Modeling transit-time microwave tubes

An excitation model for TM-mode cylindrical cavities driven by a modulated electron beam with application in transit-time microwave tubes is developed. It is shown that the slowly time varying amplitude of the self-sustained oscillations is described by a reduced form of the classical van der Pol equation. Encompassing the relevant parameters of the dynamics of the rf field-electron beam interaction (nonlinear conductance of the electronic beam, the ratio R/Q of the resonant cavity, dc beam resistance), an excitation equation is then obtained from which expressions for the starting current and saturation amplitude are readily derived.