A Tutorial on the One-Dimensional Theory of Electron-Beam Space-Charge Effect and Steady-State Virtual Cathode

The space-charge effects of pulsed high-current electron beams are very important to high-power particle beam accelerators and high-power microwave devices. The related physical phenomena have been studied for decades, and a large number of informative publications can be found in numerous scientific journals over many years. This review article is aimed at systematically summarizing most of the previous findings in a logical manner. Using a normalized one-dimensional mathematical model, analytical solutions have been obtained for the space-charge-limited current of both planar diode and drifting space. In addition, in the case of a beam current higher than the space-charge-limited current, the virtual cathode behavior and beam current reflection are quantitively studied. Furthermore, the criteria of steady-state virtual cathode formation are investigated, which leads to the physical understanding of the unstable nature of the virtual cathode. This review article is expected to serve as an integrated source of related information for young researchers and students working on high-power microwaves and pulsed particle beams.