Most of the models for Thin Film Transistors take no account of the way that electrons move across grain boundaries. The purpose of this work is to produce an analytical model, physically based, that takes account of both the change in height of the potential barriers that separate the grains, and the change of carrier density in the grains with gate voltage. The high current and subthreshold regions are treated. The model enables the change of field effect mobility with grain size and temperature to be determined. Two closed form expressions are provided which should be of value to both device and circuit designers, as well as providing insight to the physical processes occurring in such devices. They form a close analogy with the electrical characteristics of MOSFETs on crystalline silicon, diffusion and drift being replaced by quasi-diffusion and primary quasi-drift.