Longitudinal coupling impedance of a particle traveling in PEC rings: A regularised analysis

Abstract

The analysis of a charged particle traveling through one, two, or infinite conducting rings is presented. The problem is formulated in the spectral domain as integral equations of Fredholm type, solved by Galerkin's method employing entire domain functions factorising the correct edge behaviour of the unknown induced current. As a result, high accuracy and numerical stability are achieved. Numerical results are presented, proving the effectiveness of the method, and the parameters relevant to accelerator physics are calculated.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings

“…A first possibility is to subtract the asymptotic behavior of the kernels, thus leading to integrals of the product of four Bessel functions and powers. Such an approach has been used, for example, in [44] in a simpler, scalar case. As a matter of fact, it can be shown that the integral of the product of four Bessel functions can be analytically evaluated as a Meijer function.…”

“…However, even more can be said: such an approach properly works, i.e., it is also possible to apply Fredholm theory when the matrix operator corresponding to the singular part is invertible (not necessarily diagonal) with a continuous two-side inverse and the residual part is a compact operator [26]. Such a procedure has been called Method of Analytical Preconditioning (MAP) [27], and has been used to solve a huge number of scattering, radiation, and propagation problems [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].…”

Guest Editorial: Method of analytical regularisation for new frontiers of applied electromagnetics

“…The analysis provided in the paper ‘Longitudinal coupling impedance of a particle travelling in PEC rings: A regularised analysis’ by Assante et al. [16], of relevance in the accelerator and collider physics, is carried out in the Hankel transform domain. The obtained first‐kind singular integral equation is discretised by means of the method of analytical preconditioning with expansion functions reconstructing the behaviour of the unknowns at the edges.…”

Analysis of the Scattering from a Two Stacked Thin Resistive Disks Resonator by Means of the Helmholtz–Galerkin Regularizing Technique

“…This is what happens when: (1) the Galerkin scheme is adopted, and (2) the selected expansion functions are orthonormal eigenfunctions of a suitable operator containing the most singular part of the integral operator at hand. Such an approach, appropriately called method of analytical preconditioning, is very effective, as clearly shown in the literature devoted to the study of the scattering, propagation, and radiation problems [23][24][25][26][27][28][29][30][31][32][33]. Another way to obtain a guaranteed-convergence consists in solving numerically the singular integral equation by means of a Nyström-type discretization scheme taking into account the singularity of the integral equation and the behavior of the unknowns at the edges [34][35][36][37][38].…”