Velocity-Modulated Reflex Oscillator

A mathematical analysis of the mechanism utilized by the microwave reflex oscillator in producing high-frequency oscillations from a direct-current beam of electrons is presented. A simplified small-signal theory is postulated in which the electrodes are assumed to be ideal parallel planes and the electron motion is rectilinear, uninfluenced by space charge. The finite transit time of the electrons in traversing the modulator gap is taken into consideration. From this theory are derived expressions for the velocity modulation and the resultant current-density modulation of the beam by action of a retarding field. An equation is derived for the fundamental-frequency component of the current induced in the tank circuit. The necessary conditions for the self-starting of oscillations are determined, and the minimum starting current is given as a function of the tank-circuit characteristics and the optimum transit-angle values. Equations are derived for the rate of change of oscillating frequency with reflector voltage and beam voltage. To determine the amount of electronic tuning possible, calculations are made of the range over which the reflector voltage can be varied for a particular mode and oscillations maintained. An efficiency curve is given for the conversion of beam-current power to high-frequency power, and optimum efficiencies are calculated for conditions in which the amplitude of oscillation is small. Efficiency curves are also presented for the case of large amplitudes when the transit angle of the modulator gap is negligibly small.