## 3 Modeling and Analysis of System  
### Modeling for Offset Coils  
The magnetic vector potential \(A\) is introduced to study the characteristics of offset coils. Unlike the potential, \(A\) is not a measurable quantity and can be directly related to the physical quantity, which can help to solve the field distribution. \(A\) can be used to represent the interconnection of two axially aligned rings, and the magnetic vector potential around the closed path needs to be linearly integrated[24]:  
\[\varphi=\int BdS=\int{(\
abla A)dS}=\oint{Adl}. \tag{10}\]  
In the WCS, when the excitation source is AC voltage, the ability of primary coil to cause an open circuit voltage in the secondary coil is called mutual inductance \(M\), which is formally defined in electromagnetism:  
\[M=\frac{N\varphi}{I}, \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Compensation inductance & \(L_{f1}\)\(L_{f2}\) & 42.4 uH \\ Compensation capacitance 1 & \(C_{f1}\)\(C_{f2}\) & 82.7 nF \\ Compensation capacitance 2 & \(C_{1}\)\(C_{2}\) & 19.1 nF \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of compensation circuit. Figure 4: Equivalent circuit of \(LCC\) compensation topology network. where \(\varphi\) is the magnetic flux connecting the two coils, which is usually a fraction of the total magnetic flux produced by the loosely coupling coils; \(N\) is the turns of coils; and \(I\) is the excitation current of the primary coil. Figure 5 shows an offset diagram of the one turn coil. The horizontal deviation of the coil is \(\Delta y\), and the magnetic vector potential at the point P on secondary coil is as follows:  
\[A=\oint\frac{\mu_{0}I_{1}dl}{4\pi R}, \tag{12}\]  
where \(\mu_{0}\) is the transmission efficiency. According to the geometric relationship, the lateral distance \(x\) and the angle \(\alpha\) can be obtained:  
\[x=\sqrt{a^{2}-\frac{\Delta y^{2}}{4}}, \tag{13}\]  
\[\alpha=2\cos^{-1}(\frac{\Delta y}{2a}). \tag{14}\]  
The effective area that captures the primary flux can be interpreted as the intersection parts of coils, \(B_{\mathit{eff}}\). The overlapping area of the primary and secondary coil is easily obtained from the relationship between \(\Delta y\) and the radius \(a\):  
\[B_{eff}=2\frac{1}{180^{\circ}}\cos^{-1}(\frac{\Delta y}{2a})\pi a^{2}-2(\frac{ \Delta y}{2a^{2}})[\sqrt{\frac{\left(\Delta y\right)^{2}}{\left(2a^{2}\right) ^{2}+1}}]a^{3}. \tag{15}\]  
Then, we can get the relationship of mutual inductance:  
\[M=(\pi a^{2})(B_{\mathit{eff}})\frac{\mu_{0}}{\pi(\sqrt{a^{2}+h^{2}})}^{3}. \tag{16}\]  
For the primary and secondary coils with \(N\) turns, the total mutual inductance can be calculated by superposition of the individual flux:  
\[M=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\mu_{0}(\pi a_{i}{}^{2})(B_{\mathit{eff}, j})}{\pi(\sqrt{a_{j}{}^{2}+h^{2}})}. \tag{17}\]  
It can be seen from Equation (17) that \(M\) is mainly related to the geometrical quantities such as shape, size and relative position of the coil. When the offset \(\Delta y\) increases, \(M\) decreases, which  
Figure 5: Offset diagram: (**a**) coordinate view; (**b**) top view. may affect the transmission efficiency. The relationship between the coupling coefficient \(k\) and the self-inductance of coils is as follows:  
\[k=\frac{M}{\sqrt{L_{1}L_{2}}}. \tag{18}\]  
According to Equation (18), when the change of self-inductance \(L_{1}L_{2}\) caused by the offset is ignored, the coupling coefficient \(k\) is proportional to the mutual inductance. When the offset increases, the ability to transmit electromagnetic fields reduces. Due to coils mounted on the floor in front of vehicle, only the lateral offset of secondary coil in the vehicle coordinate system is considered. Coil models with full alignment, 10%, 20%, 30%, 40% offset and 50% offset are built in Maxwell, as shown in Figure 4:0 The self-inductance of the primary coil and secondary coil, and the mutual inductance between two coils are obtained by simulation. Thus, the coupling coefficient can be calculated.