The proposed project aims to answer the question whether it is feasible to phase-lock a harmonic gyrotron oscillator by estimating the input power needed for such an operation and studying the influence of a number of parameters on the locking conditions. Since available microwave sources, which can be used to provide the input (locking) signal, are limited in frequency and power, it is proposed to premodulate the electron beam in an input cavity at the fundamental of the gyrofrequency and to use the harmonics of the ac beam current to lock the output of a second cavity at the harmonics of the gyrofrequency. This will reduce considerably the required frequency of the input signal and will facilitate the choice of the source to provide it. The design of a third harmonic gyrotron will be given and a general theory for phase-locking a harmonic gyrotron oscillator via a premodulated electron beam will be developed. Competition from other harmonics and especially from the fundamental component of the ac beam current will be considered. A small-signal theory will be developed to study the influence of peniotron interaction on harmonic gyrotron operation. Spread in electron velocity and guiding center will be included. A preliminary design of a phase-locked harmonic gyrotron will be attempted to illustrate the results of the developed theory and to study the feasibility of this kind of microwave tube.