The phase-locked loop (PLL) is a core component for grid synchronization of power converters. Existing technologies commonly employ the PLL based on the synchronous reference frame (SRF-PLL). The introduction of SRF-PLL complicates the model of grid-connected converters and makes stability analysis challenging. Typically, numerical verification analysis is conducted using Bode plots, making it difficult to obtain analytical solutions and effectively reveal the underlying mechanisms of stability. Therefore, an algebraic operation-based PLL (AO-PLL) control technology is proposed, which offers faster synchronization and reduces the order of the converter model under this PLL control mode. Based on this, a closed-form analytical solution for system stability is presented, and necessary and sufficient conditions for system stability are obtained. On this basis, the impact of active and reactive power control coupling on system stability is analyzed; the different mechanisms of the proportional controller and integral controller of the current loop on system stability are revealed; and the connotation of using short-circuit ratio to describe the system stability margin is explained. Furthermore, the intrinsic relationship between the proposed AO-PLL and the conventional SRF-PLL is elaborated, as well as the applicability of stability conclusions under conventional SRF-PLL control. Finally, numerical examples are provided to verify the correctness of the aforementioned theoretical analysis.