pith. sign in

arxiv: 2512.09574 · v3 · submitted 2025-12-10 · 📡 eess.SY · cs.SY

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

C\'esar Garc\'ia-Veloso , Mario Paolone , Federico Milano This is my paper

Pith reviewed 2026-05-16 23:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords instantaneous complex phaseinstantaneous frequencyanalytic signalsspace vectorspower system transientsphase definitionsfrequency estimation
0
0 comments X

The pith

The analytic signal and space vector definitions of instantaneous complex phase and frequency are equivalent under balanced positive-sequence operation without harmonics or interharmonics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents the two core ways to define instantaneous complex phase and frequency in power systems and demonstrates that they produce identical results when the network is balanced, contains only electro-mechanical dynamics, and lacks harmonics. One definition relies on analytic signals; the other uses space vectors. Showing their direct relationship removes ambiguity that arises when different engineering communities apply the same concepts. The authors also introduce consistent notation to replace the scattered terms currently in use.

Core claim

Under the assumptions of balanced positive-sequence operation, sole presence of electro-mechanical transient dynamics, and absence of harmonics and interharmonics, the instantaneous complex phase and frequency obtained from analytic-signal methods are identical to those obtained from space-vector methods. The paper derives both formulations from first principles, states the modeling premises required for each, and exhibits the algebraic equivalence that follows from those premises.

What carries the argument

Direct algebraic mapping between the analytic-signal formulation and the space-vector formulation of instantaneous complex phase and frequency.

If this is right

  • Either formulation can be used interchangeably for phasor and frequency tracking in standard transmission-system models.
  • Existing software and hardware implementations based on one approach remain valid for the other under the stated conditions.
  • A single set of equations and symbols can replace the multiple notations now scattered across literature.
  • Real-time monitoring algorithms can adopt the simpler of the two equivalent expressions without changing numerical results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence supplies a common reference point for comparing frequency-estimation methods that currently cite different origins.
  • Extension to unbalanced or harmonic-rich conditions would require explicit correction terms derived from the same mapping.
  • The unified notation could reduce implementation errors when porting algorithms between signal-processing and power-engineering toolboxes.

Load-bearing premise

The system must remain balanced positive-sequence with only electro-mechanical transients and no harmonics or interharmonics.

What would settle it

A measurable difference between the analytic-signal and space-vector values of instantaneous phase or frequency when the voltage is unbalanced or contains harmonics would disprove the claimed equivalence.

Figures

Figures reproduced from arXiv: 2512.09574 by C\'esar Garc\'ia-Veloso, Federico Milano, Mario Paolone.

Figure 1
Figure 1. Figure 1: Sample cases where equivalence conditions are not met. Each column [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper presents the two core definitions of instantaneous complex phase and frequency—one via analytic signals and one via space vectors—along with their underlying premises. It derives the exact equivalence between these formulations under the explicit assumptions of balanced positive-sequence operation, purely electro-mechanical transients, and the absence of harmonics or interharmonics. The manuscript concludes by proposing unified notation and terminology to reduce confusion in the literature.

Significance. If the algebraic equivalence holds, the work supplies a useful conceptual clarification for phasor-based modeling in transmission-system dynamics. By stating the domain of validity explicitly and deriving the mapping through the positive-sequence component without fitted parameters or circular definitions, the paper strengthens consistency across analytic-signal and space-vector approaches. This is particularly relevant for transient-stability studies where instantaneous frequency definitions must be unambiguous.

minor comments (3)
  1. [§2] §2 (analytic-signal definition): the transition from the Hilbert transform to the instantaneous phase expression would benefit from an explicit intermediate step showing how the quadrature component is obtained under the balanced positive-sequence premise.
  2. [§4] §4 (equivalence derivation): while the algebraic steps are consistent, adding a short numerical verification (e.g., a two-machine swing trajectory) would make the domain of exact equivalence more tangible for readers.
  3. [Notation proposal] Notation table (proposed unified symbols): the mapping between existing symbols and the new unified set should be presented in a single compact table rather than scattered across paragraphs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the summary of the equivalence derivation under balanced positive-sequence assumptions and the recommendation for minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; algebraic equivalence is independently derived

full rationale

The paper presents the analytic-signal and space-vector definitions as separate starting points, then shows their equivalence through direct algebraic mapping under the explicitly listed domain assumptions (balanced positive-sequence, electro-mechanical transients only, no harmonics). These assumptions delimit the validity region rather than serving as hidden premises that force the result. No parameter fitting occurs, no self-citation chain carries the central claim, and the derivations do not reduce to renaming or self-definition. The work is a standard clarification with self-contained algebraic content.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard domain assumptions of power-system modeling rather than new free parameters or invented entities.

axioms (3)
  • domain assumption balanced positive sequence operation
    Invoked as the operating regime for transmission systems in the abstract.
  • domain assumption sole presence of electro-mechanical transient dynamics
    Stated as a modeling premise limiting the dynamics considered.
  • domain assumption absence of harmonics and interharmonics
    Explicitly listed as a condition for the equivalence to hold.

pith-pipeline@v0.9.0 · 5398 in / 1261 out tokens · 29097 ms · 2026-05-16T23:49:10.215715+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Tax- onomy of power converter control schemes based on the complex frequency concept,

    D. Moutevelis, J. Rold ´an-P´erez, M. Prodanovic, and F. Milano, “Tax- onomy of power converter control schemes based on the complex frequency concept,”IEEE Trans. Power Syst., vol. 39, no. 1, pp. 1996– 2009, 2024

    work page 1996

  2. [2]

    Dy- namic complex-frequency control of grid-forming converters,

    R. Domingo-Enrich, X. He, V . H ¨aberle, and F. D ¨orfler, “Dy- namic complex-frequency control of grid-forming converters,” in Procs. IECON, pp. 1–6, 2024

    work page 2024

  3. [3]

    Com- plex frequency locked loop for estimations of RoCoV and frequency,

    X. Quan, J. Tang, T. Xu, Z. Wu, W. Gu, S. Pan, and A. Q. Huang, “Com- plex frequency locked loop for estimations of RoCoV and frequency,” IEEE Trans. Power Syst., vol. 40, no. 6, pp. 5476–5479, 2025

    work page 2025

  4. [4]

    Local synchronization of power system devices,

    I. Ponce and F. Milano, “Local synchronization of power system devices,”IEEE Trans. Power Syst., vol. 40, no. 5, pp. 4194–4204, 2025

    work page 2025

  5. [5]

    Complex frequency,

    F. Milano, “Complex frequency,”IEEE Trans. Power Syst., vol. 37, no. 2, pp. 1230–1240, 2022

    work page 2022

  6. [6]

    A note concerning instantaneous frequency,

    D. Linden, “A note concerning instantaneous frequency,”Procs. IRE, vol. 46, 1958

    work page 1958

  7. [7]

    The instantaneous complex frequency concept and its application to the analysis of building-up oscillations in oscillators,

    S. L. Hahn, “The instantaneous complex frequency concept and its application to the analysis of building-up oscillations in oscillators,” Procs. Vib. Problems, vol. 1, pp. 24–46, 1959

    work page 1959

  8. [8]

    Complex variable frequency electric circuit theory,

    S. L. Hahn, “Complex variable frequency electric circuit theory,”Procs. IEEE, vol. 52, no. 6, pp. 736–737, 1964

    work page 1964

  9. [9]

    Hilbert transforms,

    S. L. Hahn, “Hilbert transforms,” inTransforms and Applications Handbook, 3rd ed.(A. D. Poularikas, ed.), pp. 300–399, Taylor & Francis, 2010

    work page 2010

  10. [10]

    Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals,

    B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals,”Procs. IEEE, vol. 80, no. 4, pp. 520–538, 1992

    work page 1992

  11. [11]

    Accurate modeling of pll with frequency-adaptive prefilter: On the positive feedback effect,

    J. Lei, X. Quan, S. Feng, J. Zhao, and W. Chen, “Accurate modeling of pll with frequency-adaptive prefilter: On the positive feedback effect,” IEEE Trans. Power Electron., vol. 37, no. 4, pp. 3747–3752, 2022

    work page 2022

  12. [12]

    On the uniqueness of the definition of the amplitude and phase of the analytic signal,

    S. L. Hahn, “On the uniqueness of the definition of the amplitude and phase of the analytic signal,”Signal Processing, vol. 83, no. 8, pp. 1815– 1820, 2003

    work page 2003

  13. [13]

    Complex-phase, data-driven identification of grid-forming inverter dy- namics,

    A. B ¨uttner, H. W ¨urfel, S. Liemann, J. Schiffer, and F. Hellmann, “Complex-phase, data-driven identification of grid-forming inverter dy- namics,”IEEE Trans. Smart Grid, vol. 16, no. 6, pp. 4854–4864, 2025

    work page 2025

  14. [14]

    Symmetrical components in the time domain and their application to power network calculations,

    G. Paap, “Symmetrical components in the time domain and their application to power network calculations,”IEEE Trans. Power Syst., vol. 15, no. 2, pp. 522–528, 2000

    work page 2000

  15. [15]

    A product theorem for Hilbert transforms,

    E. Bedrosian, “A product theorem for Hilbert transforms,”Proceedings of the IEEE, vol. 51, no. 5, pp. 868–869, 1963

    work page 1963

  16. [16]

    Beyond phasors: Modeling of power system signals using the Hilbert transform,

    A. Dervi ˇskadi´c, G. Frigo, and M. Paolone, “Beyond phasors: Modeling of power system signals using the Hilbert transform,”IEEE Trans. Power Syst., vol. 35, no. 4, pp. 2971–2980, 2020

    work page 2020

  17. [17]

    Functional basis analysis for the characterization of power system signal dynamics: Formulation, implementation, and validation,

    A. Karpilow, A. Dervi ˇskadi´c, and M. Paolone, “Functional basis analysis for the characterization of power system signal dynamics: Formulation, implementation, and validation,”IEEE Trans. Instr . Meas., vol. 74, pp. 1–14, 2025

    work page 2025

  18. [18]

    A new approach to the definition of power components in three-phase systems under nonsinusoidal condi- tions,

    A. Ferrero and G. Superti-Furga, “A new approach to the definition of power components in three-phase systems under nonsinusoidal condi- tions,”IEEE Trans. Instrum. Meas., vol. 40, no. 3, pp. 568–577, 1991

    work page 1991

  19. [19]

    A geometrical interpretation of frequency,

    F. Milano, “A geometrical interpretation of frequency,”IEEE Trans. Power Syst., vol. 37, no. 1, pp. 816–819, 2022

    work page 2022

  20. [20]

    Jacewicz,Multivector and Clifford Algebra in Electrodynamics

    B. Jacewicz,Multivector and Clifford Algebra in Electrodynamics. Singapore: World Scientific, 1989

    work page 1989

  21. [21]

    Applications of the Frenet frame to electric circuits,

    F. Milano, G. Tzounas, I. Dassios, and T. K ¨erc ¸i, “Applications of the Frenet frame to electric circuits,”IEEE Trans. Circ. Syst. I: Regular Papers, vol. 69, no. 4, pp. 1668–1680, 2022

    work page 2022