Results of a systematic study of the current vs. voltage characteristics of ungated AIGaN/GaN heterostructures grown on sapphire substrates are presented. It is experimentally observed that the saturation current nearly doubles as the source-to-drain channel lengths decrease from 11.8 to 1.7μm. The average electric field at which current saturation occurs is 10 to 30kV/cm, i.e. much less than the electron velocity saturation field. The experimental data is interpreted in the framework of a new model that takes into account the non-uniformity of the electron density in the channel, electron velocity saturation, and thermal effects. The temperature dependent electron transport characteristics of the model are based on Monte Carlo simulations of electron transport in GaN. It is shown that appreciable contact resistance, which leads to partial channel depletion near the source, and significant self-heating of the devices under high drain-to-source bias are the main reasons for the observed current saturation. The effective ambient temperature in the channel of the devices is calculated from a two-dimensional thermal model of heat dissipation through the sapphire substrate. Equilibrium channel carrier concentrations and low-field mobilities are determined from Hall effect data. The ungated structures are demonstrated to provide much useful materials and process characterization data for the development of AIGaN/GaN heterostructure field effect transistors.