Harmonic instability occurs frequently in the power electronic converter system. This paper leverages multi-resolution dynamic mode decomposition (MR-DMD) as a data-driven diagnostic tool for the system stability of power electronic converters, not requiring complex modeling and detailed control information. By combining dynamic mode decomposition (DMD) with the multi-resolution analysis used in wavelet theory, dynamic modes and eigenvalues can be identified at different decomposition levels and time scales with the MR-DMD algorithm, thereby allowing for handling datasets with transient time behaviors, which is not achievable using conventional DMD. Further, the selection criteria for important parameters in MR-DMD are clearly defined through derivation, elucidating the reason for enabling it to extract eigenvalues within different frequency ranges. Finally, the analysis results are verified using the dataset collected from the experimental platform of a low-frequency oscillation scenario in electrified railways featuring a single-phase converter.