A Novel Approach for Calculations of Helical Toroidal Coil Inductance Usable in Reactor Plasmas

In this paper, formulas are proposed for the self- and mutual-inductance calculations of a helical toroidal coil by direct and indirect methods at superconductivity conditions. The direct method is based on the Neumann's equation, and the indirect approach is based on the toroidal and the poloidal components of the magnetic flux density. Numerical calculations show that the direct method is more accurate than the indirect approach at the expense of its longer computational time. Implementation of some engineering assumptions in the indirect method is shown to reduce the computational time without loss of accuracy. Comparison between the experimental, empirical, and numerical results for inductance, using the direct and the indirect methods, indicates that the proposed formulas have high reliability. It is also shown that the self-inductance and mutual inductance could be calculated in the same way, provided that the radius of curvature is greater than 0.4 of the minor radius and that the definition of the geometric mean radius in the superconductivity conditions is used. Plotting contours for the magnetic flux density and the inductance show that the inductance formulas of the helical toroidal coil could be used as the basis for coil optimal design. Optimization target functions such as maximization of the ratio of stored magnetic energy with respect to the volume of the toroid or the conductor's mass, the elimination or the balance of stress in certain coordinate directions, and the attenuation of leakage flux could be considered.

I. Introduction

Recent research work on superconducting magnetic energy storage (SMES) systems, nuclear fusion reactors, and plasma reactors such as the Tokamak has suggested the use of advanced coil with a helical toroidal structure [1]–[4]. The main reason for this suggestion is the ability to implement special target functions for this coil in comparison with other structures such as the toroidal, the solenoid, and the spherical coils [5], [6]. The structure of this coil is shown in Fig. 1. In this coil, the ratio of the major to the minor radius , the number of turns in a ring , and the number of rings in a layer are called aspect ratio, poloidal turns (or the pitch number), and helical windings, respectively. For example, the coil in Fig. 1 is composed of five helical windings with nine poloidal turns . The inductance formulas show that parameters , , and of the helical toroidal coil can be used as design parameters to satisfy special target functions. With respect to the fact that each ring of the coil generates both toroidal and poloidal magnetic fields simultaneously, the coil can be regarded as a combination of coils with toroidal and solenoid fields. Furthermore, the coil can be designed in a way to eliminate the magnetic force component in both the major and minor radius directions. These are called force- and stress-balanced coils, respectively. In addition, the coils that utilize the virial theorem to balance these two force components are called virial-limited coils [7]–[9]. In some applications, helical toroidal coils are used in a double-layer manner with two different winding directions (respectively with different or the same current directions in each layer) to reduce the poloidal leakage flux being compared to the toroidal leakage flux or vice versa. In this paper, the investigation is focused on the one-layer helical toroidal coil. Fig. 1.

Structure of a monolayer helical toroidal coil with five rings of nine turns.