Analysis and Optimization of the Stray Capacitance of Rogowski Coils

Abstract

In this work, the lumped model of Rogowski coils is briefly reviewed to illustrate that the reduction in stray capacitance expands the bandwidth and improves the high-frequency performance. Then, a network model as well as explicit formulas for the stray capacitance of Rogowski coils are established, and the influence of geometrical parameters of the skeletons on the stray capacitance is investigated. It is found that the stray capacitance of Rogowski coils is approximately proportional to the perimeter of the skeleton cross-section. Based on the above discussion, optimization of the shape of the skeleton cross-section, aiming at minimizing the perimeter without affecting the magnetic flux and mutual inductance of the coils, is carried out. The widely adopted circular and rectangular cross-sections are discussed first. Then, the cross-section of an arbitrary smooth and convex shape is optimized by solving a constrained variational problem, leading to an explicit equation for the optimal skeleton cross-section of Rogowski coils. Numerical results demonstrate that, compared with the common circular and rectangular skeleton cross-sections, the proposed optimal cross-section exhibits the shortest perimeter and thus the highest upper cutoff frequency under fixed magnetic flux. The optimization method developed by this work can provide a theoretical basis and guidance for the design of Rogowski coils.

1. Introduction

Rogowski coils, also known as air-cored coils, have been widely adopted as current transducers due to their many favorable properties over the conventional iron-cored current sensors, e.g., compact structure, low cost, wide bandwidth, simplicity, and the fact that they are not affected by core saturation [1,2,3,4].

Since Rogowski coils are frequently used in the measurement of pulsed currents with wide spectrum, their high-frequency performance is a major concern. Parasitic influences can deteriorate the high-frequency response of the coils; hence, many works have been devoted to the investigation of the stray parameters. The extraction and influence of stray inductance have been thoroughly investigated [5,6,7], yet less attention is paid to the stray capacitance.

In [8], it is demonstrated that the stray capacitance is one of the main factors affecting the high-frequency performance of Rogowski coils. It is proposed in [9] that the stray capacitance can be obtained through the electromagnetic energy in the surrounding volume of Rogowski coils, but the inter-turn capacitances are ignored, leading to inaccurate results. In addition, in this work the influence of the geometrical parameters of the skeletons on the high-frequency response of the coils are analyzed, and optimization of the skeletons of three typical shapes is further put forward. However, only the stray inductance is taken into account in the optimization. In [10], each turn of Rogowski coils and the metallic shielding shell are modelled as two electrodes of a capacitor, and the stray capacitance is estimated by using the cylindrical capacitor model. The above works typically involve troublesome finite element analysis, or analytical formulations for simple geometry, which are not practical and flexible. Therefore, in [11], a simple equivalent circuit model of the stray capacitance of high-voltage capacitive dividers is constructed, and experimental results are incorporated to extract circuit parameters.

To the best of the authors’ knowledge, so far, there are few studies on the influence of the skeletons on the frequency characteristics of the Rogowski coils from the perspective of the stray capacitance. The existing works of [12,13] often adopt standard rectangular cross-sections and determine geometrical parameters via optimization algorithms such as the particle swarm method [13], which require a dense coding effort, but the effect may be limited by the simple shape of the skeleton cross-section. Therefore, to fill this research gap, we investigate the optimization of the shape of the skeleton cross-section to minimize the stray capacitance without compromising other performance indicators, e.g., the mutual inductance. The optimization approach to be presented yields analytical expressions of the optimal convex boundary of the skeleton cross-section of Rogowski coils. Thus, it is straightforward to implement and provides a theoretical limit of the reduction in stray capacitance of Rogowski coils.

The remainder of this manuscript is structured as follows. In Section 2, the lumped model of Rogowski coils is briefly reviewed to illustrate that the stray capacitance is one of the main factors affecting the high frequency performance of Rogowski coils. In Section 3, a network model of the stray capacitance of Rogowski coils is established, and the influence of the geometrical parameters of the skeletons on the stray capacitance is investigated. Based on the discussion in Section 3, Section 4 first presents the optimization of the skeleton of circular and rectangular shapes of cross-sections. Then for skeletons of arbitrary smooth and convex shape, a general optimization approach to minimizing the stray capacitance of the skeleton is proposed, which can provide theoretical basis and guidance for the design 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 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 $e\left(t\right)$ with integrator circuits to accurately extract the time profile of the measured current $i_1\left(t\right)$. 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 $u_0\left(t\right)$ 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]

$$f_{h1}={\displaystyle \frac{1}{2\pi R_s{C}_c}}, f_{l1}={\displaystyle \frac{R_s+R_c}{2\pi L_c}}.$$

(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 $R\ge 10R_s$. The upper cut-off frequency $f_{h2}$ and the lower cut-off frequency $f_{l2}$ of the measurement circuit can be calculated as [17]

$$f_{h2}={\displaystyle \frac{1}{2\pi \sqrt{L_c{C}_c}}}, f_{l2}={\displaystyle \frac{1}{2\pi RC}}.$$

(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. 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.

3. Stray Capacitance of Rogowski Coils

To extract the stray capacitance of Rogowski coils, we regard each turn of the winding as a single coil represented by lumped elements, and present a network model of the equivalent circuit of Rogowski coils in Figure 4, where $R_i$ and $L_i$ are the internal resistance and self-inductance of the i-th turn of the winding, respectively. $M_i$ represents the mutual inductance of the i-th turn and the current-carrying conductor, and $M_{i,k}$ denotes the mutual inductance of the i-th and k-th turns. Similarly, $C_{i,0}$ is the capacitance of the i-th turn and the shielding shell, while $C_{i,k}$ is the capacitance of the i-th and k-th turns of the winding.

3.1. A Network Model of the Stray Capacitance of Rogowski Coils

Based on Figure 4, the capacitance network of Rogowski coils is depicted in Figure 5, in which the edge effect of the winding at both ends is ignored [18]. In Figure 5, each marked node represents a single turn of the winding, and the grounding shield is modeled as a reference node S. The stray capacitance $C_c$ is defined as the lumped stray capacitance between the two terminal nodes A and B of Rogowski coils; thus, $C_c$ is referred to as $C_{AB}$ hereinafter. After (N-2) times of the star-delta transformation, the intermediate nodes are eliminated step by step, and the π-type equivalent circuit of the stray capacitance network can be obtained, and the stray capacitance $C_{AB}$ can be calculated accordingly. However, when the number of nodes is large, this approach is very cumbersome.

3.2. 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

$$C_{AB}(2)=C_{tt}+{\displaystyle \frac{C_{ts}}{2}}.$$

(3)

When there are three nodes, we derive $C_{AB}(3)$ as

$$C_{AB}(3)={\displaystyle \frac{C_{tt}}{2}}+{\displaystyle \frac{C_{ts}}{2}}+C_{tt}.$$

(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 ${C^{\prime }}_{AB}(2)={\displaystyle \frac{C_{tt}}{2}}+{\displaystyle \frac{C_{ts}}{2}}$, and ${C^{\prime }}_{AB}(3)={\displaystyle \frac{C_{tt}}{2}}+{\displaystyle \frac{C_{ts}}{2}}$, the following recursive formula can be obtained

$${C^{\prime }}_{AB}(N)={\displaystyle \frac{{C^{\prime }}_{AB}(N-2)\times \frac{C_{tt}}{2}}{{C^{\prime }}_{AB}(N-2)+\frac{C_{tt}}{2}}}+{\displaystyle \frac{C_{ts}}{2}}, N>3.$$

(5)

Obviously, there is

$$C_{AB}(N)={C^{\prime }}_{AB}(N)+C_{tt}={\displaystyle \frac{{C^{\prime }}_{AB}(N-2)\times \frac{C_{tt}}{2}}{{C^{\prime }}_{AB}(N-2)+\frac{C_{tt}}{2}}}+{\displaystyle \frac{C_{ts}}{2}}+C_{tt}.$$

(6)

Let $\eta =C_{ts}/C_{tt}$, then define

$${\xi }_{AB}(N)={\displaystyle \frac{C_{AB}(N)}{C_{tt}}}={\displaystyle \frac{{C^{\prime }}_{AB}(N)}{C_{tt}}}+1=\{\begin{array}{l}{\displaystyle \frac{1}{2+\frac{C_{tt}}{{C^{\prime }}_{AB}(N)}}}+{\displaystyle \frac{\eta }{2}}+1 (N>3)\\ {\displaystyle \frac{1}{4}}\left(\eta +\sqrt{{\eta }^2+4\eta }\right)+1 (N\to \infty )\end{array}.$$

(7)

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 ${\displaystyle \frac{1}{4}}\left(\eta +\sqrt{{\eta }^2+4\eta }\right)+1$. According to Equation (7),

$$C_{AB}(N)\approx C_{AB}(\infty )\cdot C_{tt}={\displaystyle \frac{1}{4}}\left(\eta +\sqrt{{\eta }^2+4\eta }\right)C_{tt}+C_{tt} (N>10).$$

(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,

$$C_{AB}(N)=C_{tt}+{\displaystyle \frac{C_{tt}}{N-1}}.$$

(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 $C_{AB}$. In the presence of a shielding case, the turn-to-turn capacitance $C_{tt}$ can be computed by

$$C_{tt}={\displaystyle \frac{{\epsilon }_0{l}_t}{2}}{\displaystyle {\int }_{{\theta }_1-\pi /2}^{\pi /2}\frac{1}{\frac{p}{2\left(r_c+\delta \right)}+\frac{1}{{\epsilon }_r}\ln \frac{r_c+\delta }{r_c}-\cos \theta }\mathrm{d}\theta },$$

(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 ${\epsilon }_r$ are the thickness and relative permittivity of the wire insulation layer, respectively.

The approximate formula for the turn-to-shield capacitance $C_{ts}$ is

$$C_{ts}=2{\epsilon }_0{l}_t{\displaystyle {\int }_0^{{\theta }_1}\frac{1}{\frac{h}{r_c+\delta }+\frac{1}{{\epsilon }_r}\ln \frac{r_c+\delta }{r_c}-\cos \theta }\mathrm{d}\theta },$$

(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 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. Validation of the stray capacitance model.

Stray Capacitance	FEM	Proposed Method	Relative Error
Cts	7.74 pF	8.05 pF	4.0%
Ctt	2.66 pF	2.81 pF	5.9%
CAB	7.52 pF	7.94 pF	5.6%

. 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.

3.3. Discussions of the Factors Affecting Stray Capacitance

In this subsection, the influencing factors of $C_{tt}$ and $C_{ts}$ is investigated, and measures to reduce the stray capacitance of Rogowski coils are put forward.

3.3.1. Investigations of Factors Affecting $C_{tt}$

From Equation (10), it is obvious that $C_{tt}$ is proportional to the length $l_t$. The influences of h, r_c, $\delta$, and ${\epsilon }_r$ on C_tt are shown by Figure 9. It can be seen that under fixed $l_t$, $C_{tt}$ can be reduced by increasing the distance p among the adjacent turns, or decreasing h, $r_c$, $\delta$, and ${\epsilon }_r$.

3.3.2. Investigations of Factors Affecting $C_{ts}$

Similarly, $C_{ts}$ is proportional to $l_t$. The influences of h, r_c, $\delta$, and ${\epsilon }_r$ on C_ts are given by Figure 10. It is seen that under constant $l_t$, an increase of h and decrease of p, r_c, $\delta$, and ${\epsilon }_r$ result in reduced $C_{ts}$.

3.3.3. Measures to Reduce Stray Capacitance

Based on the above results, there exists a tradeoff regarding the selection of the turn-to-turn distance p and the distance h between the turns and shielding shell. Increasing p or decreasing h can reduce $C_{tt}$ but increase $C_{ts}$. On the other hand, adjustment of p changes the number of turns N, which is directly related to the self-inductance $L_c$ and mutual inductance $M$. As for h, this parameter is limited by space; thus, it cannot be chosen freely. Similarly, small winding conductor radius $r_c$, wire insulation thickness $\delta$, and ${\epsilon }_r$ are favorable for reduction of both capacitance $C_{tt}$ and $C_{ts}$, but they are restricted in practical applications. For example, reducing $r_c$ leads to increased internal resistance and spoils the performance of Rogowski coils.

Last but not least, $C_{tt}$, $C_{ts}$, and $C_{AB}$ are proportional to $l_t$; hence, shortening the length of the single-turn winding $l_t$ to simultaneously reduce these three parameters is favorable from the perspective of minimizing the stray capacitance. Since $l_t$ is approximately equal to the perimeter of the skeleton cross-section l, $C_{AB}$ can be optimized by deliberate design of the cross-section of the skeleton of Rogowski coils so that $l_t\approx l$ is minimized.

It is worth emphasizing that the optimization must be implemented under the condition of invariant magnetic flux through the cross-section, or in other words, guaranteeing constant mutual inductance, so that the low-frequence performance of Rogowski coils is not affected.

4. Optimization of the Skeleton Cross-Section

In this section, the optimization of the shape of the cross-section of Rogowski coils is discussed. The goal is to minimize the perimeter of the cross-section under constant magnetic flux, so that the stray capacitance $C_{AB}$ is reduced without compromising other performance indicators of Rogowski coils, e.g., the mutual inductance M.

4.1. Optimization of Skeletons of Three Common Shapes

4.1.1. Rectangular Cross-Section

A schematic of rectangular cross-section is depicted by Figure 11, in which a and b are the lengths of the edges of the cross-section, i is the measured current flowing through the center of the coil, and R₁ is the inner radius of the skeleton. In the case of uniform winding, the magnetic flux through each turn of the winding is

$${\Phi }_{\mathrm{m}}={\displaystyle \frac{{\mu }_{0}i}{2\pi }}b\ln {\displaystyle \frac{R_1+a}{R_1}}.$$

(12)

The optimization problem for the dimensions of a rectangular cross-section skeleton can be expressed as follows. Given fixed inner radius R₁, measured current i, and perimeter l = 2(a + b), determine the relation between a and b, such that the magnetic flux ${\Phi }_{\mathrm{m}}$ through the rectangular cross-section is maximized.

In [20], a straightforward derivation is presented as follows. Substituting

$$b={\displaystyle \frac{l}{2}}-a$$

(13)

into Equation (12) yields

$${\Phi }_{\mathrm{m}}={\displaystyle \frac{{\mu }_{0}i}{2\pi }}({\displaystyle \frac{l}{2}}-a)\ln {\displaystyle \frac{R_1+a}{R_1}}.$$

(14)

The maximum ${\Phi }_{\mathrm{m}}$ is achieved when

$${\displaystyle \frac{\partial {\Phi }_{\mathrm{m}}}{\partial a}}=0,$$

(15)

namely

$${\displaystyle \frac{{\mu }_{0}i}{2\pi }}\left[-\ln {\displaystyle \frac{R_1+a}{R_1}}+({\displaystyle \frac{l}{2}}-a){\displaystyle \frac{1}{R_1+a}}\right]=0.$$

(16)

Recall Equation (13), when a and b satisfy

$$b=(R_1+a)\ln {\displaystyle \frac{R_1+a}{R_1}},$$

(17)

The rectangular cross-section is optimized.

4.1.2. Circular Cross-Section

The magnetic flux through the circular cross-section is

$${\Phi }_{\mathrm{m}}={\displaystyle \frac{{\mu }_{0}i{\left(\sqrt{R_1+d}-\sqrt{R_1}\right)}^2}{2}},$$

(18)

where d is the diameter of the circular cross-section.

If $A_0={\displaystyle \frac{2\pi {\Phi }_{\mathrm{m}}}{{\mu }_{0}i}}$ is fixed, the perimeter of the circular section is given by

$$l=\pi d=\pi \left(A_0+2\sqrt{A_0{R}_1}\right),$$

(19)

The minimum perimeter l does not exist in this case. Although shortening the circumference makes the stray capacitance smaller, it also reduces the magnetic flux and thus the mutual inductance M. Therefore, a circular cross-section skeleton should be avoided from the perspective of keeping the mutual inductance unchanged while only minimizing the stray capacitance.

4.1.3. Oval Cross-Section

One of the anonymous reviewers reminded us of the fact that the adoption of rectangular cross-section imposes a sharp bending in the enameled-copper wire, inducing potential enamel rupture and wire shorting. Therefore, in practical applications, it is favorable to adopt shapes with rounded corners, referred to as oval cross-section. Figure 12 presents an example of oval cross-section, which consists of a rectangular domain and two half-circles. The magnetic flux through the oval cross-section is

$${\Phi }_{\mathrm{m}}={\displaystyle \frac{{\mu }_{0}i{\left(\sqrt{R_1+d}-\sqrt{R_1}\right)}^2}{2}}+{\displaystyle \frac{{\mu }_{0}i}{2\pi }}b\ln {\displaystyle \frac{R_1+d}{R_1}}.$$

(20)

The perimeter of this cross-section is $l=2b+\pi d$; thus,

$$b={\displaystyle \frac{l}{2}}-\pi d.$$

(21)

The magnetic flux can be recast as

$${\Phi }_{\mathrm{m}}={\displaystyle \frac{{\mu }_{0}i{\left(\sqrt{R_1+d}-\sqrt{R_1}\right)}^2}{2}}+{\displaystyle \frac{{\mu }_{0}i}{2\pi }}({\displaystyle \frac{l}{2}}-\pi d)\ln {\displaystyle \frac{R_1+d}{R_1}}.$$

(22)

Similar to the optimization of rectangular cross-section, the oval cross-section is optimized when ${\displaystyle \frac{\partial {\Phi }_{\mathrm{m}}}{\partial d}}=0$, which results in a nonlinear equation (in terms of d) that can be efficiently solved by software such as MATLAB version 2022b and Mathematica version 12.0.

4.2. Optimal Design of the Skelenton Cross-Section

In this subsection, we consider an improved cross-section of the skeleton of Rogowski coils with smooth and convex boundary, as presented in Figure 13, where R₁ and R₃ are the inner and outer radii of the skeleton, respectively. Note that R₁ is usually fixed to a pre-given value, there are four geometrical parameters remaining to be determined, namely z₀, z_max, R₂, and R₃. The goal is to optimize these parameters to obtain a skeleton with the minimal perimeter under constant magnetic flux. To achieve this goal, a constrained variational problem is defined and solved.

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 B at a point within skeleton cross-section

$$B={\displaystyle \frac{\mu i}{2\pi r}},$$

(23)

where $\mu$ is the magnetic permeability of the skeleton. Obviously, 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 (R₁, 0) and (R₃, 0), respectively. Another point is that since the magnitude of B is proportional to 1/r, at least a portion of the boundary of the skeleton should lie on the straight-line r = R₁ to yield as large magnetic flux as possible with unit perimeter. Assume that this portion is the line segment from (R₁, 0) to (R₁, z₀). In the interval $\left[R_1<r\le R_3\right]$, 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 $\left[R_1\le r\le R_3\right]$, given the magnetic flux ${\Phi }_{\mathrm{m}}/2$ 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 R₁ 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 R₁ decreases, this approximate circumference gradually compresses along the r-direction and partially coincides with the straight line r = R₁, 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 $\left(R_1,z_0\right)$ and $\left(R_3,0\right)$, i.e., ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_1}=+\infty$ and ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_3}=-\infty$, respectively. In addition, the first-order derivative at a point $R_2$ on the convex curve is equal to zero, namely ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_2}=0$, and its corresponding z-coordinate is $z_{\max }=z(R_2)$.

The magnetic flux through the area surrounded by the upper part of the curve $z\left(r\right)$ and the segment $\overline{R_1{R}_3}$ is

$${\displaystyle \frac{{\Phi }_{\mathrm{m}}}{2}}={\displaystyle {\int }_{R_1}^{R_3}\frac{\mu i}{2\pi r}}z\left(r\right)\mathrm{d}r.$$

(24)

With the definition $A={\displaystyle \frac{\pi {\Phi }_{\mathrm{m}}}{\mu i}}$, Equation (24) can be rewritten as

$${\displaystyle {\int }_{R_1}^{R_3}\frac{1}{r}}z\left(r\right)\mathrm{d}r=A.$$

(25)

In the interval $\left[R_1\le r\le R_3\right]$, the length of the upper half of the curve $z\left(r\right)$, which is denoted by ${\displaystyle \frac{l}{2}}$, is calculated as

$${\displaystyle \frac{l}{2}}=z_0+{\displaystyle {\int }_{R_1}^{R_3}\sqrt{1+{\left[z^{\prime }\left(r\right)\right]}^2}\mathrm{d}r}.$$

(26)

So far, the shortest perimeter problem has been attributed to the problem of determining the extreme value of the functional (26) under the constraints described by Equation (25). In summary, it can be formulated as the following conditional variational problem

$$\mathrm{minimize} I\left[z\left(r\right)\right]=z_0+{\displaystyle {\int }_{R_1}^{R_3}\sqrt{1+{\left[z^{\prime }\left(r\right)\right]}^2}\mathrm{d}r} \mathrm{under} \{\begin{array}{l}A={\displaystyle {\int }_{R_1}^{R_3}\frac{1}{r}}z\left(r\right)\mathrm{d}r=\mathrm{const}\\ \underset{r\to R_1^{+}}{\lim }z\left(r\right)=z_0 , z\left(R_2\right)=z_{\max } , z\left(R_3\right)=0\\ z^{\prime }\left(R_1\right)=+\infty , z^{\prime }\left(R_2\right)=0 , z^{\prime }\left(R_3\right)=-\infty \end{array}.$$

(27)

4.2.2. Solution to the Constrained Variational Problem via Euler’s Method

A Lagrange multiplier $\lambda \left(r\right)$ is introduced to transform the original constrained variational problem into a free variational problem [21,22]. The resultant free variational problem is

$$I^{\ast }\left[z\left(r\right),\lambda \right]={\displaystyle {\int }_{R_1}^{R_3}\left\{\sqrt{1+{\left[z^{\prime }\left(r\right)\right]}^2}+\frac{z_0}{R_3-R_1}+\lambda \frac{1}{r}z\left(r\right)-\lambda \frac{A}{R_3-R_1}\right\}\mathrm{d}r}.$$

(28)

Applying the Euler method to Equation (28) yields

$${\displaystyle \frac{\lambda }{r}}-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}\left({\displaystyle \frac{z^{\prime }\left(r\right)}{\sqrt{1+{\left[z^{\prime }\left(r\right)\right]}^2}}}\right)=0.$$

(29)

Integrate Equation (29) to obtain

$$z^{\prime }\left(r\right)=\pm {\left\{{\lambda }^{-2}{\left[\ln \left(C_{1}r\right)\right]}^{-2}-1\right\}}^{-1/2},$$

(30)

where C₁ is a constant. Based on the previous discussion of the nature of the curve z(r), Equation (30) can be rewritten as

$$z^{\prime }\left(r\right)=\{\begin{array}{l}\begin{array}{c}\end{array}{\left\{{\lambda }^{-2}{\left[\ln \left(C_{1}r\right)\right]}^{-2}-1\right\}}^{-1/2} \left(R_1<r\le R_2\right)\\ -{\left\{{\lambda }^{-2}{\left[\ln \left(C_{1}r\right)\right]}^{-2}-1\right\}}^{-1/2} \left(R_2\le r<R_3\right)\end{array}.$$

(31)

Integrating Equation (31) yields

$$z\left(r\right)=\{\begin{array}{l}\begin{array}{c}\end{array}{\displaystyle {\int }_{R_1}^r{\left\{{\lambda }^{-2}{\left[\ln \left(C_{1}r\right)\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}+C_2 , \left(R_1<r\le R_2\right)\\ -{\displaystyle {\int }_{R_2}^r{\left\{{\lambda }^{-2}{\left[\ln \left(C_{1}r\right)\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}+C_3 , \left(R_2\le r\le R_3\right)\end{array},$$

(32)

where C₂ and C₃ are constants introduced during the integration. Obviously, from $\underset{r\to R_1^{+}}{\lim }z\left(r\right)=z_0$ and $z\left(R_2\right)=z_{\max }$, we conclude that $C_2=z_0$ and $C_3=z_{\max }$. Next, we sequentially determine C₁, R₂ satisfying $z\left(R_2\right)=z_{\max }$, the Lagrange multiplier $\lambda$, z_max and z₀.

Since z(R₂) = z_max, the derivatives of z(r) satisfy ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_2^{+}}={{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_2^{-}}=0$, which can be substituted into Equation (31) to conclude ${\lambda }^2{\left[\ln \left(C_1{R}_2\right)\right]}^2=0$, i.e., C₁R₁ = 1. Therefore, we have

$$C_1={\displaystyle \frac{1}{R_2}}.$$

(33)

Thus, Equation (32) can be recast as

$$z\left(r\right)=\{\begin{array}{l}{z}_0+{\displaystyle {\int }_{R_1}^r{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r} \left(R_1<r\le R_2\right)\\ z_{\max }-{\displaystyle {\int }_{R_2}^r{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r} \left(R_2\le r\le R_3\right)\end{array}.$$

(34)

When r = R₁ and r = R₃, the first-order derivatives of z(r) are ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_1}=+\infty$ and ${{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}r}}|}_{r=R_3}=-\infty$, respectively. Then, it is straightforward to see from Equation (31) that

$$\{\begin{array}{c}{\lambda }^2{\left[\ln \left(C_1{R}_1\right)\right]}^2=1\\ {\lambda }^2{\left[\ln \left(C_1{R}_3\right)\right]}^2=1\end{array}.$$

(35)

The only possibility is

$$\ln \left(C_1{R}_1\right)=-\ln \left(C_1{R}_3\right)\Rightarrow C_1={\displaystyle \frac{1}{\sqrt{R_1{R}_3}}}.$$

(36)

The combination of Equations (33) and (36) leads to

$$R_2=\sqrt{R_1{R}_3}.$$

(37)

Substituting Equations (36) and (37) into Equation (35) gives

$${\lambda }^2={\left[\ln \sqrt{{\displaystyle \frac{R_3}{R_1}}}\right]}^{-2}.$$

(38)

From Equation (34) and z(R₃) = 0, it is deduced that

$$z_{\max } ={\displaystyle {\int }_{R_2}^{R_3}{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}.$$

(39)

It is also obtained from Equation (34) and the continuity of z(r) at r = R₂ that

$$z_{\max } =z_0+{\displaystyle {\int }_{R_1}^{R_2}{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}.$$

(40)

$$z_0={\displaystyle {\int }_{R_2}^{R_3}{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}-{\displaystyle {\int }_{R_1}^{R_2}{\left\{{\lambda }^{-2}{\left[\ln \frac{r}{R_2}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}.$$

(41)

Finally, by substituting Equations (33), (37), (38), (40), and (41) into Equation (34), the explicit formulation for z(r) of the optimal skeleton cross-section can be obtained as

$$z\left(r\right)=\{\begin{array}{l}{z}_0+{\displaystyle {\int }_{R_1}^r\left\{{\left[\ln \sqrt{\frac{R_3}{R_1}}\right]}^2{\left[\ln \frac{r}{\sqrt{R_1{R}_3}}\right]}^{-2}-1\right\}\mathrm{d}r} \left(R_1<r\le \sqrt{R_1{R}_3}\right)\\ z_{\max }-{\displaystyle {\int }_{\sqrt{R_1{R}_3}}^r\left\{{\left[\ln \sqrt{\frac{R_3}{R_1}}\right]}^2{\left[\ln \frac{r}{\sqrt{R_1{R}_3}}\right]}^{-2}-1\right\}\mathrm{d}r} \left(\sqrt{R_1{R}_3}\le r\le R_3\right)\end{array},$$

(42)

where

$$z_0={\displaystyle {\int }_{\sqrt{R_1{R}_3}}^{R_3}{\left\{{\left[\ln \sqrt{\frac{R_3}{R_1}}\right]}^2{\left[\ln \frac{r}{\sqrt{R_1{R}_3}}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}-{\displaystyle {\int }_{R_1}^{\sqrt{R_1{R}_3}}{\left\{{\left[\ln \sqrt{\frac{R_3}{R_1}}\right]}^2{\left[\ln \frac{r}{\sqrt{R_1{R}_3}}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r},$$

(43)

and

$$z_{\max } ={\displaystyle {\int }_{\sqrt{R_1{R}_3}}^{R_3}{\left\{{\left[\ln \sqrt{\frac{R_3}{R_1}}\right]}^2{\left[\ln \frac{r}{\sqrt{R_1{R}_3}}\right]}^{-2}-1\right\}}^{-1/2}\mathrm{d}r}.$$

(44)

4.3. 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. Comparison of different shapes of skeleton cross-section.

Shape of the Cross-Section	Perimeter (cm)	Rc ($\mathsf{\Omega }$)	Lc ($\mathsf{\mu }\mathbf{H}$)	Cc (pF)	${\mathit{f}}_{\mathit{h}\mathbf{1}}$	${\mathit{f}}_{\mathit{h}\mathbf{2}}$
Rectangular	4.22	0.0760	0.1044	1.230	1	1
Circular	3.93	0.0711	0.1044	1.151	1.069	1.034
Oval	3.85	0.0697	0.1044	1.128	1.090	1.044
Proposed	3.71	0.0671	0.1044	1.086	1.133	1.064

, 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, ${\epsilon }_r=5$, and $h=1.1$ mm.

First, it is observed that under fixed magnetic flux (namely identical L_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_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_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 $f_{l1}={\displaystyle \frac{R_s+R_c}{2\pi L_c}}$ decreases under the reduced R_c. To justify this conclusion, we start with the transfer function of the self-integration coil in Figure 2

$$G\left(s\right)={\displaystyle \frac{U_0\left(s\right)}{I_1\left(s\right)}}={\displaystyle \frac{Ms}{L_c{C}_c{s}^2+\left(\frac{L_c}{R_s}+R_c{C}_c\right)s+\left(\frac{R_c}{R_s}+1\right)}}.$$

(45)

The frequency response $\left|G\left(\mathrm{j}\omega \right)\right|$ around the lower cut-off frequency $f_{l1}\approx 15$ MHz and the uppercut-off frequency $f_{h1}\approx 0.14$ THz of the presented example are plotted in Figure 15a and Figure 15b, respectively. It is seen that in both low-frequency and high-frequency ranges, the proposed cross-section yields the highest transmission gain, while the rectangular cross-section exhibits the worst performance. To further evaluate the dynamic performance of the coils, Figure 16 depicts the step response of the self-integration coil. As expected, the proposed cross-section corresponds to the quickest response, indicating the largest overall bandwidth of this design. For comparison, the oval and circular cross-sections exhibit very close comparisons, while the rectangular cross-section leads to the longest rising time and thus the most unfavorable performance.

Last but not least, there are some extra benefits due to the reduction of the perimeter of the cross-section, including lighter weight and more compact structures.

5. Conclusions

The stray capacitance $C_c$(=$C_{AB}$) is one of the main factors affecting the high frequency performance of Rogowski coils. Reducing the value of $C_c$ can expand the bandwidth and improve the high frequency performance of the coils.

The value of the lumped stray capacitance $C_{AB}$ is approximately proportional to the length of a single-turn winding $l_t$ (which is approximately equal to the perimeter of the skeleton cross-section $l$); thus, the stray capacitance can be reduced via minimizing $l$. Therefore, this manuscript is devoted to the optimization of the shape of the skeleton cross-section of Rogowski coils, so that l is minimized without affecting the total magnetic flux as well as the low frequency performance of the coils.

The stray capacitance of a Rogowski coils of rectangular cross-section skeleton is minimized when the length $b$ and the width $a$ satisfy $b=\left(R_1+a\right)\ln (1+a/R_1)$, where $R_1$ is a pre-given inner radius of the skeleton. Similar optimization approach can be applied to the more realistic oval cross-section. On the other hand, the shortest perimeter that minimizes the stray capacitance under fixed magnetic flux does not exist for a circular cross-section skeleton.

Finally, under the condition that the total magnetic flux and the inductance of the coil are fixed, explicit formulations of the boundary of the optimal cross-section of the skeleton are deduced by solving a conditional variational problem. Numerical results demonstrate that the proposed optimization of the cross-section can effectively reduce the stray capacitance and resistance, leading to optimal bandwidth and dynamic response of Rogowski coils. The oval cross-section, as a combination of the rectangular and circular cross-sections, surprisingly outperforms the latter two options and exhibits very close performance to that of the proposed optimal cross-section.