## 3 Global Loss Model of PMSM
### Fundamental Motor Loss  
The fundamental motor loss of PMSM can be divided into two parts: the fundamental copper loss and fundamental iron loss. The fundamental copper loss is caused by the fundamental stator current through the stator armature winding. The fundamental iron loss consists of stator iron loss and rotor iron loss. Compared with the stator iron loss, the rotor iron loss is so small that it will have little effect on the loss optimization. Hence this paper only considers the stator iron loss and ignores the rotor iron loss. The fundamental mathematical model of PMSM in the \(d\)-axis and \(q\)-axis is shown in Figure 4.  
The voltage equation and current equation of a PMSM can be expressed as:  
\[\left\{ \begin{array}{c}u_{d}=L_{1d}\frac{d_{i}}{dt}+R_{s\_f}i_{d}+L_{md}\frac {d_{i}}{dt}-n_{p}\omega_{r}L_{md}i_{olo}\\ u_{q}=L_{1q}\frac{d_{i}}{dt}+R_{s\_f}i_{d}+L_{md}\frac{d_{i}}{dt}+n_{p}\omega_{ r}\psi_{f}+n_{p}\omega_{r}L_{md}i_{od}\\ u_{od}=L_{md}\frac{d_{i}}{dt}-n_{p}\omega_{r}L_{md}i_{ol}\\ u_{qq}=L_{md}\frac{d_{i}}{dt}+n_{p}\omega_{r}\psi_{f}+n_{p}\omega_{r}L_{md}i_{ od}\\ \end{array} \right. \tag{22}\]  
\[\left\{ \begin{array}{c}i_{d}=i_{cd}+i_{od}\\ i_{q}=i_{cq}+i_{oq}\\ i_{cd}=\frac{1}{R_{c}}u_{od}\\ i_{cq}=\frac{1}{R_{c}}u_{qq}\end{array} \right. \tag{23}\]  
where \(L_{md,q}\) and \(L_{ld,q}\) are the stator's self-inductance and leakage-inductance in the \(d\)-axis and \(q\)-axis, respectively; \(R_{s\_f}\) is defined as the fundamental stator's resistance without consideration of skin effects and proximity effects, \(R_{c\_f}\) is the iron loss resistance for the fundamental iron loss and \(n_{\rm p}\) is the pole pairs of the PMSM; \(\omega_{r}\) is the motor rotation speed; \(u_{d,q}\) and \(i_{d,q}\) are the stator's voltage and the  
Figure 4: Fundamental mathematical model of PMSM. (**a**) Fundamental motor model in the \(d\)-axis; (**b**) fundamental motor model in the \(q\)-axis. stator's current, respectively; \(i_{od}\) and \(i_{qq}\) are the magnetizing currents of the PMSM in the \(d\)-axis and \(q\)-axis; \(i_{cd}\) and \(i_{cq}\) are the exciting current of PMSM in the \(d\)-axis and \(q\)-axis. \(\Psi_{f}\) is the flux linkage of permanent magnet. From the mathematical model of PMSM, the fundamental copper loss can be shown as:  
\[P_{Cu\_f}=\frac{3}{2}R_{s\_f}\Big{(}i_{d}^{2}+i_{q}^{2}\Big{)} \tag{24}\]  
where the transformation coefficient \(\frac{3}{2}\) is caused by the \(d\)-\(q\) axis current calculation with CLARKE and PARK transmission in the principle which remains the motor flux value invariable. The fundamental iron loss can be described as:  
\[P_{Fe\_f}=\frac{3}{2}R_{c}\Big{(}i_{cd}^{2}+i_{cq}^{2}\Big{)} \tag{25}\]  
The stator iron loss consists of the eddy current loss and hysteresis loss. From the Bertotti iron loss formula [24], the eddy current loss and hysteresis loss per volume can be expressed as:  
\[dP_{Fe}=dP_{Fe\_h}+dP_{Fe\_d}=k_{h}B_{mf}^{2}+\frac{\pi^{2}\sigma k_{d}^{2}}{ 6}B_{mf}^{2} \tag{26}\]  
where \(dP_{Fe\_h}\) is the eddy current loss per volume and \(dP_{Fe\_d}\) is the hysteresis loss per volume; \(k_{h}\) is the coefficients of the hysteresis loss; \(\sigma\) is the conductivity of the material; and \(k_{d}\) is the thickness of the lamination. These three parameters are material characteristics of the motor core;\(f\) is the frequency, and \(B_{m}\) is the peak value of the magnetic flux density. Equation (26) shows that there is a linear relationship between hysteresis loss and the frequency of the magnetic field. In addition, the relationship between the eddy current loss and the square of magnetic field frequency is a direct ratio. Therefore, the iron loss resistance can only describe the eddy current loss precisely. To acquire an accurate iron loss, the iron loss resistance \(R_{c}\) is a variable of the rotor speed. From Equations (24) and (26), the fundamental motor loss of PMSM drive system can be shown as:  
\[