We have developed a statistical model for the flashover of a $45\ifmmode^\circ\else\textdegree\fi{}$ vacuum-insulator interface (such as would be found in an accelerator) subject to a pulsed electric field. The model assumes that the initiation of a flashover plasma is a stochastic process, that the characteristic statistical component of the flashover delay time is much greater than the plasma formative time, and that the average rate at which flashovers occur is a power-law function of the instantaneous value of the electric field. Under these conditions, we find that the flashover probability is given by $1\ensuremath{-}\mathrm{exp}(\ensuremath{-}{E}_{p}^{\ensuremath{\beta}}{t}_{\mathrm{eff}}C/{k}^{\ensuremath{\beta}})$, where ${E}_{p}$ is the peak value in time of the spatially averaged electric field $E(t)$, ${t}_{\mathrm{eff}}\ensuremath{\equiv}\ensuremath{\int}[E(t)/{E}_{p}{]}^{\ensuremath{\beta}}dt$ is the effective pulse width, $C$ is the insulator circumference, $k\ensuremath{\propto}\mathrm{exp}(\ensuremath{\lambda}/d)$, and $\ensuremath{\beta}$ and $\ensuremath{\lambda}$ are constants. We define $E(t)$ as $V(t)/d$, where $V(t)$ is the voltage across the insulator and $d$ is the insulator thickness. Since the model assumes that flashovers occur at random azimuthal locations along the insulator, it does not apply to systems that have a significant defect, i.e., a location contaminated with debris or compromised by an imperfection at which flashovers repeatedly take place, and which prevents a random spatial distribution. The model is consistent with flashover measurements to within 7% for pulse widths between 0.5 ns and $10\text{ }\ensuremath{\mu}\mathrm{s}$, and to within a factor of 2 between 0.5 ns and 90 s (a span of over 11 orders of magnitude). For these measurements, ${E}_{p}$ ranges from 64 to $651\text{ }\text{ }\mathrm{kV}/\mathrm{cm}$, $d$ from 0.50 to 4.32 cm, and $C$ from 4.96 to 95.74 cm. The model is significantly more accurate, and is valid over a wider range of parameters, than the J. C. Martin flashover relation that has been in use since 1971 [J. C. Martin on Pulsed Power, edited by T. H. Martin, A. H. Guenther, and M. Kristiansen (Plenum, New York, 1996)]. We have generalized the statistical model to estimate the total-flashover probability of an insulator stack (i.e., an assembly of insulator-electrode systems connected in series). The expression obtained is consistent with the measured flashover performance of a stack of five 5.72-cm-thick, 1003-cm-circumference insulators operated at 100 and $158\text{ }\text{ }\mathrm{kV}/\mathrm{cm}$. The expression predicts that the total-flashover probability is a strong function of the ratio ${E}_{p}/k$, and that under certain conditions, the performance improves as the capacitance between the stack grading rings is increased. In addition, the expression suggests that given a fixed stack height, there exists an optimum number of insulator rings that maximizes the voltage at which the stack can be operated. The results presented can be applied to any system (or any set of systems connected in series) subject to random failures, when the characteristic statistical delay time of a failure is much greater than its formative time.
