## 3 Analysis of Parameter for Designing Pickup Coil in PCB
### Distance between Conductor and Coil  
Figure 7 and Table 3 presents the FEM results of mutual inductance and inductive coupling index according to the distance between the coils (Model-A and Model-C) and the conductor through which the primary current flows. From the results of both structures, it is confirmed that the \(L_{c}/M\) of the coil increased as the distance between the coil and the conductor increased in the two coil models in the given \(d\) range. In the vertical structure, \(d\) is determined as much as the height of the layer; hence, the range that can be varied is limited. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Coil Type** & \multicolumn{4}{c}{**Vertical Structure (Model-A)**} & \multicolumn{4}{c}{**Horizontal Structure (Model-C)**} \\ \hline a \(\times\) b (mm \(\times\) mm) & 3.25 \(\times\) 0.4 & 3.25 \(\times\) 0.8 & 3.25 \(\times\) 1.2 & 3.25 \(\times\) 1.6 & 2.92 \(\times\) 2.18 & 2.92 \(\times\) 3.2 & 2.92 \(\times\) 3.95 & 2.92 \(\times\) 4.63 \\ \hline Area \(S\) (mm\({}^{2}\)) & 1.3 & 2.6 & 3.9 & 5.2 & 6.36 & 9.34 & 11.5 & 13.5 \\ \hline Inductive coupling & 0.009 & 0.02 & 0.034 & 0.037 & 0.09 & 0.129 & 0.155 & 0.174 \\ \hline Mutual inductance \(M\) [nH] & 0.28 & 0.38 & 0.45 & 0.58 & 0.9 & 1.35 & 1.75 & 2.01 \\ \hline Self inductance \(L_{s}\) [nH] & 4.51 & 5.17 & 5.71 & 6.58 & 6.29 & 7.62 & 8.69 & 9.63 \\ \hline \(L_{c}/M\) & 16.1 & 15 & 12.6 & 11.3 & 6.9 & 5.6 & 4.96 & 4.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Influence of the inductance under different magnetic flux incident areas \(S\) with the FEA method. Figure 7: FEM results under different distances: (**a**) vertical structure coil model and (**b**) horizontal structure coil model. A pickup coil is a typical non-contact insulation sensor, but the coupling noisecurrent affects the sensor coil owing to the coupling capacitance, \(C_{coup}\), between conductors formed by the close distance between the conductor and sensor coil. In this case, particularly for a WBG device such as a GaN semiconductor, which has a high dv/dt characteristic, the sensitivity of coupling noise in the power semiconductor affects the sensor coil during switch operation, hence the overshoot or ringing phenomena due to noise can occur at the coil output. \[i_{c}=C_{coup}\frac{\mathrm{d}}{\mathrm{dt}}v_{ext}(t) \tag{18}\]  
where \(u_{ext}\) is the voltage gradient of the drain to source voltage of the power semiconductor, \(i_{c}\) is the coupling current in the pickup coil, and \(v_{e}\) is EMF of the coil and the arrows on the circuit show the flow of the coupling noise current. Figure 8 shows the sensor coil equivalent circuit, considering the formation of the coupling capacitance [23]. In the coil output stage, the original coil EMF \(v_{coil}\) and the voltage to which the voltage drops due to coupling noise \(v_{coup}\) is added. \[v_{o}=v_{coil}+v_{coup} \tag{19}\]  
Figure 9 and Table 4 presents the results of analyzing the influence of the coupling capacitance according to the distance between the pickup coil and the conductor and through which the primary current flows, using the finite element method. From the Table 4 results, it was confirmed that the smaller \(d\), the larger the value of the coupling capacitance in the two coil models, and this shows that it is necessary to consider the coupling noise according to \(d\). Moreover, the value of the coupling capacitance is larger in the vertical structure (Model-A) than in the horizontal structure (Model-C). Figure 8: Equivalent circuit of a pickup coil with coupling capacitive interferences.