Solutions of the low-frequency plasma sheath circuit equations

We derive a relation between the time derivatives of the current and voltage of the low-frequency plasma sheath. This relation is used to derive a first order differential equation for the electrical current in a driven series resistor, capacitor, and sheath circuit. Analytic and semi-numeric solutions are obtained for pulse and periodic excitations. We use these solutions to analyze the Langmuir probe response in some common diagnostic applications: the pulse excitation (\v{S}amara et al, 2012)\footnote{\v{S}amara et al. A dc-pulsed capacitively coupled planar Langmuir probe for plasma process diagnostics and monitoring, Plasma Sources Sci. Technol. 21 (2012) 065004} and AC Bias (Van Nieuwenhove \& Van Oost, 1988)\footnote{Van Nieuwenhove \& Van Oost, Novel Langmuir probe technique for the real-time measurement of the electron temperature, Rev.Sci.Instrum. 59(7),July 1988} methods.