Calculation of magnetic forces and torques on the Kibble coil

TL;DR

This paper presents simplified analytical methods for calculating magnetic forces and torques on a coil in yoke-based magnet systems, emphasizing energy-based calculations over line integrals, and introduces a measurement technique for coil positioning.

Contribution

It introduces a simplified energy-based approach for force and torque calculation on coils in yoke-based magnets and a method to determine coil position and orientation from measurements.

Findings

Simplified equations for forces and torques in yoke-based systems

Energy method is more straightforward than line integral approach

Method to determine coil position from three measurements

Abstract

Analytically the forces and torques on a coil in a field of magnetic flux density can be calculated one of two ways. The line integral can be conducted along the coil's wire, summing up the differential force contribution. For each differential line segment, the force is obtained as a cross product with the magnetic flux density. Alternatively, the coil's energy in the field is the current times the flux threading the coil. Hence, the energy is obtained by executing a surface integral over the coil's open surface. Here, a dot product of the differential surface element with the magnetic flux density is executed under the integral sign. The forces and torques can then be obtained from the negative derivative of the energy with respect to the appropriate coordinate. For yoke-based Kibble balances, the latter method is much simpler since most of the flux is contained in the inner yoke of the magnet and can be written as a simple equation. Here, we use this method to provide simple equations and their results for finding the torques and forces that act on a coil in a yoke-based magnet system. We further introduce a simple method that allows the calculation of the position and orientation difference between the coil and the magnet from three measurements.