Differing from synchronous generators, there are lack of physical laws governing the synchronization dynamics of voltage-source converters (VSCs). The widely used phase-locked loop (PLL) plays a critical role in maintaining the synchronism of current-controlled VSCs, whose dynamics are highly affected by the power exchange between VSCs and the grid. This paper presents a design-oriented analysis on the transient stability of PLL-synchronized VSCs, i.e., the synchronization stability of VSCs under large disturbances, by employing the phase portrait approach. Insights into the stabilizing effects of the first- and second-order PLLs are provided with the quantitative analysis. It is revealed that simply increasing the damping ratio of the second-order PLL may fail to stabilize VSCs during severe grid faults, while the first-order PLL can always guarantee the transient stability of VSCs when equilibrium operation points exist. An adaptive PLL that switches between the second-order and the first-order PLL during the fault-occurring/-clearing transient is proposed for preserving both the transient stability and the phase tracking accuracy. Time-domain simulations and experimental tests, considering both the grid fault and the fault recovery, are performed, and the obtained results validate the theoretical findings.