A generalized external circuit model for electrostatic particle-in-cell simulations

A fully self-consistent second order accuracy model for coupling a generalized external circuit and a one-dimensional bounded electrode-driven plasma was proposed in this paper. The plasma was embedded in the circuit as a non-linear element with a potential difference. Based on Kirchhoff's voltage law and the definition of current, the problem of solving the generalized external circuit was transformed into an initial value problem of a set of first-order ordinary differential equations which can be discretized numerically by the second-order backward differential formula. The charge conservation equation at the electrode plate was coupled to the above equation, and the voltage, current, and surface charge density at the next moment were solved by a differential equation solver. Dirichlet boundary conditions of Poisson's equation were obtained through the surface charge density σ0 of the generalized external circuit equation and the plasma density ρ of the Particle-in-Cell (PIC) model. The spatial distribution of plasma potential was solved by using the second-order central difference scheme. The obtained potential at the electrode plate can be used as the Robin boundary condition of the system of generalized external circuit equations. In this model, the loose coupling between the generalized external circuit and plasma was realized by the boundary condition, and the system was fully self-consistent based on the charge conservation law and energy conservation law. We simulated the capacitively coupled plasmas (CCP) under different external circuits as the example and verified the performance of the model. This model can be used to study the influence of different external circuit structures and parameters on plasma discharge, and can be used for any plasma sources driven by electrodes, like CCP, some vacuum electronic devices, and Z-Pinches.