## 3 Analysis of Parameter for Designing Pickup Coil in PCB  
### Pickup Coil Configuration in PCB  
PCB-embedded pickup coils have various mutual inductance values according to their position and shape. The distance between the coil and conductor through which the primary current flows, the shape, and the turn number of the coil determines the amount of magnetic flux incident on the closed-loop area of the coil formed by the primary current, and thus affects the EMF value. Each of the five models in Figure 3 depicts five structures in which a single-turn rectangular pickup coil can be placed on the PCB. These five models are classified according to the number of traces through which the primary current flows adjacent to the pickup coil. In all coil models, the self-inductance value can be flexibly determined by varying the number of turns and coil size. The pickup coil depicted in Figure 3a is the most basic coil model, and it consists of the trace primary current flowing on the bottom layer of the PCB and a coil composed of pattern and vias, where the magnetic flux incident surface is located in the vertical direction to the trace, i.e., the magnetic flux formed by one side of the conductor trace through which the primary current is flowing enters the magnetic flux incident area of the pickup coil [30; 34]. In this case, the distance \(d\) is determined as a small value equal to the PCB layer thickness level. According to Maxwell-Faraday law, the amount of total magnetic flux \(\phi_{A}\) through the flux incident area \(S\) is the integral value of the magnetic flux density B, as shown below. All mathematical analysis methods assumed the shape of the virtual wire, not the shape of the trace. \[\phi_{A}=\int_{S}B\cdot ds=\frac{\mu_{o}\cdot i_{1}}{2\pi}\int_{d}^{b+d}\frac {1}{b}\cdot a\cdot db=\frac{\mu_{o}\cdot a\cdot i_{1}}{2\pi}\ln\frac{(b+d)}{d} \tag{7}\]where \(\mu_{o}\) is the air permeability. The EMF voltage \(v_{A}\) is induced at the coil output terminal, as follows:  
\[v_{A}(t)=-M_{1}\frac{di_{1}}{dt}=-\frac{d\phi}{dt}=-\frac{\mu_{o}\cdot a}{2\pi} \ln\frac{(b+d)}{d}\frac{di_{1}}{dt} \tag{8}\]  
In addition, the mutual inductance \(M_{A}\) between the conductor and the coil is as follows:  
\[M_{A}=\frac{\mu_{o}\cdot a}{2\pi}\ln\frac{(b+d)}{d} \tag{9}\]  
Figure 3b shows the coil structure in which the magnetic flux incident surface of the pickup coil is positioned perpendicular to the trace in the situation of the primary current flowing through the via and two traces in the upper and lower layers. The magnetic flux generated by the primary current flowing at the top and bottom is incident inside the coil in the same direction [30]. The amount of magnetic flux incident \(\phi_{B}\), in this case, is doubled compared to the previous case. \[\phi_{B}=\frac{\mu_{o}\cdot a\cdot i_{1}}{\pi}\ln\frac{(b+d)}{d} \tag{10}\]  
The mutual inductance \(M_{B}\) between the conductor and coil is as follows:  
\[M_{B}=\frac{\mu_{o}\cdot a}{\pi}\ln\frac{(b+d)}{d} \tag{11}\]  
Unlike Model-A and B, in Model-C, D, and E, the coil is positioned horizontally-concerning the trace through which the primary current flows. Therefore, to construct a multi-turn coil, the number of coil turns must be increased or decreased according to the number of PCB layers. The horizontal direction coil can receive a wider magnetic flux per coil area because the magnetic flux incident surface \(S\) by the primary current of the pickup coil is largely based on the same self-inductance compared to the previous vertical coil. This guarantees a higher measurement sensitivity and reduces the number of turns, reducing overheating and bottlenecks resulting from a reduced number of vias. In the structure shown in Model-C, the pickup coil is located on the same plane near one trace through which the primary current flows so that the magnetic flux can be incident.