In this paper, we analyze the interaction between electron beam and microwave in the coaxial cavity, and obtain the coupling coefficient of the coaxial cavity and the electronic load conductance. The X-band tri-axial relativistic klystron amplifier is designed and simulated. Simulation results show that coaxial cavity can increase the efficiency. With an input microwave power of 70 kW, electron beam voltage of 600 kV, and electron beam of 5 kA, we obtain a microwave with a power of 1.1 GW, frequency of 9.37 GHz, and an efficiency of 37%.