A general, self-consistent multi-loop circuit model with multiple electrode surfaces and simultaneous potential solution for particle-in-cell simulations

In this work, we present a generalized second-order algorithm for coupling external circuits of arbitrary complexity to a particle-in-cell plasma simulation code. With multi-loop external circuits consisting of linear (resistor, inductor, capacitor) components attached to the plasma chamber, convective currents are collected on each plasma-bounding electrode surface. The voltage delivering and charge conservations on electrode surfaces are derived by applying Gauss's theorem on each surface and Kirchhoff's current/voltage laws on each node/path. The boundary condition of the Poisson equation on each electrode is now coupled with potentials from all other electrode surfaces, and they are solved simultaneously with plasma potential using the potential decomposition method. The displacement current, which is part of charge balance and crucial in RF discharges, is also expressed in terms of electrode potentials and included consistently, allowing an accurate and stable solution.