The steady state oscillations in the generator of the backward wave in the framework of the two-dimensional model

The processes of establishing steady-state oscillations in the backward wave generator has been considered in the superregenerative mode within the framework of the linear theory. In this case, the electron current exceeds the starting value. In the analysis, a complex frequency is introduced, the imaginary part of which characterizes the increase of oscillations in time. The consideration is carried out in the framework of a two-dimensional model, taking into account the influence of a focusing magnetic field of finite magnitude. In this case, in addition to the longitudinal high-frequency displacements, the appearance of transverse ones is possible in the electron beam. The results are obtained using the method of successive approximations. The conditions for obtaining the characteristic responsible for the wave growth are derived, the dependences of this quantity on the ratio of the electron beam current and the starting current value for various values of the magnetic field are given. Also the dependencies for the time of the establishment of oscillations for different parameters of the system in a linear approximation have been represented. In limiting cases, the results obtained agree well with the well-known results of the one-dimensional theory. The case of cyclotron resonance is considered separately, when only a slow cyclotron wave is excited in the electron beam, which noticeably simplifies the original problem and the initial relations.