A self-consistent relativistic field theory for a helix traveling wave tube (TWT) is presented for a configuration in which a magnetized pencil beam propagates through a tape helix enclosed with a loss-free well. A linear analysis of the interaction is solved subject to the boundary conditions imposed by the beam, helix, and wall. The wave equation for the fields within the electron beam corresponds to the Appleton-Hartree magnetoionic wave modes that are of mixed electrostatic/electromagnetic polarization. Hence, the determinantal dispersion equation that is obtained implicitly includes beam space-charge effects without recourse to a heuristic model of the space-charge field. This dispersion equation includes azimuthal variations and all spatial harmonics of the tape helix. Solutions that correspond to both the extraordinary (X) and ordinary (O) solutions for the Appleton-Hartree modes are found numerically.
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