## 3 Global Loss Model of PMSM
### Harmonic Motor Loss  
The harmonic motor loss model of a PMSM is shown in Figure 5.  
The harmonic voltage equation and harmonic current equation can be described as:  
\[\left\{ \begin{array}{l}u_{az\_mn}=u_{oa\_mn}+i_{a\_mn}.R_{s\_mn}\\ u_{bz\_mn}=u_{ob\_mn}+i_{b\_mn}.R_{s\_mn}\\ u_{cz\_mn}=u_{oc\_mm}+i_{c\_mn}.R_{s\_mn}\\ u_{oz\_mn}=i_{oa\_mn}.j\omega_{mn}L_{ms}\\ u_{obz\_mn}=i_{ob\_mn}.j\omega_{mn}L_{ms}\\ u_{ocz\_mn}=i_{oc\_mn}.j\omega_{mn}L_{ms}\\ u_{nz\_mn}=\frac{1}{3}(u_{az\_mn}+u_{bz\_mn}+u_{cz\_mn})\\ \end{array} \right. \tag{31}\]  
\[\left\{ \begin{array}{l}i_{a\_mn}=i_{oa\_mn}+i_{ca\_mn}\\ i_{b\_mn}=i_{ob\_mn}+i_{cb\_mn}\\ i_{c\_mn}=i_{oc\_mn}+i_{cc\_mn}\\ i_{ca\_mn}=\frac{1}{k_{c\_mn}}u_{oz\_mn}\\ i_{cb\_mn}=\frac{1}{k_{c\_mn}}u_{obz\_mn}\\ i_{cc\_mn}=\frac{1}{k_{c\_mn}}u_{ocz\_mn}\\ \end{array} \right. \tag{32}\]  
where \(u_{az\_mn}\), \(u_{bz\_mn}\) and \(u_{cz\_mn}\) are the A-phase, B-phase and C-phase output voltages of the inverter in the \(n\)-th fundamental signal and the \(m\)-th carrier signal, respectively. \(u_{oz\_mm}\), \(u_{obz\_mn}\) and \(u_{ocz\_mn}\) are the three-phase magnetizing voltages of PMSM in the \(n\)-th fundamental signal and the \(m\)-th carrier signal. \(i_{az\_mn}\)\(i_{az\_mn}\) and \(i_{az\_mn}\) are the three-phase motor currents in the \(n\)-th fundamental signal and the \(n\)-th fundamental signal and the \(m\)-th carrier signal, respectively. \(R_{c\_mn}\) is the harmonic iron loss resistance of the \(n\)-th fundamental signal and the \(m\)-th carrier signal, and the  
Figure 5: Harmonic motor loss model of a PMSM. \(\omega_{mn}\) is the angular frequency of the \(n\)-th fundamental signal and the \(m\)-th carrier signal, which can be expressed as:  
\[\omega_{mn}=n\cdot n_{p}\omega_{r}+2\pi m\cdot f_{sw} \tag{33}\]  
\(R_{s\_mm}\) is the stator's resistance in the \(n\)-th fundamental signal and the \(m\)-th carrier signal which has been considered about skin effect and proximity effect:  
\[R_{s\_mn}=\alpha\cdot R_{s\_f} \tag{34}\]  
where \(\alpha\) is the coefficient of skin effect. \(\alpha\) is affected by the frequency of harmonic current, which can be described as:  
\[\alpha=\frac{d}{2\sqrt{2}\delta}\frac{\mathrm{ber}_{0}\left(\frac{d}{\sqrt{2 }\delta}\right)\cdot\mathrm{bei}_{0}^{\prime}\left(\frac{d}{\sqrt{2}\delta} \right)-\mathrm{ber}_{0}^{\prime}\left(\frac{d}{\sqrt{2}\delta}\right)\cdot \mathrm{bei}\left(\frac{d}{\sqrt{2}\delta}\right)}{\left[\mathrm{ber}_{0}^{ \prime}\left(\frac{d}{\sqrt{2}\delta}\right)\right]^{2}+\left[\mathrm{bei}_{ 0}^{\prime}\left(\frac{d}{\sqrt{2}\delta}\right)\right]^{2}} \tag{35}\]  
where \(d\) is the copper wire diameter of stator windings, and \(\delta\) is the depth of penetration. The depth of penetration is affected by the current frequency, which is shown in Figure 6.  
The terms \(\mathrm{ber}_{0}\) and \(\mathrm{bei}_{0}\) are the zero order Kelvin function, which can be expressed as:  
\[\left\{ \begin{array}{l}\mathrm{ber}_{0}(u)-j\mathrm{bei}_{0}(u)=J_{0}\left(ue ^{-j\frac{3\pi}{4}}\right)\\ \mathrm{ber}_{0}{}^{\prime}(u)-j\mathrm{bei}_{0}{}^{\prime}(u)=-e^{-j\frac{3 \pi}{4}}\cdot J_{1}\left(ue^{-j\frac{3\pi}{4}}\right)\end{array} \right. \tag{36}\]  
From Equations (31) and (32), the amplitude of three-phase currents of PMSM can be described as:  
\[I_{s\_mn}=\left|\frac{(2\cdot u_{az\_mn}-u_{bz\_mn}-u_{cz\_mn})\cdot j\omega_ {mn}L_{ms}\cdot R_{c}}{3[R_{c}\cdot R_{s\_mn}+j\omega_{mn}L_{ms}\cdot(R_{c}+ R_{s\_mn})]}\right| \tag{37}\]  
where \(I_{s\_mn}\) is the amplitude of three-phase magnetizing currents, and \(I_{s\_ms}\) is the amplitude of three-phase currents. Figure 6: Penetration depth of copper wire with current frequency. Therefore, the harmonic copper loss can be expressed as:  
\[P_{Cu\_h}=3\left[\sum_{n=2}^{\infty}I_{s\_0n}^{2}\cdot R_{s\_mn}+\sum_{m=1}^{ \infty}\sum_{ \begin{subarray}{c}n=-\infty\\ n\
eq 0\end{subarray}