A lumped circuit model for transient arc discharges

Electrical breakdown of highly charged insulating systems can result in an arc discharge, i.e., a sudden, intense pulse of current. We model such arcs by a simple circuit: the discharge of a capacitor C (related to the initial charge reservoir) through a series inductor L and resistor R. For R=V*/‖Ia‖, where V* is a positive constant and Ia is the arc current, an essentially arbitrary dependence for L=L(Ia), a constant capacitance, and a circuit starting voltage V0, we establish four remarkable results for the subsequent arc discharge: (1) no discharge occurs at all unless ‖V0‖>V*; (2) if n is the largest non-negative integer for which ‖V0‖≥(2n+1)V*, then the arc current will reverse sign precisely n times and will decline in amplitude by 2V* at each extreme; (3) the discharge stops abruptly at a final voltage Vf=(−1)n+1[V0−(n+1)2V* sgn V0]; (4) maxima and minima in Ia occur at voltages V=±V*. Results (1) and (3) provide the threshold condition and finite final potential necessary for any realistic arc discharge theory, while result (2) suggests an experiment to look for a finite number of current oscillations in a highly driven arc. Result (4) suggests an experimental method for determining V*. Finally, the empirical areal scaling laws for arcs are reproduced with this model. The usual phenomenological treatments of arc start and stop voltages, current ringing, and areal scaling are thus modeled by a single parameter, V*. These results are generalized to voltage-dependent capacitance, C(V).