# Analysis and Optimization of the Stray Capacitance 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 finiteelement 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.