On the evaluation of Pierce parameters C and Q in a traveling wave tube

A study of an exactly solvable model of a traveling wave tube (TWT) shows that Pierce gain parameter C and space charge parameter Q generally depend on wavenumber k in addition to frequency ω. The choice of k at which C and Q are evaluated may strongly affect their values and, consequently, the values of the small signal gain obtained from 3- and 4-wave Pierce theory. In order to illustrate this effect, we calculate the spatial amplification rate, ki, from the exact dispersion relation for a dielectric TWT model which is exactly solvable. We compare this exact value of ki with approximate values obtained from Pierce's classical 3-wave and 4-wave dispersion relations, obtained by making various assumptions on k in the evaluation of C and Q. We find that the various ways to approximate C and Q will have a significant influence on the numerical values of ki. For our dielectric TWT example, Pierce's 4-wave TWT dispersion relation generally yields the most accurate values of ki if Q is evaluated for k = ω/v0, where v0 is the beam velocity, and if the complete frequency and wavelength dependence of C is retained. Pierce's 3-wave theory also yields accurate values of ki using a different form of Q from the 4-wave theory. The implications of this result for TWT design are explored.