## 2 Lumped Parameter Model of V-Shaped IPMSM
### PM Flux Linkage  
Figure 1 describes the MEC to be solved for the estimation of PM flux linkage which contributes to magnetic torque. This MEC only considers linear reluctances of air gap and flux barrier of magnet cavity with flux sources from PM and a bridge that is a narrow part in rotor between air gap and magnet cavity. To estimate PM flux linkage, flux density on air gap must be calculated from MEC by finding \(\phi_{g}\) flowing on \(r_{g}\) using superposition principle. The individual elements can be calculated as below.  
\[r_{m}=\frac{l_{c}}{\mu_{0}A_{c}},\ r_{g}=\frac{l_{g}}{\mu_{0}A_{g}} \tag{2}\]  
\[\phi_{b}=B_{sat}A_{b},\ \phi_{m}=B_{r}A_{m} \tag{3}\]  
\[B_{g}=\frac{r_{m}}{r_{m}+r_{g}}\frac{\phi_{m}-\phi_{b}}{A_{g}} \tag{4}\]  
\[B_{1}=\frac{4}{\pi}B_{g}(\sin(\alpha_{0})-\sin(\alpha_{1})) \tag{5}\]  
\[\lambda_{PM}=\frac{\sqrt{2}rlB_{1}N_{a}k_{w}}{p} \tag{6}\]  
Figure 1: Magnetic equivalent circuit (MEC) for d-axis permanent magnet (PM) flux linkage (**a**) d-axis linear MEC for PM flux linkage. (**b**) Geometric definition for reluctances.  
where \(r_{m,g}\): linear reluctances of magnet cavity and airgap, [H\({}^{-1}\)], \(l_{c,g}\): length of magnet cavity and airgap, [m], \(A_{c,g}\): area of magnet cavity and airgap, [m\({}^{2}\)], \(\phi_{b}\): leakage flux passing through bridge, [Wb], \(\phi_{m}\): total remanent flux from PM, [Wb], \(A_{b}\): area of magnetic path on bridge, [m\({}^{2}\)], \(A_{m}\): area of PM contacted on core, [m\({}^{2}\)], \(B_{sat}\): flux density of the saturated bridge, [T], \(B_{m}\): remanent flux density of PM, [T], \(B_{g}\): flux density on airgap, [T], \(B_{1}\): fundamental amplitude of flux density distributed on airgap, [T], \(\alpha_{0,\,1}\): electrical angular positions depicted in Figure 1, [rad], \(\lambda_{PM}\): flux linkage at no-load, [Wb], \(r\): inner radius of stator, [m], \(l\): stack length, [m], \(N_{a}\): serial turns per phase, \(k_{w}\): winding factor. Air gap reluctance, \(r_{g}\), is distributed from \(\alpha_{0}\) to \(\alpha_{1}\). \(\phi_{b}\) is evaluated on the assumption that a bridge is fully saturated at \(B_{sat}\) by PM flux. Flux density on air gap from \(\alpha_{0}\) to \(\alpha_{1}\) is calculated from \(\phi_{g}\) and \(A_{g}\) as in Equation (4), and flux density from \(\alpha_{1}\) to \(\alpha_{2}\) is assumed to be 0 according to the results from FEA. Thus, flux density distributed on air gap at no load will be of rectangular shape. The amplitude of fundamental component of flux density is calculated with Equation (5). Finally, PM flux linkage is obtained via Equation (6).  
Analyzing the circuit and formulations, as the included angle between two wings of magnet cavity becomes narrower, the area of magnet cavity becomes larger. It is expected that its reluctance \(r_{m}\) in Equation (1) be diminished and more PM be inserted for stronger PM flux linkage. In an opposite case, the wider included angle causes higher reluctance, Rm, and weaker PM flux linkage.  
In the conventional LPM of the literature, it was assumed that flux density is distributed up to the middle of bridge [21]. However, FEA results shows better agreement with the span inside the magnet cavity suggested in Figure 2. Figure 2 depicts flux density and flux lines over one pole of single layered magnet cavity in FEA.