Absolute instability at the band edges in linear beam traveling wave tubes

We use the Briggs-Bers criterion to examine the absolute instability at the lower and upper band edges of a coupled cavity TWT. We find that the lower band-edge is not subjected to absolute instability. At the upper band-edge, we find a threshold beam current beyond which absolute instability is excited. In general, an absolute instability would occur in a linear beam tube if the cold-tube circuit dispersion curve in the frequency-wavenumber (ω-k) plane is locally convex, and would not occur if the cold circuit dispersion curve is locally concave.