We extend a quasi two-dimensional model for the channel potential at threshold as a function of an arbitrary distribution of trapped charge. The model agrees well with detailed numerical simulations, as long as the effective channel length is adjusted to model the uneven turn-on in the device. This formulation of the channel potential reveals that the trapped charge affects the threshold voltage through an averaging over lengths long enough that it should reduce the effect of variations in the density of trapped charge that is expected in memory devices at small dimensions. (c) 2005 Elsevier Ltd. All rights reserved.