Mutual Inductance Calculation Between Circular Filaments Arbitrarily Positioned in Space: Alternative to Grover's Formula

In this paper, we present the full derivation of a new formula for calculating the mutual inductance between inclined circular filaments arbitrarily positioned with respect to each other. Although such a formula was already proposed by Grover more than 50 years ago, the formula presented here is slightly more general and simpler to use, i.e., it involves only a sequential evaluation of expressions and the numerical resolution of a simple numerical integration. We derived the new formula using the method of vector potential, as opposed to Grover's approach, which was based on the Neumann formula. We validated the new formula through a series of examples, which are presented here. Finally, we present the relationship between the two general formulas (i.e., Grover's and our new one) explicitly (Example 12).

I. Introduction

THE mutual inductance calculation between coaxial circular filaments has been thoroughly treated by a number of authors since the time of Maxwell, and an accuracy exceeding anything required in practice is nowadays possible [1]–[7]. A formula for two circles whose axes intersect was first given by Maxwell [1]. Formulas for circular loops with parallel axes have later been derived by Butterworth [2] and Snow [5]. Unfortunately, these formulas are slowly convergent and are applicable only within a restricted range of parameters. Using Butterworth's formula [2], Grover developed a general method to calculate the mutual inductance of inclined circular filaments in any desired positions in the form of a single integral [3], [4].