# Analysis and Optimization of the Stray Capacitance of Rogowski Coils
## 2 Analysis of Dynamic Characteristics of Rogowski Coils  
As shown in Figure 1, a Rogowski coil refers to a coil uniformly wound on a skeleton made of non-ferrous material. When a time-varying current \(i(t)\) flows along the central axis of the skeleton, a terminal voltage \(e(t)\) is induced, and \(i(t)\) can be recovered from \(e(t)\).  
An equivalent circuit model of Rogowski coils is presented in Figure 2, where \(L_{C}\), \(R_{C}\), \(C_{c}\), and \(R_{S}\) are the self-inductance, internal resistance, stray capacitance, and external sampling resistance of the coil, respectively. \(M\) represents the mutual inductance of the coil  
Figure 1: The structure and measurement principle of Rogowski coils.  
and the current-carrying wire. The current to be measured and the current flowing through the coil are denoted by \(i_{1}(t)\) and \(i_{2}(t)\), respectively [14].  
In practical applications, it is necessary to integrate \(e(t)\) with integrator circuits to accurately extract the time profile of the measured current \(i_{1}(t)\). According to different types of external signal processing circuits, the working states of Rogowski coils are usually divided into two self-integration and external integration [15; 16].  
Given sufficiently small \(R_{s}\) or large \(\mathrm{d}i_{2}/\mathrm{d}t\), the voltage \(u_{0}(t)\) is approximately proportional to the measured current \(i_{1}(t)\). Under this condition, Rogowski coils work under the self-integrating state, and a sampling resistor \(R_{s}\) can be directly connected to complete a current measurement loop. It is straightforward to obtain the upper cut-off frequency \(f_{h1}\) and the lower cut-off frequency \(f_{l1}\) of the measurement circuit [8]  
\[f_{h1}=\frac{1}{2\pi R_{s}C_{c}},\ f_{l1}=\frac{R_{s}+R_{c}}{2\pi L_{c}}. \tag{1}\]  
When the self-integrating condition is not satisfied, an RC integral circuit is added to the measurement circuit, as shown in Figure 3, where typically \(R\geq 10R_{s}\). The upper cut-off frequency \(f_{h2}\) and the lower cut-off frequency \(f_{l2}\) of the measurement circuit can be calculated as [17]  
\[f_{h2}=\frac{1}{2\pi\sqrt{L_{c}C_{c}}},\ f_{l2}=\frac{1}{2\pi RC}. \tag{2}\]  
From Equations (1) and (2), it is seen that the bandwidth of Rogowski coils is determined by the internal resistance \(R_{c}\), sampling resistance \(R_{s}\), self-inductance \(L_{c}\) or mutual inductance \(M\) (\(L_{c}=NM\), where \(N\) denotes the number of turns) and stray capacitance \(C_{c}\).  
In general, the internal resistance \(R_{c}\) and the mutual inductance \(M\) have considerable influence on the low frequency characteristics of the measurement circuit yet little effect on the high frequency characteristics. Increasing \(R_{c}\) brings little benefit of the amplitude-frequency gain, but it leads to worse low-frequency characteristics and narrower bandwidth. Therefore, \(R_{c}\) should be reduced as much as possible. On the other hand, increased \(M\) can improve the sensitivity and expand the bandwidth; hence, a larger \(M\) is usually favorable.  
Figure 3: The equivalent circuit diagram of external integral working state of Rogowski coil.  
Figure 2: The equivalent circuit model of Rogowski coils.  
As for the choice of the sampling resistance \(R_{s}\), there exists a tradeoff, since large \(R_{s}\) leads to oscillation, while decrease of \(R_{s}\) reduces the gain and signal-to-noise ratio.  
The stray capacitance \(C_{c}\), which is the focus of this work, is one of the dominant factors affecting the high frequency response. From Equation (2), it is obvious that reducing the stray capacitance expands the bandwidth and improves the high frequency performance of Rogowski coils.