# A Novel Approach to Reach Impedance Matching in Wireless Power Transfer Systems
## 2 Circuit Theory Analyzing  
Figure 1 shows the simplified circuit model of the four-coil WPT system. It is composed of four coils: Drive loop, transmitting (\(Tx\)) coil, reserving (\(Rx\)) coil, and load loop. Each of these can be equivalent to lumped circuit elements (\(L\), \(C\), and \(R\)) in series. \(R_{i}\)\(C_{i}\) and \(L_{i}\) (\(i\) = 1, 2, 3, 4) are equivalent to the internal resistance, compensation capacitor and self-inductance of the coils, respectively. The drive loop is excited by a source \(V_{S}\) with output impedance \(R_{S}\). \(R_{L}\) represents the pure resistance load linked to the load loop. \(M_{ij}\) is the mutual inductance of \(i\) coil and \(j\) coil (\(i\), \(j\) = 1, 2, 3, 4, \(i\
eq j\)). To simplify the analysis and calculation, the cross mutual inductance (\(M_{13}\), \(M_{14}\) and \(M_{24}\)) is neglected. \(K_{ij}\) is the coupling coefficient between two adjacent coils. The coupling coefficient is related to the mutual inductance as follows. \[k_{ij}=\frac{M_{ij}}{\sqrt{L_{i}L_{j}}} \tag{1}\]  
Based on mutual coupling theory and the Kirchhoff voltage law (KVL), the matrix equation of the four-coil WPT system can be derived as \[\left[ \begin{array}{cccc}Z_{11}&-j\omega M_{12}&0&0\\ -j\omega M_{12}&Z_{22}&-j\omega M_{23}&0\\ 0&-j\omega M_{23}&Z_{33}&-j\omega M_{34}\\ 0&0&-j\omega M_{34}&Z_{44}\end{array} \right]\left[ \begin{array}{c}I_{1}\\ I_{2}\\ I_{3}\\ I_{4}\end{array} \right]=\left[ \begin{array}{c}V_{s}\\ 0\\ 0\\ 0\end{array} \right] \tag{2}\]  
where \(Ii\) is the current of each loop of coils, \(\omega\) is the angular frequency of the operating WPT system, \(\omega=2\pi f\), showed in Figure 1. \(Z_{11}\), \(Z_{22}\), \(Z_{33}\) and \(Z_{44}\) are subject to the following relations. \[\left\{ \begin{array}{l}Z_{11}=R_{s}+R_{1}+j\omega L_{1}+\frac{1}{j\omega C_{1 }}\\ Z_{22}=R_{2}+j\omega L_{2}+\frac{1}{j\omega C_{2}}\\ Z_{33}=R_{3}+j\omega L_{3}+\frac{1}{j\omega C_{3}}\\ Z_{44}=R_{L}+R_{4}+j\omega L_{4}+\frac{1}{j\omega C_{4}}\end{array} \right. \tag{3}\]  
To further simplify the analysis, \(Z_{12}\), \(Z_{23}\), and \(Z_{34}\) are defined as follows. \(Z_{12}\) is the reflect impedance from load loop to Rx coil. \(Z_{23}\) is the reflect impedance from \(Rx\) coil to \(Tx\) coil. \(Z_{34}\) is the reflect impedance from \(Rx\) coil to driven loop. \[\left\{ \begin{array}{l}Z_{34}=\frac{\left(\omega M_{34}\right)^{2}}{Z_{34}} \\ Z_{23}=\frac{\left(\omega M_{23}\right)^{2}}{Z_{33}+Z_{34}}\\ Z_{12}=\frac{\left(\omega M_{12}\right)^{2}}{Z_{22}+Z_{23}}\end{array} \right. \tag{4}\]  
Substituting (3) and (4) into (2), the currents in the drive loop and the load loop can be expressed as  
\[ \begin{array}{l}I_{1}=\frac{V_{s}}{Z_{11}+Z_{12}}\\ I_{4}=\frac{j\omega^{3}M_{12}M_{23}M_{34}}{(Z_{11}+Z_{12})(Z_{22}+Z_{23})(Z_{33 }+Z_{34})Z_{44}}V_{s}\end{array}  \tag{5}\]  
According to the traditional expression of electrical transfer efficiency, \(\eta\), the efficiency of which is equal to the ratio of the effective power load \(P_{load}\) and the input power \(P_{in}\). It is calculated as follows. \[ \begin{array}{l}P_{in}=\text{Real}(I_{1}V_{s})\\ P_{load}=|I_{4}|^{2}R_{L}\\ \eta=\frac{P_{load}}{P_{in}}=\frac{\text{Real}(Z_{11}+Z_{12})}{|Z_{11}+Z_{12}|} \cdot\left|\frac{Z_{12}}{Z_{11}+Z_{12}}\right|\cdot\left|\frac{Z_{33}}{Z_{22}+ Z_{23}}\right|\cdot\left|\frac{Z_{34}}{Z_{33}+Z_{34}}\right|\cdot\left|\frac{R_{L}}{Z_{ 44}}\right|\end{array}  \tag{6}\]  
Also from [24], the squared magnitude of \(S_{21}\) is simply the ratio of delivered output power to maximum available source power, which can be expressed as follows:  
\[\left|S_{21}\right|^{2}=\left(2\Big{|}\frac{V_{L}}{V_{s}}\Big{|}\sqrt{\frac{R_ {S}}{R_{L}}}\right)^{2}=4\cdot\left|\frac{Z_{12}}{Z_{11}+Z_{12}}\right|\cdot \left|\frac{Z_{23}}{Z_{22}+Z_{23}}\right|\cdot\left|\frac{Z_{34}}{Z_{33}+Z_{34} }\right|\cdot\left|\frac{R_{L}}{Z_{44}}\right|\cdot\left|\frac{R_{L}}{Z_{11}+ Z_{12}}\right| \tag{7}\]  
where the load \(R_{L}\) is equal to the equivalent internal resistance of the source \(R_{S}\).