## 3 Resonator Design
### Data Coil Design
By incorporating flux components of opposite sign, the effective flux in the data coil structure can be minimized. Zero coupling can be achieved by appropriate selection of \(r_{\text{c2,i}}\) and \(s_{\text{b,j}}\). 3 and 4 indicate the primary and secondary data coil, respectively. It is therefore of interest to compute \(|M_{34}|\) for various coil distances \(d\) and lateral displacements \(l\) (see Figure 1a) to spot the regions of limited data coil coupling, as well as the ratio \(|M_{14}|/|M_{34}|\) to obtain the strength of interference from primary energy to secondary data coil and \(|M_{23}|/|M_{34}|\) to analyze interference from secondary energy to primary data coil. To compute the mutual inductance even for the case of these non-regular coils, the Neumann formula describing \(M_{ik}\) based on line integrals along the conductor traces of both coils \(i\) and \(k\) (see, e.g., [29]) is approximated by  
\[M_{ik} = \frac{\mu_{0}}{4\pi}\oint_{l_{i}}\oint_{l_{j}}\frac{d\vec{r}_{i} \cdot d\vec{r}_{k}}{|\vec{r}_{i}-\vec{r}_{k}|} \tag{5}\] \[\approx \frac{\mu_{0}}{4\pi}\sum_{j=1}^{z_{i}-1}\sum_{l=1}^{z_{i}-1} \frac{(\vec{r}_{i(j+1)}-\vec{r}_{ij})\cdot(\vec{r}_{k(l+1)}-\vec{r}_{kl})}{| \vec{r}_{ij}-\vec{r}_{kl}|}. \tag{6}\]  
This means that both coil trajectories are discretized into a series of \(z_{i}\) points with coordinates \(\vec{r}_{ij}\), so that an infinitesimal line element of the Neumann equation becomes a vector between two subsequent trajectory points of a coil (such as indicated by the small black arrows in Figure 3) and that the distance of the line elements to be integrated becomes the distance of the vectors' end points. Self-inductance can also be approximated with this discretization by computing the mutual inductance of a coil to itself (\(i=k\)); a small offset \(|\Delta\vec{r}_{o}|\ll|\vec{r}_{ij}-\vec{r}_{il}|\) must, however, be included in order to avoid singularities:  
\[L_{\rm i}\approx\frac{\mu_{0}}{4\pi}\sum_{j=1}^{z_{i}-1}\sum_{l=1}^{z_{i}-1} \frac{(\vec{r}_{i(j+1)}-\vec{r}_{ij})\cdot(\vec{r}_{i(l+1)}-\vec{r}_{il})}{| \vec{r}_{ij}-\vec{r}_{il}+\Delta\vec{r}_{o}|}. \tag{7}\]  
In all simulations of this work, we used 1000 discretization points per turn, so that \(z_{i}=1000\times N_{i}\). The coordinates \(\vec{r}_{ij}\) of the coil trace vectors of Figure 3 were parametrized in a custom program written in _Wolfram Mathematica 12.1;_ mutual inductances and self-inductances were then obtained by numerically evaluating the summations in (6) and (7) among all points \(\vec{r}_{ij}\). With this numerical approach, both coil topologies were optimized for minimal interference to the on-substrate energy coils by selecting \(r_{\rm c2,i}\) or \(s_{\rm b,j}\) to yield \(M_{13}=M_{24}=0\). In all cases, an outer radius of 17 mm for the primary and 7 mm for the secondary data coil (slightly larger than the energy coils) and three turns (yields inductance values that allow a suitable impedance matching to 100 \(\Omega\), see the next section) are assumed. Subsequently, \(|M_{34}|\) characterizing the data interface as well as \(|M_{14}|/|M_{34}|\) and \(|M_{23}|/|M_{34}|\), characterizing the interference to the WPT system, are computed from (6) for various distances \(d\) and lateral misalignments \(l\) (cf. Figure 3 for coordinates). The results are shown in Figure 4: A data interface consisting of the C-shaped coils reveals a zero of mutual inductance for a coil separation distance of \(d=4\) mm and axial alignment (\(l=0\) mm), so that data transfer is strongly limited at this spatial arrangement. The limited mutual inductance in this region also facilitates potential interference from the primary energy coil due to a high ratio of \(|M_{14}|/|M_{34}|\). When being axially aligned, the butterfly-shaped coils show a monotonic decrease of the data coils' mutual inductance for an increasing distance and feature a particularly small mutual inductance to the energy coils (low interference), so that they are selected for the system implementation due to these favorable characteristics.