The effect of beam cross-sectional velocity variation on backward-wave-oscillator current

Low-voltage helix-type backward-wave oscillators require a starting current that rises to infinity toward the low-frequency end of the tuning range. The effect has been attributed to the raising of the space-charge parameter QC by the dc space-charge-induced velocity spread. H. R. Johnson has calculated the velocity-spread effect on starting current qualitatively, but the predicted nonoscillation frequency is generally much lower than the observed one. As a further analysis, space-charge wave propagation in an electron beam having an actual cross-sectional variation of dc velocity is investigated. It is shown that the RF current modulation in the slow space-charge wave is concentrated in the region of the slowest-moving electrons. In a helix-type backward-wave oscillator using a hollow beam, the slower electrons are farther away from the RF circuit, so that the effective impedance for the slow space-charge wave may be considerably reduced. The use of an impedance reduction factor therefore provides better agreement between theory and experiment with regard to the starting-current phenomenon. Theoretical and experiment results of the investigation are presented.