotti iron loss formula as  
\[P_{Fe\_h}=3\left(\sum_{m=0}^{\infty}\sum_{ \begin{array}{c}n=-\infty\\ n\
eq 0\end{array} }^{\infty}k_{hd\_mm}\left(L_{s}I_{s_{mn}}\right)^{2}+k_{ep\_ mm}\left(L_{s}I_{s_{mn}}\right)^{1.5}\right) \tag{32}\]  
where \(L_{s}\) is the phase self-inductance of the PMSM; \(k_{hd\_mm}\) is the coefficient of the equivalent iron hysteresis and eddy losses in the \(n\)th fundamental signal and \(m\)th carrier signal; and \(k_{ep\_mn}\) is the coefficient of equivalent iron excess loss in the \(n\)th fundamental signal and \(m\)th carrier signal, which are shown as  
\[\left\{