Numerical solution to an electromagnetic model with Neumann boundary conditions, for a microwave-driven plasma reactor

Abstract

This work deals with the two-dimensional electromagnetic modelling of a microwave-driven plasma reactor, operated by an axial injection torch (AIT). The model solves Maxwell's equations, which are discretized using a finite difference scheme within staggered grids, adopting a time-harmonic description at fixed 2.45 GHz excitation frequency. The study focuses upon azimuthal axis-symmetric situations, which can be described by a single second-order Helmholtz-type differential equation for the transverse magnetic field. Perfect-conductor boundary conditions are imposed at metal walls, corresponding to zero derivatives for the magnetic field. In situations where convergence requires a more restrictive framework, these Neumann boundary conditions are better replaced by equivalent Dirichlet conditions, whose boundary values depend on the problem solution. Here, we propose a simple numerical algorithm to manage these situations, by tailoring the Dirichlet boundary values to satisfy the physical Neumann conditions. The algorithm is applied to an air-filled circular wave-guide (as a test system) and to the AIT-reactor device (in the presence of plasma). Solution benchmarking checks its accuracy with respect to the corresponding analytical solution (for the circular wave-guide), and analyses its numerical precision by using different integral expressions to calculate the power transmission coefficient (for the AIT-reactor). Results show a 99% accuracy and precision errors lower than 0.1%, for a mesh with 104 grid points.