Along with de-carbonization, the penetration of power electronic converters increases, and the power system becomes a power electronic-based power system. In such a situation, the stability of the power system faces great challenges. In the event of a large disturbance, the power grid will lack the ability to maintain a stable voltage and frequency. In order to improve the stability of the power grid, the traditional grid-following control is needed to be converted to the grid-forming control. This paper analyzes the control schemes of grid-following converter and grid-forming converter by investigating their state-space models, and eigenvalue trajectories of both control schemes are shown to analyze the stability of the systems. Moreover, a case study is exemplified to compare the performance of two control strategies responding to frequency disturbances. Finally, a time-domain simulation model of 15 kW grid-connected converter is built in Matlab/Simulink to benchmark the performance of the grid-following and grid-forming converters under different working conditions. The result reveals that the grid-following converter may encounter some instabilities applied in the power electronic-based systems, while the grid-forming converter is more suitable for the weak power grid.