## 2 Machine Topology and Simulation Results
### Machine Topology  
The three models in this article are all 8-pole and 48-slot PMA-SynRM with three layers of U-shaped flux barriers. The stator core and rotor core were stacked with 50 JN1300 (Sourced from JFE Steel, Tokyo, Japan) silicon steel sheets, and the quantity of silicon steel sheets in the three models was very similar. The permanent magnet used was an NMX-42F (Sourced from Hitachi, Tokyo, Japan) NdFeB permanent magnet, with a residual magnetism of 1.28 T. For the accuracy of the simulation, the three models were configured with the same stator structure and winding. The stator and rotor topology of the basic model and proposed models one and two are shown in Figure 1. The arrow direction in Figure 1 represents the magnetization direction of the permanent magnet. It should be noted that proposed model one is very similar to proposed model two in terms of rotor structure. However, proposed model two has an additional permanent magnet in the flux barrier at the top of the rotor compared to proposed model one. Figure 1: (**a**) Topological diagram of stator core structure. (**b**) Basic model rotor core topology structure. (**c**) Proposed model 1 rotor core topology structure. (**d**) Proposed model 2 rotor core topology structure. Table 1 shows the same dimensional parameters for the three models. The volumes of the permanent magnets for the basic model and proposed models one and two are 24,800 mm\({}^{3}\), 32,000 mm\({}^{3}\) and 37,200 mm\({}^{3}\), respectively. The arrangement of permanent magnets in the basic model has a certain magnetic focusing effect, but the asymmetric structure causes a blockage on the other side, resulting in a decrease in the utilization rate of permanent magnets. By utilizing the advantages of its magnetic focusing effect, proposed model one is proposed, which enables the magnetic circuits on both sides of the rotor to have magnetic focusing effect in a single stage. Proposed model one and proposed model two form a permanent magnet arrangement of Halbach arrays as shown in Figure 2, further increasing the \(q\)-axis magnetic flux. According to Figure 3, a planar expansion diagram of proposed model two can provide a clearer view of the permanent magnet arrangement and Halbach array magnetic focusing effect. The pink arrow in Figures 2 and 3 is the magnetization direction of the permanent magnet, and the permanent magnets in the adjacent rotor barrier ribs can form a series. The permanent magnet in the middle of the rotor barrier and the permanent magnet in the rib of the barrier form the arrangement of the Halbach array. Such a permanent magnet arrangement can effectively use the magnetic circuit inside the rotor, the magnetic flux aggregation effect is obvious, and the excitation effect is better. This paper will use finite element simulation results to prove the theory. The vector diagram of proposed model two was established according to Figure 3, as shown in Figure 4. According to Figure 4, the voltage equation, the magnetic chain equation and the electromagnetic torque equation can be listed as shown in Equations (1)-(3), respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Unit** & **Basic Model/Prop. Model 1/Prop. Model 2** \\ \hline Pole/Slot & / & 8/48 \\ Stator outer diameter & mm & 98 \\ Stator inner diameter & mm & 61 \\ Rotor outer diameter & mm & 60 \\ Rotor inner diameter & mm & 15 \\ Air gap length & mm & 0.5 \\ Motor axial length & mm & 100 \\ Remanence of magnet & T & 1.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of PMA-SynRM models. Figure 3: Proposed model 2: flat, unfolded view. Figure 2: Halbach array magnetization method. \[\left\{ \begin{array}{ll}u_{d}=\frac{d\psi_{d}}{dt}-\omega\psi_{q}+Ri_{d}\\ u_{q}=\frac{d\psi_{q}}{dt}+\omega\psi_{d}+Ri_{q}\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{ll}\psi_{d}=L_{d}i_{d}+\psi_{PM}\\ \psi_{q}=L_{q}i_{q}\end{array} \right. \tag{2}\]  
\[