Variational Self-Consistent Theory for Beam-Loaded Cavities

A variational theory is presented for beam loading in microwave cavities. The beam-field interaction is formulated as a dynamical interaction whose stationarity according to Hamilton’s principle will naturally lead to steady-state solutions that indicate how a cavity’s resonant frequency, $Q$, and optimal coupling coefficient will detune as a result of the beam loading. A driven cavity Lagrangian is derived from first principles, including the effects of cavity-wall losses, input power, and beam interaction. The general formulation is applied to a typical klystron input cavity to predict the appropriate detuning parameters required to maximize the gain (or modulation depth) in the average Lorentz factor boost, $⟨\mathrm{\Delta }\gamma ⟩$. Numerical examples are presented, showing agreement with the general detuning trends previously observed in the literature. The developed formulation carries several advantages for the analysis and design of beam-loaded cavity structures. It provides a self-consistent model for the dynamical (nonlinear) beam-field interaction, a procedure for maximizing gain under beam-loading conditions, and a useful set of parameters to guide cavity-shape optimization during the design of beam-loaded systems. Enhanced clarity of the physical picture underlying the problem seems to be gained using this approach, allowing straightforward inclusion or exclusion of different field configurations in the calculation and expressing the final results in terms of measurable quantities. Two field configurations are discussed for the klystron input cavity, using finite magnetic confinement or no confinement at all. Formulating the problem in a language that is directly accessible to the powerful techniques found in Hamiltonian dynamics and canonical transformations may potentially carry an additional advantage in terms of analytical computational gains, under suitable conditions.

Figure 1

Illustration of a single-port cavity with an arbitrary shape. The port connects to a feeding transmission line (e.g., waveguide) and the walls are highly conductive. A current density $\mathbf{J}$ may exist inside the cavity. The surface normal vector points outwards and the port’s reference plane, $z=z_p$, is chosen for convenience at the $E$-field standing wave’s minimum (dip) nearest to the cavity walls.

Figure 2

Simplified drawing of the geometry of a klystron input cavity. The typical reentrant cavity shape forms a gap (here ungridded) where the electric field interacts with the beam. The electron gun is represented as a conceptual block.

Figure 3

Maximum detuning gain, ${\mathcal{G}}_{\max }$, as a function of beam-loaded cavity parameters $\overline{X}$ and $\overline{Y}$.

Figure 4

Geometry and dimensions of the x-band cavity analyzed in the first example. The figure shows the upper half of the cavity’s cross section. Electric field streamlines are shown in gray. Dimensions are in mm.

Figure 5

Geometries and dimensions of the c-band cavities analyzed in the second (a), third (b), and fourth (c) examples. The figures show the upper halves of the cavity cross sections. The cavity in (a) is similar to the one studied in Refs. [2, 3, 4]. Profiles of the “mushroom” and “inverted-mushroom” cavities in (b) and (c) are drawn with piecewise-elliptic functions, with guiding dimensions shown (to-scale). The cavity in (b) gives highest maximum gain, ${\mathcal{G}}_{\max }$, upon pretuning compared to the ones in (a) and (c). Electric field streamlines are shown in gray. Dimensions are in mm.

Figure 6

Frequency shift experienced by the c-band cavity shown in Fig. 5 under fixed beam voltage $V_b=240$ kV and varying perveance (current), as predicted by the presented theory, with sample current values chosen to match those presented by Ref. [2]. The almost linear trend is in agreement with general trend reported in Ref. [2], but with a lower slope. See Table 3 and text for details.