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A time-dependent slow-time-scale theory is developed for gyrotron oscillators driven by an external signal. The signal is introduced either directly into the cavity output or via a beam prebunching cavity. The theory is applied to a high voltage gyrotron configuration and the numerical, nonlinear calculations are compared with simple analytical estimates of the frequency bandwidth for phase-locked operation. For the case of direct injection, the non- linear slow-time-scale calculations are in good agreement with Adler's relation. Two approaches are investigated for the case of phase-locking with a prebunching cavity. In the first the induced as current density due to the prebunching cavity is treated as a small perturbation of the ac current density in the oscillator. In this approach the equations for the time-dependent wave amplitude and phase are similar in structure to the equations for gyrotrons driven by direct injection and lead to a simple analytical estimate of the locking bandwidth. The accuracy of the perturbation approach is investigated by comparing it with the results of an alternate approach used in the analysis of gyro-klystrons. The maximum phase-locking bandwidth obtainable with the prebunching cavity approach is discussed.