Abstract: To improve the stability of frequency and phase of the output microwave on condition of repetitive operation, an S-band, long-pulse relativistic klystron amplifier (RKA) is investigated by theoretical modeling, numerical simulation and experimental measurement. Description and analysis are made on the physical origins of self-oscillation, output microwave pulse shortening, the instability in the repetitive operation and the phase fluctuation of output microwave in the RKA. Measures are taken to solve or alleviate these problems, which are demonstrated by the simulation and experiment. It is confirmed that the electrons reflected from the gaps of idler cavity and output cavity of RKA result in the self-oscillation and the output microwave pulse shortening, and the electron dispersion and bombardment on the noses of the output cavity and idler cavity affect the stability of repetitive operation. It is validated that enlarging the drifting tube radius, adding some microwave absorber in the drifting tube between those cavities and optimizing the structure of the input cavity, output cavity and collector of the electron beams can alleviate the problem of self-oscillation and pulse shortening obviously and improve the stability of repetitive operation and phase fluctuation of output microwave. Using an 830 kV, 7.7 kA, 190 ns electron beam and an input microwave of 80 kW, an output power of 1.55 GW, pulse width of 163 ns, and phase fluctuation of 18° are generated by the S-band three-cavity RKA operating at 25 Hz.