Zones of soft and hard self-excitation in gyrotrons: Generalized approach

It is known that the gyrotron theory is developed in a general form that allows one to draw many important conclusions about gyrotron operation, which are valid for gyrotrons operating in arbitrary modes, at arbitrary frequencies, and driven by electron beams with different voltages and currents. One of important issues in this theory is the analysis of possible start-up scenarios, i.e., the methods allowing, first, to excite the desired mode prior to competitors in the region of soft self-excitation of this mode and, then, drive it into the zone of hard self-excitation where, as a rule, the operation with high efficiency is possible. So far, in all studies, these zones of soft and hard self-excitation were defined for specific voltages. In the present paper, it is shown how one can determine these zones in a more general manner that makes the results applicable to gyrotrons operating at arbitrary voltages. The study also includes consideration of the no-start-current zones and the role of electron velocity spread.