This paper focuses on analyzing the impact of different DC-side characteristics on control parameter design for grid-following (GFL) converters. Based on the state-space averaging theory, an averaged model of the GFL converter is established. The DC-side terminal in typical application scenarios is equivalently modeled as a constant power source and a constant current source. Linearized modeling methods for phase current control, phase-locked loop (PLL), and DC voltage control are derived, and controller parameters are quantitatively calculated for both operating modes. For grid connection under a constant current source, the right-half plane (RHP) poles introduced by the DC voltage control lead to open-loop instability and potential operational instability. To address this, the control bandwidth and phase margin of the DC voltage control are designed to balance stability and dynamic response speed, with controller parameters being calculated and verified based on frequency characteristics. Finally, a detailed switching model for the GFL converter is constructed, and simulation analyses with different DC-side sources are conducted. The results validate the accuracy and effectiveness of the proposed controller parameter design scheme, ensuring control parameter adaptability in varying operational conditions.