## 5 Optimization of the FRM Design
### Optimization Procedure  
The Nelder-Mead method is used in this work to minimize the objective function according to the criterion merits described before:  
\[F=K_{1}K_{2}\sqrt{K_{3}}K_{4}^{\frac{1}{10}} \tag{8}\]  
This expression means that the following improvements of the objectives are equally valuable: Decreasing \(K_{1}\) by 1%, decreasing \(K_{2}\) by 1%, decreasing \(K_{3}\) by 0.5%, and decreasing \(K_{4}\) by 0.1%. Implementing the optimization procedure based on the one-criterion method is justified by significant efforts required for computing the objective function (8): Three modes must be calculated for each objective function call, and the large number of the rotor position must be taken into account. Not all criteria will be improved after carrying out the optimization procedure. The value of a criterion may increase if it is considered to be a fair trade-off for lowering other criteria. This does not mean that the presence of increasing criteria in the objective function is useless, as their absence in the objective function could have led to their even greater increase. This paper describes simulation and optimization of a three-phase FRM (4.5 kW, 28,000 rpm) consisting of three single-phase motors of 1.5 kW each. Table 2 shows the geometrical parameters of the FRM, which are constant during optimization process. In order to reduce eddy loss in the permanent magnets, each magnetic pole is divided into 8 electrically insulated segments in the azimuthal direction. The residual flux density (\(B_{r}\)) and the coercivity (\(H_{\textit{cj}}\)) of the permanent magnet are 1.2 T and \(H_{\textit{cj}}\) = 1000 kA/m, respectively. In practice, PMs having a greater residual flux density, e.g., \(B_{r}\) = 1.3 T, may be necessary due to the presence of insulating gaps between magnets. The M350-50A steel grade is selected for both the stator and the rotor. Steel sheet thickness is 0.5 mm. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline The stator outer diameter, mm & 54 \\ The rotor shaft diameter, mm & 7 \\ Single-phase FRM stack length, mm & 40 \\ The rotor slot bottom diameter, mm & 14 \\ The magnet thickness, mm & 1.4 \\ The air gap, mm & 0.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Fixed geometrical parameters of the FRM. Figure 5 displays the detailed slot geometry. The slot opening is smoothed with small segments 1-2 and 1\({}^{\prime}\)-2\({}^{\prime}\). The segments 4-5 and 4\({}^{\prime}\)-5\({}^{\prime}\) represent the teeth legs and are radii. The lines 2-3-4 and 2\({}^{\prime}\)-3\({}^{\prime}\)-4 approximate an Archimedes helix. Let's introduce the following notation:  
1. Radius values \(R_{1}\), \(R_{2}\), \(R_{3}\), \(R_{4}\), \(R_{5}\) of the stator slot points 1, 2, 3, 4, 5;
2. \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\), \(\alpha_{5}\) are the angular distances between points 1 and 1\({}^{\prime}\), 2 and 2\({}^{\prime}\), 3 and 3\({}^{\prime}\), 4 and 4\({}^{\prime}\), 5 and 5\({}^{\prime}\) of the slot;
3. \(\alpha_{6}\) is the angular distance between points 6 and 6\({}^{\prime}\) of the rotor teeth. Therefore, the following constraints are imposed on the parameters:  
\[ \begin{array}{c}R_{2}=R_{1}+0.7 mm,\\ R_{3}=(R_{2}+R_{4})/2\\ \alpha_{2}=\alpha_{1}\\ \alpha_{3}=(\alpha_{2}+\alpha_{4})/2\\ \alpha_{4}=\alpha_{5}\end{array}  \tag{9}\]  
The geometrical parameters \(R_{1}\), \(R_{4}\), \(R_{5}\), \(\alpha_{1}\), \(\alpha_{5}\), \(\alpha_{6}\), which are independent according to (9), are adopted as optimization parameters. In this paper, the current angle, which is the angle between the current vector and the rotor position, is assumed to be zero when at rotor position of phase A, illustrated in Figure 4. The phase current A takes its maximum value, and the current sign is chosen so as to provide the positive torque. In the underload modes (Mode 2 and Mode 1), the current angle is chosen to be zero.