## 3 Theoretical Modeling by LPMEC Method
### LPMEC Model
Noting the reciprocal relationship between reluctance and permeance, \(R_{\mathit{ff}}\) can be given as  
\[R_{\mathit{ff}}=\frac{1}{P_{\mathit{tf}}} \tag{6}\]  
Next, an improved LPMEC model taking the magnetic flux leakage in the RCDMC into consideration is obtained by adding \(R_{\mathit{mf}}\), \(R_{\mathit{oc}}\) and \(R_{\mathit{ff}}\) to the LPMEC model in Figure 2, as shown in Figure 6. This improved model can yield accurate analytical results of the MFD in long air gaps. The \(N+7\) closed loops \(L_{i}\) (\(i=1,2,\ldots\)\(N+7\)) are defined with the direction marked in Figure 6. According to [PERSON]'s law, the following equations can be obtained by  
\[\left\{ \begin{array}{l}-\phi_{1}R_{L_{1}}+\phi_{2}R_{1}=-F_{m}\\ \phi_{1}R_{1}-\phi_{2}R_{L_{2}}+\phi_{3}R_{2}=0\\ \vdots\\ \phi_{k-1}R_{k-1}-\phi_{k}R_{L_{k}}+\phi_{k+1}R_{k}=0\\ \vdots\\ \phi_{N+5}R_{N+5}-\phi_{N+6}R_{L_{N+6}}+\phi_{N+7}R_{N+6}=0\\ \phi_{N+6}R_{N+6}-\phi_{N+7}R_{L_{N+7}}=F_{m}\end{array} \right. \tag{7}\]  
where, \(F_{m}\) is the magneto-motive force of the PMs, and \(F_{m}=B_{r}h_{m}\)\(/\mu_{m}\); \(\phi_{i}\) (\(i=1,2,\ldots\)\(/N+7\)) is the magnetic flux passing through the \(i\)th closed loop; \(R_{L_{i}}\)(\(i=1,2,...,N+7\)) is the sum of reluctances in the \(i\)th closed loop, i.e., \(R_{L_{1}}=R_{L_{N+7}}=R_{m}+R_{1}=R_{m}+R_{N+6}\), \(R_{L_{2}}=R_{L_{N+6}}=R_{30}+R_{41}+R_{1}+R_{2}=R_{3N}+R_{42}+R_{N+5}+R_{N+6}, R_{L_{3}}=R_{L_{N+5}}=R_{10}+R_{2}+R_{3}=R_{1N}+R_{N+4}+R_{N+5},R_{L_{4}}=R_{L_{N+4 }}=R_{3}+R_{4}\)\(=R_{N+3}+R_{N+4}\), \(R_{L_{i}}=R_{1(i-4)}+R_{3(i-4)}+R_{i-1}+R_{i}\)( \(i=5\), \(6\),...\(/N+3\)); \(R_{i}\) (\(i=1,2,\ldots\)\(/N+6\)) is the intersected reluctances of the \(i\)th and \((i+1)\)th closed loops, i.e., \(R_{1}=R_{N+6}=R_{\mathit{mf}}\),\(R_{2}=R_{N+5}=R_{\mathit{oc}}\), \(R_{3}=R_{N+4}=R_{\mathit{ff}}\), \(R_{i}=R_{2(i-3)}\) (\(i=4,5,\ldots\)\(/N+3\)). Herein \(R_{\mathit{m}}\) is the reluctance of the PMs; \(R_{1i}\), \(R_{3i}\) (\(i=0,\)\(1,\ldots\)\(/N+1\)) are the reluctances of the \(i\)th axial segment of the center yoke and outer yoke, respectively; \(R_{2i}\) (\(i=1,2,\ldots\)\(/N\)) is the reluctance of \(i\)th segment of the air gap; \(R_{41}\), \(R_{42}\) are the reluctances of the end yokes. The various reluctances in the improved LPMEC model can be calculated with equations listed in Table 1, in which \(\mu_{y}\) is the permeability of yokes which varies at different positions of yokes. Figure 6: Improved LPMEC model for the RCDMC. Thus, the matrix equation that gives the relation between reluctance, magnetic flux and magneto-motive force is expressed as  
\[\mathbf{R}\times\mathbf{I}^{\mathrm{T}}=\mathbf{F}^{\mathrm{T}}, \tag{8}\]  
where \(\mathbf{R}\) is the reluctance matrix, the dimension of which is \((N+7)\times(N+7)\), and can be expressed as  
\[\mathbf{R}=\left[ \begin{array}{ccccc}-R_{L_{1}}&R_{1}&0&\ldots&0\\ R_{1}&-R_{L_{2}}&R_{2}&0&\ldots\\ 0&\ldots&\ldots&\ldots&0\\ \ldots&0&R_{N+5}&-R_{L_{N+6}}&R_{N+6}\\ 0&\ldots&0&R_{N+6}&-R_{L_{N+7}}\end{array} \right] \tag{9}\]  
\(\mathbf{I}\) is the magnetic flux vector, the dimension of which is \(N+7\), and can be expressed as \(\mathbf{I}=\left[ \begin{array}{ccccc}\phi_{1}&\phi_{2}&\ldots&\phi_{N+6}& \phi_{N+7}\end{array} \right]\); \(\mathbf{F}\) is the magneto-motive force vector, the dimension of which is \(N+7\), and can be expressed as \(\mathbf{F}=\left[ \begin{array}{ccccc}-&F_{m}&0&\ldots&0&F_{m}\end{array} \right]\). Finally, the MFD in the long air gap can be expressed as  
\[B_{i}=\frac{\mathbf{I}(i+3)-\mathbf{I}(i+4)}{L_{g}/N},i=1,2,\ldots,N \tag{10}\]