# Analysis and Optimization of the Stray Capacitance of Rogowski Coils
## 4 Optimization of the Skeleton Cross-Section
### Numerical Example  
Equation (42) gives an explicit formulation for the perimeter of the optimal skeleton cross-section of Rogowski coils, yet the complex integral terms are not easy to compute in a pure analytical way. Therefore, the integral terms are calculated via some numerical approaches, e.g., Gaussian integration and trapezoidal rule. Without losing generality, we assume \(R_{1}=1\) cm and \(R_{3}=2\) cm. Firstly, from Equations (30) and (31), \(R_{2}=\sqrt{2}\) cm and \(\lambda^{2}=\left(2/\ln 2\right)^{2}\) are obtained, respectively. Substitution of these two values into Equations (34) and (36) yields \(z_{0}=0.2708\) cm and \(z_{\max}=0.6453\) cm. Then, the right-hand side of Equation (42) is evaluated via Gaussian integration to obtain the optimal skeleton cross-section of Rogowski coils, which is plotted in Figure 14. It is worth noting that instead of Equation (42), an alternative solution approach to the constrained variational problem, i.e., Equation (27), is the finite element method (FEM), which is more general yet troublesome to implement. Detailed implementation of the FEM solution to Equation (27) is included in the Appendix A, and the results are also presented in Figure 14. It is observed that the results via Equation (42) agree well with those obtained by the FEM; thus, the correctness and accuracy of Equation (42) is justified. To further demonstrate the advantage of the optimized cross-section over the widely adopted rectangular, circular, and oval cross-sections, we present a brief performance comparison of the four shapes of cross-sections under identical \(R_{1}=1\) cm and total magnetic flux in Table 2, where \(f_{h1}\) denotes the upper cut-off frequency normalized by the value corresponding to the rectangular cross-section under self-integration state, while \(f_{h2}\) corresponds to the external-integration state. It is worth noting that the geometry parameters of the rectangular and oval cross-sections have been optimized to make a fair comparison. The number of turns of the considered Rogowski coil is 70. The reference values of the parameters are \(r_{c}=0.465\) mm, \(\delta=0.035\) mm, \(\varepsilon_{r}=5\), and \(h=1.1\) mm. First, it is observed that under fixed magnetic flux (namely identical \(L_{\text{C}}\)) and \(R_{1}=1\) cm, the rectangular skeleton requires the longest perimeter, and therefore exhibits the largest stray capacitance and the lowest upper cutoff frequencies. The proposed improvement of the cross-section of the skeleton results in the shortest perimeter and highest \(f_{h1}\) and \(f_{h2}\), especially under the self-integration state. As for the external integration working state, the proposed cross-section leads to a cutoff frequency 6.4% higher than that of the rectangular cross-section. Second, the performance of the oval cross-section is expected to be somewhere between those of the circular cross-section and rectangular cross-section, yet surprisingly this shape turns out to be the second-best choice. The stray capacitance and resistance are very close to those of the proposed optimized cross-section. Third, the resistance \(R_{\text{c}}\) is also approximately proportional to the perimeter of the skeleton cross-section \(l\). Thus, the proposed method for minimization of \(l\) also leads to a reduction in \(R_{\text{c}}\) and the expansion of the bandwidth of the Rogowski coils under a self-integration state, because from Equation (1), it can be concluded that the lower cut-off frequency \(f_{l1}=\frac{R_{\text{c}}+R_{\text{c}}}{2 mL}\) decreases under the reduced \(R_{\text{c}}\).