## 2 Design Process Using Lumped-Parameter Method (LPM)
### D, Q-Axis Inductance  
Figure 3 shows the magnetomotive force (MMF) formed by the D, Q-axis current, and the flow of the flux. Similarly to calculating the magnetic circuit of Q-axis PM flux linkage, calculating the flux linkage for Q-axis inductance of the MEC in Figure 3a considered only linear reluctance. In general, it was assumed that Q-axis inductance was constant, because linear reluctance is dominant under the influence of the magnetic flux barrier in PMSM when having a two-layer structure or more [21]. Therefore, Q-axis inductance was calculated through the MMF caused by the Q-axis current generated in the winding of the stator and the linear MEC composed of the magnetic resistance component of the air-gap and the magnetic flux barrier. \[F_{q}=\frac{m}{2}\frac{4}{\pi}\frac{N_{a}k_{w}p}{2}\sqrt{2}I_{q} \tag{9}\]  
\[f_{qs1}=F_{q}\frac{\cos(\alpha_{0})-\cos(\alpha_{1})}{\alpha_{1}-\alpha_{0}} \tag{10}\]  
\[f_{qs2}=F_{q}\frac{\cos(\alpha_{1})-\cos(\alpha_{2})}{\alpha_{2}-\alpha_{1}} \tag{11}\]  
where \(F_{q}\): magnitude of fundamental wave of MMF by the Q-axis (AT), \(f_{qs1}\): magnitude of MMF between \(\alpha_{0}\) and \(\alpha_{1}\) (AT), \(f_{qs2}\): magnitude of MMF between \(\alpha_{1}\) and \(\alpha_{2}\) (AT). Since MEC analysis aims to obtain the magnetic flux and inductance flowing in the air gap, it could be formulated as Equation (12) by applying the principle of superposition. Therefore, air-gap magnetic flux density was obtained by dividing the magnetic flux for each section by the air-gap area, and the magnitude of the fundamental component of the air-gap magnetic flux density was calculated through Equations (13) and (14). Lastly, the Q-axis inductance was obtained through the calculated air-gap flux density from Equation (15). \[\varnothing_{qg1}=\frac{f_{qs1}R_{q1}+f_{qs2}R_{eq2}}{r_{g1}},\ \varnothing_{qg2}=\frac{f_{qs1}R_{eq3}+f_{qs2}R_{eq4}}{r_{g2}} \tag{12}\]  
\[B_{qg1}=\varnothing_{qg1}/A_{g1},\ B_{qg2}=\varnothing_{qg2}/A_{g2}, \tag{13}\]  
\[B_{q1}=B_{qg1}(\alpha_{1}-\alpha_{0})+B_{qg2}(\alpha_{2}-\alpha_{1}) \tag{14}\]  
Figure 3: Magnetic equivalent circuit for (**a**) Q-axis current (**b**) D-axis current. \[\lambda_{q}=\frac{2 lrB_{q1}N_{a}k_{w}}{p},\ L_{q}=\frac{\lambda_{q}}{I_{q}} \tag{15}\]  
where \(\varnothing_{qg1}\), \(\varnothing_{qg2}\): magnetic flux flowing through air-gap in each section (Wb), \(B_{qg1}\), \(B_{qg1}\): air-gap magnetic flux density in each section (T), \(B_{q1}\): magnitude of the fundamental air-gap magnetic flux density (T), \(\lambda_{q}\): flux linkage through D-axis current (Wb), \(L_{q}\): Q-axis inductance (H). In addition to Q-axis inductance, D-axis inductance is a major variable that determines reluctance torque, which is the second term in Equation (1). To improve efficiency, the difference between the two inductances should be maximized. Figure 3b shows the MEC for D-axis inductance, which was similar to the process of calculating the Q-axis inductance, but had to be calculated considering the saturation of iron-core characteristics. The D-axis flux flowed along the iron flux-path of the rotor by MMF generated by the D-axis current, as shown in Figure 3b. As a result, the magnitude of magnetic resistance was greatly variable due to the nonlinear saturation characteristic of the flux-path of the iron-core with high permeability, according to the magnitude of the current, and the range of variability in the D-axis inductance was also large. In the D-axis MEC, the MMF source was located in three branches, and nonlinear magnetic resistance was present in the teeth of the stator, the back iron, and two yokes of the rotor; air-gap magnetic resistance was located in each branch. The magnitude of the fundamental component of the D-axis MMF was calculated using Equation (16), and the magnitude of MMF corresponding to each section was calculated using Equation (17).