Advances in Electrical Engineering

Volume 2016, Article ID 3654021, 6 pages

http://dx.doi.org/10.1155/2016/3654021

## Mutual Inductance and Magnetic Force Calculations for Bitter Disk Coil (Pancake) with Nonlinear Radial Current and Filamentary Circular Coil with Azimuthal Current

^{1}Département de Génie Physique, École Polytechnique, CP 6079, Succ. Centre Ville, Montréal, QC, Canada H3C 3A7^{2}Département de Génie Électrique, École Polytechnique, CP 6079, Succ. Centre Ville, Montréal, QC, Canada H3C 3A7

Received 12 June 2016; Revised 8 August 2016; Accepted 22 August 2016

Academic Editor: Gorazd Stumberger

Copyright © 2016 Slobodan Babic and Cevdet Akyel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Bitter coils are electromagnets used for the generation of extremely strong magnetic fields superior to 30 T. In this paper we calculate the mutual inductance and the magnetic force between Bitter disk (pancake) coil with the nonlinear radial current and the circular filamentary coil with the azimuthal current. The close form expressed over complete elliptic integrals of the first and second kind as well as Heuman’s Lambda function is obtained for this configuration either for the mutual inductance or for the magnetic force. The results of this method are compared with those obtained by the improved modified filament method for the presented configuration. All results are in an excellent agreement.

#### 1. Introduction

In the literature and scientific papers the calculation of the mutual inductance and the magnetic force between ordinary circular coils with the uniform current densities have been given by different methods either analytical or numerical [1–15]. However, there are not a lot references about the mutual inductance and the magnetic force calculation between circular coils in which the current densities are not uniform. The coils with this characteristic are Bitter coils or Bitter solenoids. Bitter coils are used in high magnetic field applications and they differ from the ordinary coils in that they have an inverse radial distribution of current [16, 17]. The interesting point is the calculation of the mutual inductance and the magnetic force between Bitter coils or between one Bitter coil and an ordinary coil. Generally, the calculation of the magnetic force between circular coils is closely related to the calculation of their mutual inductance. Since their mutual energy is equal to the product of their mutual inductance and the currents in the coils, the component of the magnetic force of attraction or repulsion in any direction is equal to the product of the currents multiplied by the differential coefficient of the mutual inductance taken with respect to that coordinate. The mutual inductance and the magnetic force are obtained over multiple integrals with the different kernel functions. Thus, in the calculation of the mutual inductance and the corresponding magnetic force between two coils we need to integrate their kernel functions depending on coil configurations. Hopefully, these kernel functions are Green functions and , where the “” is the distance between two coils. This integral approach, much easier than the differential approach, leads to relatively simple expressions which are incorporated in these two physical quantities. In this paper we calculate the mutual inductance between the Bitter disk coil with nonlinear radial current and a circular filamentary coil with the azimuthal current (ordinary coil). Coils are coaxial and in air. Either the mutual inductance or the magnetic force is obtained in closed form expressed over complete elliptic integrals of the first and second kind as well as over Heuman’s Lambda function [18, 19]. The results of these calculations will be compared to those obtained by the improved filament method for concerned configuration. The results obtained by these two methods are in an excellent agreement.

#### 2. Basic Expressions

The mutual inductance and magnetic force between a disk coil (pancake) with the uniform azimuthal current density and a filamentary coil with the azimuthal current (see Figure 1) can be calculated, respectively, by [1, 2]whereand is the number of turns of the disk coil (pancake).