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A linearized field theory of a grating-coupled Smith-Purcell traveling-wave tube (TWT) is presented. The configuration describes the propagation of a magnetized, sheet electron beam through a parallel-plate waveguide in which two opposing faces of the waveguide consists of either 1) a pair of identical diffraction gratings (referred to as a double-sided configuration) or 2) a diffraction grating along one face and a flat plate along the opposing face (referred to as a single-sided configuration). This is a two-dimensional formulation since we assume the configuration is uniform in the direction parallel to the slots in the grating. For each of these configurations, we derive the linearized dispersion equations for the vacuum structures and the wave-particle interaction in both the infinitely strong magnetic field limit and for an arbitrary (solenoidal) magnetic field. A discussion of the extraction efficiency based on phase-trapping arguments is also presented. An example of a W-band Smith-Purcell TWT is discussed.