A Small Amplitude Theory for Magnetrons†

Abstract

The impedance of the electron cloud in a magnetron is defined in a manner suitable for electronic calcula t ions, as the ratio between r.f. electric and magnetic field component s at the surface of the electron cloud. Ali expression for this quantity is first derived by the indirect method of calculating the impedance seen by the electrons, i.e. transforming the load impedance, via, the output elements, the 'resonator system and the space between the anode and the electron-cloud, to the surface of the latter. The direct calculation of electron impedance from the physics of the electron cloud is then undertaken. By the introduction of hydrodynamical concepts the equations governing non-steady, non-symmetrical, single-stream electron motion in a self-consistent field are rcduced to a single partial ifferential equation for' the velocity potential. This equation is solved by considering the r.f. velocity potential as it small perturbation of the d.c. velocity potential associated with the single-stream steady state. Radio-frequency field components are derived from the velocity potential and the impedance looking into the ele ctron-cloud is calculated. The impedance is shown to possess a negative resistive component under the conditions described in C.V.D. report Mag. :H, and, under other conditions, corresponding to lower d.c. anodc voltages, to consist of a pure reactance which, unlike common reactances, decreases with increasing frequency. The latter property is shown to be,1. sufficient condition for the spontaneous start-up of oscillations. By matching this electron reactance to the reactance previously calculated from the load, operating condit ions are derived as functions of the load, which are in close agreement with those observed in experiments (cf. C.V.D. Mag. 35). The spontaneous start-up 'of oscillations in strapped valves is thus explained and the gap filled which was left :in the theory of C.V.D). Mag. :n. A slight modification of the criterion given in that report is proposed.