A numerical approach based on a cavity/drive circuit model and a numerical calculation of the beam impedance is outlined that self-consistently determines the modulation of a klystron input cavity for an arbitrary coupling of the rf source to the cavity and arbitrary cavity parameters. The model is fully defined by the cavity R/Q factor, resonant frequency, rf field distribution as determined by an rf cavity simulation, unloaded cavity Q, and externally-loaded Q without beam. We find an expression for the power required to maintain a desired steady-state input cavity modulation with beam for both the case when the coupling is perfectly matched to the rf drive, and also for the case of arbitrary coupling. In addition, we find expressions for the cavity detuning and externally-loaded Q that, if satisfied, lead to the optimum matching condition. This formalism is especially useful for high-power, relativistic klystrons, where the beam loading can be large and nonlinear. Examples are given for an X-band cavity. An extension of this approach to accelerator cavities is presented.
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