## 2 Fundamental Loss Model of PMSMs
### Fundamental Iron Loss  
In the PMSM, the iron loss can be accurately calculated by finite element analysis (FEA) which can estimate both the fundamental iron loss and harmonic iron loss. Unfortunately, FEA is too time-consuming to be practical for efficiency optimization control. This paper presents an accurate and fast approach for iron loss calculation. The Bertotti iron loss formula [18] is the most common evaluation method of iron loss, which calculates the iron loss per volume as  
\[dP_{Fe}=k_{h}B_{m}^{2}f+\frac{\tau^{2}\alpha k_{d}^{2}}{6}B_{m}^{2}f^{2}+k_{e }B_{m}^{1.5}f^{1.5} \tag{9}\]  
where \(k_{h}\) and \(k_{e}\) are the coefficients of the hysteresis loss and excess loss, respectively; \(\sigma\) is the conductivity of the material; and \(k_{d}\) is the thickness of the lamination. These four parameters are material characteristics of the motor cores;\(f\) is the frequency, and \(B_{m}\) is the peak value of the magnetic flux density. To calculate the iron loss in the stator core, it is crucial to determine the accurate average flux density in the stator tooth and stator yoke, which is shown in Figure 2.  
The \(S_{t}\) and \(S_{y}\) are the cross-section area of the stator tooth and stator yoke in Figure 2, respectively. Therefore, the average flux density in the stator tooth and stator yoke in the \(d\)-axis and \(q\)-axis can be obtained as  
\[\left\{ \begin{array}{c}B_{td}=\frac{2n_{t}\Phi_{d}}{\alpha_{i}S_{t}Q}\\ B_{tq}=\frac{2n_{t}\Phi_{q}}{\alpha_{i}S_{t}Q}\\ B_{yd}=\frac{\Phi_{q}}{2S_{y}}\\ B_{yq}=\frac{\Phi_{q}}{2S_{y}}\end{array} \right. \tag{10}\]  
where \(\Phi_{d}\) and \(\Phi_{q}\) are the flux in the \(d\)-axis and \(q\)-axis, \(\alpha_{i}\) is the pole arc factor, and \(Q\) is the number of slots. The \(d\)-axis and \(q\)-axis fluxes can be derived from the flux linkage as  
\[\left\{ \begin{array}{c}\Phi_{d}=\frac{\mathrm{b}\phi_{d}}{N_{1}K_{\mathrm{ d}\mathrm{l}}}\\ \Phi_{q}=\frac{\mathrm{b}\phi_{q}}{N_{1}K_{\mathrm{d}\mathrm{l}}}\end{array} \right. \tag{11}\]  
where \(N_{1}\) is the number of turns per phase, and \(K_{\mathrm{d}\mathrm{p}1}\) is the fundamental winding factor. Therefore, from Equations (9)-(11), the iron loss caused by the \(d\)-axis and \(q\)-axis flux linkages or currents can be calculated as  
\[ \begin{array}{rl}P_{Fe\_f}&=dP_{Fetd,q}V_{t}+dP_{Fetd,q}V_{y}\\ &=\frac{3}{2}k_{hd}\left(\psi_{d}^{2}+\psi_{q}^{2}\right)+\frac{3}{2}k_{ep} \left(\psi_{d}^{2}+\psi_{q}^{2}\right)\\ &=\frac{3}{2}k_{hd}\left[\left(L_{dmi}i_{dm}+\psi_{f}\right)^{2}+\left(L_{qm}i _{qm}\right)^{2}\right]+\frac{3}{2}k_{ep}\left[\left(L_{dm}i_{dm}+\psi_{f} \right)^{1.5}+\left(L_{qm}i_{qm}\right)^{1.5}\right]\end{array}  \tag{12}\]  
where \(P_{Fetd,q}\) is the fundamental iron loss of the stator tooth in the \(d\)-axis and \(q\)-axis, and \(P_{Fetd,q}\) is the fundamental iron loss of the stator yoke in the \(d\)-axis and \(q\)-axis. \(V_{t}\) and \(V_{y}\) are the total volumes of the stator tooth and yoke, respectively,  
\[\left\{ \begin{array}{c}V_{t}=Qh_{t}S_{t}\\ V_{y}=\pi(D_{1}-h_{y})S_{y}\end{array} \right. \tag{13}\]  
where \(h_{t}\) and \(h_{y}\) are the heights of the stator tooth and stator yoke, respectively. \(D_{1}\) is the outer diameter of the stator. Figure 2: Diagram of stator tooth and stator yoke. Further, \(k_{hd}\) is the coefficient of the equivalent iron hysteresis and eddy losses, and \(k_{ep}\) is the coefficient of the equivalent iron excess loss, i.e.,  
\[\left\{ \begin{array}{c}k_{hd}=\left(k_{h}f_{0}+\frac{\pi^{2}\alpha k_{d}^{2}}{ f_{0}^{2}}\right)\left[\frac{\left(2n_{p}\right)^{2}V_{t}}{\left(\alpha_{i}S_{i}Q \right)^{2}}+\frac{V_{y}}{\left(2s_{y}\right)^{2}}\right]\\ k_{ep}=k_{e}B_{m}^{1.5}f_{0}^{1.5}\left[\frac{\left(2n_{p}\right)^{2}V_{t}}{ \left(\alpha_{i}S_{i}Q\right)^{1.5}}+\frac{V_{y}}{\left(2s_{y}\right)^{1.5}} \right]\end{array} \right. \tag{14}\]  
where \(f_{0}\) is the frequency of the stator fundamental current.