  Engineering of the eigenmode dispersion of slow-wave structures (SWSs) to
achieve desired modal characteristics, is an effective approach to enhance the
performance of high power traveling wave tube (TWT) amplifiers or oscillators.
We investigate here for the first time a new synchronization regime in TWTs
based on SWSs operating near a third order degeneracy condition in their
dispersion. This special three-eigenmode synchronization is associated with a
stationary inflection point (SIP) that is manifested by the coalescence of
three Floquet-Bloch eigenmodes in the SWS. We demonstrate the special features
of "cold" (without electron beam) periodic SWSs with SIP modeled as coupled
transmission lines (CTLs) and investigate resonances of SWSs of finite length.
We also show that by tuning parameters of a periodic SWS one can achieve an SIP
with nearly ideal flat dispersion relationship with zero group velocity or a
slightly slanted one with a very small (positive or negative) group velocity
leading to different operating schemes. When the SIP structure is synchronized
with the electron beam potential benefits for amplification include (i) gain
enhancement, (ii) gain-bandwidth product improvement, and (iii) higher power
efficiency, when compared to conventional Pierce-like TWTs. The proposed theory
paves the way for a new approach for potential improvements in gain, power
efficiency and gain-bandwidth product in high power microwave amplifiers.
