Cavity Excitation Described by the van der Pol Equation in Transit-Time 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 (nonlinear conductance of the electronic beam, ratio of the resonant cavity, and dc beam resistance) of the dynamics of the RF field-electron beam interaction, an excitation equation is then obtained from which expressions for the starting current and saturation amplitude are readily derived.