## 2 Design Process Using Lumped-Parameter Method (LPM)
### Q-Axis PM Flux Linkage  
In LPM, performance is analyzed through the magnetic flux generated by the shape of the stators and rotors on the basis of the magnetic circuit. Figure 2 shows the magnetic circuit of the PMa-SynRM, which can be regarded as a linear MEC consisting of three magnetic flux sources and four magnetic resistances in a rotor shape with a two-layered magnetic flux barrier. The first magnetic flux source was generated from the permanent magnet, and the total magnetic flux could be calculated by determining the residual magnetic flux density of the permanent magnet from the area in which the permanent magnet and the iron core come into contact. The second magnetic flux source was located on a bridge at the end of the second-layer magnetic flux barrier that connected the inner core of the rotor and the outer core in front of the permanent magnets. The third magnetic flux source was located at the bridge at the end of the first-layer flux barrier with opposite polarity. The magnitude of the magnetic flux source was calculated as the product of the saturation magnetic flux density of the iron core and the bridge area, and the formula is shown in Equation (2). In addition, the magnetic resistance due to the gap between the stator and the rotor was determined as in Equation (3), and the magnetic resistance of the magnetic flux barrier inserted with the permanent magnet was a linear value, as shown in Equation (4).  
\[\varnothing_{m}=B_{r}A_{m},\ \varnothing_{b}=B_{sat}A_{b} \tag{2}\]  
\[r_{g1}=\frac{g}{\mu_{0}A_{g1}},\ r_{g2}=\frac{g}{\mu_{0}A_{g2}} \tag{3}\]  
\[r_{m1}=\frac{l_{c1}}{\mu_{0}A_{c1}},\ r_{m2}=\frac{l_{c2}}{\mu_{0}A_{c2}} \tag{4}\]  
where \(\varnothing_{m}\): magnetic flux by permanent magnet (Wb), \(B_{r}\): residual magnetic flux density of the permanent magnet (T), \(A_{m}\): contact area between the iron core and the permanent magnet (m\({}^{2}\)), \(A_{b}\): cross-sectional area of bridge (m\({}^{2}\)), \(\varnothing_{b}\): magnetic flux through the bridge (Wb), \(B_{sat}\): iron core saturation magnetic flux density (T), \(r_{m1}\), \(r_{m2}\): magnetic resistance of PM (H\({}^{-1}\)), \(l_{c1}\), \(l_{c2}\): thickness of magnetic flux barrier (m), \(A_{c1}\), \(A_{c2}\): cross-sectional area of magnetic flux barrier (m\({}^{2}\)), \(A_{g1}\), \(A_{g2}\): air-gap cross-sectional area (m\({}^{2}\)).  
The air-gap magnetic flux density was calculated through Equations (5) and (6) on the basis of the principle of superposition. Using air-gap flux density for each section, the superimposed fundamental wave air-gap magnetic flux density was obtained through Equation (7) using the Fourier series. Here, the fundamental wave denotes the frequency of the air gap MMF corresponding to the number of pole pairs of the motor. Therefore, the no-load flux linkage could be obtained through Equation (8).  
\[B_{g1}=\varnothing_{m}/A_{eq1}+\varnothing_{b1}/A_{eq2}+\varnothing_{b2}/A_{eq3} \tag{5}\]  
\[B_{g2}=\varnothing_{m}/A_{eq4}+\varnothing_{b1}/A_{eq5}+\varnothing_{b2}/A_{eq6} \tag{6}\]  
\[B_{1}=\frac{4}{\pi}\big{[}B_{g1}(\sin(\alpha_{0})-\sin\alpha_{1})+B_{g2}(\sin (\alpha_{1})-\sin(\alpha_{2}))\big{]} \tag{7}\]  
\[\lambda_{PM}=\frac{\sqrt{2}rlB_{1}N_{a}k_{w}}{p} \tag{8}\]  
where \(B_{g1}\), \(B_{g2}\): air-gap flux density in the first and second sections (T), \(A_{eq1}\), \(A_{eq2}\): equivalent area affected by magnetic flux source (m\({}^{2}\)), \(B_{1}\): magnitude of fundamental wave of air-gap magnetic flux density (T), \(r\): inner diameter of stator (m), \(l\): lamination length (m), \(\lambda_{PM}\): no-load flux linkage (Wb).  
Figure 2: Magnetic equivalent circuit (MEC) for PM flux linkage. (**a**) Geometric definition for reluctances; (**b**) Q-axis linear MEC for PM flux linkage.