An analytic model for the thin-film field-emission triode has been developed which allows fast and accurate calculation of the electric field distribution on the surface of the thin-film cathode for a wide range of device dimensions and operating voltages. The model can also be used to calculate the potential distribution throughout the device. The model is used together with the Fowler-Nordheim equation and with the published field-emission data on a carbon nanotube film to calculate the profiles of emission current along the surface of the thin-film cathode for a number of geometries. Non-uniform emission is shown to be a problem, which will result in undesirably large gate currents. Increasing the anode voltage to bias the cathode to just below its emission threshold even when the gate is off is shown to be an effective means of providing uniform emission when the gate is turned on while preserving a sufficiently high on/off ratio of emitted current.