Instantaneous Complex Phase and Frequency: Conceptual Clarification and Equivalence between Formulations

TL;DR

This paper clarifies the definitions of instantaneous complex phase (ICP) and instantaneous complex frequency (ICF) by contrasting analytic-signal (Hahn) and space-vector (Milano/Lei) formulations. It derives the precise conditions under which these formulations are equivalent and shows how rotation frame changes relate the two perspectives. The authors argue that equivalence is not universal and propose unified terminology, including instantaneous planar phase/frequency (IPP/IPF) and links to geometric frequency, to reduce confusion in three-phase power systems. The work aims to standardize notation and interpretation to improve communication and application in practice.

Abstract

This letter seeks to clarify the different existing definitions of both instantaneous complex phase and frequency as well as their equivalence when specific hypotheses hold. To achieve this, the two fundamental definitions, i.e., those based on either the use of (i) analytic signals or (ii) space vectors, together with the premises used for their formulation, are presented and their relationship shown. Lastly, an unified notation and terminology to avoid confusion is proposed.