Response to Comment on Origin of the Curievon Schweidler law and the fractional capacitor from time-varying capacitance J. Power Sources 532 2022 231309

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arXiv:2210.13209v2 [cond-mat.mtrl-sci] 26 Oct 2022

Response to “Comment on ‘Origin of the Curie–von Schweidler law and

the fractional capacitor from time-varying capacitance

[J. Power Sources 532 (2022) 231309]’ ”

Vikash Pandey

School of Interwoven Arts and Sciences, Krea University, Sri City, India

Abstract

We welcome Allagui et al.’s discussions about our recent paper that has proposed revisions to the

existing theory of capacitors. It gives us an opportunity to emphasize on the physical underpinnings

of the mathematical relations that are relevant for modeling using fractional derivatives. The concerns

raised by Allagui et al. are found to be quite questionable when examined in light of the established

standard results of fractional calculus. Consequently, the inferences that they have drawn are not true.

Finally, we would like to thank Allagui et al. for their Comment because this subsequent Response has

actually led to a further consolidation of our results that are supposed to be signiﬁcant for materials

science as well as for fractional control systems and engineering.

Keywords: Curie–von Schweidler law, universal dielectric response, fractional capacitor, fractional

calculus, power-laws, memory

2010 MSC: 26A33, 94C60

NOTE: The peer-reviewed version of this Response is published in the Journal of Power

Sources, Vol. 551, 232167, 2022.

DOI: https://doi.org/10.1016/j.jpowsour.2022.232167.

The published version of the manuscript is available online at

https://www.sciencedirect.com/science/article/pii/S0378775322011442.

This document is an e-print which may diﬀer in, e.g. pagination, referencing styles,

ﬁgure sizes, and typographic details.

Email address: vikash.pandey@krea.edu.in (Vikash Pandey)

Preprint submitted to Jounal of Power Sources October 27, 2022

In our recent publication [1], we gave a physical interpretation to the century-old Curie–von Schwei-

dler law of dielectrics which interestingly may also be seen as an electrical analogue of Nutting’s law

from rheology [2, 3]. The derivation of the Curie–von Schweidler law required a revision of the classical

charge–voltage relation of capacitors, Q(t) = C0V(t), where Qis the charge, C0is the constant capac-

itance of the capacitor, Vis the voltage, and tis the time. It is worthwhile to note that the classical

relation is more than two-centuries old [4] and it was originally proposed for a constant capacitance

capacitor. In contrast most capacitors of practical applications exhibit a time-varying capacitance

but in lack of a better relation the same-old classical relation has been inadvertently used. In order

to overcome this lacking we proposed a new charge–voltage relation which is equally applicable for a

constant capacitance capacitor as well as for a capacitor with a time-varying capacitance. The relation

is expressed by Eq. (9) in our paper as [1]:

Q(t) = C(t)∗˙

V(t),(1)

where ∗represents the convolution operation and C(t) is the time-varying capacitance of the capacitor.

In their Comment on our paper, Allagui et al. [5] have raised certain concerns with the use of the

convolution integral of linear time-varying capacitance with the time-derivative of voltage. In this

Response, for the sake of clarity, completeness, and correctness, we prove that each of their observations

and the inferences that they have drawn from them are quite questionable and therefore not valid.

1. According to Allagui et al. capacitances being added in parallel is a frequency domain assertion,

and that they cannot be added in the time domain [5]. Moreover, they have the opinion that

circuit synthesis is not done in the time domain.

Response: We would like to emphasize that as long as the linearity of the system holds, circuit

synthesis could be done in the time domain as well as in the frequency domain. We transform

a problem from one domain to another domain depending upon the domain in which ﬁnding a

solution to the problem is relatively easier. For example, modeling a digital ﬁlter is often easier

in the frequency domain as it facilitates a convenient framework for the mathematical analysis

of its properties. In contrast, a sampling process is meaningful in the time domain. The two

domains should be seen as the two diﬀerent viewpoints from which to understand the problem.

It is also worthwhile to respect the fact that the time domain is the domain of physical reality

and it is probably more intuitive than the frequency domain.

It has already been shown that our expression for current matches exactly with the predictions

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arXiv:2210.13209v2[cond-mat.mtrl-sci]26Oct2022Responseto“Commenton‘OriginoftheCurie–vonSchweidlerlawandthefractionalcapacitorfromtime-varyingcapacitance[J.PowerSources532(2022)231309]’”VikashPandeySchoolofInterwovenArtsandSciences,KreaUniversity,SriCity,IndiaAbstractWewelcomeAllaguietal.’sdiscussion...

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Response to Comment on Origin of the Curievon Schweidler law and the fractional capacitor from time-varying capacitance J. Power Sources 532 2022 231309

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