Adjoint Equations for Beam-Wave Interaction and Optimization of TWT Design

By formulating and solving the adjoint equations governing the beam–wave interaction in a traveling-wave tube (TWT), we show the partial derivatives with respect to design parameters of various TWT figures of merit (FoMs) may be efficiently calculated. FoMs include average gain, gain flatness, and gain–bandwidth product, and design parameters include beam voltage and circuit geometry. We use these derivatives in an optimization algorithm that finds parameter values that minimize or maximize the desired FoM. We show that a 1-D large-signal simulation code, such as CHRISTINE, may be easily modified to compute the adjoint solutions. We further show that only three runs of the modified code suffice to compute the partial derivatives of the output power and phase at a specified frequency with respect to an arbitrary number of parameters, resulting in potentially large savings in computing time compared with direct, finite difference calculation of the partial derivatives. We illustrate the method by optimizing the beam voltage and gap spacing of a <inline-formula> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula>-band folded-waveguide (FWG) TWT. The formulation given here applies only to TWT slow wave structures, such as coupled-cavity and FWGs, and to klystrons, composed of discrete gaps followed by drift spaces; it does not apply to helix structures, which may be the subject of a future paper.