Theoretical study of the initiation of oscillations in electron streams through crossed fields

The purpose of this study is to find the general conditions under which spontaneous, approximately sinusoidal oscillations will build up in an idealized system consisting of a stream of electrons flowing between two admittance sheets in crossed electric and magnetic fields. The motivation for a small-signal, greatly oversimplified theory is as usual the hope that the results after comparison with experimental data nay make it possible to reach a better qualitative understanding — at least — of the behavior of realizable systems and to show the way towards further improvement of the performance of these systems. The independent variables that are given particular attention are the admittances of the boundary surfaces, i.e., the “circuit” characteristics of the oscillator. The very general statement of the problem makes it possible to make some comparisons between the mechanisms of oscillation in travelling-wave and magnetron oscillators. The theoretical investigation presented studies the mode of oscillation in a re-entrant system formed by an ideal laminar stream of electrons partly filling the space between two plane boundaries of specified wave admittances. Comparisons are made between propagation parallel and perpendicular to the constant magnetic field. The particular purpose of the study is to survey the conditions under which modes exist that in the small-signal range show exponential growth of amplitude with time, in the hope that the result provide some better understanding of the limits for spontaneous initiation of oscillations in travelling-wave oscillators and magnetrons. When at least one of the boundary admittances contain inductance components, the wave guide formed by the admittance walls and the space between them is a propagating structure, and the classical small-signal travelling-wave-amplifier theory applies. More interesting are the cases where this structure is operated under cut-off conditions and attenuates only, in the absence of the beam. The wall admittances are then resistive and capacitive, and the analysis indicates that within certain ranges of the parameters spontaneous oscillations can still occur. Each root to the characteristic equation obtained for a specific case of boundary admittances leads to a relation between the radian wave numbers α and γ, on the one hand, and the complex frequency on the other hand, with beam velocity and dimensions as parameters. A study of this relation reveals the range of the parameters for which self-excited oscillations can be obtained, the electronic timing obtainable by varying the parameters, primarily the beam voltage, and the mode selection rules when a set of different wave numbers are permitted. The significance of the lower boundary (y = 0) and of its admittance is apparent in a magnetron, where the cathode surface forms a boundary which may have considerable wave resistance and, in case of a helical shape, also appreciable wave reactance. When a potential minimum exists, it is a rather complicated problem to state the boundary condition realistically. The analysis shows definitely, however, that even for very small perturbations the space charge is by no means a perfect shield around the cathode, and that the finite cathode admittance has an appreciable effect on the natural modes of oscillation of the system. The wide-range operation of voltage-tunable magnetrons may possibly be explained on the basis of this analysis. It is too early to present such a detailed interpretation here; only a general outline may be in order. When the anode admittance is large, the resistivity of the cathode may provide the necessary condition for the existence of growing waves even at frequencies where the mode is primarily capacitive, thus making the oscillation only to a minor extent depending on the anode circuit. The mode selection and the variation of output power with frequency is still likely to be determined by the anode circuit and its matching network, even if the frequency range over which oscillations occur is not to any appreciable degree.