This paper describes the self-oscillation phenomenon which was detected in the course of hot-test of an S-band 22kW Average-Power-Klystron with 7.14% relative bandwidth [1], the oscillation frequency is in the vicinity of 8.9GHz and in accord with the high-order mode of TE311 of cylindrical cavity by the means simulation of the double-gap output cavity. In order to eliminate the self-oscillation, an absorber method was taken into klystron, and a new klystron was fabricated and tested, then the self-oscillation disappeared, and a kind of phenomenological analysis theory explanation was put forward.