General Steady-state Theory of Cylindrical Magnetrons

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Abstract
This paper presents a general steady-state theory of cylindrical magnetrons. Space charge and the Maxwellian velocity distribution of the emitted electrons are taken into account, but the collisions between individual electrons are neglected. The paper explains the mathematical basis of the calculations and gives expressions for the volume density of the electrons and for the two components of the electron current density. No calculations of the actual potential distributions are given but the problem is briefly discussed at the end of the paper. It is shown however that for certain important ranges of potential distributions the following conclusions concerning the steady-state theory of cylindrical magnetrons can be drawn (1)Brillouin's single-stream flow is not possible when the emission velocities are taken into account (2)In a well ‘ cut-off ’ magnetron the azimuthal component of the current density may be several hundred times larger than the radial component. Thus, even slight imperfections in the geometry of the valve or in the homogeneity of the magnetic field may cause large contributions to the anode current from those electrons which nominally should only graze the anode (3)The appearance of a potential minimum between the electrodes does not necessarily limit the amount of current drawn by the anode (4)A now regime of operation of the valve called the ‘ magnetic-field-limited ’ region must be introduced (5)There exists a critical potential distribution which separates the ‘ magnetic-field-limited ’ and the ‘ space-charge-limited ’ regions of operation of the valve (6)The usual difficulties of transition between the mathematical expressions derived for plane and cylindrical magnetrons disappear when all initial velocities of the emitted electrons are taken into account (7)Introduction of the initial velocities of emission removes the theoretical difficulty of establishing a steady-state flow for certain special values of the cathode to anode ratio, quoted in the past.