# Efficient calculation of the mutual inductance of arbitrarily oriented circular filaments via a generalisation of the Kalantarov-Zeitlin method

## TL;DR

This paper introduces a new analytical formula for calculating the mutual inductance between arbitrarily oriented circular filaments, addressing singularity issues and extending applicability compared to previous methods.

## Contribution

It generalizes the Kalantarov-Zeitlin method to derive a flexible, efficient formula for mutual inductance between circular filaments in any orientation, improving upon prior approaches.

## Key findings

- The new formula resolves singularity issues in existing methods.
- Validation confirms high accuracy compared to literature and FastHenry software.
- The method extends to calculating mutual inductance with tilted plane projections.

## Abstract

In this article, we present a new analytical formulation for calculation of the mutual inductance between two circular filaments arbitrarily oriented with respect to each other, as an alternative to Grover [1] and Babic [2] expressions reported in 1944 and 2010, respectively. The formula is derived via a generalisation of the Kalantarov-Zeitlin method, which showed that the calculation of mutual inductance between a circular primary filament and any other secondary filament having an arbitrary shape and any desired position with respect to the primary filament is reduced to a line integral. In particular, the obtained formula provides a solution for the singularity issue arising in the Grover and Babic formulas for the case when the planes of the primary and secondary circular filaments are mutually perpendicular. The efficiency and flexibility of the Kalantarov-Zeitlin method allow us to extend immediately the application of the obtained result to a case of the calculation of the mutual inductance between a primary circular filament and its projection on a tilted plane. Newly developed formulas have been successfully validated through a number of examples available in the literature, and by a direct comparison with the results of calculation performed by the FastHenry software.