At present, there is a lack of systematic research on the impact of phase lag introduced by proportional-resonant (PR) controllers on the passivity of the output admittance of grid-connected inverters. Moreover, existing phase compensation methods are largely based on trial-and-error tuning and lack analytical design criteria. To address these issues, this paper proposes a boundary-frequency-based passivity optimization method. First, an inverter output admittance model that accounts for the phase-lag effect of the PR controller is established, and two types of boundary frequencies defining the passive region are identified. Based on the relative position of the two boundary frequencies, a first-order phase compensator cascaded with the PR controller is designed to extend the passivity region up to the Nyquist frequency. Furthermore, without requiring additional sensors, a point-of-common-coupling (PCC) voltage feedforward is introduced to enhance passivity robustness at the boundary frequencies. Finally, experiments on a 4.8 kW inverter prototype confirm that the proposed approach maintains stable operation under inductive/capacitive weak-grid conditions, as well as in the presence of filter parameter variations.