Generalized Paschen’s Law for Overvoltage Conditions

According to Paschen’s law for the gas discharge ignition, the static self-breakdown voltage$U_{\text {st}} $depends on the gas pressure$P$product by interelectrode spacing$d$. The law is described by classic Paschen’s curves$U_{\text {st}} ({\textit {Pd}})$with one characteristic minimum. Also, it is known that with increasing high-voltage waveform (HVW) steepness the achieved voltage amplitude$U({\textit {Pd}})$increases above$U_{\text {st}} ({\textit {Pd}})$. Analyzing the$U_{\text {st}} ({\textit {Pd}})$curves from a point of view of the electron runaway, we predicted and observed experimentally that$U({\textit {Pd}})$not only increases above$U_{\text {st}} ({\textit {Pd}})$, but, in addition, the$U({\textit {Pd}})$minimum position displaces to the higher$\textit {Pd}$magnitudes with increasing HVW steepness. Thereby, Paschen’s law is generalized: in general, the breakdown voltage$U$depends not only on$\textit {pd}$but also on the HVW rise time in the open-circuit mode${\tau }_{{o{-}c}} $; accordingly, Paschen’s curve$U_{\text {st}} ({\textit {Pd}})$of a particular gas is a degenerate case of a general family of curves$U({\textit {Pd},{\tau }_{{o{-}c}}})$.