Modification of Pierce’s Classical Theory of Traveling-Wave Tubes

For a realistic case of an electron beam interacting with the electromagnetic fields supported by a thin perfectly conducting tape helix, we derive the exact dispersion relation, from which an unambiguous determination of Pierce's space-charge parameter Q is made. In the process of doing so, we discover that the circuit mode in the equivalent three-wave theory of Pierce must be modified at high beam current, an aspect not included in Pierce's original analysis. We quantify this circuit mode modification by a new parameter that we call q, introduced here for the first time in traveling-wave tube theory. In a realistic example, we find that the effect of q is equivalent to a modification of the circuit phase velocity by as much as two percent, which is a significant effect equivalent-to a detune of two percent.