The high-impedance relativistic klystron based on the hot cathode, of high power, long pulse, high efficiency and stable amplitude and phase, is one of the promising candidates for the high-power microwave coherent power combing. As a high-quality, small radius electron beam plays a crucial role in an effective beam-wave interaction, a high-convergence electron optics system is required for an 11.424 GHz high-impedance relativistic klystron, with an output power of 105 MW. In the design, to alleviate the cathode loading due to high-convergence, the uniformity of the cathode emittance and the laminarity of beam are both sacrificed. Then serious space charge effect is likely to occur with a rather high current density. To minimize the scalloping, it is difficult to provide the best overlap of the beam trajectories and magnetic force lines with priority to the amplitude of magnetic flux at the cathode. With the synthesis of Pierce’s electron gun and the cylindrical focusing electrode, a particle in cell model of the electron gun is constructed to obtain the electrostatic characteristics of the electron beam. Based on the electrostatic beam trajectories, a beam-focusing system is designed, with which electron beam can be well focused by adjusting the magnetic field profile especially in the transient region. The PIC simulation results indicate that the balance radius of the reasonably fluctuated beam is about 3 mm ,with a fine laminarity, and the transmission, the perveance and the area convergence are 100%, 0.9 µP and 97 : 1 respectively, which satisfy the requirement of the 11.424 GHz high- impedance relativistic klystron. Moreover, with the comparison between thermionic emission and explosive emission in PIC code, the micro-characteristics of beam shows the latter is tend to be more reliable.