# A Novel Approach to Reach Impedance Matching in Wireless Power Transfer Systems
## 3 Calculating Couple Coil Parameters  
In this paper, a novel method has been proposed to calculate the self and mutual inductance of multiple-turn printed spiral coils (PSC), as shown in Figure 4a. The PSC is a square coil with a rectangular cross-section. As described in [25], the PSC can be equivalent to a set of concentric and coplanar single-turn coils, connected in series, as shown in Figure 4b. The error under this assumption can be neglected. Moreover, the computed results were verified using the finite simulation software Ansys Maxwell 16.0. In [26], many formulas and tables used to calculate the self-inductance and mutual inductance of various cross-sections and shape conductors were presented, but it is not apparent that the listed expressions can be directly applied to the PSC in this paper. Combining [26] with the coil model in this paper produced an expression that exhibits a good approximation for the self-inductance of a one-turn inductor in air:  
\[L_{one-self}=\frac{2\mu_{0}a}{\pi}\left(\ln\frac{2a}{1+\sqrt{a}}+\sqrt{2}-2- \ln 0.2232(b+c)\right) \tag{10}\]  
where \(\mu_{0}\) is the permeability of the vacuum, \(a\) is the length of the side of one turn coil, and \(b\), \(c\) are the width and thickness of rectangular cross-section copper conductors. According to superposition theorem, the self-inductance value of the PSC can be viewed as the sum of the self-inductance of the single-turn coils and the mutual inductance between two non-adjacent coils. The general self-inductance of the PSC can be written as:  
\[L_{Mul-self}=\sum_{i=1}^{N}L_{i-self}+\sum_{j=1}^{N}\sum_{i=1}^{N}M_{ij},\ (j \
eq i) \tag{11}\]  
where \(L_{i\text{-}self}\) is the self-inductance of the \(i\)th turn of an inductor with \(N\) turns. \(M_{ij}\) is the mutual inductance between the \(i\)th turn and the \(j\)th turn of the square spiral coils, and \(i\) is not equal to \(j\). The mutual inductance of two parallel coaxials can be obtained by summing the mutual inductance between two turns. The structure diagram is shown in Figure 5a. In this WPT system, the impedance matching is achieved by varying the angle of the inner coils, therefore, the mutual inductance of two coils in different planes is a key factor in the paper. A sketch of two non-plane coils is presented in Figure 5b. Figure 4: Equivalent model of printed spiral coils: (**a**) physical truth, (**b**) equivalent mode. It is crucial to calculate the mutual inductance of single-turn coils. When analyzing the proposed position relation of coils, every turn of coils can be deemed as connecting with four line segments. The segments of different coils have three position relations: parallel, vertical or non-coplanar. According to the Neumann formula (12), which represents the mutual inductance of two line segments, we found that the mutual inductance of vertical line segments is zero. Therefore, the position relation of the line segments can be expressed as disjointed while considering the mutual inductance. \[M=\frac{\mu_{0}}{4\pi}\underset{l_{1}}{\int}\underset{l_{2}}{\int}\frac{ \overset{\rightarrow}{\mathrm{d}\;l_{1}\;\mathrm{d}\;l_{2}}}{\mathrm{D}} \tag{12}\]  
Figure 6a shows two non-intersecting lines in space, two planes \(\theta_{1}\) and \(\theta_{2}\) where the prescribed lines inside them can always be found. \(a\) and \(b\) are the length of these lines, and the inclined angle of the presented lines is \(\varphi\) in the range of \(0\leq\varphi\leq\pi\). The vertical distance between the two planes is \(h\). \(x\) and \(y\) are the coordinate axis and the orientation consistent with the current direction of wires, respectively. \(O_{1}\) and \(O_{2}\) are original points of the coordinate axis, and \(x_{1}\), \(x_{2}\), \(y_{1}\) and \(y_{2}\) are the beginning and end terminals of the lines, respectively. Figure 5: Relative position relation of coils, (**a**) parallel coaxial coils, (**b**) concentric coils. Figure 6: (**a**) General line segment position relation; (**b**) Diagrammatic drawing of single-turn coils.