Droop-controlled voltage-source converters (VSCs) can provide frequency and voltage support to power grids. However, during a grid fault, VSCs may experience transient instability, which can be significantly affected by both the control parameters and fault conditions. This mechanism has not been fully elucidated in previous studies. In particular, grid-voltage faults are commonly accompanied by a grid voltage phase-angle jump (VPAJ), which is typically ignored in the evaluation of the transient stability of VSCs. To address this issue, this study comprehensively assesses the impact of the VPAJ and key control parameters on the transient characteristics of VSCs. Furthermore, the critical clearing angle and critical clearing time are quantitatively calculated to define the transient stability boundary. In addition, a transient stability-enhancement control method that considers the transient stability constraints is proposed. Finally, simulations and experimental tests are conducted to validate both the theoretical analysis and proposed method.