## 3 Analysis of Parameter for Designing Pickup Coil in PCB
### Pickup Coil Configuration in PCB
Figure 3: Geometry of (**a**) Model-A: vertical structure of a pickup coil with conductor trace on one side; (**b**) Model-B: vertical structure of a pickup coil with conductor trace on two sides; (**c**) Model-C: horizontal structure of a pickup coil with conductor trace on one side; (**d**) Model-D: horizontal structure of a pickup coil with conductor trace on two sides; and (**e**) Model-E: horizontal structure of a pickup coil with conductor trace on three sides. Figure 3d,e illustrates cases in which the numbers of traces of the primary current flowing around the pickup coil on the same plane are two and three, respectively. In these three cases, the magnetic flux incident on the coil flows in a direction perpendicular to the PCB surface, and the direction of the magnetic flux incident on each trace surface is the same where \(d\) represents the distance between the coil and the conductor and was set to the same value for the models shown in Model-C, D, and E. The magnetic flux of Model-C is expressed as follows:  
\[\phi_{\text{C}}=\frac{\mu_{o}\cdot a\cdot i_{1}}{2\pi}\ln\frac{(b+d)}{d} \tag{12}\]  
The mutual inductance \(M_{\text{C}}\) between the conductor trace and the coil is as follows:  
\[M_{\text{C}}=\frac{\mu_{o}\cdot a}{2\pi}\ln\frac{(b+d)}{d} \tag{13}\]  
In Model-D, the pickup coil is surrounded by the coil traces on both sides where the primary current flows; therefore, the magnetic flux incident in the coil is given as follows:  
\[\phi_{D}=\frac{\mu_{o}\cdot i_{1}}{\pi}\bigg{\{}\frac{a}{2}\ln\frac{(b+d)}{d} +\frac{b}{2}\ln\frac{(a+d)}{d}\bigg{\}} \tag{14}\]  
In addition, the mutual inductance between the coil and trace is given as follows:  
\[M_{D}=\frac{\mu_{o}}{2\pi}\bigg{\{}a\ln\frac{(b+d)}{d}+b\ln\frac{(a+d)}{d} \bigg{\}} \tag{15}\]  
The pickup coil structure in Model-E is surrounded by three traces through which the primary current flows on three sides. The magnetic flux by the coil and the mutual inductance between the coils and current trace are given by the following equations, respectively:  
\[v_{A}(t)=-M_{1}\frac{di_{1}}{dt}=-\frac{d\phi}{dt}=-\frac{\mu_{o}\cdot a}{2\pi} \ln\frac{(b+d)}{d}\frac{di_{1}}{dt} \tag{8}\]0  
\[v_{A}(t)=-M_{1}\frac{di_{1}}{dt}=-\frac{d\phi}{dt}=-\frac{\mu_{o}\cdot a}{2\pi} \ln\frac{(b+d)}{d}\frac{di_{1}}{dt} \tag{8}\]1  
Figure 4 shows the magnetic flux analysis results using the finite element method (FEM) for the five models. From the diagram of the magnetic flux, it can be derived that the magnetic flux density increases at the crossover position between the coil and the trace. For the relative comparison of the magnetic flux, the maximum value of the magnetic flux was set based on the 0.009 magnetic flux (Tesla) value. Table 1 lists the parameters measured by the Ansoft Maxwell and Q3D Extractor, Canonsburg, PA, USA, a specialized simulation tool for finite element analysis (FEA). From the results of Table 1, quantitatively, the increasing trend of the \(M\) value according to the number of conductors was confirmed the same. On the other hand, the results of the mathematical results do not exactly match the actual data in \[v_{A}(t)=-M_{1}\frac{di_{1}}{dt}=-\frac{d\phi}{dt}=-\frac{\mu_{o}\cdot a}{2\pi} \ln\frac{(b+d)}{d}\frac{di_{1}}{dt} \tag{8}\]2 This is because the skin effect of the conductor through which the primary current flows is not considered and the exact shape of the actual trace is not reflected. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Coil Model** & **Model-A** & **Model-B** & **Model-C** & **Model-D** & **Model-E** \\ \hline Inductive coupling & 0.03 & 0.14 & 0.11 & 0.147 & 0.2 \\ Self-inductance \(L_{s}\) [nH] & 5.8 & 5.8 & 6.48 & 6.49 & 6.48 \\ Mutual-inductance \(M\) [nH] & 0.42 & 0.71 & 1.04 & 1.3 & 1.7 \\ Trace inductance [nH] & 8.32 & 8.65 & 6.29 & 6.32 & 6.41 \\ \(L_{c}/M\) & 13.8 & 8.16 & 6.23 & 4.99 & 3.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the dependent coil structures using finite element method with the FEA method.