Existing studies on large-signal stability of grid-following (GFL) converters predominantly focus on the AC current control timescale, often assuming a constant DC-link voltage or relying on chopper control. In contrast, this paper addresses the overlooked dynamics within the DC-link voltage control timescale, where the cumulative effects of multiple control loops critically influence system stability. First, a novel dynamic model of the GFL converter is proposed. This model uniquely captures the transient trajectory of the internal voltage, enabling a physics-based quantification of stability through average acceleration during a swing. The transient stability margin of the GFL converter is then quantified based on whether the average acceleration of the internal voltage swing is positive or negative. This trajectory-based analysis provides a unique method for investigating transient stability. Furthermore, the large-signal stability of the GFL converter connected to a weak grid is analyzed under various physical scenarios. This is achieved by analyzing the average acceleration during the swing process in conjunction with phase portraits, which together provide valuable insights into the transient stability mechanism of this nonlinear system. The research reveals that, in a weak grid, the large-signal stability of the GFL converter diminishes as the DC-link voltage control bandwidth approaches the phase-locked loop (PLL) bandwidth. Conversely, stability improves as these two bandwidths diverge. Time-domain simulations and experimental tests are conducted to validate the theoretical analysis.