# Impact of Coil Misalignment in Data Transmission over the Inductive Link of an EV Wireless Charger
## 3 Theoretical Analysis of the Coil Misalignment Effects on the Wireless Channel Capacity  
Coil misalignment refers to the condition in which the two elements do not keep a relative distance equal to the one considered in the design process for any part constituting them. There are three main types of misalignments: vertical, horizontal and joint vertical-horizontal. Figure 3 illustrates these misalignments for square coils, but it is also valid for other kind of coil. As in practical EV wireless charger implementations, the primary coil (with dimensions \(a1\times b1\)) is bigger than the secondary coil, which has a size equal to \(a2\times b2\). In vertical misalignment, the centers of the two coils (\(Center1\) and \(Center2\)) are placed in a line perpendicular to the planes where the primary and the secondary coils are. In the design process \begin{table}
\begin{tabular}{c c c} \hline \hline
**Topology** & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) \\ \hline Series–Series & \(\dfrac{1}{\omega^{2}L_{1}}\) & \(\dfrac{1}{\omega^{2}L_{2}}\) \\ \hline Series–Parallel & \(\dfrac{L_{2}^{2}C_{2}}{L_{1}L_{2}-M^{2}}\) & \(\dfrac{1}{\omega^{2}L_{2}}\) \\ \hline Parallel–Series & \(\dfrac{L_{2}C_{2}}{L_{1}+\dfrac{M^{4}}{L_{1}L_{2}C_{2}R_{p}^{2}}}\) & \(\dfrac{1}{\omega^{2}L_{2}}\) \\ \hline Parallel–Parallel & \(\dfrac{\left(L_{1}L_{2}-M^{2}\right)L_{2}^{2}C_{2}}{\dfrac{M^{4}R_{1}^{2}C_{2} }{L_{2}}+\left(L_{1}L_{2}-M^{2}\right)^{2}}\) & \(\dfrac{1}{\omega^{2}L_{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the capacitors in the single-resonant compensation topologies. Figure 3: Illustration of the types of coil misalignments that may occur in an EV wireless charger: (a) vertical misalignment; (b) horizontal misalignment; (c) vertical and horizontal misalignment. of the coils to determine their features (numbers of turns, the dimensions, the cable section, etc. ), the centers of the coils are assumed to be separated at a distance equal to the gap of the design process (\(gd\)). However, when vertical misalignment happens, this separation (\(gap\)) is modified. This could be greater or smaller than the gap of the design process (\(gd\)). Conversely, in horizontal misalignment, the minimum distance between the center and the plane of the opposite coil is the \(gd\). However, the centers are not aligned and the separation between \(Center1\) and the reflection of \(Center2\) in the plane of the primary coil are \(xm\). Finally, it is possible to have the two misalignments, as illustrated in Figure 3c. The promotion of wireless chargers and their extended use in dynamic or stationary scenarios [13] inevitably imposes coil misalignments in most cases. Thus, it is necessary to analyse the performance of the wireless charger taking into account this circumstance. As previously explained, coil misalignment provokes changes in the magnetic flux that the coils concatenate. As a consequence, the mutual inductance and the induced voltage in the secondary coil are altered [25]. This variation impacts the performances of the power transfer and the communication system. Concerning the effects on the power transfer, [14] demonstrated that the compensation networks have a relevant influence on the capability to cope with coil misalignments. However, extension of this previous analysis is required to determine the consequences for communication performance. The appropriateness of a system to transfer data can be quantified by its channel capacity. This parameter represents the limit of the bit rate that a data source should generate with advanced modulation techniques. According to the Shannon-Hartley theorem, this magnitude is computed as:  
\[C=B\cdot log_{2}\bigg{(}1+\frac{P_{L}}{N_{o}B}\bigg{)} \tag{1}\]  
where \(B\) and \(P_{L}\) correspond to the channel bandwidth and the power perceived in the receiver, respectively. The bandwidth is defined as the double the difference between the central frequency and the frequency at which the power is half of the power received at the central frequency.