Interpreting Frame Transformations as Diagonalization of Harmonic Transfer Functions

Analysis of ac electrical systems can be performed via frame transformations in the time-domain or via harmonic transfer functions (HTFs) in the frequency-domain. The two approaches each have unique advantages but are hard to reconcile because the coupling effect in the frequency-domain leads to infinite dimensional HTF matrices that need to be truncated. This paper explores the relation between the two representations and shows that applying a frame transformation on the input-output signals creates a direct equivalence to a similarity transformation to the HTF matrix of the system. Under certain conditions, such similarity transformations have a diagonalizing effect which, essentially, reduces the HTF matrix order from infinity to two or one, making the matrix tractable mathematically without truncation or approximation. This theory is applied to a droop-controlled voltage source inverter as an illustrative example. A stability criterion is derived in the frequency-domain which agrees with the conventional state-space model but offers greater insights into the mechanism of instability in terms of the negative damping (non-passivity) under droop control. Therefore, the paper not only establishes a unified view in theory but also offers an effective practical tool for stability assessment.