where γi is the secondary emission coefficient (dimensionless), N is the number density (SI unit: 1/m3), α is the reduced Townsend growth or decay coefficient (SI unit: m2), s is the arc length along the particle trajectory, and D is the distance from the source boundary to any destination boundary. Using this, the following 3 regimes are defined.

Rearranging Equation 10-1, it is obvious that no discharge will occur if the following condition is true:

When the left side of Equation 10-1 is greater than 1, a self-sustaining discharge can occur. Another way of writing this condition is that a self-sustained discharge can form when the Townsend condition is met:

When the exponential of the left side is above around 108, a streamer will form across the gap. Mathematically, the streamer condition is given by:

where d is the gap distance in cm.

The Electrical Breakdown Detection physics interface defines a variable, ebd.bi, which takes the value of 0 for the no-discharge case, 1 for the sustained discharge, and 2 for the streamer. This variable is plotted by default when running a study.

All the information about in which regime the system will operate is embedded in the reduced Townsend coefficient, α. The reduced Townsend coefficient is a strong function of the reduced electric field:

where E is the electric field parallel to the streamlines (SI unit: V/m). The software computes the integral by solving an ordinary differential equation along the test particle trajectories:

where Nstp is the number density at standard temperature and pressure. Another quantity of interest is the pressure multiplied by the path length. This is also computed by solving the following ordinary differential equation: