Magnetic stiffness calculation for the corresponding force between two current-carrying circular filaments arbitrarily oriented in the space

TL;DR

This paper derives analytical formulas for calculating the magnetic stiffness components between two arbitrarily oriented current-carrying circular filaments using multiple methods, validated through mutual comparison.

Contribution

It introduces new integral and differentiation-based formulas for magnetic stiffness components, enhancing accuracy and validation for complex filament configurations.

Findings

Formulas expressed via elliptic integrals for magnetic stiffness.

Validation shows high agreement between different calculation methods.

Provides comprehensive tools for magnetic force analysis in complex filament arrangements.

Abstract

In this article, sets of analytical formulas for calculation of nine components of magnetic stiffness of corresponding force arising between two current-carrying circular filaments arbitrarily oriented in the space are derived by using Babic's method and the method of mutual inductance (Kalantarov-Zeitlin's method). Formulas are presented through integral expressions, whose kernel function is expressed in terms of the elliptic integrals of the first and second kinds. Also, we obtained an additional set of expressions for calculation of components of magnetic stiffness by means of differentiation of Grover's formula of the mutual inductance between two circular filaments with respect to appropriate coordinates. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.