Thin-film field-effect transistors in hydrogenated amorphous silicon are notoriously unstable due to the formation of silicon dangling bond trapping states in the accumulated channel region during operation. Here, we show that by using a source-gated transistor a major improvement in stability is obtained. This occurs because the electron quasi-Fermi level is pinned near the center of the band in the active source region of the device and strong accumulation of electrons is prevented. The use of source-gated transistors should enable stable analog circuits to be made in amorphous silicon.