## 4 Electromagnetic Torque Modeling and Torque Density Optimization
### Torque Density Optimization
The motor designed in this paper uses 1]22 material, and the magnetic flux density in the yoke should be set between 1.35 and 1.45 T, while the magnetic flux density in the teeth should be set between 1.75 and 1.85 T. The optimization constraints are defined as follows:  
\[\left\{ \begin{aligned} 0.65\leq K_{\text{sf}}\leq 0.8\\ 0.35U_{\text{N}}\leq\sqrt{3}E_{\text{L,max}}\leq 0.45U_{\text{N}}\\ 1.35\text{T}\leq B_{\text{sy}}\leq 1.45\text{T}\\ 1.75\text{T}\leq B_{\text{st}}\leq 1.85\text{T}\end{aligned} \right. \tag{23}\]  
where \(K_{\text{sf}}\) represents the slot full factor of the motor, \(E_{\text{L,max}}\) is the maximum counter electromotive force under rated load conditions, \(U_{\text{N}}\) is the motor's rated voltage (48 V in this paper), \(B_{\text{sy}}\) denotes the magnetic flux density of the stator yoke, and \(B_{\text{st}}\) denotes the magnetic flux density of the stator teeth. The optimized motor model was established using Maxwell software. The transient simulation of the new motor finite element model was conducted, and the electromagnetic \begin{table}
\begin{tabular}{c c c} \hline
**Parameter Name** & **Parameter Range** & **Unit** \\ \hline _Width of stator notch_\(b_{0}\) & 4.5–5.5 & mm \\ \hline Stator groove depth \(h_{\text{s}}\) & 4.9–5.9 & mm \\ \hline Thickness of stator yoke \(h_{\text{sy}}\) & 1.6–2.6 & mm \\ \hline _Thickness of permanent magnet_\(t_{\text{m}}\) & 4.5–5.5 & mm \\ \hline Motor air gap length \(h_{\text{g}}\) & 0.3–0.7 & mm \\ \hline \end{tabular}
\end{table} 
Table 3: Optimize the range of values of variables. Figure 27: Schematic diagram of some structural parameters of axial flux permanent magnet motor. torque simulation results before and after optimization are compared in Figure 28. The results indicate that the average electromagnetic torque increases from 2.7 N\(\cdot\)m before optimization to 3.2 N\(\cdot\)m after optimization. The effective mass of the motor decreases from 0.606 kg to 0.599 kg, as summarized in Table 4. Based on these results and using formula (22), the torque density is calculated. Before optimization, the motor torque density is 4.45 N\(\cdot\)m/kg, while after optimization, it improves to 5.34 N\(\cdot\)m/kg, representing a 20% increase in torque density. Transient simulation calculations were conducted on the new motor finite element model, resulting in the optimized motor's rated load counter electromotive force waveform. Taking the counter electromotive force of phase A as an example, a comparison of the counter electromotive force under rated load conditions before and after optimization is shown in Figure 29. The maximum counter electromotive force under rated load conditions was 11.3 V before optimization and increased to 11.9 V after optimization, satisfying the specified constraint requirements. Static magnetic field simulations were also performed on the new motor finite element model. The optimized magnetic flux density of the stator y  
The motor parameters and performance before and after optimization using the genetic algorithm based on the electromagnetic torque analytical model of the motor are summarized in Table 5. Following optimization, the torque density of the axial flux permanent magnet motor improves by 20%, with all constraint variables meeting the specified requirements. These results validate the effectiveness of the optimization algorithm.