Features of the Application of the Virial Theorem for Magnetic Systems with Quasi-Force-Free Windings

Abstract

The article shows that in a magnetic system with a thin-walled balanced winding close to a force-free one, a significant increase in the parameter θ = W_Mγ/Mσ_M, is possible, which, according to the virial theorem, characterizes the ratio of the energy of the magnetic system W_M to the weight of equipment with a material density γ, where under the action of electromagnetic forces there appears a mechanical stress σ_M. In a quasi-force- free magnetic system, the main part of the winding is in a state of local equilibrium, and only a relatively small part of the equipment is subject to stress. This part determines the weight of the entire system, and this weight can be minimized. The configurations of balanced thin-walled windings are developed, at the boundaries two boundary conditions are fulfilled simultaneously—the absence of the induction component normal to the boundary and the constancy of the product of induction and radius. The authors consider an example of a system consisting of a main part—a sequence of balanced “transverse” modules in the form of flat discs and end parts, consisting of a combination of “transverse” modules and “longitudinal” ones, having the form of rings elongated along the axis with balanced end parts. It is shown that in the system under consideration, the characteristic dimensionless parameter θ with an unlimited increase in the number of elements of the main part can reach a value of about 24, and when the number of these elements changes within 20–40, it changes from 6 to 9.

Appendices

APPENDIX

Construction of a Free Boundary by the Iteration Method

The iteration method is used to solve a problem with two boundary conditions (2), (4), specified at the boundary with an unknown configuration (the free boundary problem [19]). Radii-vectors of boundary points can be conveniently presented as complex numbers: p(s) = z(s) + ir(s), where s is the parameter—arc length counted from the beginning of the free boundary with specified coordinate p(0) = p₀. The magnetic field vector for poloidal field B(s) at this point is also written in the complex form. The iteration process consists of two stages.

Stage 1. Each iteration step includes a calculation of the magnetostatic problem with the specified boundary condition (2) through solving of an integral equation of the first kind. The boundary is assigned at the first step by an initial approximation, the boundary calculated at the previous step is used at each subsequent step. Magnetic field vector B₁(s) = B₁(s)exp(iα₁) is directed along a tangent line towards the conductor boundary, where α₁ is the tangle line direction angle. Then, according to the equilibrium condition, we adopt |B_φ(s)| = B₁(s) and calculate constant C₂ by averaging the function r|B_φ(s)| at a boundary with length l:

$${{C}_{2}} = \frac{1}{l}\int\limits_l^{} {r{\text{|}}{{B}_{\varphi }}(s){\text{|}}dl(s).} $$

Stage 2. The integral equation of the second kind is solved to calculate the magnetostatic field with boundary condition (4), where constant C₂ was determined at the first stage. The magnetic field vector at the boundary takes on a new value B₂ = B₂exp(iα₂) and changes its direction by angle Δα = α₂ – α₁.

A change of the angle means a violation of boundary condition (2), according to which the magnetic field vector must be directed along a tangent line towards the boundary. To meet condition (2), we choose a new value for the angle of tangent line slope at each point of boundary α' = α + λΔα, where λ is the iteration parameter that accelerates process convergence: 1 < λ < 2 A new coordinate for a free boundary node with number k + 1 at the given value of the angle of tangent line slope towards boundary dp/ds = exp(iα'(s)) can be calculated using the trapezoid formula: p_{k + 1} = p_k + h_kexp[0.5i($\alpha _{k}^{'}$ + $\alpha _{{k + 1}}^{'}$)], k = 0, ..., N – 1, where N is number of free boundary elements, $\alpha _{k}^{'}$, $\alpha _{{k + 1}}^{'}$ is tangent line slope angle at the initial and final point of each element, h_k is its length. h_k was chosen so that the node points displaced in the given direction (usually in parallel to one of the coordinate axes).

The iteration process is continued until the following condition is met with the specified accuracy: α₂ – α₁ = 0. The examples of plane problem, which allow a comparison with an analytical solution, have confirmed the convergence of the iteration process on the basis of on an initial approximation which is far from the final one [20].