Understanding Small-Signal Impedance Matrices in Different Reference Frames

This paper systematically analyzes the relationships among the $dq$-domain, $\alpha \beta$-domain, and sequence-domain representations used in small-signal impedance modeling of voltage-source converters (VSCs). It is shown that the AC impedance matrix expressed with $dq$-complex and $\alpha \beta$-complex variables leads to different formulations in the sequence domain. The study demonstrates that asymmetric systems exhibit different physical phenomena in the rotating and stationary reference frames; therefore, the transformations between these frames are not physically consistent in such cases. It is also demonstrated that the so-called modified sequence-domain impedance is equivalent to the universal impedance model in the frequency domain. The analysis clarifies several notational inconsistencies found in the literature. Finally, a physical interpretation is presented highlighting the implications of using stationary and rotating reference frames for stability analysis of power converters.