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

TL;DR

This paper clarifies and unifies the definitions of instantaneous complex phase and frequency in transmission systems, demonstrating their equivalence under standard assumptions.

Contribution

It presents a comprehensive comparison of analytic signal and space vector formulations, proposing a unified notation to prevent confusion.

Findings

Established the equivalence of different formulations under standard assumptions

Provided a unified notation and terminology for clarity

Clarified the conceptual differences and relationships between definitions

Abstract

This letter seeks to clarify the different existing definitions of both instantaneous complex phase and frequency as well as their equivalence under standard modeling assumptions considered for transmission systems, i.e. balanced positive sequence operation, sole presence of electro-mechanical transient dynamics and absence of harmonics and interharmonics. 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, a unified notation and terminology to avoid confusion is proposed.