Nonlinear Spark Resistance and Capacitive Circuit Models of Electrostatic Discharge

Over the past century, several models of spark discharge have been developed that approximate the effect of a spark channel in a circuit as a nonlinear resistance. In this article, we review the physical derivations of a set of spark models, and give analytical and numerical solutions to spark discharges in simple capacitive circuits. In electrostatic discharge (ESD), we find the drop in spark resistance is limited by the initial stored charge in the system. For a simpler ESD model (spark resistance in series with the electrode capacitance), we obtain the analytical solutions for a spark resistance model of a general integral form, consisting of the time integral of the current raised to an arbitrary power. Herein, we also consider a circuit with an additional fixed “victim” resistance in series with the spark resistance, and capacitance. Here, although the general case requires numerical solution, it is possible to find asymptotic limits for the case of the Rompe and Weizel spark model. Our analysis provides insights into system response that can, for example, be used to evaluate risk zones of parameter space for ESD protection.