# Analysis and Optimization of the Stray Capacitance of Rogowski Coils
## 3 Stray Capacitance of Rogowski Coils
### Simplified Model of the Stray Capacitance Network
In the presence of a shielding case, the turn-to-turn capacitance \(C_{tt}\) can be computed by  
\[C_{tt}=\frac{\varepsilon_{0}l_{t}}{2}\int_{\theta_{1}-\pi/2}^{\pi/2}\frac{p}{2 (r_{c}+\delta)+\frac{1}{\varepsilon_{r}}\ln\frac{r_{c}+\delta}{r_{c}}-\cos \theta}\mathrm{d}\theta, \tag{10}\]  
where \(\theta_{1}=\arctan(p/2h)\), \(l_{t}\) is the length of a single turn of the winding, \(p\) is the distance between turns, \(h\) is the distance between turns and the shielding shell, \(r_{c}\) is the radius of the winding conductor, and \(\delta\) and \(\varepsilon_{r}\) are the thickness and relative permittivity of the wire insulation layer, respectively. The approximate formula for the turn-to-shield capacitance \(C_{ts}\) is  
\[C_{ts}=2\varepsilon_{0}l_{t}\int_{0}^{\theta_{1}}\frac{1}{\frac{h}{r_{c}+ \delta}+\frac{1}{\varepsilon_{r}}\ln\frac{r_{c}+\delta}{r_{c}}-\cos\theta} \mathrm{d}\theta, \tag{11}\]  
where the definitions of all parameters are the same as those in Equation (10). The simplified model of the stray capacitance as well as Equations (10) and (11) is validated by finite element (FEM) analysis using the commercial software COMSOL Multiphysics version 6.0. The geometry and mesh are depicted in Figure 8. The inner and outer radii of the skeleton are 56 mm and 64 mm, respectively. Both the coils and the shield are made of copper. The thickness of the grounded shield is 0.5 mm. The total number of turns is 100. It is worth noting that we intentionally disconnect adjacent turns of coils by including a small air gap, so that each turn can be regarded as an independent  
Figure 7: Variation of coefficient \(\xi_{AB}\) value with number of turns. conductor. After exciting Coil1 with a static electric potential and setting other turns grounded, electrostatic simulation is carried out and total electric charge of each conductor can be calculated. Then by straightforward post processing, the stray capacitances of Coil1 and all other turns of the coils as well as the shield can be extracted. Due to the symmetry of the model, all mutual capacitances among the turns of coils capacitances can be obtained by exciting only Coil1. The comparison of the proposed model against the FEM results is given in Table 1. Note that the FEM results in this table is grid-independent solutions, or in other words, we gradually increase the mesh density so that the values of are almost invariant. The resultant number pf tetrahedron elements are 638,331. It is seen that the results from the proposed method agree well with FEM solutions; thus, the accuracy of the simplified model of the stray capacitance as well as Equations (10) and (11) is justified. In addition, we must point out that substitution of the accurate \(C_{ts}\) and \(C_{tt}\) from the FEM simulation into Equation (8) leads to a relative error of 0.7% in \(C_{AB}\). Therefore, the main error source in \(C_{AB}\) is the approximation of \(C_{ts}\) and \(C_{tt}\) by Equations (10) and (11), and our simplification of the capacitance network by ignoring coupling among non-adjacent turns of coils has very little influence on the accuracy.