A large-signal, relativistic theory of the electron-field interaction in a new class of microwave amplifiers is presented and applied to the analysis of a high-power, 450-MHz amplifier for accelerator applications. The analysis indicates that electronic efficiencies in excess of 90 percent are obtainable and that overall efficiencies of 90 percent are possible. The amplifier is unique in several respects; the electron velocity is perpendicular to the circuit energy flow, the device uses a fast-wave circuit, and the electron beam is deflection modulated.