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Crossed-field space-charge limited current (CF-SCLC) represents the maximum stable current that can be produced in crossed-field devices (CFDs); prominent CFDs include magnetrons and cyclotrons, critical technologies in vacuum electronics. While planar crossed-field geometries have the most developed theoretical models, cylindrical geometries are far more common and useful in practical applications. CFDs are characterized by the strength of the externally applied magnetic field B, orthogonal (crossed) to the electric field; this is often normalized to the magnetic insulation condition, the Hull cutoff field BH. We apply variational calculus to derive a new theoretical model for CF-SCLC for planar and cylindrical devices below and above BH. The variational model offers a concise derivation for planar results without transforming to the time domain and gives the first analytic results from first principles for cylindrical CF-SCLC. We implement a fully three-dimensional simulation in CST Particle Studio which, in addition to additional derived simple theoretical models, explains the often overlooked experimental current scaling ∝ [1 − (B/BH)2]1/2 which decreases to zero current as. These additional simple models reduce the maximum mismatch magnitude between theory and experiments or simulations by up to 68% compared to the variational model, with the most improvement at the critical limit . Justification for the variational model and future applications are discussed.

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Darr, A.M., Bhattacharya, R., Browning, J., & Garner, A.L. (2021). Space-Charge Limited Current in Planar and Cylindrical Crossed-Field Diodes Using Variational Calculus. Physics of Plasmas, 28(8), 082110.

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