# A Novel Approach to Reach Impedance Matching in Wireless Power Transfer Systems
## 2 Circuit Theory Analyzing
In order to achieve the resonance condition, the operating frequency \(f_{0}\) of the system will be guaranteed to be the same as the resonance frequency \(f_{i}\) of the coils. Moreover, the input impedance \(Z_{in}\) is approximately equal to \(Z_{12}\) because \(R_{1}\) is very small. For the characterization of the scattering parameters, the prerequisite is to ensure that the maximum power output, that is, \(Z_{12}\) matches the internal resistance of the source. Then, the square of \(\left|\,S_{21}\,\right|\) is equal to two times of \(\eta\), as in Figure 2c,d. In terms of the traditional electric, aimed at gaining maximum transfer efficiency, we can make \(Z_{12}\) large enough by adjusting the relevant parameters, as shown in Figure 2a, so that the modulus of \(Z_{12}/(Z_{12}+Z_{11})\) is close to one. At this time, \(\eta\) will be close to \(\left|\,S_{21}\,\right|^{2}\) according to Figure 2b,d. Thus, \(\left|\,S_{21}\,\right|^{2}\) can be regarded as the maximum transfer efficiency that the system may achieve. In the following analysis and experiment,will be used as the main analysis parameter of the system. A detailed description of the simulated parameters is listed in Table 1.  
Figure 1: Equivalent circuit model of the four-coil WPT system. Figure 2: Numerical analysis results of the four coil system, (**a**) \(Z_{12}\) versus \(k_{12}\) and \(k_{34}\), (**b**) transfer efficiency versus \(k_{12}\) and \(k_{34}\), (**c**) is the y-z view of (**b,d**) \(\mid\)\(S_{21}\)\(\mid\)\({}^{2}\) versus \(k_{12}\) and \(k_{34}\). Based on the matching conditions of the scattering parameters, \(R_{S}\), \(R_{L}\) and \(Z_{in}\) are equal. Equation (7) can be simplified as follows:  
\[|S_{21}|^{2}=\left|\frac{Z_{23}}{Z_{22}+Z_{23}}\right|\cdot\left|\frac{Z_{34}}{Z _{33}+Z_{34}}\right| \tag{8}\]  
The four-coil WPT system can be equivalent to a two-coil system, as shown in Figure 3. The drive coil is equivalent to the voltage source, and the load coil is represented as \(Z_{34}\). According to the previous literature [16], in order to obtain the maximum \(S_{21}\), the derivative of Expression (8) with respect to \(Z_{34}\) is taken, and then the derivative result is set to zero. The optimal \(Z_{34}\) which maximizes \(S_{21}\) can be gained and the optimal \(k_{34\text{-}opt}\) will also be attained by transformation. At the same time, \(k_{34\text{-}opt}\) is substituted into the expression of \(Z_{12}\), and the optimal \(k_{12\text{-}opt}\) can also be attained. \[ \begin{split} k_{34\text{-}opt}&=\frac{1}{\omega} \sqrt{\frac{R_{L}R_{3}}{L_{3}L_{4}}\sqrt{1+\frac{\left(\omega k_{23}\sqrt{L_{ 2}L_{3}}\right)^{2}}{R_{2}R_{3}}}}\\ k_{12\text{-}opt}&=\frac{1}{\omega}\sqrt{\frac{R_{L }R_{2}}{L_{1}L_{2}}\sqrt{1+\frac{\left(\omega k_{23}\sqrt{L_{2}L_{3}}\right)^{ 2}}{R_{2}R_{3}}}}\end{split}  \tag{9}\]  
As shown in Figure 2b,c, the red line denotes the transfer efficiency when \(k_{34}\) is fixed and consistent with the calculation result of Equation (9). It is worth mentioning the complete symmetry four-coil WPT system when the transmission distance is fixed, existing the same \(k_{12}\) and \(k_{34}\) values, that maximize the transfer efficiency. This result is consistent with Expression (9), and it is also confirmed by the simulation, as shown in Figure 2d. Finally, \(k_{23}\) is a constant value, which means that the transmission distance is fixed, \(k_{34}\) is a variable to achieve the optimal load, and \(k_{12}\) can be changed to match the internal resistance of the source in the simulation. Thus we found that the impedance matching of the overall system is necessary to achieve the maximum transfer efficiency.