Nonlinear self-consistent theory for crossed-field devices.

Abstract

A closed, nonlinear set of fluid equations that is based on the electron guiding-center orbits and is generally applicable to the analysis of crossed-field, slow-wave devices, is developed. The equations are used to model the behavior of the crossed-field amplifier. The dielectric response from the spoke charge is self-consistently included. A mean-field approximation is introduced to express the effect of the spoke charge on the rf mode profile. The dielectric modifications are then parametrized by an average amplitude factor \ensuremath{\Lambda}^ and an average phase shift \ensuremath{\psi}^ from the vacuum values. In the synchronous with the rf signal frame of reference the streamlines follow the equipotential surfaces of the transformed fields. In the steady state, the flow is incompressible. A uniform-density, constant-height electron hub feeds the current spokes. The secondary electron production at the cathode is computed self-consistently through the secondary-emission coefficient and the average impact energy. The spoke current is determined by the difference of the E\ifmmode\times\else\texttimes\fi{}B drift at the top of the hub from the rf phase velocity. At small space-charge density relative to the Brillouin density, the dielectric corrections enter as a rotation ${\mathit{e}}^{\mathit{i}\mathrm{\ensuremath{\psi}}\mathrm{^}}$ of the complex growth rate relative to the growth without spoke self-fields. The numerical solutions for arbitrary space charge are compared with previous results without spoke-field effects. An increase in the rf gain and in the anode dc current is observed at any given operation point, despite a small drop in the efficiency. It is concluded that the increase in the rf field strength in the anode-cathode space, caused by the spoke self-field, compensates for the detuning effects from the modifications on the rf mode profile. In reentrant devices, the recycling of the space charge further increases the output power and the level of noise generated by the output-input feedback.