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This article re-examines the Brillouin flow solutions in crossed-field diodes, with applications to magnetrons, magnetically insulated line oscillators (MILOs), and magnetically insulated transmission lines (MITLs). The Brillouin flow solutions are constructed for various geometries, including planar magnetrons, MILOs, and MITLs, cylindrical magnetrons with electrons flowing in the azimuthal direction, cylindrical MITLs and MILOs with electrons flowing in the axial direction, and radial MITLs and MILOs with electrons flowing in the radial direction. A common theme of this analysis is that two main external parameters are used to characterize the Brillouin flow: the anode&#x2013;cathode voltage (<inline-formula> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula>) and the total magnetic flux within the crossed-field diodes (<inline-formula> <tex-math notation="LaTeX">$A_{a}$ </tex-math></inline-formula>). These two parameters are equivalent to the gap voltage and a specification of the degree of magnetic insulation, which is approximately equal to the ratio of the magnetic field to the Hull cutoff (HC) magnetic field. The magnetic flux may be provided externally by a magnet (as in a magnetron) or by the wall currents without an external magnet (as in a MILO or MITL), or by some combination of the two, as in the intermediate case of a magnetron&#x2013;MILO hybrid. Once these two parameters are specified, the electron flow speed at the top of the Brillouin hub is uniquely determined. This immediately yields the Buneman&#x2013;Hartree (BH) condition according to the Brillouin flow model, whether it be a planar magnetron or a cylindrical MILO. In so doing, we have obtained, for the first time using the Brillouin flow model, the BH condition for a cylindrical MILO, and we show that the same condition is obtained from the single-particle orbit model. We also found that, in general, the electron current within the Brillouin hub contributes only to a very small fraction of the magnetic flux <inline-formula> <tex-math notation="LaTeX">$A_{a}$ </tex-math></inline-formula>, regardless of the gap voltage <inline-formula> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula>, thereby correcting an erroneous notion that the electron flow within the crossed-field gap could be responsible for the magnetic insulation. Another counter-intuitive finding is that, for a given degree of magnetic insulation, the Brillouin hub height decreases as the gap voltage <inline-formula> <tex-math notation="LaTeX">$V_{a}$ </tex-math></inline-formula> increases. These conclusions, and other results, were based on the simple, explicit analytic expressions that we have obtained for the Brillouin flow profiles, including the velocity, electron density, as well as the self-magnetic field and the self-electric field profiles due to the Brillouin hub electrons. From these analytical expressions, we deduce useful scaling laws that are applicable to the prevailing cases where the magnetic field exceeds 1.5 times the HC magnetic field, and they are valid in both relativistic and non-relativistic regimes. Thus, these scaling laws show the contrast between magnetrons and MILOs, and a ready assessment of the viability of building a moderate-current MILO, a low-voltage MILO, and a magnetron&#x2013;MILO hybrid which might combine the advantages of a magnetron and a MILO. The Brillouin flow profiles in a radial MITL are explicitly calculated. Additional issues are addressed.

Explicit Brillouin Flow Solutions in Magnetrons, Magnetically Insulated Line Oscillators, and Radial Magnetically Insulated Transmission Lines