Adjoint Equations and Their Application to Helix TWT Design

We derive and apply a set of adjoint equations to the problem of determining the effect of small changes of the values of various design parameters on the output power and phase of a helix TWT, using a simple 1-D disk model for the beam.  The set of design parameters includes the phase velocity, attenuation, and Pierce impedance profiles as functions of axial location, and the beam voltage, current, and radius.  Such calculations are useful in determining the sensitivity of the TWT output to small variations in parameters and may be used therefore to analyze the effects of manufacturing tolerances, for example.  The adjoint method may also be used to compute derivatives of various TWT figures of merit in derivative-based optimization algorithms such as steepest descent.  The important feature of this method is that, once the adjoint solutions are obtained (by running just two simulations with particular initial conditions, as we will show), they may be used over and over again in the formula we derive to evaluate the effects of any perturbation or combination of perturbations of the design parameters without running any additional simulations.  The adjoint solutions therefore serve as a type of Green’s function for derivatives of the TWT output with respect to design parameters. The method applies both to small- and large-signal operation.