This paper presents the results of mathematical modeling of heat transfer in the field emission process in a conic cathode of small dimensions with its possible melting considered. It is shown that the possibility of melting is determined by the cathode vertex angle. The melting is modeled in the framework of the phase field system using the proposed methods for the formation of the liquid phase zone.
The TWT model based on placed in parallel impedance electrodes is used to analyze effectiveness of a sheet e-beam interaction with the in-phase mode of a slow wave excited in the impedance electrodes. The impedance approximation was used for presentation fields of E- and H-waves by zero and ±1 space harmonics, respectively. The generalized formula for the coupling coefficient, characterizing the interaction effectiveness, was used to demonstrate the its dependence on the e-beam thickness and the gaps between the impedance and screen electrodes. The obtained dependences on the e-beam tunnel geometry and the operating frequency were compared with that calculated for a cylindrical beam in a conventional helix.