Calculation of Pierce’s Parameters <inline-formula> <tex-math notation="LaTeX">${C}$ </tex-math> </inline-formula> and <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math> </inline-formula> by Differentiation of the TWT “Hot” Dispersion Relation

Abstract

A new method of calculating the values of Pierce’s parameters ${C}$ and ${Q}$ , which appear in the fourth-order polynomial TWT characteristic equation for the phase constants, is derived and illustrated. The use of this method avoids the need to compute the power flow by integration of the Poynting flux in the absence of the beam, which can be difficult to do analytically. By introducing the coupling coefficient ${K}_{c}$ and the depression coefficient $\Gamma$ in place of ${C}$ and ${Q}$ , it becomes possible to analyze the TWT characteristic equation in the limit of zero density of the e-beam when parameter ${C}$ is equal to zero. It is shown that ${K}_{c}$ and $\Gamma$ can generally be determined by differentiation of the dispersion relation in the presence of the beam with respect to the transverse constant $\gamma$ , evaluated in the limit that the beam current vanishes. ${K}_{c}$ (and therefore impedance ${K}$ ) is shown to be determined by the first derivative and $\Gamma$ (and therefore ${Q}$ ) is shown to be determined by the combination of the first and second derivatives with respect to $\gamma$ . This method is applied to two TWTs based on two different types of slow-wave structure, a “comb” and a “sheath helix.”