## 3 Design and Optimization of the Proposed Extended DD Coil
### Design of the Extended DD Coil  
After the coil size has been determined, the extended DD coil can be optimally designed. The purpose of the optimized design is to minimize the cross-coupling of \(L_{in}\) to \(L_{p}\) and \(L_{s}\) when the system is in an aligned state. This results in a mutual decoupling effect and the achievement of a high-efficiency magnetic coupling structure. Furthermore, the optimized extended DD coil increases the system output power by maximizing the coupling between \(L_{in}\) and \(L_{s}\) when the system is in a misaligned state. Particularly, the extended DD coil comprises two oppositely electrically connected unipolar coils (and \(D_{2}\)), and the distance between them is used as a variable for the optimal design in this paper. The magnetic coupling structure designed was simulated by the FEM tool. Specifically, the door-to-door direction was set as the \(X\)-axis, and the front-to-rear direction was the \(Y\)-axis. First, the DD coil is adopted as the compensation inductor in the conventional integration method; the distance between \(D_{1}\) and \(D_{2}\) is 0 mm. Then, based on the DD coil, when the design variable is changed to shift the distance between \(D_{1}\) and \(D_{2}\) along the \(\pm X\)-axis or \(\pm Y\)-axis direction, the variation in the coupling coefficient between \(L_{p}\) and \(L_{s}\) is as shown in Figure 5a. It can be observed that \(D_{1}\) and \(D_{2}\) shift 0 to 80 mm on the \(\pm X\)-axis or 0 to 15 mm on the \(\pm Y\)-axis direction; \(K_{p-s}\) only changes by a small value of 0.0005 from 0.2160 to 0.2165 in the aligned state. Simultaneously, \(K_{in-p}\) and \(K_{in-s}\) are close to zero, which can be ignored; therefore, the changes are not shown here. Since the transmitter coil is rectangular, the maximum misalignment distances for the \(Y\)-axis and \(X\)-axis are set to 100 mm and 150 mm, respectively. When the system is an \(X\)-axis direction misalignment of 150 mm, the \(K_{p-s}\) decreases due to changes in the magnetic field distribution. After the \(D_{1}\) and \(D_{2}\) move along the \(\pm X\)-axis or \(\pm Y\)-axis, \(K_{in-p}\) is still  
Figure 4: 3-D geometry and dimensions of the designed magnetic coupling structure. Figure 5: (a) \(K_{p-s}\) changes at different distances between \(D_{1}\) and \(D_{2}\) of the extended DD coil of the system in the aligned state and (b) \(K_{in-s}\) changes at different distances between \(D_{1}\) and \(D_{2}\) in the 150 mm \(X\)-axis misaligned state. close to 0, but the value of \(K_{in-s}\) changes, as shown in the simulation results in Figure 5b. As \(D_{1}\) and \(D_{2}\) move along the \(\pm\)X-axis, the value of \(K_{in-s}\) increases. At this point, the second magnetic coupling of the system is generated, according to (11); hence, the designed magnetic coupling structure can slow down the decline in the \(P_{out}\) of the system in the misaligned state. If the distance moves along the \(\pm\)X-axis and then the \(\pm\)Y-axis, the second coupling will first increase and then decrease. For example, as the distance between \(D_{1}\) and \(D_{2}\) moves by 80 mm along the \(\pm\)X-axis, \(K_{in-s}\) reaches its maximum of 0.0683. If the distance between \(D_{1}\) and \(D_{2}\) moves along the \(\pm\)Y-axis by 15 mm, the value of \(K_{in-s}\) is reduced to 0.0562. As a result, \(D_{1}\) and \(D_{2}\) move 80 mm along the \(\pm\)X-axis, with the \(\pm\)Y-axis position remaining unchanged as it is the best position for \(D_{1}\) and \(D_{2}\), as shown in Figure 3a. Therefore, it can be seen from the above discussion that the proposed fully integrated LCCL-S topology has two coupling coefficients, and consequently, two magnetic flux transfer paths. Precisely, one path is from \(L_{p}\) to \(L_{s}\), forming the first magnetic coupling, and the other path is from \(L_{in}\) to \(L_{s}\), forming the second magnetic coupling.