## 3 Design of a CMG for SWT
### Equivalent Magnetic Circuit (EMC) Modeling  
The equivalent magnetic circuit (EMC) was selected to provide an analytical overview of the magnetostatic field of the CMGT, especially for the magnetic flux density in the air-gap. Using EMC modeling, the calculating time will become faster than the finite-element simulation [17]. Besides, the EMC model provides enough insight and clear relationships among the geometric dimensions, magnet material properties, and magnetostatic field, which is suitable for the preliminary design of a CMGT. Two assumptions for the EMC model to simplify the analysis are as follows:  
1. There is no magnetic saturation occurring in the CMGT;
2. The rotor back iron has infinite permeability compared with the air-gap. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Values** \\ \hline Number of magnet pole pairs on the outer rotor & \(p_{o}\) & 22 \\ Number of magnet pole-pairs on the inner rotor & \(p_{i}\) & 4 \\ Number of pole-pieces on the carrier & \(N_{c}\) & 26 \\ Outer diameter of the outer rotor (mm) & \(D_{o}\) & 140 \\ Inner diameter of the inner rotor (mm) & \(D_{i}\) & 54 \\ Outer air-gap length, inner air-gap length (mm) & \(g_{1},g_{2}\) & 1 \\ Radial length of outer rotor iron yoke (mm) & \(A\) & 5 \\ Thickness of permanent magnet on the outer rotor (mm) & \(B\) & 6 \\ Thickness of pole-piece (mm) & \(C\) & 4 \\ Thickness of permanent magnet on the inner rotor (mm) & \(D\) & 6 \\ Radial length of inner rotor iron yoke (mm) & \(E\) & 20 \\ Axial length (mm) & \(L_{a}\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Preliminary design parameters of a CMGT. Figure 3: Design parameters of the CMGT: (**a**) Sectional view; (**b**) Axial view. Due to the geometrical symmetry of a CMGT, it is only necessary to model one such pair [18], as shown in Figure 4a. The outer rotor and inner rotor steel areas are modeled as magnetic reluctance \(R_{or}\) and \(R_{ir}\), respectively. The permanent magnets on the outer rotor and inner rotor are modeled as the magnetic flux sources \(\varphi_{or}\) and \(\varphi_{ir}\), respectively, and as magnetic reluctance \(R_{m1}\) and \(R_{m2}\), respectively, with the direction of the source dictating the magnet polarity. The air-gaps \(g_{1}\), \(g_{2}\), and \(g_{3}\) are considered as magnetic flux tubes with magnetic reluctance \(R_{g1}\), \(R_{g2}\), and \(R_{g3}\), respectively. The magnetic reluctance of the pole-piece on the carrier is modeled as \(R_{p}\), whereas the reluctance modeling the leakage fluxes in the inner magnets, outer magnets and pole-pieces are \(R_{li}\), \(R_{lo}\) and \(R_{lm}\), respectively. The related equivalent magnetic circuit is shown in Figure 4b. Figure 5 shows the simplification of the EMC. In Figure 5a, \(\phi\), \(\phi_{l1}\), and \(\phi_{l2}\) represent the magnetic flux leaving the magnet, flux leakage in the outer air-gap, and flux leakage in the inner air-gap, respectively. The magnetic flux in the outer and inner air-gaps can be written as \(\phi_{g1}=K_{l1}\phi\) and \(\phi_{g2}=K_{l2}\phi_{g1}\), where \(K_{l1}\) and \(K_{l2}\) represent leakage factors in the outer and inner air-gaps. The leakage factors are typically less than one [19]. Moreover, the circuit is simplified by eliminating the leakage reluctance, as shown in Figure 5b. The leakage reluctance is eliminated because the distribution ranges of the magnetic leakage fluxes are always very small. Hence, the leakage reluctance of the magnet poles is much larger than the internal reluctance of the magnet poles [20]. Therefore, it is difficult to express the \(R_{l1}\) and \(R_{l2}\). In order to obtain the \(\phi_{gn}\), the solution for \(\varphi\) will be multiplied by an estimation of \(K_{l}\) between 0.8 to 0.9 [21] to compensate for the flux that follows the leakage path. When the leakage reluctance is eliminated, the magnetic reluctance is in series to be lumped.