The harmonic state-space, the dynamic phasor, and the generalized dq modeling are three methods developed for linearization of ac power electronic systems. This paper reveals explicitly mathematical relationships between the three modeling methods in both time and frequency domain. Representations of linearized models in different reference frames and from time domain to frequency domain, as well as their transformations are elaborated step by step. Case studies on a three-phase voltage-source converter that is connected to an unbalanced grid verify the theoretical findings.

Grid-forming voltage-source converters (VSCs) tend to suffer from sideband oscillations in stiff grids. Conventional state-space eigenvalue analysis or impedance-based stability analysis based on generalized Nyquist stability criterion can only analyze how specific control parameters affect the overall system stability, which ends up with less insight into a controller design-oriented analysis. To tackle this challenge, this article starts with the impedance modeling and stability analysis of grid-forming VSCs in stiff grids. Then, an impedance decomposition approach to characterizing the impacts of multiple control loops is proposed, which benefits in a controller design-oriented analysis for stability enhancement. Finally, simulation and experimental results are provided to validate the approach.