Local field calculations for electrical trees in point-plane geometry

In the discharge-avalanche model for electrical trees branching is caused by time-dependent fluctuations in the local fields at the tree tips. Space charges of both polarities are produced by tube-discharges and polymer avalanches and make a major contribution to these fields. The fluctuations have thus been associated with rearrangements of the space-charge distribution brought about by the tube-discharges, avalanches in the polymer, recombination and diffusion. In calculating the local fields to be used in the model the space-charge distribution both within the tubular structure and around the tree periphery must be taken into account. In order to satisfy the boundary conditions, i.e. H.V. applied to a needle electrode with a grounded plane electrode, it is necessary to include the image charges (or equivalent surface charges) in the electrodes. These have a significant effect upon the local fields. Here we report field calculations for realistic tree structures incorporating space charges and their images. The walls of the tree tubules are assumed to hold adsorbed positive ions, while regions around the tree contain varying amounts of negative and positive charge. This work is intended to be a preliminary investigation prior to a full incorporation of the space charge fields into the discharge-avalanche model.