## 2 Proposed WPT System with the MMs
### Theoretical Analysis of the Four-Coil WPT System  
Figure 2 shows the equivalent circuit diagrams of the four-coil WPT system. All the parameters are considered in the complete model. As a contrast, the equivalent model can be simplified to ignore the less influential quantities. The internal resistance of the power source and the load equivalent resistance are \(R_{S}\) and \(R_{L}\), respectively. The signal source is denoted by Vs. The distance of the \(Tx\) coil and \(Rx\) coil is defined as the transfer distance in the four-coil WPT system. The electrical parameters of each coil are denoted by \(R_{i}L_{i}C_{i}\) (\(i,j=1,\,2,\,3,\,4\)). \(M_{ij}\) is the mutual inductance of two adjacent coils, and the cross coupling parameters are negligible (\(M_{13}=M_{24}=M_{14}=0\)). Compared with the \(Rs\) and \(R_{L}\), the coil equivalent resistance of the drive loop and the load loop are so tiny so are neglected. In addition, the four coils have same operating resonance frequency. The simplified circular model is illustrated in Figure 2b. Figure 1: The architecture of the Wireless Power Transfer (WPT) system based on the metamaterials (MMs). Figure 2: The equivalent circuit model of the four-coil WPT system. (**a**) The complete model. (**b**) The simplified model. When the proposed four coils work at same resonant frequency of 2.80 MHz, according to the Kirchhoff voltage law (KVL) and the mutual coupling theories, the voltage and current in each coil can be illustrated as follows:  
\[\left\{ \begin{array}{l}V_{s}=R_{s}I_{1}+j\omega M_{12}I_{2}\\ 0=R_{2}I_{2}+j\omega M_{12}I_{1}+j\omega M_{23}I_{3}\\ 0=R_{3}I_{3}+j\omega M_{23}I_{2}+j\omega M_{34}I_{4}\\ 0=R_{L}I_{4}+j\omega M_{34}I_{3}\end{array} \right. \tag{1}\]  
Similarly, the input impedance, \(Z_{in}\), is a very significant parameter for the design of Class-E, and it can be expressed as:  
\[Z_{in}=\frac{V_{s}-R_{s}I_{1}}{I_{1}} \tag{2}\]  
Solving (1) and (2), the \(Z_{in}\) can be derived [7], which is:  
\[Z_{in}=R_{s}\cdot\frac{{k_{12}}^{2}Q_{1}Q_{2}(1+{k_{34}}^{2}Q_{3}Q_{4})}{1+{k_{ 23}}^{2}Q_{2}Q_{3}+{k_{34}}^{2}Q_{3}Q_{4}} \tag{3}\]  
where \(Q_{i}\) is the quality factor of the \(i\)th coil. \(Q_{1}\) is equal to \(Q_{4}\) and \(Q_{2}\) is the same as the \(Q_{3}\) due to the symmetry of the proposed system. \(k_{12}\), \(k_{23}\), and \(k_{34}\) are the coupling coefficients of the drive loop and the \(Tx\) coil, the \(Tx\) coil and the \(Rx\) coil, and the load loop and the \(Rx\) coil, respectively. When the distance between the coils changes, the coupling coefficient varies correspondingly, and eventually the \(Z_{in}\) changes, as shown in the Figure 3. It can be seen that as \(k_{12}\) and \(k_{34}\) increase, the efficiency also is improved. For a traditional two-coil WPT system, the secondary coil can be reflected as the pure resistance when two coils are in resonance. By adjusting the load resistance value, the system can achieve the maximum efficiency output, namely, there is an optimal load in the system. The optimal load can be expressed as [23]:  
\[R_{L}=\sqrt{1+k^{2}Q_{1}Q_{2}}\cdot R_{2} \tag{4}\]  
in which \(k\) is the coupling coefficient, \(Qi\) is the quality factor of the \(i\)th coil, and \(R_{2}\) is the resistance of the secondary coil. Similarly, the four-coil WPT system can be equivalent to the traditional two-coil resonator. The drive loop can be reflected to the \(Tx\) coil and the load loop can be reflected in the \(Rx\)  
Figure 3: The relationship (**a**) between input impedance \(Z_{in}\) and \(k_{12}\), and (**b**) between input impedance \(Z_{in}\) and \(k_{34}\). coil, as shown in Figure 4. _Resqt_, _Vseqt_, and _RLeqt_ is the equivalent resistance, voltage supply, and the load resistance, respectively.