## 3 Theoretical Modeling by LPMEC Method
### FEM Verification  
The FEM method is used to verify the accuracy of the proposed modeling method. A FEM simulation is conducted using commercial FEM software Maxwell. FEM simulation results and LPMEC model analysis results of MFD in the 1.4 m long air gap, which is depicted as the blue line P1P2 in Figure 1b, are compared.  
The results given by FEM and the proposed LPMEC model are shown in Figure 9. The parameters used for comparison are \(l_{g}\) = 1400 mm, \(h_{m}\) = 50 mm, \(T_{m}\) = 150 mm, \(T_{y}\) = 75 mm, \(T_{s}\) = 25 mm, \(l_{se}\)= 150 mm  
Figure 8: Procedures of the proposed iteration solving method.  
Figure 7: Polynomial fitting result of the H-\(\mu_{r}\) curve of pure iron material.  
and \(\delta=20\) mm. For the mathematical modeling process of the RCDMC structure, N is set to 1400, which means the 1.4 m air gap is divided equally into 1400 segments with each spaced 1 mm. It can be seen that the results given by the LPMEC model agree with the FEM results very well both in curve shape and values. Only a little difference at the end regions of curves is observed owing to the fringing effect [14]. The calculation results based on the LPMEC model without considering the magnetic flux leakage, which are shown in Figure 2, are also given for comparison. It can be seen that the results agree with the FEM results in curve shape, while the values are \(\sim\)8% higher than the FEM results. Therefore, it can be concluded that the modeling of the magnetic flux leakage in the RCDMC is very successful and brings significant improvement in precision to the LPMEC model.  
A few cases with different parameters \(\{h_{m},T_{s},l_{se},\delta\}\) are further investigated in order to evaluate the validity of the improved LPMEC model. Several evaluation parameters including the working point of PM \(B_{m}\), average MFD along air gap \(B_{ave}\), and non-uniformity of MFD along with air gap \(B_{u}\), which are important parameters of main interest during the development of long-stroke CDMC, are compared.  
The \(B_{ave}\) and \(B_{u}\) are defined as  
\[B_{ave}=\frac{1}{l}\int_{l_{g}}B(l)dl \tag{12}\]  
where \(B(l)\) is the MFD in the air gap along the \(x\)-direction.  
The cases with different geometric parameters \(\{h_{m},T_{s},l_{se},\delta\}\) are listed in Table 2. The corresponding results of the FEM and the proposed LPMEC model are listed in Table 3. It can be seen that the deviations of \(B_{m}\), \(B_{ave}\) and \(B_{u}\) between FEM and LPMEC models are all less than 1%. This further proves that the improved LPMEC model has high accuracy, and is valid and suitable for the design and optimization of ultra-long stroke RCDMC.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & \multicolumn{4}{c}{**Geometrical Dimensions**} \\ \cline{2-5}  & \(h_{m}\) **(mm)** & \(T_{s}\) **(mm)** & \(l_{se}\) **(mm)** & \(\delta\) **(mm)** \\ \hline
1 & 50 & 25 & 150 & 20 \\
2 & 30 & 25 & 130 & 20 \\
3 & 50 & 15 & 150 & 20 \\
4 & 50 & 25 & 100 & 20 \\
5 & 50 & 25 & 150 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cases with different parameters \(\{h_{m},T_{s},l_{se},\delta\}\).  
Figure 9: Magnetic flux density (MFD) distributions given by finite element method (FEM) and theoretical models.  
Further research shows that the deviations increase as \(T_{y}\) decreases, e.g., when \(T_{y}\) = 50 mm, and all other comparison parameters are the same, the deviations of \(B_{m}\), \(B_{ave}\) and \(B_{u}\) are 5%, 6.5%, 10.5%, respectively. The reason is that during the calculation of \(R_{m\!f}\), \(R_{\infty}\) and \(R_{\!f}\) in Section 3.1, it is assumed that there is no saturation occurring in the yokes, and the permeability of yokes is far bigger than that of air. When \(T_{y}\) decreases to a certain extent, this assumption is not valid anymore. During the optimization design in Section 5, saturation in yokes is by all means avoided.