Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Comments and Suggestions for Authors 
This manuscript presents a comprehensive and rigorous framework for analyzing 
electrochemical impedance spectroscopy (EIS) data using multiple equivalent circuit models. 
The methodology is thorough, combining analytical expressions, hybrid optimization, 
statistical validation, and automated Kramers–Kronig checks. The study is timely and relevant, 
and the quantitative approach is well executed. However, a few areas could be clarified or 
refined to further improve readability, consistency, and the practical impact of the work. Please 
consider the following suggestions: 
1. Comment on Abbreviations: 
The manuscript uses several abbreviations (e.g., Cdl, Rct, Rs, CPE, ZW, EIS, AIC, BIC). It 
would improve clarity if each abbreviation were defined the first time it appears. Consistently 
using the same notation throughout the paper will also help readers follow the text more easily. 
2. Comment on Equations: 
Please ensure that all variables and symbols used in equations are clearly defined at their first 
occurrence. This will make the equations self-contained and easier to follow for readers. 
3. Reviewer Comment on Results Section: 
The Results are clear, well-organized, and backed by extensive data in Figures 1–35 and Tables 
1–8. The authors provide useful quantitative metrics (RMSE, R², AIC/BIC, confidence 
intervals, condition numbers) and include residuals and Kramers–Kronig checks, which 
strengthens the reliability of the fits. 
A few minor improvements could make the section even clearer: 
1. Use precise figure references (e.g., Fig. 6a–b) instead of “left/right panels.” 
2. Clarify whether residuals are raw or normalized. 
3. Provide a numerical tolerance for the Kramers–Kronig validation. 
The results convincingly show that simpler models perform well at low noise, while partial 
models like Randles+CPE, (Rct+ZW)∥CPE, and Randles+Warburg offer better stability and 
fidelity under higher noise or non-ideal conditions. The full Randles model captures both non
ideality and diffusion but is only justified when the spectrum clearly exhibits both. 
4. Reviewer Comment on Conclusions: 
The Conclusions clearly summarize the work and highlight the main achievements, including 
the robust framework, accurate parameter estimation, and use of CPE/Warburg elements for 
non-ideal systems. Automated Kramers–Kronig checks are a strong point. 
Minor suggestions: 
1. Briefly mention practical implications for experiments or industry. 
2. Add one sentence on when simpler vs. more complex models are preferred.

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 1

1. Abbreviations

Comment: Please define abbreviations (Cdl, Rct, Rs, CPE, ZW, EIS, AIC, BIC) at first occurrence and ensure consistency.

Response: I have defined all requested abbreviations at their first occurrence and standardized the notation throughout the manuscript. Electrochemical parameters are now consistently written with subscripts: C_dl (double-layer capacitance), R_ct (charge-transfer resistance), R_s (solution resistance), and Z_W (Warburg impedance). I also introduced all acronyms at first use, including EIS, CPE, Z_W, DE, LM, RMSE, AIC, BIC, KK, LTI, ECM, and RLC.

2. Comment on Equations

Comment: Please ensure that all variables and symbols used in equations are clearly defined at their first occurrence. This will make the equations self-contained and easier to follow for readers.

Response: All symbols are now defined at first use. The specific edits in the manuscript are:

• Parallel combination equation: after the formula for the equivalent impedance of N parallel branches, I appended: “where N is the number of parallel branches and Z_k is the impedance of branch k.”
• Charge-transfer resistance definition: I replaced the kinetic n by n_e and explicitly defined all constants: R (Jmol⁻¹ K⁻¹), T (K), n_e (dimensionless number of electrons), F (Cmol⁻¹), A (m²), and j₀ (Am⁻²). Consistency note: throughout this subsection, every kinetic “n” was changed to “n_e” to avoid conflict with the CPE exponent n.
• Constant-phase element (CPE): I added “with Q in Fsⁿ⁻¹ and 0 < n < ”
• Warburg impedance: I added “here A_W has units Ωs^−1/2.”
• Randles (characteristic frequency): I reported the units explicitly as ω_max = 1/(R_ctC_dl) in rads⁻¹ and f_max = ω_max/(2π) in Hz.

3. Results Section

Comment 1: Improve clarity by: (i) using precise figure references (e.g., Fig. 6a–b)

Response: I appreciate the Reviewer’s suggestion. I have revised the manuscript to replace all generic panel references (e.g., “left/right panel”, “upper/lower panel”) with explicit subfigure citations throughout the text and captions. For consistency, I now use the form “Fig. Xa–b”, “Figs. Xa–b and Xc–d”, etc., aligned with the journal style. This change has been applied across the Results section and all figure captions.

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Comment 2: Clarify whether residuals are raw or normalized.

Response: Thank you for this observation. All residuals reported in this work are raw complex residuals, computed as the direct difference between the experimental and fitted impedance, r_i = Z_exp(ω_i) − Z_fit(ω_i), without normalization by |Z| or by the noise level. This choice avoids altering the statistical distribution of errors and ensures direct comparability across models. We have explicitly clarified this point in the caption of the residual figures (Figs. ??, ??, ??, etc.) and in the Results section.

Comment 3:. Provide a numerical tolerance for the Kramers–Kronig validation. The results convincingly show that simpler models perform well at low noise, while partial models like Randles+CPE, (Rct+ZW)CPE, and Randles+Warburg offer better stability and fidelity under higher noise or non-ideal conditions. The full Randles model captures both non ideality and diffusion but is only justified when the spectrum clearly exhibits both.

Response: In all cases, the Kramers–Kronig (KK) validation was performed via Hilbert transform of Re(Z), and the reconstructed imaginary part was compared to the measured Z^′′(ω). The validation successful when the relative deviation satisfied

|ZKK′′ (ω) − Z′′(ω)|

                                                                < 5% ∀ω,

|Z(ω)|

which is a commonly used tolerance threshold in impedance spectroscopy. This tolerance has now been explicitly stated in the Methodology section and in the figure captions for the KK validation.

Comment on Conclusions: (1) Briefly mention practical implications for experiments or industry. (2) Add one sentence on when simpler vs. more complex models are preferred.

Response: Thank you for these constructive suggestions. In the revised Conclusions, I have added: (i) a brief statement on the practical implications of the proposed framework for experimental electrochemistry and industrial diagnostics, and (ii) a sentence clarifying under which conditions simpler versus more complex equivalent circuit models are recommended. These additions strengthen the applicability and readability of the Conclusions without altering their original structure.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript describes an analytical–computational framework for electrochemical impedance spectroscopy (EIS) that combines closed-form impedance derivations for seven most frequently used equivalent circuits with a hybrid fitting approach. The methodology incorporates, among others, bootstrap-based uncertainty estimates and Kramers–Kronig validation. Model performance is assessed on synthetic spectra with added random noise to identify the conditions under which different circuit architectures are most applicable.

The paper has several notable strengths. First, the seven equivalent circuits most commonly used in EIS are clearly derived, and the scope of their application to real systems is well described, which will be especially useful for readers new to the field. Second, the proposed methodology provides an objective procedure for model selection while simultaneously checking the self-consistency of the data, thereby reducing the subjectivity that often affects EIS interpretation.

As a pedagogical suggestion, the authors could consider including in the Introduction a brief (2–3 sentences) discussion comparing the phenomenological models described in the paper with physics-based EIS models derived from transport and kinetic equations. This would help readers, particularly novices in the field of EIS, to understand the limitations of interpreting parameters from phenomenological models, which are often over-interpreted in the literature, and would enhance the educational value of the paper.

One significant limitation of the paper is that, apart from a few very general descriptions of the Python computational platform used for simulation, fitting, and validation of EIS data, no details are provided regarding the software architecture or the algorithms implemented. Based on my more than 30 years of experience in writing code for simulation and analysis of scientific data, this lack of information makes it impossible for readers to fully assess the reproducibility of the published results, critically evaluate the analytical methodology, or implement their own code based on the described approach. While the authors note that “the datasets and code are available from the corresponding author on reasonable request,” this statement is problematic, as access depends entirely on the authors’ discretion and does not align with current open science standards.

To address this issue, it is strongly recommended that the authors either make the code publicly available (in line with contemporary open science practices, particularly those promoted by the MDPI publisher) or provide a detailed description of the software architecture and computational procedures in the Supplementary Information, including references for any standard numerical methods used. Without such transparency, the paper’s scientific value is substantially limited, and it risks being perceived more as a description or advertisement of the authors’ software rather than a contribution to reproducible scientific methodology.

Overall, the manuscript presents a methodologically sound and potentially useful framework for EIS analysis, with clear pedagogical value and well-structured evaluation of different equivalent circuits. However, the lack of transparency regarding the computational implementation and algorithms significantly limits reproducibility and scientific credibility. Major revisions are required to either make the code publicly available or provide detailed description of the software architecture and computational procedures. In my opinion, addressing these points is essential for the manuscript to meet the standards required for publication.

Author Response

Response to Reviewer 2

I sincerely thank the reviewer for the time and effort devoted to evaluating my manuscript. The comments have been very helpful in improving both the clarity and the scientific rigor of the work. Below I provide a point-by-point response to the reviewer’s observations. All modifications have been incorporated into the revised manuscript, as detailed below.

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Reviewer Comment 1

As a pedagogical suggestion, the authors could consider including in the Introduction a

brief (2–3 sentences) discussion comparing the phenomenological models described in the paper with physics-based EIS models derived from transport and kinetic equations. This would help readers, particularly novices in the field of EIS, to understand the limitations of interpreting parameters from phenomenological models, which are often over-interpreted in the literature, and would enhance the educational value of the paper.

Author Response

I thank the reviewer for this insightful suggestion. In the revised manuscript, a new paragraph has been added to the Introduction to highlight the distinction between phenomenological equivalent-circuit models and physics-based approaches. The added text reads:

“Although equivalent-circuit models offer a convenient and widely adopted phenomenological representation of impedance data, they differ from physicsbased approaches derived from transport and kinetic equations. While the latter provide parameters with a direct mechanistic interpretation, equivalent circuits mainly capture effective behaviors and may lead to over-interpretation if treated as physical constants. This distinction highlights the pedagogical value of our framework: it clarifies the scope and limitations of phenomenological modeling while offering objective criteria for model selection.”

This addition clarifies the scope of the study and enhances its pedagogical value, as suggested.

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Reviewer Comment 2

One significant limitation of the paper is that, apart from a few very general descriptions of the Python computational platform used for simulation, fitting, and validation of EIS data, no details are provided regarding the software architecture or the algorithms implemented.

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...... Major revisions are required to either make the code publicly available or provide detailed description of the software architecture and computational procedures.

Author Response

I fully agree with the reviewer that transparency and reproducibility are essential. In response, I have made the complete Python implementation openly available through Zenodo, which provides a permanent DOI and ensures long-term accessibility. The Data Availability section of the manuscript has been updated to read:

“The full Python implementation (eis.py) is openly available at Zenodo (https://doi.org/10.5281/zenodo.17109696). This ensures reproducibility and facilitates adaptation of the methodology by other researchers. Further clarifications regarding the implementation can be obtained by contacting the corresponding author at the email address provided in the manuscript.”

This update guarantees full compliance with open science standards and allows independent verification and reuse of the proposed framework.

Sincerely,

Francisco Augusto Núñez Pérez

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

In the revised manuscript, the authors have implemented all of the revisions suggested by the referee. By making their code publicly available on Zenodo, they have significantly enhanced the transparency, scientific credibility and long-term value of their work.
I recommend acceptance of the manuscript.

Author Response

Response to Reviewer 2

I sincerely thank the reviewer for the time and effort devoted to evaluating my manuscript. I greatly appreciate the very positive assessment and the recommendation for acceptance. The comments acknowledging the improvements in transparency and reproducibility, particularly through the public release of the code, are highly encouraging. Since no further corrections were requested, only minor editorial polishing was performed to ensure clarity and consistency.

 

Reviewer Comment:
In the revised manuscript, the authors have implemented all of the revisions suggested by the referee. By making their code publicly available on Zenodo, they have significantly enhanced the transparency, scientific credibility and long-term value of their work. I recommend acceptance of the manuscript.

Response
I sincerely thank the reviewer for this very positive evaluation and recommendation of acceptance. I am grateful for the recognition of my efforts to improve transparency, scientific rigor, and reproducibility by making the complete code openly available through Zenodo. As no additional revisions were requested, only minor editorial polishing was carried out.

 

Sincerely,
Francisco Augusto Núñez Pérez

Author Response File: Author Response.pdf