## 2 Basics of a Pickup Coil  
While measuring the switch current of power semiconductors in discrete packages or measuring the current flowing through the PCB trace, there are many difficulties in configuring the existing toroidal Rogowski coil; these difficulties can be addressed via a pickup coil configuration. Figure 1 shows the pickup coil structure in two cases that can be embedded in the PCB. Figure 1a shows the case of a coil configured at a vertical position, concerning the trace through which the primary current \(i_{1}\) flows and the coil thickness corresponds to the height at which the magnetic flux is received; it is possible to configure a coil with a large number of turns. In the case of Figure 1a, it corresponds to the model of the existing pickup coil [30; 33]. Figure 1b shows a coil configured in a horizontal position to trace the primary current \(i_{1}\) flows. This coil is configured in a horizontal position to receive the magnetic flux of the primary current on the same PCB layer [34]. The number of turns is limited according to the number of layers because it increases or decreases in size according to the number of layers. For both coil models, the magnetic flux formed by the primary current \(i_{1}\) passes through the inside of the closed-loop area in the coil, and an EMF is induced at the output end of the loop following Maxwell-Faraday law.  
Therefore, it is necessary to consider the influence of the position of the external conductor in situations in which accurate switch current measurement in case ofdesigning the pickup coil in PCB is required.  
Figure 2 is an equivalent circuit analyzed with a lumped model for a pickup coil and \(L_{c}\) is the magnetization inductance of the coil, \(R_{c}\) is the self-resistance value of the coil, \(C_{c}\) is the self-capacitance value of the coil, and \(M\) denotes the mutual inductance between the coil and conductor.  
\[G_{coil}(s)=\frac{v_{coil}(s)}{i_{1}(s)}=\frac{M\cdot s}{(L_{c}C_{c})s^{2}+s( \frac{L_{c}+R_{c}R_{c}C_{c}}{R_{d}})+\frac{R_{c}+R_{d}}{R_{d}}} \tag{1}\]  
When the primary current \(i_{1}\) flows through the trace, the density \(B\) of the flux density around conductor trace by the Biot-Savart law is basically formed by the magnetic field intensity \(H\) as follows:  
\[B=\mu\cdot H=\frac{\mu\cdot i_{1}}{2\pi r} \tag{2}\]  
Figure 1: Schematic of the pickup coil in PCB: (**a**) vertical structure and (**b**) horizontal structure.  
Figure 2: Lumped circuit of the pickup coil.  
where \(r\) is away from the conductor to the measurement point, and \(\mu\) is relative magnetic permeability. The generated flux passes through the \(S\) area, the amount of generated magnetic flux \(\mathbf{\phi}\) in that area is as follows:  
\[\phi=\int_{S}B\cdot ds \tag{3}\]  
where \(\mu_{o}\) is the relative magnetic permeability, \(ds\) is the smallest section of area S, and the EMF induced at the output stage of the coil is,  
\[v_{e}(t)=-\frac{d\phi}{dt}=-M\frac{di_{1}}{dt} \tag{4}\]  
where \(M\) represents the mutual inductance between the coil and the conductor, itdetermines the sensitivity of the sensor. If \(M\) is increased by increasing the self-inductance, the sensitivity of the sensor also increases; however, at the same time, the bandwidth of the coil is reduced due to the high \(L_{c}\). This can reduce accuracy when measuring the switch current for fast power semiconductor devices such as GaN. At this time, the bandwidth of the sensor is as follows [31]:  
\[f_{c}\leq\text{Bandwidth}\leq f_{col\_pole} \tag{5}\]  
\[f_{col\_pole}=\frac{1}{2\pi}\cdot\sqrt{\frac{R_{c}+R_{d}}{L_{c}R_{d}C_{c}}} \approx\frac{1}{2\pi\sqrt{L_{c}C_{c}}} \tag{6}\]  
where \(f_{col\_pole}\) is the pole frequency of \(G_{col}\) in Equation (1).  
To configure a current sensor using a coil in a specific bandwidth with high sensitivity for a fast device such as GaN, a coil design capable of significantly lowering the \(L_{c}/M\) value is required. Therefore, the analysis of the coil in this paper proceeds based on the performance parameter \(L_{c}/M\).