This paper presents a new approach to the analysis and understanding of beam wave interaction taking place in fast-wave devices (beamlet theory). The theory takes fully relativistic effects into account, and, as it turns out, is applicable to slow-wave interactions as well. This is achieved by defining a generalized electromagnetic field whose polarization, phase velocity and origin need not be known to solve the relativistic equation of motion under small-signal conditions. As a result, two distinct but fundamental classes of interactions can be defined, referred to herein as synchronous and resonant. The characteristics of both classes of interactions are discussed in details and exemplified by the interactions taking place in gyrotrons and peniotrons. As a consequence, a systematic classification of commonly used high power microwave tubes is presented, and two new TM interactions occurring close fo waveguide cut-off are identified. The Pierce gain parameter is defined in terms of coupling and beam impedances, showing the gain of synchronous and resonant interactions to be respectively proportional to the cube and square root of the beam current.
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