# Simplified Design of Magnetic Gear by Considering the Maximum Transmission Torque Line
## 3 Maximum Transmission Torque Line and Design Parameters of Magnetic Gear
### Relationship of Design Parameters  
Based on Equation (3), which represents the MTTL, the total volume of PMs is estimated from any required transmission torque. Since the main purpose of this study was to develop a simplified MG that can meet the required transmission torque, the parameters of the structure of MG need to be obtained by relating the total volume of PMs with the parameters of the MG. Firstly, the total volume of PMs is derived as the sum of volume of PMs at high-speed and low-speed rotors, as expressed in Equation (8). \[V_{\rm pm}=V_{\rm m1}+V_{\rm m2} \tag{8}\]  
where \(V_{\rm m1}\) and \(V_{\rm m2}\) are the volume of PMs at low-speed rotor and high-speed rotor, respectively. Figure 4: The transmission torque influenced by the thickness ratio of (**a**) \(t_{\rm pm}\) and \(t_{\rm p}\), (**b**) \(t_{\rm pm}\) and \(t_{\rm o}\) and (**c**) \(t_{\rm pm}\) and \(t_{\rm i}\). After separating both volume of PMs at both high-speed and low-speed rotor, each rotor equation is derived separately. By relating the volume of cylinder formula, the volume of PMs at low-speed rotor can be related to the parameters at low-speed rotor, consisting of the outer radius of outer rotor, the thickness of outer yoke, and the thickness of PMs at the low-speed rotor, as written in Equation (9). As shown in Equation (10), the thickness of the outer yoke, \(t_{\rm o}\), is eliminated by substituting Equation (6) into Equation (9). \[V_{\rm m1} =\pi(r_{\rm o}-t_{\rm o})^{2}\ell-\pi(r_{\rm o}-t_{\rm o}-t_{\rm pm })^{2}\ell \tag{9}\] \[V_{\rm m1} =\pi(r_{\rm o}-4t_{\rm pm})^{2}\ell-\pi\left(r_{\rm o}-5t_{\rm pm} \right)^{2}\ell \tag{10}\]  
where \(V_{\rm m1}\) is the volume of PMs at low-speed rotor, \(r_{\rm o}\) is the outer radius of outer yoke of the magnetic gear, and \(\ell\) is the length of the magnetic gear. Same as the low-speed rotor, the volume of PMs at high-speed rotor can be related to the parameters at high-speed rotors which are the inner radius of inner rotor, thickness of inner yoke, and thickness of PMs at the high-speed rotor, given as Equation (11). By substituting Equation (7) into Equation (11), the thickness of inner yoke, \(t_{\rm i}\), can be eliminated as shown in Equation (12). \[V_{\rm m2} =\pi(r_{\rm i}+t_{\rm i}+t_{\rm pm})^{2}\ell-\pi(r_{\rm i}+t_{\rm i })^{2}\ell \tag{11}\] \[V_{\rm m2} =\pi(r_{\rm i}+6t_{\rm pm})^{2}\ell-\pi(r_{\rm i}+5t_{\rm pm})^{2}\ell \tag{12}\]  
where \(V_{\rm m2}\) is the volume of PMs at high-speed rotor, while \(r_{\rm i}\) is the inner radius of the inner yoke. Equation (13) shows the combination of fixed value of air gap, \(g\), with the thickness of pole pieces, \(t_{\rm pm}\), as \(c\), which represents the space between the PMs of high-speed and low-speed rotor. \[c=t_{\rm p}+2g;c=2.5t_{\rm pm}+1 \tag{13}\]  
where \(g\) is the air gap between both high-speed and low-speed rotors and the pole pieces, while c is the combination of \(g\) with the thickness of pole pieces, \(t_{\rm p}\). To simplify Equations (10) and (12), an equation can be formed by relating the inner radius with the outer radius, as written in Equations (14) and (15). \[r_{\rm i}=r_{\rm o}-t_{\rm o}-t_{\rm m2}-c-t_{\rm m1}-t_{\rm i} \tag{14}\]  
\[r_{\rm i} =r_{\rm o}-13.5t_{\rm pm}-1 \tag{15}\]  
Subsequently, the \(V_{\rm m2}\) value is as shown in Equation (16). \[V_{\rm m2} =\pi(r_{\rm o}-7.5t_{\rm pm}-1)^{2}\ell-\pi(r_{\rm o}-8.5t_{\rm pm }-1)^{2}\ell \tag{16}\]  
The net \(V_{\rm pm}\) is as shown in Equation (17), and its quadratic representation is in Equation (18), where the thickness of PMs, \(t_{\rm pm}\), is unknown. Figure 4:0 Figure 4:1  
To solve the quadratic in Equation (18), the quadratic formula is used, resulting in Equation (19). Thus, the thickness of PMs is obtained from the estimated total volume of PMs that corresponds to the desired transmission torque through the MTTL. The radius and the length of magnetic gear are chosen according to the applications.