## 3 Design of PMa-WFSM  
The proposed topology presented in this article involves integrating assisted permanent magnets into the rotor yoke to augment the magnetic flux along the d axis direction and bolster the magnetic torque. One aspect worth exploring further is the impact of different permanent magnet materials on the overall performance of the motor. The selection of materials for these permanent magnets includes ferrite, neodymium, and samarium-cobalt. Ferrite permanent magnets are notably economical but exhibit lower coercivity and magnetic flux density compared with other materials. Table 2 shows the properties of the magnets. Therefore, when employed as assisted magnets, they may enhance the magnetic flux along the d axis while concurrently elevating magnetic resistance, thereby potentially leading to a reduction in torque. On the other hand, samarium-cobalt permanent magnets offer the advantage of possessing a high knee point and robust temperature stability, rendering them less susceptible to irreversible demagnetization. However, they are comparatively pricier than neodymium magnets and demonstrate lower remanence and magnetic flux density. As a result, the neodymium magnet (N42) made by ARNOLD location in New York, which provides a balanced combination of cost-effectiveness, high coercivity, and magnetic flux density, was deemed suitable for the intended application. The torque equation for the PMa-WFSM is calculated by adding the magnetic flux linkage (\(\Psi_{m}\)) of the permanent magnets to the magnetic torque term of Equation (1). \[T\text{=}T_{r}+T_{m}\text{=}1.5p\{(\Psi_{f}+\Psi_{m})i_{a}\text{cos}\beta+0.5(L _{q}-L_{d})i_{a}{}^{2}\text{sin}2\beta\} \tag{4}\]  
Due to the addition of permanent magnets, there is a change in inductance, requiring recalculations. Figure 6 represents the magnetic equivalent circuit of the proposed model. Additionally, the addition of permanent magnets results in changes to the inductance, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \multicolumn{4}{c}{**Value**} \\ \hline  & BR [T] & BH max. [KJ/m\({}^{3}\)] & HCB min. [kA/m] & TEMP max. [\({}^{\circ}\)C] \\ Ferrite magnet (Y20) [ARNOLD] & 0.2 & 18 & 135 & 250 \\ Samarium–cobalt magnet (24 HE) [ARNOLD] & 1.02 & 195 & 765 & 350 \\ Neodymium magnet (N42) [ARNOLD] & 1.315 & 334 & 955 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specification of permanent magnet. Figure 5: Comparison of experimental and FEM analysis torque values. necessitating a recalculation. \(F_{pm}\) is permanent magnet flux, \(R_{pmd}\) and \(R_{pmq}\) represent the permanent magnet reluctances of the d axis and q axis components. \[L_{d}=\frac{N\big{(}F_{d}+F_{pm}\big{)}}{i_{d}}=\frac{N^{2}}{R_{gd}+R_{sd}+R_{rd }+R_{pmd}}\,\ L_{q}=\ \frac{NF_{q}}{i_{q}}=\frac{N^{2}}{R_{gq}+R_{sq}+R_{sq}+R_{mq}+R_{pmq}} \tag{5}\]  
Using a magnetic equivalent circuit, inductance variation and torque were calculated, and based on this, FEM analysis was performed using the fixed permeability method. First, the torque of the PMa-WFSM was classified into three categories as shown in Figure 7. The reluctance torque due to the inductance difference was derived by applying only the stator current, the magnetic torque (winding) was derived via the DC current supply only to the rotor winding, and the magnetic torque (magnet) was derived by inserting only the permanent magnet without any current supply. The sum of the three torque components yields the total torque, with the maximum torque occurring at a current phase angle of 15 degrees [18]. Additionally, a reduction in \(I_{dc}\) is required due to the saturation of the rotor teeth. With the addition of permanent magnets, as demonstrated in Figure 8, saturation of the rotor teeth occurs. Therefore, to prevent non-linear output performance, \(I_{dc}\) was adjusted to 3.5 \(A_{dc}\). The arrangement of the permanent magnets with barriers on the shaft side serves to enhance the magnetic field concentration in the core region, thereby optimizing motor performance.