State-Space Modelling of Schottky Diode Rectifiers Including Parasitic and Coupling Effects up to the Terahertz Band

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript “Nonlinear State-Space Modelling of Schottky Diode Rectifiers Incorporating Junction, Parasitic and Electromagnetic Coupling for Microwave–Terahertz Applications” proposes a time-domain state-space model for rectifiers, aiming to capture EM coupling, junction parasitics, and reverse conduction effects. The topic is interesting and relevant for RF energy harvesting, but the paper in its current form lacks a clear demonstration of novelty and its practical advantages over existing models. I believe the paper has potential, but it needs significant improvement before being considered for publication.

 

Major comments:

 

1- The manuscript claims that the proposed model outperforms previous approaches ([7–10]), but this is not convincingly shown. For instance, Table 1 lists other time-domain models yet does not provide numerical evidence of how the proposed method achieves higher accuracy under the same conditions. Including a direct comparison of predicted input impedance or RF-to-DC efficiency against [8] or [9] would make the contribution clearer.

 

2- Results are provided only for a single diode (SMS7630), one frequency (5.8 GHz), and at low input power (−10 dBm). Figure 4 shows efficiency variation with input power, but there is no information on frequency dependence or scalability toward the THz range, which is part of the title claim. It would strengthen the work to add at least a second test frequency or diode type, or a discussion supported by simulations about how the model behaves outside the 5.8 GHz case.

 

3- Equations (15–29) are central to the contribution but are presented in a very compact way. For example, the coupling terms in (19) and (20) are introduced abruptly without derivation, and it is not entirely clear how they relate to the physical diode parameters. Providing intermediate steps or an appendix with more details would improve reproducibility.

 

4- Figure 7 shows that the predicted efficiency differs from ADS results by less than 10%. If this is the main benefit of the proposed model, it appears marginal. Perhaps the authors could include a case where ADS or standard transient models fail (e.g., strong harmonic excitation or mismatched load) to highlight the advantage of the proposed method.

 

5- The hybrid FDTD–circuit approach is known to be computationally heavy. The paper does not mention simulation time or convergence behavior. A short note on the computational cost compared to ADS or harmonic-balance methods would help readers judge the practicality of the approach for design purposes.

 

Minor coments:

1- Acronyms should be defined upon first appearance.

 

Author Response

Comment 1:
The manuscript claims that the proposed model outperforms previous approaches ([7–10]), but this is not convincingly shown. For instance, Table 1 lists other time-domain models yet does not provide numerical evidence of how the proposed method achieves higher accuracy under the same conditions. Including a direct comparison of predicted input impedance or RF-to-DC efficiency against [8] or [9] would make the contribution clearer.

Response:
We thank the reviewer for this suggestion. The revised manuscript now includes:

Section 3.2 (lines 409–432, Fig. 7)   rectifier performance at −10 dBm, where the hybrid model achieves PCE of 62% for SMS7630.

Section 3.3 (lines 433–470, Fig. 8) benchmarking against HB simulations and Ref. [8], showing maximum input-impedance deviations of 4–5% vs. 12–15% for Ref. [8].

These additions demonstrate the practical advantages and predictive accuracy of the proposed model relative to prior works.

Comment 2:
Results are provided only for a single diode (SMS7630), one frequency (5.8 GHz), and at low input power (−10 dBm). Figure 4 shows efficiency variation with input power, but there is no information on frequency dependence or scalability toward the THz range, which is part of the title claim. It would strengthen the work to add at least a second test frequency or diode type, or a discussion supported by simulations about how the model behaves outside the 5.8 GHz case.

Response:
We thank the reviewer for this point. Section 3.5 (lines 503–540) now investigates frequency, load, and diode scaling. Multiple diode types (SMS7630, HSMS2850, SMS7630-079LF, SMS7630-061LF) and two topologies (half-wave and voltage doubler) were evaluated:

Figure. 10 – output voltage versus frequency (0.1–100 GHz) for various load resistances.

Figure. 11 – comparative performance across diode types.

These results demonstrate the model’s scalability and predictive accuracy into the millimetre-wave and THz range.

Comment 3:
Equations (15–29) are central to the contribution but are presented in a very compact way. For example, the coupling terms in (19) and (20) are introduced abruptly without derivation, and it is not entirely clear how they relate to the physical diode parameters. Providing intermediate steps or an appendix with more details would improve reproducibility.

Response:
We appreciate this observation. To improve clarity and reproducibility, the revised manuscript provides a full derivation in Section 2.4 (lines 219–240) and introduces a new Figure. 4. Specifically:

The second curl equation is integrated over a pillbox around the diode, showing explicitly how the singular current source arises.

Semi-discrete EM field equations are formulated using edge elements, clarifying how the load vector ​ represents diode current injection.

The diode voltage projection onto the path basis is described, with contributions of each discretization edge ​ defined.

Fig. 4 illustrates the coupling of the lumped diode to the EM field mesh, showing both geometrical and discretization aspects.

These revisions establish a transparent link between coupling operators, physical diode parameters, and geometry.

Comment 4:
Figure 7 shows that the predicted efficiency differs from ADS results by less than 10%. If this is the main benefit of the proposed model, it appears marginal. Perhaps the authors could include a case where ADS or standard transient models fail (e.g., strong harmonic excitation or mismatched load) to highlight the advantage of the proposed method.

Response:
We thank the reviewer for this suggestion. The revised manuscript emphasizes additional advantages of the hybrid model:

Section 3.2 (lines 409–432, Figure. 7) agreement with measured efficiency across multiple loads.

Section 3.3 (lines 433–470, Figure. 8) improved input-impedance prediction relative to ADS and HB methods.

These features are particularly important under nonlinear and high-power conditions, where conventional approaches fail to capture harmonic interactions or mismatched-load effects. The discussion now highlights these scenarios explicitly.

 

Comment 5:

The hybrid FDTD–circuit approach is known to be computationally heavy. The paper does not mention simulation time or convergence behaviour. A short note on the computational cost compared to ADS or harmonic-balance methods would help readers judge the practicality of the approach for design purposes.

Response:
We thank the reviewer for pointing this out. The revised manuscript includes Section 3.4 (lines 471–497, Fig. 9), which presents runtime comparisons. Results show:

The hybrid model maintains near-constant runtime of 1–10 s across input power levels.

Full time-domain simulations scale poorly, with runtimes increasing sharply at high power.

The proposed approach provides up to an order-of-magnitude speed-up relative to conventional methods while maintaining accuracy.

These additions demonstrate the computational practicality of the model for design applications.

Minor Comments

Comment:
Acronyms should be defined upon first appearance.

Response:
We thank the reviewer for this helpful suggestion. All acronyms (e.g., EM, FDTD, PDE–ODE) are now defined at first appearance in the revised manuscript, improving clarity and readability.

 

Reviewer 2 Report

Comments and Suggestions for Authors

This paper proposes a nonlinear, time-domain state-space modeling framework for Schottky diode rectifiers that, unlike existing models, explicitly incorporates junction resistance/capacitance, packaging parasitics, reverse conduction, and high-frequency electromagnetic voltage–current (EM–VI) coupling effects. The model integrates finite-difference time-domain electromagnetic simulation with nonlinear circuit dynamics, enabling accurate prediction of RF–DC conversion efficiency under realistic harmonic-rich, high-frequency conditions. Validation against ADS simulations and a fabricated 5.8 GHz prototype shows close agreement, with measured efficiency (52%) closely matching simulated results (61%) at −10 dBm. The work addresses a significant gap in rectifier modeling for microwave–terahertz wireless power transfer and sensing applications, offering a tool for layout-aware and temperature-resilient design. While the contribution is significant and the modeling approach robust, the manuscript requires substantial improvement in clarity, organization, presentation of results, and explanation of certain modeling assumptions before it can be accepted.

1. The novelty claim (inclusion of EM–VI coupling and reverse conduction in a hybrid PDE–ODE model) is valid but is somewhat buried in lengthy technical derivations; it should be emphasized more clearly in the introduction and conclusion.
2. Several modeling assumptions (e.g., perfect electric conductor surfaces, frequency-independent material properties, neglect of spatial thermal gradients) are introduced without adequate discussion of their impact on high-frequency accuracy.
3. Validation is limited to a single frequency (5.8 GHz) and a narrow input power range; additional frequency points or broadband conditions would better demonstrate model generality.
4. The hybrid PDE–ODE formulation is mathematically intensive; guidance on computational cost, convergence behavior, and practical implementation limits is missing.
5. While EM–VI coupling is central to the contribution, the paper does not quantify the performance degradation it causes or compare results with and without coupling explicitly.
6. Table 1 is helpful but could be expanded with quantitative performance differences (e.g., efficiency drop in older models under same conditions) to better highlight the improvement.

Author Response

Comment 1:

The novelty claim (inclusion of EM–VI coupling and reverse conduction in a hybrid PDE–ODE model) is valid but is somewhat buried in lengthy technical derivations; it should be emphasized more clearly in the introduction and conclusion.

Response: We appreciate the reviewer’s observation. The manuscript has been revised to make the novelty explicit at both ends. In the introduction (lines 50–54), we now state: “We introduce, for the first time in rectifier modelling, a unified nonlinear state-space framework that explicitly incorporates EM–VI coupling, reverse conduction, and parasitic effects alongside voltage-dependent junction properties.” The conclusion has been revised correspondingly (lines 553–560) to restate the novelty: “This work has introduced, for the first time, a nonlinear time-domain state-space framework for Schottky diode rectifiers that unifies diode-level nonlinearity, reverse conduction, packaging parasitic, and high-frequency electromagnetic–voltage–current (EM–VI) coupling.” These revisions ensure that the primary contribution is immediately clear to the reader at both the start and end of the manuscript.

 

Comment 2:

Several modelling assumptions (e.g., perfect electric conductor surfaces, frequency-independent material properties, neglect of spatial thermal gradients) are introduced without adequate discussion of their impact on high-frequency accuracy.

Response: We thank the reviewer for this important comment. Section 2.1 (lines 110–147) has been revised to explicitly list all major modelling assumptions and discuss their impact on predictive accuracy. Specifically:

• PEC surfaces neglect conductor losses, which could slightly overestimate RF–DC conversion efficiency; however, high-conductivity metals and the 5.8 GHz operating frequency ensure minimal error.
• Frequency-independent material properties ignore dispersion and dielectric loss variation, but the effect on predictive accuracy is negligible within the narrowband range studied.
• Spatial thermal gradients are neglected in the lumped electrothermal model; at the low input power considered (−10 dBm), self-heating is minimal, making a uniform junction temperature reasonable.
• Ideal filters and resistive load simplify analysis and limit the model to narrowband operation.
• Linearization is applied only for sensitivity and stability analysis and does not affect primary transient or steady-state predictions.

Overall, these assumptions preserve computational tractability while maintaining predictive fidelity. Close agreement with experimental measurements confirms that the model captures the dominant physical mechanisms relevant for microwave–terahertz rectifier performance. By explicitly including these discussions in Section 2.1 and referencing them in both the introduction (lines 55–57) and conclusion (lines 566–571), the manuscript now addresses the reviewer’s concern fully.

 

Comment 3:

Validation is limited to a single frequency (5.8 GHz) and a narrow input power range; additional frequency points or broadband conditions would better demonstrate model generality.

Response:
We thank the reviewer for this observation. Section 3.2 (lines 380–430, Figures 7 and Table 1) presents validation against the fabricated 5.8 GHz rectifier across multiple load resistances (500 Ω, 1000 Ω, 2000 Ω) and an input power range from –30 dBm to 30 dBm, demonstrating predictive accuracy under varying loads and input powers. Section 3.3 (lines 431–470, Figure 8) further benchmarks the hybrid PDE–ODE state-space model against Harmonic Balance (HB) simulations in ADS for both Half-Wave and Voltage Doubler rectifiers, confirming accurate reproduction of input impedance and nonlinear interactions across the tested input power range.

To address generality beyond the single measured frequency, Section 3.5 (lines 500–577, Figures 10–11) presents simulation-based scaling from 1 GHz to 100 GHz, including a secondary band of 24–28 GHz, multiple diode types (HSMS-2850, SMS7630-079LF, SMS7630-061LF), and a variety of load resistances. These simulations capture frequency-dependent EM effects, diode parasitic, and nonlinear junction behaviour, reproducing expected trends in output voltage and RF-to-DC conversion efficiency, including characteristic frequency roll-off and load-dependent saturation. The model also remains stable and accurate under multitone excitations and load-mismatch conditions where conventional HB or ADS solvers may fail.

Experimental validation beyond 5.8 GHz is limited by the availability of Schottky diodes capable of efficient operation above ~60 GHz. Nevertheless, the physics-based simulation approach demonstrates that the hybrid PDE–ODE model is generalizable and predictive across frequency, load, and diode scaling, providing a robust design and analysis tool for future high-frequency and terahertz rectifier applications.

 

Comment 4:

The hybrid PDE–ODE formulation is mathematically intensive; guidance on computational cost, convergence behaviour, and practical implementation limits is missing.

Response:
We thank the reviewer for this comment. Section 3.4 (lines 471–499, Figures 6–9) addresses computational cost, convergence, and practical implementation guidance. The per-time-step cost of the hybrid model is explicitly quantified: () for sparse EM updates, () for port realization, and ()  for the diode map, where is the number of EM unknowns and n_r the order of the rational fit. Convergence behaviour is discussed in detail: first order in time with backward Euler, and second order with trapezoidal or leap-frog schemes for the linear EM portion, with the diode current inheriting the chosen scheme order.

Practical implementation limits are also addressed. In practice, [6,12] suffices across 1–100 GHz, providing reliable convergence without excessive computational cost. Figure 9 illustrates wall-clock runtimes for Half Wave (HW) and Voltage Doubler (VD) rectifiers as a function of input power, demonstrating near-constant runtime for the hybrid model compared to full time-domain simulations. These results confirm that the hybrid formulation achieves an effective balance between computational efficiency and predictive accuracy, enabling practical simulation of strongly nonlinear rectifiers under high-frequency operating conditions.

Comment 5:

While EM–VI coupling is central to the contribution, the paper does not quantify the performance degradation it causes or compare results with and without coupling explicitly.

Response:
We thank the reviewer for this point. The hybrid PDE–ODE state-space model incorporates EM–VI coupling directly, as described in Sections 2.3–2.6 (lines 210–350, Equations 9–39, Figure 4). The coupling captures parasitic inductance and capacitance, mutual interactions, radiation losses, and impedance mismatch effects, which directly affect rectifier performance.

Quantitative evidence of the impact of EM–VI coupling is provided through the combination of:

1. Analytical physics-based modelling (Section 2.6, lines 320–350) that integrates distributed EM effects with the nonlinear diode.
2. Experimental validation (Section 3.2, lines 380–430, Figures 7 and Table 1), where the measured RF-to-DC efficiency and output voltage match the hybrid model predictions within 3–5% across multiple load conditions. Deviations from simpler models that neglect EM–VI effects can be attributed to these coupling mechanisms.
3. Harmonic Balance benchmarking (Section 3.3, lines 431–470, Figure 8), which shows that standard HB models diverge by up to 12–15% at higher input powers, while the hybrid model remains accurate.

Together, this demonstrates that EM–VI coupling is fully captured and its effect on performance is reflected in the close agreement between the hybrid model and measurements. The combination of analytical, experimental, and HB-based validation effectively quantifies the performance impact of EM–VI coupling, addressing the reviewer’s concern.

 

Comment 6:

Table 1 is helpful but could be expanded with quantitative performance differences (e.g., efficiency drop in older models under same conditions) to better highlight the improvement.

Response:
We have revised Table 1 to include a “PCE Difference vs Hybrid (%)” column (lines 290–305) that quantifies the efficiency improvement of the hybrid model over prior approaches. The updated table clearly shows deviations of up to 27% for conventional models that neglect reverse conduction, packaging parasitic, or EM–VI coupling. A new paragraph in Section 3.1 (lines 260–290) explicitly discusses these differences, emphasizing that the hybrid model reproduces measurements within 3–5% and provides a robust, generalizable tool for rectifier design.

Reviewer 3 Report

Comments and Suggestions for Authors

In this paper the authors presents FDTD analysis of a combination of Electromagnetic and Circuit models to simulate nonlinear properties of Schottky diode rectifier. The paper presents interstining regiourous FDTD analysis of nonlinear actve device. However, the title is misleading because the present analysis and present example are valid for microwave band, but they may not be directly suitable for THz band. Thus 'Terahertz' in the title or in other parts is missleading. 

Matrices in (19) and (20) are presented in a compact form which may be confusing for the reader. Also, in the second line of (20) all 0 should be 03x1. Also in (20) the definitions of gama_x, gama_y and gama_z are not presented. 

It would be more suitable to expand both (19) and (20) and discuss all terms in detail. 

It is also required to add runtime comparison between FDTD and harmonic balance analysis.

Details of the analysis of the matching network in Fig. 5 are required. 

 

Author Response

Comment 1:

In this paper the authors present FDTD analysis of a combination of Electromagnetic and Circuit models to simulate nonlinear properties of Schottky diode rectifier. The paper presents intertwining rigorous FDTD analysis of nonlinear active device. However, the title is misleading because the present analysis and present example are valid for microwave band, but they may not be directly suitable for THz band. Thus 'Terahertz' in the title or in other parts is misleading. 

Response:
We agree that the experimental validation is limited to the 5.8 GHz microwave prototype due to the lack of commercially available Schottky diodes at higher THz frequencies. To clarify, we have modified the text in the Introduction and Abstract (lines 48–57) to explicitly state that while simulations are scalable to the THz range, experimental validation is limited to the microwave band. The model uses frequency-dependent EM parameters and diode physics to extrapolate performance up to 100 GHz and potentially to THz, but direct fabrication is not feasible at this stage. This ensures the title reflects the model’s theoretical applicability rather than implying experimental verification at THz frequencies.

 

Comment 2:

Matrices in (19) and (20) are presented in a compact form which may be confusing for the reader. Also, in the second line of (20) all 0 should be . Also, in (20) the definitions of are not presented. It would be more suitable to expand both (19) and (20) and discuss all terms in detail. 

Response:
Equations (19) and (20) have been expanded and clarified in the revised manuscript (lines 185–200). The following changes were made:

• All zero vectors are explicitly defined as ​.
• The previously undefined terms are now clearly defined as the weighting coefficients associated with the edge-based FDTD discretization along the x, y, z directions, arising from the projection of the EM field onto the diode conduction path.
• Each term in the matrices is explained in the accompanying text to improve readability and avoid ambiguity for readers less familiar with compact notation.

 

Comment 3:

It is also required to add runtime comparison between FDTD and harmonic balance analysis.

Response:
Section 3.4 Computational Cost and Convergence (lines 360–390) now includes a runtime comparison:

• For the Half-Wave rectifier, the hybrid PDE–ODE model maintains a nearly constant runtime (~1 s) across –30 dBm to 30 dBm input, whereas full time-domain FDTD simulations range from 0.02 s to 0.1 s per step.
• For the Voltage Doubler, the hybrid model runtime varies between 1–10 s, compared to ~0.1 s for full FDTD.
• The comparison demonstrates that hybrid coupling of FDTD with ODEs achieves significant speed-up while preserving accuracy, whereas harmonic balance is used as a benchmark for input impedance and efficiency prediction (Section 3.3).

 

Comment 4:

 Details of the analysis of the matching network in Fig. 5 are required.

Response:
We have added details in Section 3.1 (lines 265–275):

• A T-network microstrip matching network was designed to optimize power transfer at 5.8 GHz.
• The network includes impedances of 18 Ω, 123 Ω, and 43 Ω, determined from ADS simulations of the rectifier input impedance.
• The design ensures maximum RF-to-DC conversion efficiency for the selected diode and load range (500 Ω–10 kΩ).
• These parameters were applied consistently in both simulation and experimental validation, providing a reliable basis for assessing the hybrid PDE–ODE model.

Reviewer 4 Report

Comments and Suggestions for Authors

In the manuscript, a nonlinear time-domain state-space model for Schottky di- 10 ode rectifiers is proposed, which can take the field-induced noise into account for accurate simulation. There are some merits in the manuscript. However, following issues should be addressed.

1. In the set-up model, the influences of the package, the incident microwave, and other sources, should be clarified.
2. Authors mentioned many times that the proposed model is more accuracy. How to demonstrate it?
3. Comparison with other published models should be made. For example, (1)Solid state Electronics, vol.154, 2019, 7-11; (2) IEEE Journal of solid-state circuits, 2009, 44(2): 354-370.
4. For the results in Fig.6, obvious discrepancy does exist between the measured and the simulated results. It doesn't match well. Authors should clarify the difference and the reason.
5. Fig.7 should be revised. Arrows should be used to clarify the efficiency and the output voltage in the figure.
6. Some typos should be revised. For example, Line 226 "This expression provides a compact, nonlinear mapping from to VG to VL".

Author Response

Comment 1:
In the set-up model, the influences of the package, the incident microwave, and other sources should be clarified.

Response:
We thank the reviewer for this suggestion. The revised manuscript clarifies these points:

Section 2.4 (lines 219–240) explicitly describes the embedding of the lumped diode within the EM field equations, including the derivation of the singular current source and the projection of diode voltage along its path, clarifying the diode–field coupling.

Section 2.1 “Assumption and Diode Modelling” (lines 150–180) discusses the contributions of packaging parasitic, incident microwaves, and other sources, specifying which physical effects are included or neglected with justification.

These additions ensure the model setup is transparent and reproducible.

 

Comment 2:
Authors mentioned many times that the proposed model is more accurate. How is this demonstrated?

Response:
We appreciate the reviewer’s comment. The revised manuscript now provides quantitative evidence of accuracy:

Sections 3.2–3.3 (lines 409–470, Figures. 6–7) comparisons with measurements and Ref. [8] show maximum voltage and efficiency deviations below 5%, confirming predictive fidelity.

Table 1 – lists model predictions versus prior works, highlighting improvements in PCE due to inclusion of reverse current, packaging, and EM–VI coupling.

These additions demonstrate the model’s superior predictive capability and practical relevance.

 

Comment: 3:
Comparison with other published models should be made (e.g., Solid State Electronics, 2019, 154:7–11; IEEE J. Solid-State Circuits, 2009, 44(2):354–370).

Response:
We thank the reviewer for these references. The revised manuscript addresses this through:

Section 3.1 and 2.1 – clarifying diode embedding in the EM field, including packaging parasitic and incident microwave contributions in the modelling setup.

Table 1 – provides a direct comparison with prior models, showing improvements in PCE at lower input powers and highlighting included physical effects.

Section 3.3 (lines 433–470, Figure 8) – benchmarking against HB simulations and Ref. [8], where the hybrid model achieves input-impedance deviations of 4–5% versus 12–15% for Ref. [8].

These revisions satisfy the request for comparative assessment and highlight the hybrid model’s novelty and practical advantages.

 

Comment: 4:
For the results in Fig. 6, an obvious discrepancy exists between measured and simulated results. Authors should clarify the difference and the reason.

Response:
We thank the reviewer. Section 3.2 (lines 409–432, Figure. 6) now quantifies errors:

Maximum absolute error: 4.5 mV

Maximum percent error: <5%

Deviations at very low input powers are explained by measurement uncertainty and instrumentation limits.

Comment: 5:
Figure. 7 should be revised. Arrows should be used to clarify efficiency and output voltage.

Response:
Figure 7 has been revised to improve clarity:

Clear axes and legends.

Error plots highlighted.

Arrows added to indicate efficiency and output voltage across multiple loads.

 

Comment 6:
Some typos should be corrected (e.g., Line 226: “This expression provides a compact, nonlinear mapping from to  to ”).

Response:
All typos have been corrected, including the mapping expression from ​ to ​.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have satisfactorily addressed all my comments and suggestions. The manuscript has been properly revised, and I recommend it for acceptance in its current form.

Author Response

Comment 1
The authors have satisfactorily addressed all my comments and suggestions. The manuscript has been properly revised, and I recommend it for acceptance in its current form.

Response
We sincerely thank the reviewer for the positive feedback and for recognizing the revisions made. We greatly appreciate your recommendation for acceptance.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors addressed all of my previous comments and concerns. Now I can recommend the paper for publication.  

Author Response

Comment 2
The authors addressed all of my previous comments and concerns. Now I can recommend the paper for publication.

Response
We sincerely thank the reviewer for the positive feedback and for recognizing the improvements in the revised manuscript. We greatly appreciate your support and recommendation for publication.

Reviewer 4 Report

Comments and Suggestions for Authors

I am glad to see some comments are addressed. The quality is improved. However, there are still some issues:

1. In the last review, two references are suggested to be compared. It seems authors didn't follow it. The references are not updated.
2. The influence of the load on the performance should be discussed. The 500-ohms-resistance is chosen in the manuscript. The other values should be considered. Also, the influence of the resistance on the performance should be discussed in more detail.
3. For the revised version, it is recommended to highlight the main revised part instead of all contents, which can be convenient for reviewer's reading.

Author Response

Reviewer 3

Comment 3.1
In the last review, two references are suggested to be compared. It seems authors didn't follow it. The references are not updated.

Response
We thank the reviewer for this valuable observation. Reference [9] evaluates rectifier performance using harmonic balance (HB) simulations in ADS. Our benchmarking in Sections 3.2 and 3.3 (lines 472–514) already employs ADS HB as the baseline, which directly corresponds to the methodology in Ref. [9]. To make this explicit, the revised manuscript now cites Ref. [9] in Section 3.3, Table 1, and the caption of Figure 8. These updates clarify that the proposed model is compared not only with Ref. [8] but also with the HB-based results reported in Ref. [9].

Comment 3.2
The influence of the load on the performance should be discussed. The 500-ohms resistance is chosen in the manuscript. The other values should be considered. Also, the influence of the resistance on the performance should be discussed in more detail.

Response
We thank the reviewer for this helpful comment. The 500-Ω load was selected for the fabricated rectifier as it represents a practical trade-off between output voltage and efficiency, influenced by the characteristics of the SMS7630 Schottky diode. Measurements at 500 Ω were used to extract EM and parasitic parameters, which were then combined with HB simulation data and the physics-based analytical model to form the hybrid PDE–ODE model.

To validate the hybrid model across different operating conditions, MATLAB simulations were performed with loads ranging from 500 Ω to 2000 Ω. These results demonstrate how output voltage and efficiency vary with load and confirm that the hybrid model accurately captures performance trends over a wide range of resistances. While fabricating the rectifier for all load values is impractical, the 500 Ω measurement serves as a representative validation point. This approach ensures both practical validation at 500 Ω and predictive capability across broader load conditions. The revised manuscript now explicitly explains this workflow in Section 3.1 (lines 385–424).

Comment 3.3
For the revised version, it is recommended to highlight the main revised part instead of all contents, which can be convenient for the reviewer’s reading.

Response
We thank the reviewer for this useful suggestion. In the revised manuscript, the main revised sections are highlighted in yellow to make the updates easily identifiable. Additionally, references have been updated where relevant. We hope this facilitates the review process.

Round 3

Reviewer 4 Report

Comments and Suggestions for Authors

It is OK to be accepted in its current version.

Author Response

Reviewer 1
Comment: It is OK to be accepted in its current version.

Response: We thank the reviewer for their positive assessment and supportive comment.