Clock Pulling Enables Maximum-Efficiency Wireless Power Transfer

Chaoqiang Jiang; Chi Kong Tse; Jianlong Zou; Ke Yin; Sheng Ren; Tianyu Dong; Xianglin Hao; Xiaosheng Wang

REVIEW 3 major objections 6 minor 29 references

Clock Pulling Enables Maximum-Efficiency Wireless Power Transfer

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims a clock-pulling feedback loop can stabilize the maximum-efficiency state of nonlinear parity-time-symmetric wireless power transfer, and demonstrates it experimentally.

desk verdict Clever experiment stabilizes the max-efficiency mode in PT-symmetric WPT, but a sign error in Eq. (4) contradicts the paper's own stability argument and must be fixed before publication. read the letter →

arxiv 2507.22907 v1 pith:VVAEBP6F submitted 2025-07-15 physics.app-ph physics.class-phphysics.opticsquant-ph

Xianglin Hao , Xiaosheng Wang , ke Yin , Sheng Ren , Chaoqiang Jiang , Jianlong Zou , Tianyu Dong , Chi Kong Tse This is my paper · ORCID

classification physics.app-phphysics.class-phphysics.opticsquant-ph

keywords wirelesspowertransferparity-timesymmetrynon-Hermitiansystemsclockpullingphase-lockedloopnonlineargainefficiencymultistability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that a nonlinear feedback mechanism it calls clock pulling can force a parity-time (PT) symmetric two-coil wireless power transfer system to settle on its conventionally unstable steady state, the one with maximum transfer efficiency. PT symmetry here means the transmitter gain and receiver loss are balanced, so the system has two symmetric modes $\tilde{\omega}_{1,2}$ and a high-gain mode $\tilde{\omega}_0 = 1$. The paper argues that stability in these nonlinear non-Hermitian systems is not permanently fixed by the principle of minimal gain, but can be reconfigured through the frequency-dependent phase of the required complex gain. If the claim is right, a wireless power link can run at its theoretical efficiency limit over a range of coil couplings without active retuning, and the same feedback principle could apply to other nonlinear non-Hermitian platforms.

What carries the argument

The mechanism is clock pulling: a phase-locked loop whose voltage-controlled oscillator provides a classical clock and forces continuous frequency variation, so feedback can act on the frequency-dependent phase of the steady-state required gain instead of being defeated by rapid transient frequency jumps. The load-bearing identity is the complex steady-state gain formula $g_{ss,c}$ (Eq. (4)); for the PT case its imaginary part changes sign across the mode $\tilde{\omega}_0 = 1$, giving the PLL a pulling polarity that increases frequency on one side and decreases it on the other. That polarity is what stabilizes the previously unstable maximum-efficiency state and destabilizes the symmetric modes.

What would settle it

Measure the phase-frequency response of the clock-pulling gain module around $\tilde{\omega}_0 = 1$ and check whether its slope matches the polarity assumed in Fig. 2(d); then flip the phase-detector operating point from $\pi$ to $0$ and see whether the stable mode switches from $\tilde{\omega}_0$ to $\tilde{\omega}_{1,2}$. If reversing the pulling polarity has no effect on which steady state is reached, the clock-pulling mechanism is not what stabilizes the maximum-efficiency mode.

Watch

Extended reading notes

Core claim

The central discovery is that the steady-state selection of a nonlinear PT-symmetric WPT dimer can be switched by a clock-pulling feedback loop. Examining the dispersion of the required gain, the authors find that the mode $\tilde{\omega}_0 = 1$, which conventionally is dismissed as unstable because it requires the highest gain, sits at an extremum of the complex required gain $g_{ss,c}$; its imaginary part has opposite polarity in the adjacent frequency bands. A phase-locked loop whose voltage-controlled oscillator provides a classical clock enforces continuous frequency variation, and near its $\phi_{\mathrm{PLL}} = \pi$ equilibrium the loop adjusts frequency with the correct polarity to pull the system to $\tilde{\omega}_0$ and hold it there, while making the symmetric modes $\tilde{\omega}_{1,2}$ unstable. The paper reports an 85 kHz two-coil prototype in which the system operates stably at $\tilde{\omega}_0$ with the theoretically maximum transfer efficiency in the strong-coupling region, consistent with coupled-mode calculations.

Load-bearing premise

The design rests on the sign of the frequency-dependent complex gain in Eq. (4), which sets the pulling polarity: if the imaginary part of the required gain does not change sign across $\tilde{\omega}_0 = 1$ in the way the paper's arrows show, the feedback would push the system away from the maximum-efficiency state instead of toward it.

Editorial extensions

If this is right

If the paper is right, two-coil wireless power transfer can operate at the theoretical maximum transfer efficiency over a range of coupling coefficients without active tuning, outperforming conventional PT-symmetric designs in the strong-coupling region.
The same clock-pulling principle should stabilize the corresponding zero-point mode in asymmetric non-Hermitian systems where $\chi_l \chi_c \neq 1$, since the phase-frequency response of the required gain remains similar.
The stability assignment reverses in the PT-broken region: the clock-pulling configuration that stabilizes $\tilde{\omega}_0$ is unstable there, so the method is confined to the strong-coupling regime.
Because the feedback reconfigures stability dynamically rather than changing the dispersion landscape, the mechanism should transfer to other nonlinear non-Hermitian platforms such as waveguide resonators, acoustic cavities, and optoelectronic oscillators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same design logic should generalize to any nonlinear non-Hermitian oscillator whose required gain has an extremum at a target mode: clock pulling turns a frequency-domain extremum into a stable attractor, so other 'maximum-gain' modes could be stabilized without changing the physical nonlinearity.
A testable extension is to sweep load resistance and coupling while recording efficiency: the clock-pulled state should track the theoretical maximum-efficiency curve up to the PT-broken boundary and then lose stability, giving a sharp quantitative prediction of the operating envelope.
The engineering limit implied by the mechanism is feedback speed: if the loop bandwidth is too low to follow the rapid transient frequency shifts reported in earlier PT-circuit experiments, the stabilizing action will fail, so measuring loop bandwidth against frequency-switch transients would set the practical range.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jackie,

Quick take: this is a paper worth reading and worth refereeing, but as written it has a load-bearing typo. The experimental part is genuinely new: the authors show that a PLL-based clock-pulling loop can stabilize the 'unstable' e0 mode—the one with maximum transfer efficiency—in a two-coil PT-symmetric WPT system. Measured frequencies and efficiencies track the coupled-mode model nicely over the coupling range, and the idea of using a classical clock to enforce continuous frequency variation so feedback can pull the system to a normally unstable steady state is clever, with clear reach beyond WPT.

The soft spot is real. Equation (4), which is the theoretical basis for the entire polarity design, does not follow from Eq. (2). For the PT case (χc=χl=1), solving Eq. (2) for the required complex gain gives Im(gss,c) = 2δ[1 − k²/(γ²+4δ²)]. The printed Eq. (4) has a plus sign in the bracket. That is not cosmetic: the whole stability argument in Fig. 2(d) relies on the Im(gss,c) slope being negative near e0 for k>γ, so that a positive frequency error produces a phase error that makes the PLL pull the frequency down. With the printed plus sign, the feedback would be positive and would destabilize e0 instead of stabilizing it. This looks like a missing minus sign in the second term of the imaginary part, but it is a central equation and the paper cannot stand as is.

Two smaller points. There is no closed-loop stability analysis; the stability is inferred from the dispersion sign plus an intuitive PLL picture. That is acceptable for a first demonstration, but a small-signal loop model would make the claim solid. And the measured waveforms show noticeable harmonic distortion; the authors acknowledge it and argue it doesn't affect the fundamental, which is fine.

If the authors fix the sign in Eq. (4) and confirm the plotted dispersion uses the corrected formula, this is a solid paper. The experimental result is the strongest evidence and would likely survive correction. I'd send it to peer review with a request for that fix, and ideally a short closed-loop stability argument.

Best,

Referee Report

3 major / 6 minor

Summary. The manuscript proposes a 'clock-pulling' feedback mechanism, realized by a phase-locked loop, to stabilize the conventionally unstable symmetric mode ω̃₀ = 1 of a nonlinear parity-time-symmetric two-coil wireless power transfer system. The authors derive a coupled-mode-theory dispersion relation for the complex steady-state gain, argue that the frequency-dependent phase response of this gain permits a feedback polarity that makes the maximum-efficiency mode robustly stable, and support the idea with an FPGA-based PLL prototype operating at 85 kHz. Measured frequency and efficiency data are compared with the CMT predictions over a range of coupling coefficients. The central claim is that this clock-pulling scheme forcibly breaks the PT symmetry and enables operation at the theoretical maximum transfer efficiency without active tuning.

Significance. If correct, the paper would offer a control-theoretic route to operate a two-coil WPT system at the maximum-efficiency mode while retaining frequency robustness, and it would generalize the clock-pulling concept to other nonlinear non-Hermitian platforms. The experimental comparison is a genuine strength: Fig. 3 reports measured frequency and efficiency versus coupling coefficient for both the clock-pulling scheme and a conventional PT-symmetric system, with the model curves evidently not fitted to the data. The proposed mechanism is falsifiable and the central claim is clearly stated. However, the written algebraic derivation of the key dispersion relation contains a sign error that, as printed, reverses the feedback polarity and would destabilize the very state the design aims to stabilize. That issue must be resolved before the theoretical mechanism can be considered established.

major comments (3)

[Eq. (4) and Fig. 2(b)-(d)] Equation (4) is algebraically inconsistent with Eq. (2) in the sign of the k² term in the imaginary part. Setting χ_c = χ_l = 1 and solving Eq. (2) for complex g_ss,c as a function of real ω̃ gives Im(g_ss,c) = 2(ω̃−1)[1 − k²/(γ² + 4(ω̃−1)²)], whereas Eq. (4) as printed has the same bracket with a plus sign. In the PT-symmetric phase k > γ, the correct sign is negative near ω̃ = 1, and the printed plus sign gives the opposite dispersion. Since the clock-pulling polarity argument in Fig. 2(d) relies on the sign of Im(g_ss,c) to determine whether a frequency error produces a stabilizing or destabilizing PLL response, the derivation as written does not support the claimed stabilization of ω̃₀. The authors should correct Eq. (4), re-plot the phase-dispersion curves in Fig. 2(b), and confirm that the polarity analysis still predicts stabilization of ω̃₀.
[Fig. 2(b)-(d) and Sec. II] The narrative around Fig. 2(b) states that the left neighborhood of ω̃₀ exhibits negative phase angles and the right neighborhood positive angles. With the corrected sign derived from Eq. (2), the opposite holds for k > γ: left of ω̃₀ (δ < 0) gives positive Im(g_ss,c) and right (δ > 0) gives negative Im(g_ss,c). The text therefore describes the polarity that follows from the erroneous plus sign in Eq. (4), not from the stated model. The stabilization arrows in Fig. 2(d) must be re-derived from the corrected phase-frequency relation; as printed, they implement positive feedback and would destabilize the maximum-efficiency state.
[Sec. II, Fig. 2(d)] The stability analysis is presented only as a heuristic red-arrow argument in Fig. 2(d) with no linearized small-signal stability calculation for the combined PLL/resonator dynamics. The claim that the clock-pulling scheme makes ω̃₀ 'uniquely robust stable' is load-bearing but is not supported by a Lyapunov or perturbation analysis. Given the sign error in Eq. (4), a proper linearized treatment is necessary to establish the claimed polarity and to define the conditions under which the continuous-time PLL loop actually converges to ω̃₀ rather than to ω̃₁ or ω̃₂.

minor comments (6)

[Abstract and Introduction] The grammar 'nonlinear parity-time (PT) symmetry ... have posed' should be 'has posed', and the term 'parity-time symmetric WPT system' is used before the abbreviation PT is fully established in the same sentence; please polish the opening paragraph.
[Fig. 1(d) caption] The caption contains a typo ('steady-state requried gain') and the axis label 'Log(gss)' should be 'Log(|g_ss,c|)' for clarity.
[Eq. (4)] The bracket structure of Eq. (4) is ambiguous in the typeset equation; the imaginary part should be written with explicit parentheses so that the reader can immediately see the term 2χ_c²ω̃ − 2χ_c⁴χ_l² and the k² term are both inside the imaginary bracket.
[Reference [19]] Reference [19] is given as 'See Supplement Material at url' with a placeholder 'url'; the actual supplementary link and document identifier should be provided before publication.
[Fig. 3(c)-(d)] The measured data in Fig. 3(c) and (d) are presented without error bars or a statement of measurement uncertainty or number of repeated trials; adding this information would strengthen the claimed agreement with the CMT calculation.
[Fig. 2(a) description] The description of the two PLL fixed-point types is confusing: for ϕ = (2n+1)π the text says the output is 'in phase' with the reference, which is unconventional for a sinusoidal phase detector; please clarify the sign convention and the relation to z_f.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: clock-pulling stabilization is derived from the stated CMT equations and compared with measurements, not fitted or assumed.

full rationale

The paper's derivation chain is self-contained. The CMT Hamiltonian (Eq. 1) and characteristic equation (Eq. 2) yield the steady-state gain conditions (Eqs. 3a-3b); the complex gain gss,c in Eq. (4) is obtained directly from Eq. (2) under the same assumptions, and the clock-pulling polarity design in Fig. 2 is an application of that dispersion relation, not an input to it. The central claim—that PLL feedback stabilizes mode ω˜0 = 1—is supported by the phase-frequency response of the derived gain and by experiments in Fig. 3 that compare measured frequencies and efficiencies with calculated values; no parameter is fitted to the stabilization claim. The authors' prior works [15, 18] are cited as background on dispersive-gain and pseudo-Hermitian WPT, but the clock-pulling mechanism and its stability analysis are new to this paper, so the self-citations are not load-bearing. The possible sign discrepancy in Eq. (4) identified by the reader is an internal-consistency/correctness issue: if the printed sign is wrong, the polarity argument fails on mathematical grounds, but that failure would not mean the argument reduces to its own conclusion. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central model is standard coupled-mode theory with a steady-state gain assumption and a PT-symmetric dimer Hamiltonian. The clock-pulling feedback is treated as a frequency-controlling phase-locked loop, whose closed-loop stability is not derived. No parameters are fitted to the reported efficiency data; circuit constants are stated. The main load-bearing theoretical issue is the sign of the dispersion relation in Eq. (4), which appears inconsistent with Eq. (2).

assumptions (3)

domain assumption The nonlinear gain gnl(a1) can be replaced by a constant steady-state gain gss when analyzing steady states.
Invoked in the derivation of Eq. (2) and the steady-state frequencies; standard in gain-saturated oscillator analysis but unstated as a stability condition.
domain assumption The PLL feedback can enforce continuous frequency variation and pull the system to the desired steady state.
Central to the clock-pulling mechanism in Fig. 2; no closed-loop stability or bandwidth analysis is provided.
ad hoc to paper The sign of the dispersion of the complex gain gss,c near e0 determines the required feedback polarity.
Used to justify the pulling polarity; the sign in Eq. (4) appears inconsistent with Eq. (2), so this assumption is not established.

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Cite this review

Pith. "Pith review of Clock Pulling Enables Maximum-Efficiency Wireless Power Transfer." pith.science (2026). https://pith.science/paper/VVAEBP6F

@misc{pith2026250722907,
  author       = {Pith},
  title        = {Pith review of: Clock Pulling Enables Maximum-Efficiency Wireless Power Transfer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVAEBP6F}},
  note         = {Machine review of arXiv:2507.22907}
}

read the original abstract

Nonlinear parity-time (PT) symmetry in non-Hermitian wireless power transfer (WPT) systems, while attracting significant attention from both physics and engineering communities, have posed formidable theoretical and practical challenges due to their complex dynamical mechanisms. Here, we revisit multistability in nonlinear non-Hermitian systems and find that the PT-symmetry state is not always stable even in PT-symmetry phase. We report a discovery on a nonlinear clock-pulling mechanism, which can forcibly break the PT symmetry. Proper implementation of this mechanism can switch the system stability, particularly in stabilizing the conventional unstable state which has the maximum transfer efficiency for WPT. Our work offers new tools for non-Hermitian physics and is expected to drive technological progress.

Figures

Figures reproduced from arXiv: 2507.22907 by the authors.

**Figure 3.** FIG. 3. (a) Photo of the experimental setup. (b) Measured steady-state waveforms of the WPT system based on clock-pulling gain when [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Solja ˇci`c, Wireless power transfer via strongly coupled magnetic resonances, Science 317, 83 (2007)

work page 2007

[2] [2]

S. Y . R. Hui, W. Zhong, and C. K. Lee, A critical review of re- cent progress in mid-range wireless power transfer, IEEE Trans. Power Electron. 29, 4500 (2013)

work page 2013

[3] [3]

Assawaworrarit, X

S. Assawaworrarit, X. Yu, and S. Fan, Robust wireless power transfer using a nonlinear parity-time-symmetric circuit, Nature 5 546, 387 (2017)

work page 2017

[4] [4]

Krasnok, D

A. Krasnok, D. G. Baranov, A. Generalov, S. Li, and A. Alù, Coherently enhanced wireless power transfer, Phys. Rev. Lett. 120, 143901 (2018)

work page 2018

[5] [5]

Jiang and B

Y . Jiang and B. Zhang, A fractional-order wireless power trans- fer system insensitive to resonant frequency, IEEE Trans. Power Electron. 35, 5496 (2019)

work page 2019

[6] [6]

V .-B. Vu, A. Ramezani, A. Triviño, J. M. González-González, N. B. Kadandani, M. Dahidah, V . Pickert, M. Narimani, and J. Aguado, Operation of inductive charging systems under mis- alignment conditions: A review for electric vehicles, IEEE Trans. Transp. Electrific. 9, 1857 (2022)

work page 2022

[7] [7]

Z. Hua, K. Chau, W. Liu, X. Tian, and H. Pang, Autonomous pulse frequency modulation for wireless battery charging with zero-voltage switching, IEEE Trans. Power Electron. 70, 8959 (2022)

work page 2022

[8] [8]

C. Zeng, Z. Guo, K. Zhu, C. Fan, G. Li, J. Jiang, Y . Li, H. Jiang, Y . Yang, Y . Sun,et al., Efficient and stable wireless power trans- fer based on the non-Hermitian physics, Chinese Phys. B 31, 010307 (2022)

work page 2022