A nonlinear Eulerian theory with phase expansion for traveling wave tubes

A nonlinear Eulerian beam wave interaction (BWI) theory with a series of phase expansion for traveling wave tubes (TWTs) has been put forward, which is based on the result that the phase expansion can obtain more accuracy by numerical comparison of the AC harmonic currents and phase expansion components in the nonlinear region. The phase equation in Lagrangian nonlinear BWI theory has been conducted with the phase expansion and equation of motion, combining with field equation. Then the nonlinear Eulerian BWI theory is established. It agrees well with Lagrangian theory in the BWI nonlinear region, when its interaction is not very deep.