## 4 Validation of Proposed Method
### Calculation Time  
In the analysis of no load and load mentioned above, the simulation steps for one electrical cycle are set to 101 steps, and the same hardware of the computer is used. The calculation time for a single 3D model under one operating condition is about 100 h. Under the same conditions, the calculation time for a single 2D model under one working condition is about 20 min. The results are calculated sequentially at each layer and processed. The overall time consumption for one single working condition using the proposed two-dimensional multi-layer linear permanent magnet synchronous motor is about 20 \(\times\) 9 min = 3 h. That is, under the premise that the calculation results are very close and the error meets the engineering requirements in the preliminary design and optimization, the time consumption of the method proposed in this article is only 3% of the 3D method. It should be noted that the 3D model is a quarter model, and the 2D model has not been simplified. If the same half model or quarter model is used, its calculation speed will be further improved.  
In addition, to demonstrate the comparison between the proposed method in this article and the traditional two-dimensional method mentioned in Section 2.2, detailed calculation time data are listed in Table 3. To demonstrate differences, there are some subjective variables in the comparison time (such as the time when the model was established and the time when the parameters were set). Among them, the modeling of the 3D model is relatively complex, taking 15 min, but only needs to be modeled once, and for fixed motor sizes and parameters, there is no need to select a parameterized model. The simulation setup process of the 3D model also only needs to be executed once, and all are objective calculation times.  
For the traditional method shown in Figure 2, although establishing a single model only takes 5 min, modeling a 2D model requires nine times longer. In addition, due to the different simulation conditions in each new 2D model with changed parameters, it is necessary to set the simulation conditions once, including the simulation step size, speed, and position of periodic boundary conditions. This process needs to be repeated nine times, taking 45 min. In addition, the solution time is about 180 min, and the result processing time is about 10 min, totaling 280 min.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time-Consuming** & **Unit** & **3D FEM** & **Existed 2D FEM** & **Proposed 2D FEM** \\ \hline Geometric modeling & min & 15 & \(5\times 9\) = 45 & 5 \\ Simulation settings & min & 3 & \(3\times 9\) = 45 & 3 \\ Solution time & min & 6000 & \(20\times 9\) = 180 & \(20\times 9\) = 180 \\ Result processing time & min & 3 & 10 & 10 \\ Total time & min & 6021 & 280 & 198 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of calculation times for three methods.  
For the improved method proposed in this article, although there is no essential difference in solving time compared to traditional methods, the 2D model of this method only needs to be modeled once. By selecting five geometric parameters to achieve each layered overlay model, the simulation conditions of all models are consistent. Therefore, the positions of rotational speed, step size, and periodic boundary conditions only need to be set once, and the total time required for the result processing process is only 188 min. Based on the necessary objective calculation time, it can be seen that the timeliness and convenience of the proposed method in this article are obvious.