# Less Rare-Earth Electromagnetic Design for a High-Performance Permanent Magnet Motor
## 4 Workflow to Obtain the Motor Electromagnetic Design Candidate
Therefore, the minimum \(D_{si}\) is limited to the one that can accommodate the initial magnet weight and shape. The reason for a parametrized optimization for the stator is to speed up the whole optimization process. The starting point is to simulate a given range of values for a fixed current phasor and speed for the parameters \(w_{slot}\), \(w_{sy}\) and \(D_{si}\) (all parameters shown in Figure 4 and the nomenclature section) in FEM. Later, the computation of DC Joule losses is carried out with the results, keeping the same filling factor for all geometry cases and targeting the initial torque value as the reference. For this, Equation (4) is used, it is true that by applying Equation (4), linearity is assumed, but for this reason, a further step is to validate the selected candidate in FEM. However, since the range selected for the stator parameters does not include wide steps from the reference value, the expected results are presumed close enough. \[P_{joule}=3\cdot R_{DC}\cdot\left(\frac{I_{s\_ref}}{\tau_{ratio}}\right)^{2}, \text{ where }\tau_{ratio}=\frac{\tau}{\tau_{ref}} \tag{4}\]  
Next, the Joule losses versus \(w_{sy}\) versus \(w_{slot}\) for each \(D_{si}\) is analyzed from where an optimum range of \(w_{sy}\) is selected. Afterward, for each \(w_{sy}\) (selected), the Joule losses versus \(w_{slot}\) for all \(D_{si}\) are analyzed from where the minimum points are segregated. Finally, the minimum Joule losses for each \(D_{si}\) are plotted from which one candidate was selected. The selected candidate also considered other mechanical aspects, such as the distance between poles and the distance between the magnet layers. However, these mechanical aspects are out of the scope of this paper. Figure 15 shows the results and the plot of minimums with  
Figure 14: Rotor optimization activity diagram. Figure 13: Stator optimization activity diagram. the selected candidate highlighted. The selected candidate operating point was validated in FEM, resulting in an output torque variation of \(-0.32\%\). The rotor optimization is the second stage of Figure 12 and it is executed at room temperature (20 \({}^{\circ}\)C) using a multi-objective genetic algorithm (MOGA) included in the software Altair HyperStudy 2020.1. The treated rotor parameters are shown in Figure 4 and Equation (5). The stack length is kept as 66 mm for the optimization process. So first, the objective is set, as seen in Equation (5). Then, the automatic iteration process in Figure 14 consists of finding the Maximum Torque per Ampere (MITPA) point using the fifth FEM-coupled algorithm from [24], done for a fixed current magnitude and each variation of parameters. Once the MTPA point is found, the point is simulated in FEM and the output parameters are extracted. \[ \begin{array}{l}\text{Objective},\\ Maximize:\ \tau([x_{parameter}])\\ Minimize:\ W_{m}([x_{parameter}])\\ where\ x_{parameter}=[l_{m1},l_{m2},Dre_{Shape},R_{l1},R_{h1},R_{l2},R_{h2},\beta_{1 },\beta_{2}]\\ \text{Constraints},\\ \tau_{ripple}\leq 6.7\%\end{array}  \tag{5}\]  
The optimization results at room temperature are shown in Figure 16. The Pareto front and the selected candidate are highlighted in the figure. In addition, images of the magnet geometries variations in three sections of the Pareto front (bottom, middle, top). It is worth remarking again that the Pareto front shown in Figure 16 is for radial magnetization. However, extensive computational time to find the optimal tilt angle for each iteration is avoided by doing this and since the final results satisfy the requirements of Table 3, the process was established as acceptable. Going to the right on the curve (Figure 16), it can be noticed that the magnet layers approach each other. Therefore, the selected candidate from the Pareto front also involves mechanical criteria for the distance between magnet poles and magnet layers.