Optimization of vacuum electronic (VE) amplifiers with respect to basic design parameters is a critical task in amplifier design. This task becomes increasingly difficult as the number of design parameters gets large, as is often the case for high performance amplifiers. Gradient based optimization methods can be applied to optimization tasks with many design parameters, but gradient methods require an efficient way to compute the multi-dimensional derivatives. In [1] it was shown that the adjoint approach, based on symplectic area conservation, can be used to compute these derivatives very efficiently. The adjoint approach for the evaluation of derivatives was implemented in the 1D large signal simulation code CHRISTINE-Z [2]. Only three runs of CHRISTINE-Z are needed to compute the partial derivatives of the output power and phase at a specified frequency with respect to an arbitrary number of parameters that characterize the beam and the circuit. The CHRISTINE-Z code was used together with a conjugate gradient optimization algorithm to find parameter values that minimize or maximize various figures of merit that characterize the performance of a folded-waveguide TWT [1].

We present our recent experience with improved thermionic emission models in the finite-element gun code MICHELLE <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,2</sup> .

We show that optimization of Traveling Wave Tube (TWT) design with respect to many design parameters can be efficiently performed by gradient based methods. The partial derivatives with respect to design parameters of various TWT figures of merit can be efficiently calculated by using the adjoint approach [1] , [2] . Several practically important figures of merit have been studied, include average gain, gain flatness, and gain-bandwidth product, and design parameters include beam voltage and circuit geometry. The derivatives are calculated by modified 1D large signal simulation code CHRISTINE-Z [3] and used in the steepest descent optimization algorithm that finds parameter values that minimize or maximize the desired figure of merit. Only three runs of the modified code are needed to compute the partial derivatives of the output power and phase at a specified frequency with respect to an arbitrary number of parameters. It results in a potentially large savings in computing time compared with direct, finite difference calculation of the partial derivatives. illustrate the method by optimizing the beam voltage and gap spacing of a W-band folded-waveguide TWT. The examples of TWT design optimization for small signal gain and output power in large signal regime will be presented and discussed.

An electromagnetic shock occurs when a charged particle strikes a conducting surface and is removed. Using a line charge model, this paper compares the induced current due to this electromagnetic shock with the classical, electrostatic induced current of Ramo and Shockley. The shock-induced current is found to be negligible for a deeply non-relativistic impact energy but becomes appreciable at mildly relativistic impact energies. The implications of this study are discussed.

When a charged particle strikes a conducting surface and is removed upon impact, an electromagnetic shock is generated at the location of impact. This purely electromagnetic effect was shown in our recent extension [1] of the classical Ramo-Shockley theorem (RS) [2] . RS gives the induced surface current assuming nonrelativistic velocities and electrostatic fields. In this paper, we provide a comparison of the electromagnetic shock-induced current with the classical RS for an infinitely long charged rod striking an infinite, perfectly conducting plate in the single and parallel plate geometries. The electromagnetic shock-induced currents are calculated assuming a constant velocity of the charge before impact, and the classical induced current in the parallel-plate geometry assumes that the charge in transit is subject to an RF voltage. We note that the induced current due to the electromagnetic shock is comparatively small for low impact energies (less than 100 eV, such as those found in multipactor discharges [3] ) but becomes significant for relativistic impact velocities. The electromagnetic shock thus could have a considerable effect on beam loading in relativistic magnetrons and in magnetically insulated line oscillators (MILOs) [4] , but is relatively unimportant in multipactor discharges [3]

The classical Ramo-Shockley theorem gives the induced current on a conductor, assuming quasi-static motion of charges and electrostatic fields. This paper presents examples of the induced current distribution that is relativistically and electromagnetically correct.

Jensen has developed a general expression for the current density J(GT F) emitted from a surface by thermal or field emission, or by a combination of these processes; he has also shown how the same expression may be analytically extended to compute the current density due to photoemission. J(GT F) is expressed as a single integral, which Jensen evaluates analytically in certain limits and numerically outside those limits. In this paper, we show how the integral may be expressed generally as convergent series that are easily evaluated numerically for all values of the physical parameters. No real or apparent singularities, like those that appear in the function Z defined by Jensen occur in the series obtained here.

We demonstrate a practical numerical approach to the analysis of the effects of finite fabrication tolerances on the performance of a folded-waveguide TWT. The approach involves the simulation of a numerically generated 'ensemble' of TWTs that differ from each other by small random variations in certain key physical circuit dimensions. By computing the ensemble averages and standard deviations of the computed values of various performance metrics (gain, phase, and output power, for example) we can predict the average values and ranges of these and other metrics to be expected in as-built devices. Such studies also can help discover those physical dimensions to which the performance is most sensitive, and therefore can help to optimize a TWT design for maximum performance and/or manufacturing yield.

We develop a sensitivity function for the design of electron guns, based on a form of reciprocity implicit in Hamilton's equations of motion. The sensitivity function allows for the determination of the effect of small, arbitrary changes in electrode potentials, electrode positions, and magnet locations on specific beam quality figures of merit with a single, time-reversed run of a beam optics code. The sensitivity function can thus be used to predict the sensitivity of a design to manufacturing errors, or it can be used in an optimization cycle of a gun's design.

This paper presents an efficient method for the calculation of impedance matrices of slow and standing wave structures used in vacuum electronic devices. These matrices can be imported by large signal parametric codes like TESLA-Z and CHRISTINE-CC for accurate small and large signal analysis of TWTs, klystrons and other VE devices [1], including effects of all geometric features and material properties included in the calculation of the impedance matrix. The technique employs both lumped (internal) and wave (modal, boundary) ports in the 3D finite element electromagnetic (EM) code ANALYST [2].

Joule heating limits the operation of most current carrying components and devices, especially in nanoscale circuits such as carbon nanofiber based field emitters, graphene electronics, and nanolasers [1]. For many materials of interest it is important to consider the temperature dependence of the thermal and electrical conductivities when calculating the effects of Joule heating. We examine the effects of linear temperature dependence of the electrical and thermal conductivities on the heating of a one-dimensional conductor by solving the coupled non-linear steady state electrical and thermal conduction equations. We find that there are conditions under which no steady state solution exists. In the special case in which the temperature dependence of the electrical conductivity may be neglected, we have obtained explicit expressions for these conditions. The maximum temperature and its location within the conductor are examined for various boundary conditions. We note that the absence of a steady state solution may indicate the possibility of thermal runaway [2].

Using the Briggs-Bers criterion, we find that the lower band edge of linear beam TWTs is not subjected to absolute instability. At the upper band edge, we find a threshold beam current beyond which absolute instability is excited. In general, an absolute instability would occur in a linear beam tube if the cold-tube circuit dispersion curve in the frequency-wavenumber (ω-k) plane is locally convex and would not occur if the cold-circuit dispersion curve is locally concave, whenever the group velocity of the circuit mode is in the same direction as the beam mode. These results are used to examine the recent studies on the start current of a Smith-Purcell source.

We use the Briggs-Bers criterion to examine the absolute instability at the lower and upper band edges of a coupled cavity TWT. We find that the lower band-edge is not subjected to absolute instability. At the upper band-edge, we find a threshold beam current beyond which absolute instability is excited. In general, an absolute instability would occur in a linear beam tube if the cold-tube circuit dispersion curve in the frequency-wavenumber (ω-k) plane is locally convex, and would not occur if the cold circuit dispersion curve is locally concave.

Applying the Briggs-Bers "pole-pinch" criterion to the exact transcendental dispersion relation of a dielectric traveling wave tube (TWT), we find that there is no absolute instability regardless of the beam current. We extend this analysis to the circuit band edges of a linear beam TWT by approximating the circuit mode as a hyperbola in the frequency-wave-number (ω-k) plane and consider the weak coupling limit. For an operating mode whose group velocity is in the same direction as the beam mode, we find that the lower band edge is not subjected to absolute instability. At the upper band edge, we find a threshold beam current beyond which absolute instability is excited. The nonexistence of absolute instability in a linear beam TWT and the existence in a gyrotron TWT, both at the lower band edge, is contrasted. The general study given here is applicable to some contemporary TWTs such as metamaterial-based and advanced Smith-Purcell TWTs.

We describe our progress on the development of a Ka-band TWT driver-booster combination to produce >1kW over a 5 GHz band centered at 35 GHz. The driver is an existing > 500 Watt broadband coupled-cavity TWT1, employing a 2 stage depressed collector. The power booster is a newly designed sever-less folded waveguide TWT designed to provide 3-4 dB of additional gain. The booster also uses a 2 stage depressed collector and a slightly modified version of the same electron gun used in the driver. A photo of the driver and booster under test is shown in Fig. 1. The principal challenges that had to be met in a booster design were: (1) achieving the required output power and bandwidth, (2) ensuring stable booster operation under both small and large signal conditions and (3) presenting a good output match to the driver so that the driver remains stable.

A voltage scale, Vs that characterizes electro-thermal runaway, is deduced from the heat conduction equation, Vs = √k/σ'0, where k-is the thermal conductivity and σ'0 is the rate of change of the electrical conductivity with respect to temperature. Vs depends only on material properties and is independent of geometry and the operating voltage. Vs measures the intrinsic tolerance of the material to electro-thermal instability. Numerical values of Vs are consistent with the well-known properties of several common materials.

Using a simple model we derive a universal condition for oscillation in a traveling wave tube for frequencies near a band edge, when the adjacent band gap is small. The condition is expressed graphically as a value for the maximum allowable (stable) small signal gain as a function of the size of the band gap. As an example of the application of the general condition, a simple upper bound on the E-plane offset of a beam tunnel in a folded-waveguide TWT is obtained.

A two stage 'driver-booster' TWT configuration designed to produce 1.8 kW in Ka-band will be described. The folded waveguide circuit used in the power booster is designed to meet bandwidth requirements while simultaneously eliminating the usual stop-band at 2π, in order to avoid drive-induced oscillation.