## 2 Machine Topology and Simulation Results
### On-Load Characteristics Analysis  
To further verify the torque performance of the three models, on-load simulations were conducted on the three structural models to verify their performance. Figure 8 shows cloud maps of magnetic flux density for each model at maximum average torque at a current RMS value of 10 A. From Figure 8a, it can be seen that due to the basic model blocking one side of the magnetic circuit, the magnetic flux density in the D and E regions is significantly lower than that of proposed model two. Proposed model two has a more sufficient and uniform magnetic flux density distribution in the D and E magnetic circuits, indicating that proposed model two has better utilization of the magnetic circuit.  
In Figure 9, the torque of the three models can be seen. The torque of proposed model two is greater than that of the base model and proposed mode one. The formulas for average torque and torque ripple are shown in Equations (4) and (5).  
\[\textit{Tave}=\frac{\sum\limits_{i=1}^{n}T_{i}}{n} \tag{4}\]  
\[\textit{Trip}=\frac{\textit{Tmax}-\textit{Tmin}}{\textit{Tave}}\times 100\% \tag{5}\]  
where, _Tave_ and _Trip_ are the average torque and torque ripple, respectively; \(n\) is the number of steps of the motor in one operating period; \(T_{i}\) is the torque of the motor at step \(i\); \(\textit{Tmax}\) and \(\textit{Tmin}\) are the maximum and minimum torque of the motor in one operating period, respectively.  
Figure 8: Cloud maps showing magnetic flux density under load conditions for the three models. (**a**) Basic model. (**b**) Proposed model 1. (**c**) Proposed model 2. D and E are applicable in all three models. D and E are the magnetic circuit areas in each of the two-layers of the flux barrier in the rotor.  
Figure 9: Electrical torque of three models.  
According to Table 3, compared with the basic model, the average torque of proposed model two increased by 24.08%, and its efficiency also increased by 1.45% at a speed of 1500 rpm, but the torque ripple increased by 18.34%. The total loss in Table 3 contains iron loss and copper loss. Among them, the iron loss contains hysteresis loss and eddy current loss. In order to raise the torque performance of proposed model two, multi-objective optimization on the rotor topology structure of proposed model two was conducted.  
The output torque of a PMA-SynRM is mainly composed of permanent magnet torque and reluctance torque. The frozen permeability method was applied to decompose the total torque of the three models at different phase angles. The permanent magnet torque, reluctance torque, and total torque for the three models are shown in Figure 10. It should be noted that the green markers in Figure 10 are the points of maximum torque. It can be seen from Figure 10 that the phase difference between the maximum value of the total torque and the maximum value of the reluctance torque for the basic model and proposed model one is zero. The phase difference between the maximum value of the total torque and the maximum value of the reluctance torque for proposed model two is also only 5\({}^{\circ}\). Because the basic model is a pole-biased PMA-SynRM, the total torque, permanent magnet torque and reluctance torque are made to peak at the same or similar phase angle by the magnetic chain offset. In Figure 10a, it is shown that the phase difference between the total torque maximum and the permanent magnet torque maximum for the base model is also only 10\({}^{\circ}\). And the phase difference between the maximum value of the total torque and the maximum value of the permanent magnet torque for proposed models one and two are 90\({}^{\circ}\) and 80\({}^{\circ}\)