Abstract: There are many impedance-based methods for analyzing oscillation of a grid-connected converter system (GCS) via dq frame, sequence frame and polar frame, which can be transformed into each other and are equivalent in assessing the stability of the system. However, the discrepancy and adaptation of these methods have not been discussed theoretically. Therefore, the impedance models of GCS are established in above three kinds of coordinates respectively. Then, the mathematical and physical characteristics of them are analyzed. Next, the decoupling of impedance models in different frames is analyzed, and the differences of the impedance criteria in stability analysis and mechanism interpretation are given. Furthermore, the sub-synchronous oscillation caused by the phase-locked loop considering the converter's four quadrant operation scenario are illustrated. The existence of open-loop transfer function right-half-plane poles and its sensitivity to uncertain parameters are analyzed, focusing on the nominal performance and robustness of the impedance criteria. On this basis, the adaptation of several impedance criteria of GCS is analyzed. Theoretical analysis and simulations confirm the validity of the proposed conclusion.