# Analysis and Optimization of the Stray Capacitance of Rogowski Coils
## 3 Stray Capacitance of Rogowski Coils
### Simplified Model of the Stray Capacitance Network  
Generally speaking, the stray capacitances among non-adjacent turns of the winding are much smaller than those of the adjacent turns [19]. In order to simplify the network, the capacitances among non-adjacent turns are excluded, and a simplified network model of the stray capacitance of Rogowski coils is obtained, which is shown in Figure 6, where \(C_{tt}\) is the inter-turn capacitance of adjacent turns of the winding, and the capacitance of each turn and the shielding is denoted by \(C_{ts}\). It should be emphasized that since the two turns located at the two ends of the winding are also adjacent in space, the inter-turn capacitance between them cannot be ignored. By repeatedly implementing the star-delta transformation, \(C_{AB}\) can be easily calculated. Firstly, consider a network with only two nodes. The lumped stray capacitance \(C_{AB}(2)\) is  
\[C_{AB}(2)=C_{tt}+\frac{C_{ts}}{2}. \tag{3}\]  
When there are three nodes, we derive \(C_{AB}(3)\) as  
\[C_{AB}(3)=\frac{C_{tt}}{2}+\frac{C_{ts}}{2}+C_{tt}. \tag{4}\]  
For models with more than three nodes, \(C_{AB}(N)\) can be seen as a superposition of two components. The first component \(C_{1,N}\) is the inter-turn capacitance between the two ends of the winding, namely \(C_{tt}\). The second component \(C^{\prime}_{AB}(N)\) is based on the simplified model in Figure 6, but of course \(C_{1,N}\) is removed. The evaluation of \(C^{\prime}_{AB}(N)\) can be achieved by adding one node to each end in turn. Note that \(C^{\prime}_{AB}(2)=\frac{C_{tt}}{2}+\frac{C_{ts}}{2}\), and \(C^{\prime}_{AB}(3)=\frac{C_{tt}}{2}+\frac{C_{ts}}{2}\), the following recursive formula can be obtained  
\[C^{\prime}_{AB}(N)=\frac{C^{\prime}_{AB}(N-2)\times\frac{C_{tt}}{2}}{C^{\prime }_{AB}(N-2)+\frac{C_{tt}}{2}}+\frac{C_{ts}}{2},N>3. \tag{5}\]  
Obviously, there is  
\[C_{AB}(N)=C^{\prime}_{AB}(N)+C_{tt}=\frac{C^{\prime}_{AB}(N-2)\times\frac{C_{ t}}{2}}{C^{\prime}_{AB}(N-2)+\frac{C_{tt}}{2}}+\frac{C_{ts}}{2}+C_{tt}. \tag{6}\]  
Let \(\eta=C_{ts}/C_{tt}\), then define  
\[\xi_{AB}(N)=\frac{C_{AB}(N)}{C_{tt}}=\frac{C^{\prime}_{AB}(N)}{C_{tt}}+1=\left\{ \begin{array}{ll}\frac{1}{2+\frac{C_{tt}}{C_{AB}(N)}}+\frac{\eta}{2}+1&(N>3) \\ \frac{1}{4}\left(\eta+\sqrt{\eta^{2}+4\eta}\right)+1&(N\rightarrow\infty) \end{array} \right.. \tag{7}\]  
Figure 6: Simplified model of the stray capacitance network of Rogowski coils. Figure 7 presents the variation of \(\xi_{AB}\) with the number of turns under various \(\eta\). It is observed that when \(N>10\), \(\xi_{AB}(N)\) is very close to the limit \(\frac{1}{4}\Big{(}\eta+\sqrt{\eta^{2}+4\eta}\Big{)}+1\). According to Equation (7),  
\[C_{AB}(N)\approx C_{AB}(\infty)\cdot C_{tt}=\frac{1}{4}\Big{(}\eta+\sqrt{\eta ^{2}+4\eta}\Big{)}C_{tt}+C_{tt}\;(N>10). \tag{8}\]  
In particular, if the shielding shell does not exist, the capacitance between the winding and the shell will not be included, i.e., the capacitance branches of all the junctions to the shielding shell in Figure 6 are disconnected; thus,  
\[C_{AB}(N)=C_{tt}+\frac{C_{tt}}{N-1}. \tag{9}\]  
From the above analysis, as long as the values of \(C_{tt}\) and \(C_{ts}\) are known, the stray capacitance \(C_{AB}\) can be easily computed with Equation (8) or Equation (9). In addition, it is straightforward to conclude that reduction in either \(C_{tt}\) or \(C_{ts}\) would be beneficial to yield low \(C_{AB}\).