The stability of a voltage source converters (VSC) system based on phase-locked loop (PLL) is very important issue during asymmetric grid faults. This paper establishes a transient synchronous stability model of a dual-sequence PLL-based VSC system during low voltage ride-through by referring to the equivalent rotor swing equation of synchronous generators. Based on the model, the synchronization characteristics of the VSC system under asymmetric grid faults are described, and the interaction mechanisms, as well as the transient instability phenomena of positive and negative sequence PLL during asymmetric faults are explained. Using the equal area criterion, the infuences of sequence control switching action, detection delay, and interaction between the positive and negative sequence PLL on the transient synchronous stability of the VSC system are analyzed, respectively. In addition, a transient stability assessment criterion based on the critical fault clearance angle and time and an enhancement control strategy based on the improved positive and negative sequence PLL are proposed. Finally, the analytical results are validated through simulation and experiments.