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It is demonstrated that, by a suitable modification of the well-known thin-film field-emission cathode (TFFEC) of C.A. Spindt (1948, 1976) into a FETRODE, a vacuum-type free-electron amplifier with a figure of merit superior to that achieved with triodes and FETs can be established. Analytical evaluation of important parameters of the FETRODE and the distributed amplifier's circuit is carried out. A formula for the RF efficiency of the FETRODE is applied to an approximate relation between the output current and the input RF voltage that gives the dependence of efficiency on the ratio of the RF output impedance and the DC beam impedance. Potential advantages of future FETRODEs over solid-state FETs are discussed, assuming already established DC current emission levels as well as improved levels resulting from low-work-function materials other than Mo cones.