Complex Frequency

The paper explicitly derives ds̄ = s̄ ◦ dζ̄ + S̄ dζ̄∗ and, when the independent variable is time, ˙̄s − s̄ ◦ η̄ = S̄ η̄∗ (its eqs. (31)–(32)) as the core link between complex power variations and the complex frequency η̄ = u̇ + j θ̇ (see ). The model’s solution reproduces exactly these identities using a compact derivation based on v̄ = e^{ζ̄}, s̄ = v̄ ◦ ı̄∗, and ı̄ ≈ Ȳ v̄, leading to S̄ with entries S̄hk = Ȳ∗hk e^{ζh+ζ∗k}. The paper provides the definitions s̄ = v̄ ◦ ı̄∗ and the QSS link ı̄ ≈ Ȳ v̄ (eqs. (2), (9), (11); see , , ) and the auxiliary identity ˙̄v = v̄ ◦ η̄ (eq. (37); see ). The model’s proof is correct and shorter (using the product rule for Hadamard products and Wirtinger-style differentials), while the paper’s proof proceeds via p, q partial derivatives and then compacts them using ζ̄. Both reach the same results; the model’s only omission is not stating explicitly that ı̄ ≈ Ȳ v̄ is the QSS approximation adopted in the paper.