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A three-dimensional (3-D) "amplifier model" describing the steady state operation of the helix plasma-assisted slow-wave oscillators (PASOTRON) backward-wave oscillator (BWO) is presented. This model contains an equation for the envelope of the wave, whose first space harmonic is synchronous with the electron beam for the case of interaction with the backward wave, and equations for the electron 3-D motion under the action of all the field components. In the latter equations, the Hamiltonian formulation is used to reduce the number of coordinates to be integrated. The results showed that electrons injected inside the helix are those that contribute most to the device electronic efficiency over those electrons injected outside the helix. It is also shown that by reducing the beam size, high efficiencies up to 55% can be achieved. Such high electronic efficiency, which is unachievable in conventional BWOs driven by magnetized electron beams with one-dimensional motion, can be explained by a favorable effect of the transverse motion of electrons.