Absolute Instability at the Lower Band Edge in a Traveling Wave Tube

We provide a close examination of the beam mode and its interaction with the circuit mode in the immediate vicinity of the lower band edge in a traveling wave tube. We model the circuit mode as a hyperbola in the (w, k) plane in the immediate vicinity of the lower band edge, where w is the frequency and k is the wavenumber. We find that an absolute instability may arise, according to the Briggs-Bers criterion [1], if the beam current is sufficiently high, even if the beam mode intersects with the circuit mode at a point in the (w, k) plane with a positive group velocity. This threshold condition is deduced analytically and confirmed by numerical calculations. When the threshold current is exceeded, the Green’s function [1], at a fixed position, exponentiates in time as t to the 1/3 power initially, but as (wi*t) at later time, where wi is the imaginary part of the unstable polepinch root of w according to the Briggs-Bers criterion [1]. This finding differs from some previous works [2, 3]on absolute instabilities at the lower band edge, and points to the vulnerability to absolute instabilities at both the upper and lower band edges of a TWT.