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A possible route towards achieving high power microwave (HPM) devices isthrough the use of novel slow-wave structures, represented by multiple coupledtransmission lines (MTLs), and whose behavior when coupled to electron beamshas not been sufficiently explored. We present the extension of theone-dimensional linearized Pierce theory to MTLs coupled to a single electronbeam. We develop multiple formalisms to calculate the k-{\omega} dispersionrelation of the system and find that the existence of a growing wave solutionis always guaranteed if the electron propagation constant is larger than orequal to the largest propagation constant of the MTL system. We verify ourfindings with illustrative examples which bring to light unique properties ofthe system in which growing waves were found to exist within finite bands ofthe electron propagation constant and also present possible means to improvethe gain. By treating the beam-MTL interaction as distributed dependent currentgenerators in the MTL, we derive relations characterizing the power flux andenergy exchange between the beam and the MTLs. For the growing wave, we showthat the beam always behaves as an energy source causing power flux along thetransmission lines. The theory developed in this paper is the basis for thepossible use of degenerate band-edges in periodic MTL systems for HPMamplifiers.

Extension of Pierce model to multiple transmission lines interacting with an electron beam