The effects of space charge in the excitation of a hollow resonator by a stationary flux of excited classical oscillators

An analytical method is proposed for studying the conditions of self-excitation of electromagnetic modes in microwave oscillators based on stimulated bremsstrahlung from classical electron oscillators. The method is a version of the Laplace transformation and takes into account the Coulomb interaction of particles. Consideration is given to the intrinsic waves in the electron-oscillator flux, the effect of negative mass, and the natural frequency of the resonator traversed by the flux of electron-oscillators. The proposed method can be extended to more complex theoretical models of microwave oscillators based on stimulated emission from excited classical electron-oscillators.