# A Novel Approach to Reach Impedance Matching in Wireless Power Transfer Systems
## 3 Calculating Couple Coil Parameters
From (12), we obtain the calculation formula for the mutual inductance of two general line segments, which can be written as:  
\[ \begin{array}{ll}M=&\frac{\mu_{0}}{2\pi}\cos\varphi\Big{(}x_{2}\text{ arctanh}\frac{b}{D_{22}+D_{21}}+y_{2}\text{arctanh}\frac{a}{D_{22}+D_{12}}\\ &-x_{1}\text{arctanh}\frac{b}{D_{11}+D_{12}}-y_{1}\text{arctanh}\frac{a}{D_{1 1}+D_{21}}+\frac{h}{\sin\varphi}A\Big{)}\end{array}  \tag{13}\]  
where \(D_{11}\), \(D_{12}\), \(D_{21}\), \(D_{22}\) are the distance of the beginning and end terminal of the lines, respectively, which can be expressed as:  
\[ \begin{array}{ll}D_{11}^{2}=x_{1}^{2}+y_{1}^{2}-2x_{1}y_{1}\cos\varphi+h^{2 }\\ D_{12}^{2}=x_{1}^{2}+y_{2}^{2}-2x_{1}y_{2}\cos\varphi+h^{2}\\ D_{21}^{2}=x_{2}^{2}+y_{1}^{2}-2x_{2}y_{1}\cos\varphi+h^{2}\\ D_{22}^{2}=x_{2}^{2}+y_{2}^{2}-2x_{2}y_{2}\cos\varphi+h^{2}\end{array}  \tag{14}\]  
\(A\) as an intermediate variable is given as:  
\[ \begin{array}{ll}A=&\text{arctan}(\frac{x_{1}+y_{1}+D_{11}}{h}\tan\frac{ \varphi}{2})+\text{arctan}(\frac{x_{2}+y_{2}+D_{22}}{h}\tan\frac{\varphi}{2}) \\ &-\text{arctan}(\frac{x_{1}+y_{2}+D_{12}}{h}\tan\frac{\varphi}{2})-\text{ arct}(\frac{x_{2}+y_{1}+D_{21}}{h}\tan\frac{\varphi}{2})\end{array}  \tag{15}\]  
For the WPT model analyzed in this paper, the existent position relation of the segments is shown in Figure 6. \(a_{n}\), \(b_{m}\) and \(d_{l}\) are the side of the single-turn coil \(i\), the side of the single-turn coil \(j\), and the distance of the center of segments (\(n\), \(m\), \(l\) = 1, 2, 3, 4), respectively. \(I_{a}\) and \(I_{b}\) are the currents of coil \(i\) and coil \(j\), respectively. Then \(M_{ij}\) can be represented as:  
\[ \begin{array}{ll}M_{ij}=&\left(M_{b_{1}a_{1}}+M_{b_{1}a_{3}}\right)_{b_{1}}+ \left(M_{b_{3}a_{1}}+M_{b_{3}a_{3}}\right)_{b_{3}}\\ &+\left(M_{b_{2}a_{2}}+M_{b_{2}a_{4}}\right)_{b_{2}}+\left(M_{b_{4}a_{2}}+M_{b _{4}a_{4}}\right)_{b_{4}}\end{array}  \tag{16}\]  
where \(M_{b_{m}a_{n}}\) is the mutual inductance of the \(b_{m}\) side and the \(a_{n}\) side. We found that \(M_{ij}\) is composed of four parts and every part can be seen as a mutual inductance of the \(b_{m}\) side with coil \(i\). \(M_{ij}\) will be further simplified while the analyzed model is symmetrical. The self-inductance of multiple-turn square spiral coils and the mutual inductance between the parallel coaxial coils or the homocentric coils with an angle can be calculated easily by MATLAB. Moreover, when \(\varphi\) is equal to zero or \(\pi\), this represents the presence of parallel segment conductors, but with different current directions inside the conductors. In addition, it is worth noting that the current direction inside the wire is of great importance while calculating the mutual inductance of the square loops. In order to verify the correctness of the above analysis, the simulation model using Ansys Maxwell 16.0 was established. The structure parameters of the coil are listed in Table 2, and the definitions of the parameters are shown in Figure 7.  
Figure 7: Simulation model diagrammatic drawing: (**a**) structure, (**b**) explanation of the symbols. The simulation and calculation results are shown in Figure 8. \(d_{23}\) is the distance between the \(Tx\) coil and the \(Rx\) coil, which is the transmission distance. \(d_{12}\) is the distance between the drive coil and the \(Tx\) coil, and \(\varphi\) is the angle between the load coil and the \(Rx\) coil. The coupling strength of the \(Rx\) coil and the \(Tx\) coil will reduce when \(d_{23}\) increases, as shown in Figure 8a. The drive loop and \(Tx\) coil follow the same rule, and their coupling strength drops with increasing distance, as shown in Figure 8b. This will be used to match the internal resistance of the source by changing \(d_{12}\). Figure 8c shows the relationship between the mutual inductance of the \(Rx\) coil and the load loop, and its angle and optimal load were attained by adjusting \(\varphi\) in the experiment.