## 3 Theoretical Modeling by LPMEC Method
### Iteration Solving Method  
The solving of the LPMEC model in the form of matrix equation is quite difficult because of the strong nonlinearity of yoke characteristics [13; 14; 15; 16], which is the main limit during the theoretical modeling of magnetic circuits. In order to solve \(\mathbf{R}\), \(\mathbf{I}\) and \(\mathbf{F}\) and thus to calculate the MFD in the air gap, the key is to determine the relative permeability of pure iron \(\mu_{r}\) at different positions of yokes. As a result that the yokes are made of pure iron material with strong nonlinear \(H\)-\(\mu_{r}\) curve, direct solving of the model equations is not possible.  
In this research, an iterative solving method is proposed to realize the rapid and accurate solution of the matrix equation through iteration calculation. The relative permeability of pure iron \(\mu_{r}\) as a function of the magnetic field strength \(H\) is tested by the National Institute of Metrology (NIM) of China and depicted in Figure 7. The specimen has an outer diameter of 50 mm, an inner diameter of 40 mm and a thickness of 10 mm. The relationship between \(\mu_{r}\) and \(H\) is expressed as follows through polynomial fitting.  
\[\left\{\begin{array}{l}\mu_{r}(H)=\sum\limits_{i=1}^{4}a_{i}H^{-(i-1)},H \leq H_{0}(\mu_{r}^{\mathrm{max}})\\ \mu_{r}(H)=\sum\limits_{i=1}^{8}b_{i}H^{-(i-1)},H>H_{0}(\mu_{r}^{\mathrm{max} })\end{array}\right. \tag{11}\]  
where \(H_{0}(\mu_{r}^{\mathrm{max}})\) is the magnetic field strength when the relative permeability \(\mu_{r}\) reaches maximum. It can be seen that the polynomial fitting result of the \(H\)-\(\mu_{r}\) curve fits the actual testing results very well.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reluctances** & **Equations** & **Reluctances** & **Equations** \\ \hline \(R_{m}\) & \(\frac{h_{m}}{\mu_{r}\mathrm{I}\pi/2}\) & \(R_{30}\), \(R_{3N}\) & \(\frac{l_{r}}{\mu_{r}\mathrm{I}\pi_{r}}\) \\ \(R_{10}\), \(R_{1N}\) & \(\frac{l_{r}-h_{m}}{\mu_{r}\mathrm{I}\pi/2}\) & \(R_{3i}\left(i=1,2,\ldots,N-1\right)\) & \(\frac{l_{g}/(N-1)}{\mu_{r}\mathrm{I}\left(T_{y}+T_{z}\right)}\) \\ \(R_{1i}\left(i=1,2,...,N-1\right)\) & \(\frac{l_{g}/N-1}{\mu_{r}\mathrm{I}\pi/2}\) & \(R_{41}\), \(R_{42}\) & \(\frac{T_{z}+\phi T_{y}/2}{\mu_{r}\mathrm{I}\pi/2}\) \\ \(R_{2i}\left(i=1,2,\ldots,N\right)\) & \(\frac{\delta}{\mu_{0}L_{g}^{2}/N}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Reluctance equations in the LPMEC model.  
The procedure of the iterative solving calculation is summarized in Figure 8. First, the dimensions and material property parameters of the RCDMC are initialized. Second, an initial value for the relative permeability of yokes is used, and the reluctance matrix **R** is calculated. Magnetic fluxes, MFDs and magnetic field strengths in each element of the model are then computed through solving the matrix equation. The values of relative permeability \(\mu_{r}\) in each element are updated for the next iteration step according to the \(H\)-\(\mu_{r}\) fitting curve expressed as Equation (11). Then the matrix of reluctances and flux densities is recomputed. The procedures above are repeated continually until the difference of MFD between two iteration steps is less than a required precision index (such as 0.1 mT), and the iteration is regarded as converged.