## 3 Harmonic Loss Model of the PMSM
### Harmonic Components of SPWM  
The typical working principle of SPWM is shown in Figure 7. The three-phase sinusoidal references displaced in time by 120\({}^{\circ}\), viz:  
\[\left\{ \begin{array}{l}U_{az}^{*}=U_{0}\text{cos}\omega_{0}t=MU_{dc}\text{ cos}\omega_{0}t\\ U_{bz}^{*}=U_{0}\text{cos}\left(\omega_{0}t-\frac{2}{3}\pi\right)=MU_{dc}\text{ cos}\left(\omega_{0}t-\frac{2}{3}\pi\right)\\ U_{cz}^{*}=U_{0}\text{cos}\left(\omega_{0}t+\frac{2}{3}\pi\right)=MU_{dc}\text{ cos}\left(\omega_{0}t+\frac{2}{3}\pi\right)\end{array} \right. \tag{23}\]  
where \(U_{0}\) is the output voltage peak magnitude, \(M\) is the modulation ratio, and the reference waveforms are defined with respect to the DC bus center point z.  
\[M=\frac{U_{0}}{U_{dc}}=\frac{\sqrt{u_{d}^{2}+u_{q}^{2}}}{U_{dc}}=f(i_{d},T_{e},\omega_{r}) \tag{24}\]  
The sinusoidal references of SPWM are the fundamental angular in the double Fourier integral analysis, and the harmonic components of the phase output voltage are obtained from Equation (18) as  
\[\left\{ \begin{array}{l}u_{az}(t) =U_{dc}+U_{dc}\text{Mcos}\omega_{0}t+\frac{4U_{dc}}{\pi}\sum\limits _{m=1}^{\infty}\frac{1}{m}l_{0}(m\frac{\pi}{2}M)\text{sin}(m\frac{\pi}{2})\text {cos}[m(\omega_{c}t+\theta_{c})]\\ +\frac{4U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n=\infty }\left\{\begin{array}{l}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n) \frac{\pi}{2}]\cdot\text{cos}[m(\omega_{c}t+\theta_{c})+n\omega_{0}t]\\ \end{array}\right.\\ n\
eq 0\\ u_{bz}(t) =U_{dc}+U_{dc}\text{Mcos}(\omega_{0}t-\frac{2}{3}\pi)+\frac{4U_{dc}}{\pi} \sum\limits_{m=1}^{\infty}\frac{1}{m}l_{0}(m\frac{\pi}{2}M)\text{sin}(m\frac{ \pi}{2})\text{cos}[m(\omega_{c}t+\theta_{c})]\\ +\frac{4U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n=\infty }\left\{\begin{array}{l}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n) \frac{\pi}{2}]\cdot\text{cos}[m(\omega_{c}t+\theta_{c})+n(\omega_{0}t-\frac{2} {3}\pi)]\end{array}\right\}\\ n\
eq 0\\ u_{cz}(t) =U_{dc}+U_{dc}\text{Mcos}(\omega_{0}t+\frac{2}{3}\pi)+\frac{4U_{dc}}{\pi} \sum\limits_{m=1}^{\infty}\frac{1}{m}l_{0}(m\frac{\pi}{2}M)\text{sin}(m\frac{ \pi}{2})\text{cos}[m(\omega_{c}t+\theta_{c})]\\ +\frac{4U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n=\infty }\left\{\begin{array}{l}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n) \frac{\pi}{2}]\cdot\text{cos}[m(\omega_{c}t+\theta_{c})+n(\omega_{0}t+\frac{2} {3}\pi)]\end{array}\right\}\\ n\
eq 0\\ n_{\
eq 0}\end{array} \right. \tag{25}\]  
Figure 7: Typical working principle of SPWM in the three-phase PMSM power system. where the constant term \(U_{dc}\) is caused by the output voltage definition with respect to the negative DC bus. \(J_{n}(k)\) is the Bessel function of the nth order. From these phase output voltage components, the line-line output voltage components can be presented as  
\[\left\{ \begin{array}{rl}u_{ab}(t)&=u_{az}(t)-u_{bz}(t)=\sqrt{3}U_{dc}M\text{ cos}(\omega_{0}t+\frac{\pi}{6})\\ &+\frac{8U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n= \infty}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n)\frac{\pi}{2}]\text{ sin}(n\frac{\pi}{3})\text{cos}[m\omega_{c}t+n(\omega_{0}t-\frac{\pi}{3})+\frac{\pi}{2}]\\ u_{lk}(t)&=u_{bz}(t)-u_{cz}(t)=\sqrt{3}U_{dc}M\text{cos}(\omega_{0}t-\frac{ \pi}{3})\\ &+\frac{8U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n= \infty}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n)\frac{\pi}{2}]\text{ sin}(n\frac{\pi}{3})\text{cos}[m\omega_{c}t+n(\omega_{0}t-\pi)+\frac{\pi}{2}]\\ u_{ca}(t)&=u_{cz}(t)-u_{az}(t)=\sqrt{3}U_{dc}M\text{cos}(\omega_{0}t+\frac{ \pi}{6})\\ &+\frac{8U_{dc}}{\pi}\sum\limits_{m=1}^{\infty}\sum\limits_{n=-\infty}^{n= \infty}\frac{1}{m}l_{n}(m\frac{\pi}{2}M)\text{sin}[(m+n)\frac{\pi}{2}]\text{ sin}(n\frac{\pi}{3})\text{cos}[m\omega_{c}t+n(\omega_{0}t+\frac{\pi}{3})+\frac{\pi}{2}] \end{array} \right. \tag{26}\]  
The harmonic components of the line-line output voltage are significantly different from the phase output voltage, since a lot of harmonic cancelation occurs between the phase legs.