## 3 Design of a CMG for SWT
### Parameter Determination  
The transmitted torque of a CMGT is highly dominated by its magnetostatic field distribution, especially for the magnetic flux densities in the outer and inner air-gaps. Hence, this study applied the magnetic flux densities in the outer and inner air-gaps as the objective function to discover the optimum value. The objective function transforms into a function that maximizes the magnetic flux density, as shown in Equation (10). The addition of the outer and inner rotor magnetic flux densities aims to have their maximum flux density:  
\[maxf(X)=B_{g1}+B_{g2} \tag{10}\]  
Six parameters were then chosen as the variables:  
\[X=\left[R1\ R2\ R3\ R4\ R5\ R6\right]^{T} \tag{11}\]  
where \(R1\) and \(R2\) are the outer and inner radius of the outer rotor permanent magnet, respectively, while the outer and inner radii of the flux modulator are denoted as \(R3\) and \(R4\), respectively, and the outer and inner radius of the inner rotor permanent magnet are denoted as \(R5\) and \(R6\), respectively. Those six variables are the parameters that can be changed to derive the optimum flux density in the proposed design. After randomly changing the value and calculating the flux density, the best combination will be determined as design parameters. The constraints are set as follows to derive a value of those variables that does not cause any conflict, due to the radius characteristic, which otherwise must be set as a different number. Equation (10), as the objective function, depends on Equations (12)-(18), which require some geometry parameters. The 4 (four) in the constraints signifies the minimum magnet size due to manufacturing possibilities. If the size is less than 4, it is difficult to fabricate. Because the outer diameter of the CMGT should be 140 mm and the outer iron part set to greater than or equal to 5 mm, the value of 65 appears to limit the maximum radius of \(R1\). The inner iron part's size is also set at 5 mm, so \(R6\) should be greater than or equal to 43 to keep the inner diameter at 76 mm. \[R1\leq 65, \tag{12}\]  
\[R2<R1-4, \tag{13}\]  
\[R3=R2-1, \tag{14}\]  
\[R4<R3-4, \tag{15}\]  
\[R5=R4-1, \tag{16}\]  
\[R6<R5-4, \tag{17}\]  
\[43\leq R6 \tag{18}\]  
Some of the parameters have been mentioned above; the other geometry parameters of magnetic gear can be seen in Table 6. This geometry value will be the initial parameters of this parameter determination. The random approach is used because the constraints that need to be set for the prototype are specific. The value of the variables is limited to ensure its manufacturing possibility. From the variable constraints, the objective function can be linearly solved. Therefore, the random approach is sufficient to find out the optimum combination to be determined as the parameters of the proposed design. The analytical calculation process is performed using MATLAB to achieve the optimum flux density. The geometry parameters shown in Table 6 were used as inputs for the analytical calculation. The variables and constraints were then randomly combined and analyzed. The best variable combination can be achieved after the flux density has reached its optimum value. After going through the above processes, the value of each variable and the flux densities were obtained, as shown in \[X=\left[R1\ R2\ R3\ R4\ R5\ R6\right]^{T} \tag{11}\]0 Only ten combinations are shown in the table, representing the calculation results. From the table, the best value is 1.9387 T, with the value of \(B_{g1}\) being 1.0220 T, and \(B_{g2}\) being 0.9167 T. Finally, the parameters were determined based on the calculation process's best result in \[X=\left[R1\ R2\ R3\ R4\ R5\ R6\right]^{T} \tag{11}\]1 This result was then selected as the final parameters of the magnetic gear to be designed.