## 3 Design of a CMG for SWT
### Equivalent Magnetic Circuit (EMC) Modeling
The outer rotor and inner rotor reluctance \(R_{or}\) and \(R_{ir}\), which are eliminated, as shown in Figure 5c, are modeled as a perturbation of the air-gap reluctance by introducing a reluctance factor \(K_{r}\) with its value between 1 to 1.2 [19]. Figure 4: EMC modeling: (**a**) one pair magnet modelled, (**b**) modelled circuit. The elimination is due to the assumption of infinite permeability of the outer and inner rotor iron yoke. Figure 5c depicts the simplified circuit with the \(K_{r}\) notation. Two half magnets connected in series are equivalent to a single block of permanent magnet material twice as long. As a result, \(\phi_{or}\) and \(\phi_{ir}\) remain constant while \(R_{m1}\) and \(R_{m1}\) double because reluctance is directly proportional to magnet length [19]. Finally, the magnets and rotor reluctance are swapped, as shown in Figure 5d. As mentioned in [19], the analytical expressions for the leakage factor \(K_{l}\) and reluctance factor \(K_{r}\) are seldom tried. These expressions are difficult to determine accurate values; that is why the amount of \(K_{l}\) and \(K_{r}\) used for this study was identified as discussed before. The magnet flux can be obtained using flux division, as in Equation (4):  
\[\phi=\frac{1}{1+K_{r}\Big{(}\frac{2R_{g1}+2R_{g2}+R_{g2}+R_{g3}}{2R_{m1}+2R_{m2 }}\Big{)}}(\phi_{or}+\phi_{ir}) \tag{4}\]  
Figure 5: Simplification of unit block circuit: (**a**) swapping the \(R_{or}\) and \(R_{ir}\), (**b**) eliminating the magnet and leakage reluctances, (**c**) eliminating the \(R_{or}\) and \(R_{ir}\), (**d**) swapping and simplifying the circuit. The general reluctance is expressed in Equations (5)-(7):  
\[R_{mn}=\frac{l_{mn}}{\mu_{r}\cdot\mu_{0}\cdot A_{mn}}, \tag{5}\]  
\[R_{p}=\frac{l_{p}}{\mu_{r}\cdot\mu_{0}\cdot A_{p}}, \tag{6}\]  
\[R_{gn}=\frac{g_{n}}{\mu_{0}\cdot A_{gn}}, \tag{7}\]  
In these equations, \(l_{mn}\), \(A_{mn}\), \(l_{p}\), \(A_{p}\), \(g_{n}\), \(A_{gn}\) are the magnet length, axial magnet area, pole-piece length, pole-piece axial area, air-gap length, and air-gap axial area, respectively, whereas \(n=1\), \(2\), \(3\). Moreover, \(\mu_{r}\) and \(\mu_{0}\)are relative permeability and permeability of free space, respectively. Furthermore, the flux density relationships are: \(B_{g1}=\vartheta_{g1}/A_{g1}\), \(B_{g2}=\phi_{g2}/A_{g2}\), and \(B_{r}=\phi_{ar}/A_{m1}=\vartheta_{ir}/A_{m2}\). The \(B_{r}\) of the two magnets is the same value because the magnetic gear uses the same permanent magnet material. Hence, the outer and inner air-gap flux density is determined as in Equations (8) and (9), respectively:  
\[B_{g1}=\frac{K_{l1}}{1+K_{r}\left(\frac{2R_{g1}+2R_{g2}+R_{p}+R_{g3}}{2R_{m1}+2 R_{m2}}\right)}\cdot\frac{A_{m1}+A_{m2}}{A_{g1}}\cdot Br, \tag{8}\]  
\[B_{g2}=\frac{K_{l1}\cdot K_{l2}}{1+K_{r}\left(\frac{2R_{g1}+2R_{g2}+R_{p}+R_{g3 }}{2R_{m1}+2R_{m2}}\right)}\cdot\frac{A_{m1}+A_{m2}}{A_{g2}}\cdot Br, \tag{9}\]  
Three cases are taken as examples to calculate the flux densities of the inner and outer air-gaps by EMC model. The design variables are the thickness of the permanent magnet on the outer and inner rotors B and D, the thickness of the pole-piece C, the radial length of the inner rotor iron yoke E and the diameter of the inner rotor D, as shown in Figure 3:0 Other design parameters of the CMGT are kept constant. These three cases refer to [PERSON] et al. (2020). These cases are chosen because of their transmitted torque values, length, and thickness parameters, which are roughly more similar to other models. Therefore, even the \(D_{i}\) value is different, and the parameters are reasonable since the \(D_{o}\), as the key parameter of the design space, remains unchanged. Table 4 shows the analytical results of the outer and inner air-gap flux densities, i.e., \(B_{g1}\) and \(B_{g2}\), of three CMGTs by the EMC model.