My Second Review for Sensors Journal

 The manuscript "Li, W.; Kewen, X.; Ling, W. Model and Experimental Study on Optical Fiber CT Based on Terfenol-D. Sensors 2020, 20, 2255" was reviewed by me for MDPI Sensors (ISSN 1424-8220) Journal.

This is an example of excellent experimental results with poor mathematical background

There are my Comments and Suggestions for Authors and their replies

Line 61. Figure 1. 1) It would be better to give description for BBS. 2) In the fragment of the optoelectronic circuit surrounded by the dashed line, the notation for two couplers "2x2" and "3x3" is used. It is not entirely clear why such notation is used, because judging by the figure, the left is "1x2" and the right is " 2x3 "

Reply: As your suggestions, there are more detail description for BBS (Broadband light source) has been given in part I. The notation of couplers in Figure 1 is incorrect in the original manuscript, The authors have corrected it in Figure 1 and the paper. Thank you for pointing out the errors.

In the work, lines 63–146 (83 lines in total) are devoted to the description of the mathematical model. I, as a person interested in this topic, really wanted to understand the essence of building a mathematical model. Unfortunately, I could not do this. In the description of the mathematical model there is no mathematical harmony and sequence of presentation.

Line 72. There is no link to the work where the described dependence (1) was taken from. It is clear that He is the effective magnetic field, but there is no description given for it. The (1) formula contains a partial derivative ∂λ/∂M, but there is no clear functional dependence for He(), H(), and M(). The λ as λ(M,σ,t) dependence is given in line 81. At this place t is used for denoting the temperature.

Reply: Based on Weiss molecular field theory and micromagnetism theory, Jiles and Atherton put forward the theoretical model of J-A ferromagnetism. In this model, the relationship between effective magnetic field He and non hysteresis magnetization M. Later researchers improved the model on the basis of experimental results, such as literature [14]. Because another expert put forward that the calculation result of this paper has no obvious improvement compared with the previous, I've modified the model according to recent literature. The modified model and simulation results are closer to the experimental results which can be seen from figure 4, especially in the low and medium magnetic fields.

Line 76. It is used Hσ,t denotation for fourth item in (1). May be it is a slip of the pen. Is it must be H σ,T? (because T is the temperature). It would be nice to use identical letter for temperature.

Reply: The denotation was incorrect in the original manuscript and we have modified and rectified the relevant content. The authors are grateful to the referees for pointing out the error.

line 81. Not enough evidence has been given to limit the empirical relationship between λ and M to just two terms (i=2).

Reply: Thank you, we have modified and rectified the relevant content in the revised manuscript.

Line 84. "In order to be simple and practical..." Not enough evidence has been given to limit Taylor formula just by line terms only.

Reply: Thank you, we have modified and rectified the relevant content in the revised manuscript.

Lines 95-98, formulas (5) - (8). The above relations are not entirely clear for people who are not familiar with the theory of micromagnetism and the J-A model. The concepts of Mirr and Mrev are introduced, but no interpretation of these terms is given. In line 104, it becomes clear that M is divided into two components: "irreversible" and "reversible", but as it follows from (5), (6), (7) and (8) they are completely incomprehensible. It is not clear why Man was introduced and what it means. Is Man some part of analytical solution, where is equation for it?

Reply: As your advices, We find that the order of the formulas (5) - (8) in the original paper was really a little unreasonable. So, the authors have adjusted the order of the four equations (Formula 3-6 in revised manuscript ) and added interpretations of , .

Line 107. It is said that "The change rate of magnetization can be solved by Newton iterative method, dM/dt..." but no any equation for it is given.

Reply: It was incorrect in the original manuscript. This part has been rectified, we have revised the to . The authors are grateful to the referees for pointing out the error.

Line 108. It is said that "The relationship between the strain of Tefenol-D material and the applied current or magnetic field can be obtained by (5) - (8) and (1) - (2)." I could not put these formulas into any equation. It would be nice if the authors present the conclusion of the final equation.

Reply: Thanks, in this paper, the (5) - (8) (Formula 3-6 in revised manuscript )were a set of iterative equations, so it is difficult to only present the conclusion of the final equation. The relationship between the strain the magnetic field could be only obtained by numerical calculation

Line 128, formula (10). There are no definitions for the V1, V2 and V3.

Reply: The authors have added the definitions of V1, V2 and V3 (， and are voltages of photodetector connected at output end of coupler).Thank you for pointing out the error.

Line 131. It is not clear why formula (11) is given. Suppose, knowing some values of V1, V2and V3 and incredible dimensionless parameters a, μ and γ (Line 132), we can calculate the changing of Δφ using formula (10), but how can it allow us to calculate the derivative d(Δφ)/d(λB). Why the (formula 11) generally is needed? Actually, the formula (10) and given in Line 130 formula are sufficient for λB calculation.

Reply: Formula(11) really doesn't need. We have deleted Formula(11) in the revised manuscript. Thank you for pointing out the error.

Line 137. It is completely unclear how to combine formulas (1), (2), (5)-(8) and (9), (11) leads to obtain (12).

Reply: In the revised paper, substitute formula (2) into formula (1), and then carry out iterative operation according to (3)-(6) , we can get M, and then calculate λ. We can get ΔλB according to formula (9), then Δφ. Maybe it is more clear in the revised paper. Thank you for pointing out the problem.

Line 138, formula (12). A new Δϕ (another drawing style ϕ instead φ) variable has been introduced, which is not defined.

Reply: It is a slip of the pen, and it has been rectified. Thank you for pointing out the error.

Line 139. It is not clear why the phrase is inserted here, that "The relation between and can be obtained by the relational expression (1) and (5) ~ (8). M H", since in (12) there is no explicit dependence from H.

Reply: The phrase Δφ inserted here is only to let readers understand the relationship between Δφ and λ directly. The authors have modified the expression of Δφ to

Bigger part of the work (lines 147-331, a total of 184 lines) is devoted to describing the experiment and analyzing the results. This part of the work looks more accurate.

Line 176. It is unclear the k variable and why it equals to zero.

Reply: k is a variable with an initial value of 0, and the k should be progressive increased 1 for each calculation until k=m.

Line 177. It would be better to give a whole equation system with the initial conditions.

Reply: This is an iterative process, it was difficult to described with a whole equation system, and the initial conditions and the iterative process are as follow: 

for k=0, H(0)=0, M(0)=0, then

He(1)= H (0)+……according to formula (1) ;

Man(k) is calculated by He(k) according to formula(3) ….. Successively calculate Mirr(k), then Mirr(k+1), Mrev(k+1), M(k+1),and so on.

Line 178. There is no description of how the M(k) depends on k and how the tk depends on k

Reply:

Mrev(k+1)= [ Man(k)+ Mirr(k) ],according to

M(k+1)=Mirr(k+1)+ Mrev(k+1), according to ……it is iterative calculation process.

Line 179. The relationship between λ and M is given by (2). How does it help us to obtain ε(k)?

Reply: Here, we consider that λ and ε are equal. And we have modified ε to λ. The authors are grateful to the referees for pointing out the error.

Line 181. How (9) can help to calculate FBG center wavelength variation without FBG center wavelength knowing?

Reply: When there is no strain, the initial center wavelength of FBG is 1550nm. The parameters of FBG was given in line 214(in the revised document)

Line 193. The "magnetostriction" word is bigger than others.

Reply: The authors have modified. The authors are grateful to the referees for pointing out the error.

The results of these experiments inspire and allow us to express moderate optimism in the development of this topic. At the same time, the dependencies shown in Figures 3 and 4 do not allow ensuring uniformity of the measurement scale.

Maybe it will be more reasonable to reduce the description of the mathematical model, limiting to the relationships that are used in the experiment with references to external works.

I wish success to authors in developing their measurement principle.

Reply: Due to limited experimental conditions, the measurement scale of Figures 3 and 4 do not uniformity. We will continue our efforts to improve the experimental conditions.

The authors have thought about to reduce the description of the mathematical model before, the reason why not reduced is to ensure the relative integrity of the research contents of the fiber current transformer system designed in this paper. Also, consider that the description of the mathematical model can facilitate readers to better read and understand the research content and results of this paper.

Thank you for your encouragement and guidance.