# Analysis and Optimization of the Stray Capacitance of Rogowski Coils
## 4 Optimization of the Skeleton Cross-Section
### Optimal Design of the Skeleton Cross-Section
#### 4.2.1 The Constrained Variational Problem  
As shown in Figure 13, the skeleton cross-section is in the \(z\)-\(r\) plane in the cylindrical coordinate system, and the center of the skeleton coincides with the coordinate origin O. The measured current \(i\) flows along the \(z\) axis. Assuming that the coil is wound uniformly and all turns are identical, their magnetic fluxes are equal due to symmetry. Now the remaining task is to find the skeleton cross-section with the shortest length of a single-turn winding under given magnetic flux. To begin with, consider the magnitude of the magnetic flux density \(\mathbf{B}\) at a point within skeleton cross-section  
\[B=\frac{\mu i}{2\pi r}, \tag{23}\]  
where \(\mu\) is the magnetic permeability of the skeleton. Obviously, \(\mathbf{B}\) is perpendicular to the \(z\)-\(r\) plane and independent of \(z\); thus, the optimal cross-section should have an axis of symmetry perpendicular to the \(z\)-axis, which in our context is assumed to be the \(r\)-axis for convenience. As shown in Figure 11, only the upper half of the cross-section needs to be discussed, and the intersection points of its boundary and the \(r\)-axis are denoted by (\(R_{1}\), 0) and (\(R_{3}\), 0), respectively. Another point is that since the magnitude of \(\mathbf{B}\) is proportional to \(1/r\), at least a portion of the boundary of the skeleton should lie on the straight-line \(r=R_{1}\) to yield as large magnetic flux as possible with unit perimeter. Assume that this portion is the line segment from (\(R_{1}\), 0) to (\(R_{1}\), \(z_{0}\)). In the interval \([R_{1}<r\leq R_{3}]\), the boundary \(z(r)\) should be a single-valued function. Based on the above discussions, obtaining the optimal skeleton cross-section can be attributed to the following tasks. In the interval \([R_{1}\leq r\leq R_{3}]\), given the magnetic flux \(\phi_{\rm m}/2\) through the region surrounded by the curve \(z(r)\) and the straight line segment \(\overline{R_{1}R_{3}}\), find a curve \(z(r)\) with the shortest length. When the inner radius \(R_{1}\) is large, the magnetic lines of force passing through the cross-section of the winding tend to be uniformly distributed, and the boundary of the optimal skeleton cross-section is close to a circle. As \(R_{1}\) decreases, this approximate circumference gradually compresses along the \(r\)-direction and partially coincides with the straight line \(r=R_{1}\), and extends along the \(z\)-axis, ensuring the shortest possible length of the cross-section circumference for a constant flux through the area it encloses. This indicates that there must exist a smooth and convex curve which is the contour of the optimal skeleton cross-section. The contour of the upper half of the optimal skeleton cross-section \(z(r)\) should have the following characteristics. Firstly, it should be composed of a straight line segment and a convex curve, and the straight line segment is tangential to the convex curve at the point of intersection. There should be, and there can only be, two such points on the convex curve, at which the first-order derivatives are infinite, and the two points should be the intersections of the convex curve with the straight lines \(r=R_{1}\) and \(r=R_{3}\) at the intersections \((R_{1},z_{0})\) and \((R_{3},0)\), i.e., \(\left.\frac{\mathrm{d}z}{\mathrm{d}r}\right|_{r=R_{1}}=+\infty\) and \(\left.\frac{\mathrm{d}z}{\mathrm{d}r}\right|_{r=R_{3}}=-\infty\), respectively. In addition, the first-order  
Figure 13: Optimal skeleton cross-section for Rogowski coils. derivative at a point \(R_{2}\) on the convex curve is equal to zero, namely \(\left.\frac{\mathrm{d}z}{\mathrm{d}r}\right|_{r=R_{2}}=0\), and its corresponding z-coordinate is \(z_{\mathrm{max}}=z(R_{2})\).