Research on Finite Permeability Semi-Analytical Harmonic Modeling Method for Maglev Planar Motors

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript contains unattributed reuse of text, equations, and figures from Yung et al. (1998), repeated formulas and text, methodological inconsistencies and unsupported claims. In its current form, it does not meet the standards required for publication.

Comments for author File: Comments.pdf

Author Response

Dear reviewer, the attachment is my response to your question. Please check it out.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript proposes a semi-analytic harmonic modeling method by integrating numerical and analytical approaches. The method reflects the characteristics of real magnets, providing an efficient and precise solution for complex magnetic field problems, particularly in the design of high-performance magnets. After reviewing the manuscript, following comments and questions has been raised to the authors:

1. It is recommended that the authors expand the introduction and literature review to include a more thorough analysis of the performance (in terms of both accuracy and efficiency) of current equivalent charge method and finite element method, so as to better contextualize and highlight the advancements made in this work.
2. In Figures 5-7, subfigure labels, such as (a), (b), (c), etc., should be clearly marked within the figures, and corresponding captions for each subfigure should be provided to improve readability. In addition, the Bx, By, and Bz should be marked in the figures.
3. The right figure in Figure 7, where is the minimum?
4. In Figure 8, what does “XY” refer to? Does it denote the X-axis, the Y-axis, or a combined representation?
5. For Figures 9 and 10, it is recommended to include simulation signals in the figures to facilitate direct comparison of their differences. Similarly, experimental results should be added to Table 2 for comprehensive comparison.

Author Response

Dear reviewer, the attachment is my response to your question. Please check it out.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper proposes a semi-analytical harmonic modeling method for planar maglev/Halbach arrays that mixes analytic harmonic expansions with Simpson-rule numerical integrals for cylindrical corrective magnets, and introduces a polynomial approximation iterative scheme to treat variable permeability. Simulation in COMSOL and a scanning Hall-sensor experiment are presented to compare the new Halbach arrangement against a prototype. The idea to combine analytic harmonic expansion with numerical correction to speed up modeling while keeping accuracy is promising, but the manuscript in its current form has some conceptual, methodological, and presentation problems which need substantial revision. The following statement express the main points to be improved.

Originality/Novelty:

• The paper lists prior works on planar actuators and Halbach arrays, but it fails to clearly describe what is new relative to the referenced literature. The manuscript repeatedly claims superiority over numerical and analytical methods but no provides a rigorous comparison in accuracy, scaling, or complexity to show its advantages. Authors must state precisely what part of the approach is novel?, and compare it with prior semi-analytic or hybrid methods.

Significance of Content:

• The application of the proposed model is significant, nevertheless, claims such as “breaks efficiency bottleneck” or “superior accuracy” are without foundation by rigorous timing/accuracy trade-off metrics. Authors present COMSOL as a tool to run and compare but do not give wall-clock times, CPU specs, mesh setting, convergence, or complexity scaling with number of magnets. Without these, the significance is speculative. 

• Experimental results are complex, and more in this kind of application. In the manuscript, experimental results are limited: no uncertainty analysis, no repeatability metrics, no statistical significance tests to show improvements are robust. This reduces the practical significance.

Scientific Soundness:

• Key derivations are sketched, but many intermediate steps are omitted, this provides opacity in the derivation.

• There is no a clear rule for truncating harmonics, polynomial approximation order for surface charge, or numerical Simpson integration. Practically, how many harmonics and what polynomial degree are needed for a given accuracy?

• The iterative mechanism for variable µr is described verbally and with a polynomial expansion, but there is no algorithm, convergence criterion, or evidence of convergence. Does it always converge? How many iterations? For what µr ranges?

The topic is relevant for the scientific community in precision positioning, maglev actuators, and magnetics modeling. However, the interest will depend on the rigor and reproducibility; as written, interested readers will find it hard to judge the method’s applicability due to missing quantitative comparisons and reproducibility details.

 

Author Response

Dear reviewer, the attachment is my response to your question. Please check it out.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised paper represents a substantially improved and ethically compliant version of the original.
It now:

1. Properly acknowledges Yung et al. (1998) and Boyer (1988),

2. Integrates multiple modeling layers (Halbach array> Dipole>Permeability>Semi-Analytical integration> Experimental validation),

3. Provides new simulation benchmarks supporting efficiency claims,

4. The paper did address the reviewers’ concerns.

Author Response

Thank you for your guidance. Please refer to the attachment for specific point-to-point answers.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have provided comprehensive and constructive responses to the reviewers' feedback. The manuscript now meets the publication standards of Magnetism in terms of novelty and rigor. It is recommended for acceptance after minor grammatical and formatting corrections.

Author Response

Thank you very much for your recognition and valuable feedback on this research manuscript! You pointed out that the author has provided a comprehensive and constructive response to the reviewer's comments, and the manuscript meets the publication standards of Magnetism in terms of innovation and rigor. It is recommended to accept it after minor grammar and formatting revisions. This evaluation not only affirms our previous revision work, but also points out a clear direction for future improvement. We are deeply honored and encouraged.

Next, we will immediately organize a team to conduct a comprehensive grammar check and standardize the format of the manuscript: on the one hand, we will check the language expression of the main text, chart titles, and references sentence by sentence, correct any possible grammar omissions, and ensure accurate and smooth expression; On the other hand, in accordance with the format requirements of the Magnetism journal, details such as chart numbering, formula layout, and reference entry format should be standardized to avoid format inconsistencies. We will complete all minor revisions with a rigorous and meticulous attitude, ensuring that the final presentation of the manuscript fully meets the publication standards of the journal and does not disappoint your recognition and expectations.

Once again, we sincerely thank the editor and reviewers for their attention and guidance on this study. Your professional opinions have provided important support for improving the quality of the manuscript. We are fully prepared and will complete the subsequent revisions and submit them as soon as possible. We look forward to the smooth publication of the manuscript in the journal Magnetism and hope that our research results can provide useful references for related research in the field of modeling magnetic levitation planar motors.

Reviewer 3 Report

Comments and Suggestions for Authors

After reviewing the cover letter and the revised version of the manuscript, I can confirm that the authors have adequately addressed my previous comments and concerns in an excellent manner. However, some methodological issues and minor text editing errors still need to be corrected before the final version is accepted.

Comments:

• Contradiction in efficiency comparison

Table 4 lists “Total time”: Equivalent charge method = 1.01 s, Proposed method = 1.45 s — i.e., the equivalent charge method is faster in that metric. Yet the text claims the equivalent charge method is less efficient. This is inconsistent.

• Insufficient error / convergence analysis for the harmonic truncation and Simpson numerical integration

The paper states a harmonic truncation which yields <0.5% error but provides no convergence plots or residual/error vs harmonic order. Also, Simpson’s rule is used for some integrals (semi-analytical step) without a quantification of numerical quadrature error or mesh/step size used. Authors can add (i) a convergence plot showing error vs harmonic order (k/l), (ii) sensitivity of results to Simpson step size / number of integration points, and (iii) justification of the quadrature accuracy relative to experimental uncertainty.

• Polynomial approximation

The authors approximate surface charge density by a polynomial (Eq. (21)) but do not: specify the polynomial degree n, show fit residuals, or quantify edge errors (they mention edge region errors qualitatively). Authors must state the methodology to choose n, show a table/plot of approximation error vs n.

• Experimental validation needs clearer uncertainty accounting

Hall sensor accuracy is quoted as ±1 mT, but reported mean absolute errors (e.g., 0.028 T = 28 mT) are much larger; the manuscript does not show confidence intervals, number of repeats per point. Also, the statement “maximum relative error ≥ 3.67%” is imprecise; relative to what baseline?

• Duplicate references and reference formatting errors

Refs. 8, 9, 10 appear duplicated (same title/volume repeated). Also, some references are missing journal names or have inconsistent formats. Inconsistent years / citations: Boyer is referenced as 1988 in text but 1998 in references. Fix all such inconsistencies.

Author Response

Thank you for your guidance. Please refer to the attachment for specific point-to-point answers.

Author Response File: Author Response.pdf