## 2 Motor Topology and Simulation Analysis
### Motor Topology  
In this paper, a conventional PMA-SynRM model is used as a conventional model and an asymmetric-rotor PMA-SynRM model combined with a Halbach array is the proposed model. Both models are eight-pole, 48-slot PMA-SynRMs with a three-layer U-shaped magnetic barrier and roughly the same amount of steel sheets. Meanwhile, the stator structure and winding configuration are the same for both models. The windings are configured as distributed windings. The amount of permanent magnets inside the rotor is also consistent between the conventional and proposed models. The topology of the stator-rotor for the conventional and proposed models in this paper is shown in Figure 1. The direction of the black arrows in Figure 1,\(c\) indicate the direction of magnetization of the permanent magnet. The specific parameters of the two models are shown in Table 1.  
In order to study the MAS effect of the proposed model, a simplified analytical model of the conventional and proposed models neglecting core saturation and cross-coupling is developed based on the single-pole planar expansion of the rotor shown in Figure 2. From Figure 2b, it can be seen that the permanent magnet flux linkage \(\psi_{pm}\) of the proposed model has been shifted, and \(\psi_{pm}\) forms an angle \(\alpha\) with the _d-axis_. The simplified vector diagram built from Figure 2 is shown in Figure 3.  
Figure 1: Stator and rotor topologies for the conventional and proposed models. (**a**) The rotor topology of the conventional model. (**b**) Stator and winding topology. (**c**) The rotor topology of the proposed model. From Figure 3a, it can be seen that the _d-axis_ magnetic flux linkage generated by the current in the stator is in the opposite direction to the permanent magnet flux linkage. Based on the constant coordinate transformation of the magnetic potential, the magnetic flux linkage in the conventional model with respect to the _d-axis_ and _q-axis_ can be expressed by Equation (1) [27]. \[\left\{ \begin{array}{l}\psi_{d}=\psi_{pm}+L_{d}i_{d}=\psi_{pm}-L_{d}i_{s}sin\beta \\ \psi_{q}=L_{q}i_{q}=L_{q}i_{s}cos\beta\end{array} \right. \tag{1}\]  
where \(\psi_{pm}\), \(\psi_{d}\), and \(\psi_{q}\) are the permanent magnet flux linkage, the _d-axis_ linkage, and the _q-axis_ linkage, respectively; \(L_{d}\) and \(L_{q}\) are the _d-axis_ inductance and the _q-axis_ inductance, respectively; \(i_{d}\) and \(i_{q}\) are the _d-axis_ current and the _q-axis_ current, respectively; \(i_{s}\) is the current in the stator; and \(\beta\) is the angle between the current \(i_{s}\) in the stator and the _d-axis_. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Unit** & **Conv. Model/Prop. Model** \\ \hline Pole/Slot & / & 8/48 \\ Stator outer diameter & mm & 120 \\ Stator inner diameter & mm & 80 \\ Rotor outer diameter & mm & 79 \\ Rotor inner diameter & mm & 30 \\ Air-gap length & mm & 0.5 \\ Motor axial length & mm & 140 \\ Permanent magnet & / & NdFeB \\ Permanent magnet volume & mm\({}^{3}\) & 52640 \\ Steel sheet & / & 50 JN1300 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensional parameters of the conventional and proposed models. Figure 3: Simplified vector diagrams of the two models. (**a**) Conventional model vector diagram. (**b**) Proposed model vector diagram. Figure 2: Single-pole planar expansion of the two model rotors. (**a**) Conventional model plan view. (**b**) Proposed model plan view. According to Equation (1), the torque formula for the conventional model can be obtained as shown in Equation (2). \[ \begin{array}{ll}T_{E1}&=\frac{3}{2}p(\psi_{d}i_{q}-\psi_{q}i_{d})\\ &=\frac{3}{2}p(\psi_{pmi}q+(L_{d}-L_{q})i_{d}i_{q})\\ &=\frac{3}{2}p\psi_{pmi}cos\beta+\frac{3}{4}p(L_{q}-L_{d})i_{s}^{2}sin(2\beta) \\ &=T_{PM1}+T_{RE1}\end{array}  \tag{2}\]  
where \(T_{E1}\), \(T_{PM1}\), and \(T_{RE1}\) are the total torque, permanent magnet torque, and reluctance torque of the conventional model, respectively, and \(p\) is the pole-pair number.