Oscillations of low-current electrical discharges between parallel-plane electrodes. III. Models

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Abstract
Simple models are developed to describe the results of measurements of the oscillatory and negative differential resistance properties of low- to moderate-current discharges in parallel-plane geometry. The time-dependent model assumes that the ion transit time is fixed and is short compared to the times of interest, that electrons are produced at the cathode only by ions, and that space-charge distortion of the electric field is small but not negligible. Illustrative numerical solutions are given for large voltage and current changes and analytic solutions for the time dependence of current and voltage are obtained in the small-signal limit. The small-signal results include the frequency and damping constants for decaying oscillations following a voltage change or following the injection of photoelectrons. The conditions for underdamped, overdamped, and self-sustained or growing oscillations are obtained. A previously developed steady-state, nonequilibrium model for low-pressure hydrogen discharges that includes the effects of space-charge distortion of the electric field on the yield of electrons at the cathode is used to obtain the negative differential resistance. Analytic expressions for the differential resistance and capacitance are developed using the steady-state, local-equilibrium model for electron and ion motion and a first-order perturbation treatment of space-charge electric fields. These models generally show good agreement with data from dc and pulsed discharge experiments presented in the accompanying papers.