## 3 Theoretical Modeling by LPMEC Method
### LPMEC Model  
According to existing research, the magnetic flux leakage inside a CDMC is inevitable, cannot be ignored and is difficult to be quantitatively analyzed [9; 13]. In this research, an LPMEC model without considering the magnetic flux leakage is first established, as shown in Figure 2. The long air gap is divided into \(N\) equal segments along the axial direction, the center yoke and outer yoke are respectively divided into \(N+1\) segments along the axial direction: \(N-1\) equal segments in the middle and two segments left at the two ends. The magnetic flux leakage is divided into three parts, as shown in Figure 3. The reluctance of part I, \(R_{m\!f}\), corresponds to PM leakage flux; the reluctance of part II, \(R_{\alpha c}\), corresponds to outer-to-center-yoke leakage flux; the reluctance of part III, \(R_{\!f}\), corresponds to tooth-to-center-yoke leakage flux. The \(R_{m\!f}\) is obtained by modeling magnetic flux flowing in the air gap, as depicted in Figure 4. For the case of \(h_{m}<T_{s}+\delta\), as shown in Figure 4a, the magnetic flux leakage \(R_{m\!f}\) is modeled with a circular-arc model, the whole magnetic flux leakage by PM, \(\phi_{ml}\), is an integral of differential flux flowing through differential width \(dx\), and can be expressed as  
\[\phi_{ml}=\int_{0}^{h_{m}}\frac{F(x)}{R^{\prime}(x)}=\!\!\int_{0}^{h_{m}}\frac {2\mu_{0}B_{r}L}{\pi\mu_{m}}dx=\frac{2\mu_{0}B_{r}Ll_{m}}{\pi\mu_{m}} \tag{1}\]  
where \(F(x)\) is the magneto-motive force and is expressed as \(F(x)=B_{r}x/\mu_{m}\); \(R^{\prime}(x)\) is the reluctance corresponding to width \(dx\), and is expressed as \(R^{\prime}(x)=\pi x/\left(2\mu_{0}Ldx\right)\). Figure 3: Magnetic flux leakage inside the RCDMC. Figure 2: Lumped parameter magnetic equivalent circuit (LPMEC) model of the RCDMC. For the case of \(h_{m}>T_{s}+\delta\), as shown in Figure 4b, the whole magnetic flux leakage \(R_{mf}\) can be divided into a circular-arc model and a straight-line model. \(R^{\prime}(x)\) can be expressed as  
\[\left\{ \begin{array}{l}R^{\prime}(x)=\frac{\pi x}{2\mu_{0}Ldx},x\leq T_{s}+ \delta\\ R^{\prime}(x)=\frac{T_{s}+\delta}{\mu_{0}Ldx},x>T_{s}+\delta\end{array} \right. \tag{2}\]  
Thus, \(\phi_{ml}\) can be calculated by the sum of two integrals  
\[ \begin{array}{l}\phi_{ml}=\int_{0}^{T_{s}+\delta}\frac{2\mu_{0}B_{L}}{\pi\mu _{m}}dx+\int_{T_{s}+\delta}^{h_{m}}\frac{\mu_{0}B_{L}x}{\mu_{m}(T_{s}+\delta)} dx\\ =\frac{2\mu_{0}B_{L}}{\pi\mu_{m}}(T_{s}+\delta)+\frac{\mu_{0}B_{L}}{2\mu_{m}( T_{s}+\delta)}\Big{(}h_{m}^{2}-(T_{s}+\delta)^{2}\Big{)}\end{array}  \tag{3}\]  
Finally, the equivalent reluctance \(R_{mf}\) can be calculated by dividing the magneto-motive force of PM by the whole flux leakage, i.e., \(R_{mf}=F_{m}/\phi_{ml}\). The \(R_{oc}\) can be easily calculated as  
\[R_{oc}=\frac{T_{s}+\delta}{\mu_{0}L(l_{ge}-h_{m}-T_{s})} \tag{4}\]  
The \(R_{tf}\) can be obtained by calculating its corresponding permeance \(P_{tf}\). The circular-arc straight-line model is one of the most satisfactory models for modeling the flux flowing in an air gap as depicted in Figure 5. \(P_{tf}\) is an integral of permeances of differential width \(dx\), the length of each differential element is \(\delta+\pi x\)/2. Thus, \(P_{tf}\) can be expressed as  
\[P_{tf}=\int_{0}^{T_{s}}\frac{\mu_{0}L}{\delta+\pi x/2}dx=\frac{2\mu_{0}L}{\pi} \ln\Big{(}1+\frac{\pi T_{s}}{2\delta}\Big{)} \tag{5}\]  
Figure 4: Models of permanent magnet (PM)-to-yoke magnetic flux leakage: (**a**) Circular-arc model for the case of \(h_{m}<T_{s}+\delta\); (**b**) circular-arc and straight-line model for the case of \(h_{m}>T_{s}+\delta\). Figure 5: A circular-arc straight-line permeance model of tooth-to-center-yoke flux leakage.