On the distributed resistor-constant phase element transmission line in a reflective bounded domain

Anis Allagui; Chunlei Wang; Enrique H. Balaguera

REVIEW 2 major objections 5 minor 40 references

On the distributed resistor-constant phase element transmission line in a reflective bounded domain

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The bounded resistor-CPE transmission line has exact impedance $s_n^{-\alpha/2}\coth(s_n^{\alpha/2})$, derived from the time-fractional diffusion equation.

desk verdict The time-domain solution is solid, but the impedance is already known and the DRT derivation has a correctable algebraic error. read the letter →

arxiv 2411.17368 v2 pith:UP6RQTHO submitted 2024-11-26 cond-mat.mtrl-sci physics.app-ph

Anis Allagui , Enrique H. Balaguera , Chunlei Wang This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.app-ph MSC 35R1126A3344A10

keywords transmissionlinemodelconstantphaseelementtime-fractionaldiffusionfinite-lengthWarburgimpedancegalvanostaticchargingMittag-Lefflerfunctiondistributionofrelaxationtimesporouselectrodes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives the exact impedance of a finite-length transmission line in which each unit length of an electrode pore carries a constant resistance in series with a constant-phase element (CPE) rather than an ideal capacitor. Solving the resulting time-fractional diffusion equation under galvanostatic charging, with a perfectly blocking end at $x=L$, gives the reduced impedance $Z_{\mathrm{TL}}(s_n)/(r_0 L)=s_n^{-\alpha/2}\coth(s_n^{\alpha/2})$. This is exactly the fractional modification of the reflective finite-length Warburg impedance that has long been used, on empirical grounds, to fit porous electrodes, supercapacitors, and insertion electrodes. The derivation supplies that empirical formula with a physical starting point, and it also yields the voltage and current step responses and an analytical distribution of relaxation times. If the derivation is right, one dispersion exponent $\alpha$ and the pore's resistance and length set the entire frequency- and time-domain response of a uniform blocking pore.

What carries the argument

The load-bearing object is the time-fractional diffusion equation for the voltage $v(x,t)$, with the Caputo fractional derivative of order $\alpha$ replacing the ordinary time derivative; the constant-phase element (CPE, impedance $1/(c_\alpha s^\alpha)$) is the energy-storage element per unit length. The equation is solved by applying a finite Fourier cosine transform in space and a Laplace transform in time, which turns the problem into algebraic equations whose inversion uses Mittag-Leffler inverse Laplace formulas. The impedance is then the ratio of the Laplace-transformed surface voltage to the Laplace-transformed input current, and the reflection at the blocked end $x=L$ produces the hyperbolic cotangent, while the fractional CPE law produces the exponent $\alpha/2$.

What would settle it

Take a well-characterized uniform single pore with a blocked far end, measure its impedance over a wide frequency range, and independently fix $\alpha$ from the low-frequency slope; if the full spectrum deviates from $s_n^{-\alpha/2}\coth(s_n^{\alpha/2})$, or if the low-frequency real-axis intercept of the normalized impedance is not $1/3$, the central derivation is falsified. A galvanostatic step measured on the same pore should also match the Mittag-Leffler series solution for the voltage at $x=0$.

Watch

Extended reading notes

Core claim

The central claim is that the impedance of a bounded, one-dimensional resistor-CPE transmission line is exactly $$\frac{Z_{\mathrm{TL}}(s_n)}{r_0 L}=$s_n^{{-\alpha/2}}$\coth\!\left($s_n^{{\alpha/2}}$\right),$$ with dimensionless frequency $s_n=s\left(r c_\alpha L^2\right)^{1/\alpha}$, obtained by solving the Caputo time-fractional diffusion equation $${}_0D_t^\$\alpha$ v=\frac{1}{r c_\$\alpha$}\frac{\$partial^{2}$ v}{\partial $x^{2}$},\quad 0<x<L,$$ with the galvanostatic boundary condition $-r_0^{-1}\partial_x v|_{x=0}=i_0$ and the blocking condition $\partial_x v|_{x=L}=0$. The same solution gives the voltage as an infinite Mittag-Leffler series, the step response through one further time derivative, and an RC relaxation-time distribution $g(\tau)$ such that the impedance is the integral of $g(\tau)/(1+s\tau)$. For $\alpha=1$ the formulas reduce to the classical reflective finite-length Warburg impedance of an ideal RC line, and the paper notes that this limit corrects missing terms in earlier galvanostatic solutions.

Load-bearing premise

The derivation assumes the electrode pore is a single uniform tube whose electrical resistance per unit length and whose capacitive response, described by a constant-phase element with a fixed exponent $\alpha$, do not vary along its length, and whose far end blocks all current; if any of these vary with position or state of charge, the exact impedance formula and time-domain solution no longer apply.

Editorial extensions

If this is right

The empirical fractional finite-length Warburg impedance now follows from a first-principles fractional diffusion equation, so porous-electrode fits using $s_n^{-\alpha/2}\coth(s_n^{\alpha/2})$ have a stated physical model behind them.
The same dispersion exponent $\alpha$ governs both the high-frequency half-order CPE branch and the low-frequency $s_n^{-\alpha}$ branch, linking the two regions of a Nyquist plot to one parameter.
The voltage solution gives the full galvanostatic charging curve, so chronopotentiometry and impedance spectroscopy can be analyzed with the same set of parameters $r$, $c_\alpha$, $L$, and $\alpha$.
The analytical distribution of relaxation times makes it possible to compute the RC time-constant spectrum of a bounded CPE pore directly, without numerical deconvolution schemes.
For $\alpha=1$, all derived formulas reduce to the classical reflective finite-length Warburg/RC transmission-line results, with the earlier ideal-capacitor galvanostatic solution corrected.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct cross-check the paper does not perform: fix $\alpha$ from the low-frequency CPE slope of a uniform pore, then predict the whole impedance and step response with no additional free parameters and compare; agreement would be a strong test of the derivation.
The same fractional diffusion equation with potentiostatic or permeable-end boundary conditions should produce analogous $\tanh$- or $\sinh$-type impedance forms, so the derivation likely extends beyond the perfectly blocking galvanostatic case.
The natural time scale $t/(r c_\alpha L^2)^{1/\alpha}$ predicts that galvanostatic voltage curves for pores of different lengths or electrolyte conductivities collapse onto one master curve when rescaled, a testable scaling law.
The model suggests interpreting $\alpha$ as a property of the distributed electrode/electrolyte interface rather than a pure fit parameter; one could test this by comparing $\alpha$ from impedance fits with $\alpha$ obtained from independent time-domain fractional-capacitance measurements on the same material.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The useful part of this paper is the time-domain solution. The authors solve the Caputo time-fractional diffusion equation for a finite-length R-CPE transmission line under galvanostatic charging and obtain the voltage series (Eq. 34), the current profile (Eq. 40), and the step response (Eq. 46). These are cleanly derived, the alpha=1 limit checks out, and the plots are sensible. If you work on impedance spectroscopy of porous electrodes, Eq. 46 is a practical closed form you might use.

The impedance result itself, Z(s_n)=s_n^{-alpha/2} coth(s_n^{alpha/2}), is not new. It is the anomalous finite-length Warburg impedance already derived by Bisquert and Compte (ref. 1), and the voltage series was given by Luchko (refs. 34-35). The paper cites these works, so it is honest, but the claim that this provides a 'physical derivation' of Cabanel et al.'s empirical form is overstated; that derivation already exists in the anomalous diffusion literature. What is new is framing it for the R-CPE transmission line and adding the time-domain formulas.

Two soft spots. First, the paper never states that r=r0. The boundary condition (Eq. 9) uses r0, but the current at x=0 from Eq. 40 is r0 i0/r. It equals i0 only if r=r0. The verification sentence mentions this in passing, but it should be a stated assumption in the model.

Second, and more serious: the DRT derivation in Section II-C contains an algebraic error. Substituting u=b^{-1/alpha}v into Eq. 51 and canceling gives the denominator b^2+2b v^alpha cos(alpha pi)+v^{2alpha}. Equation 52 instead has the denominator multiplied by b^{1/alpha-1}. So Eq. 53 does not follow from Eq. 51, and the 'excellent agreement' with Eq. 57 in Fig. 4(c) cannot hold as written. This is an internal inconsistency, not a disagreement with prior literature. The H-function expression (Eq. 57) may well be the correct DRT, but the Mittag-Leffler form (Eq. 53) is wrong.

Bottom line: the paper is worth sending to a referee because the time-domain solution is a legitimate contribution and the impedance derivation is correct. But the DRT section needs to be redone, and the r=r0 assumption needs to be explicit. As written, I would not cite Eq. 53, and I would treat Fig. 4(c)'s agreement claim with caution. With the algebra fixed, the paper would be a reasonable contribution to impedance spectroscopy.

Recommendation: send to peer review, major revision required.

Referee Report

2 major / 5 minor

Summary. The paper derives the analytical solution of the time-fractional diffusion equation governing a finite-length resistor-CPE transmission line under galvanostatic charging. The main claimed result is the reduced impedance Z_TL(s_n)/(r_0 L)=s_n^{-alpha/2} coth(s_n^{alpha/2}) (Eq. 44), which the authors present as a physical derivation of Cabanel et al.'s empirical fractional finite-length Warburg impedance. The paper also gives an infinite-series expression for the voltage, a step-response formula, and a distribution of relaxation times g(tau) claimed to correspond to this impedance.

Significance. If the central derivation is correct, the paper provides a useful physical justification for a widely used empirical impedance model and supplies explicit closed-form series for the time-domain response of porous electrodes. The impedance part is based on standard Laplace and finite Fourier cosine transform steps and is likely sound once the r0=r issue is resolved. The distribution-of-relaxation-times part, however, contains an algebraic error that invalidates Eq. (53) as written. The paper's contribution is therefore significant but needs a substantive correction before the DRT claims can be accepted.

major comments (2)

[Section II.C, Eq. (52)] The substitution stated in the text does not produce Eq. (52). Starting from Eq. (51) with x=b s^alpha and u=b^{-1/alpha}v gives b^{1-1/alpha}s^{alpha-1}E_{alpha,alpha}(-b s^alpha) = (sin(alpha pi)/pi) integral_0^infty [b v^alpha e^{-s v}]/[b^2+2b v^alpha cos(alpha pi)+v^{2alpha}] dv, so that s^{alpha-1}E_{alpha,alpha}(-b s^alpha) = (sin(alpha pi)/pi) integral_0^infty [b^{1/alpha} v^alpha e^{-s v}]/[b^2+2b v^alpha cos(alpha pi)+v^{2alpha}] dv. The denominator in Eq. (52) is b^{1/alpha-1} times this correct denominator, and the numerator lacks the b^{1/alpha} factor; the two expressions agree only when b=1. Consequently Eq. (53) is not the distribution of relaxation times corresponding to Eq. (44)/(46), and the agreement with Eq. (57) shown in Fig. 4(c) cannot validate Eq. (53). Please correct the substitution and Eq. (53), or remove the DRT claim.
[Section II.A, Eqs. (9), (39), (40)] The model introduces r0 in the boundary condition Eq. (9) while the current-voltage relation Eq. (39) uses r, and the two are never related. Combining Eq. (40) at x=0 with Eq. (9) gives i(0,t)=(r0/r)i0, so the physical current at the pore mouth equals the applied current i0 only when r0=r. Since the impedance in Eq. (41) is defined through i0/s, the derivation of Eqs. (43)-(44) implicitly requires this equality. The paper should state r0=r explicitly or use a single resistance-per-unit-length parameter throughout.

minor comments (5)

[Eq. (26)] For integer k, sin(k pi)=0, so the displayed expression can be simplified to the single term -r0 i0 L^2/(k^2 pi^2); presenting the simplification would make the k=0 case less opaque.
[Fig. 4 caption and text] The caption groups magnitude and phase in panel (a) and labels the Nyquist plot as (b) and the DRT plot as (c), while the text refers to the phase plot as Fig. 4(b); these panel labels should be aligned.
[Eq. (57)] The notation H_{p,q}^{m,n} is used without specifying p, q, m, and n; either define these parameters or explicitly state that the notation follows Ref. [39].
[Reference [38]] Reference [38] is a bare URL without a full bibliographic entry; please supply author, title, and access date.
[Title] The title as posted contains an OCR artifact ('transm ission') that should be corrected to 'transmission'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounded R-CPE impedance is derived from the stated fractional diffusion problem, not assumed or fitted.

full rationale

The claimed derivation is not circular. The load-bearing result, Eq. 44, is obtained by solving the time-fractional diffusion initial-boundary-value problem (Eqs. 8–11) with constant r and c_alpha, taking the Laplace transform of v(0,t), and dividing by i0/s; the CPE parameters (c_alpha, alpha) are inputs of the constitutive model, and the impedance is a function of them rather than a quantity used to define them. The experimental fit in Fig. 1 is motivational, not a fitted parameter relabeled as a prediction. References [15] and [39] are prior work by co-authors, but they are used for context, for a dimensionless rescaling, and for an auxiliary H-function representation of g(tau); the central bounded-domain solution and the impedance expression are derived in this paper from standard transforms and the externally cited Prabhakar formula and Mittag-Leffler integral representation. The skeptic's algebraic objection to Eq. 52 is a correctness concern about an intermediate substitution, not a circularity: an incorrect denominator would make Eq. 53 wrong, but it would not make the result equivalent to its inputs. The uniformity assumption r,c_alpha = const is a modeling limitation, not a circular definition. Therefore, no circular step is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the CPE constitutive law (Eqs. 4-6) which is assumed, not derived. The mathematical machinery (Caputo derivative, Laplace/Fourier transforms, Mittag-Leffler functions) is standard. No new entities are invented. The free parameters alpha, c_alpha, r, L are model inputs, with alpha and the time constant fitted to the single supercapacitor dataset in Fig. 1.

free parameters (3)

alpha (dispersion coefficient) = 0.94 (from supercapacitor fit in Fig. 1)
The CPE order alpha is the key parameter in the derived impedance s^{-alpha/2} coth(s^{alpha/2}); it is not derived from first principles but is an empirical input of the CPE model, fitted to data in the experimental section.
c_alpha (pseudocapacitance per unit length) = 0.56 F s^{alpha-1} (Rs-CPE fit) or from Eq. 2 fit
Scales the fractional diffusion coefficient; the ratio r c_alpha L^2 enters the normalized frequency s_n.
r (electrolyte resistance per unit length) = Derived as R_d/L in fits, not independently measured
Appears in the PDE and in the time constant r c_alpha L^2; its value is taken from fitting, not measured separately.

assumptions (5)

domain assumption The energy storage element is a CPE with impedance z_c(s) = 1/(c_alpha s^alpha) with constant (c_alpha, alpha)
This is the defining physical model (Eqs. 4-6); it is not derived and is the input that produces the fractional order.
standard math Caputo fractional derivative and its Laplace transform property L[D^alpha f] = s^alpha F(s) - s^{alpha-1} f(0)
Used throughout; standard in fractional calculus.
domain assumption The pore is homogeneous and one-dimensional; no Faradaic processes; zero initial voltage
Eqs. 8 and 11; this restricts applicability.
domain assumption Zero current at x=L (reflective boundary)
Eq. 10; this defines the bounded reflective domain.
standard math The series representation of coth(z) and the Prabhakar/Mittag-Leffler Laplace inversion formulas
Used to convert the series solution into the closed-form impedance (Eqs. 42-44).

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Pith. "Pith review of On the distributed resistor-constant phase element transmission line in a reflective bounded domain." pith.science (2026). https://pith.science/paper/UP6RQTHO

@misc{pith2026241117368,
  author       = {Pith},
  title        = {Pith review of: On the distributed resistor-constant phase element transmission line in a reflective bounded domain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UP6RQTHO}},
  note         = {Machine review of arXiv:2411.17368}
}

abstract

In this work we derive and study the analytical solution of the voltage and current diffusion equation for the case of a finite-length resistor-constant phase element (CPE) transmission line (TL) network that can represent a model for porous electrodes in the absence of any Faradic processes. The energy storage component is considered to be an elemental CPE per unit length of impedance $z_c(s)={1}/{(c_{\alpha} s^{\alpha})}$ with constant parameters $(c_{\alpha},\alpha)$ instead of the ideal capacitor of impedance $z(s)={1}/{(c\, s)}$ usually assumed in TL modeling. The problem becomes a time-fractional diffusion equation for the voltage that we solve under galvanostatic charging, and derive from it a reduced impedance function of the form $z_{\alpha}(s_n)=s_n^{-\alpha/2}\coth({s_n^{\alpha/2}})$, where $s_n = j\omega_n$ is a normalized frequency. We also derive the system's step response, and the distribution function of relaxation times associated with it. The analysis can be viewed and used as a support for the fractal finite-length Warburg model.

Figures

Figures reproduced from arXiv: 2411.17368 by the authors.

**Figure 2.** FIG. 2. Schematic of an electrified pore of a porous electrode [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Time-domain voltage and current response of an [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Plot of the impedance function given by Eq. 44 for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bisquert and A

J. Bisquert and A. Compte, Theory of the electrochem- ical impedance of anomalous diﬀusion, . Electroanal. Chem. 499, 112 (2001)

work page 2001

[2] [2]

Bisquert, G

J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, N. S. Ferriols, P. Bogdanoﬀ, and E. C. Pereira, Doubling exponent models for the analysis of porous ﬁlm electrodes 9 by impedance. relaxation of tio2 nanoporous in aqueous solution, J. Phys. Chem. B 104, 2287 (2000)

work page 2000

[3] [3]

Song and M

J. Song and M. Z. Bazant, Electrochemical impedance imaging via the distribution of diﬀusion times, Phys. Rev. Lett. 120, 116001 (2018)

work page 2018

[4] [4]

A. A. Moya, Low-frequency approximations to the ﬁnite- length warburg diﬀusion impedance: The reﬂexive case, J. Energy Storage 97, 112911 (2024)

work page 2024

[5] [5]

D. R. Franceschetti and J. R. Macdonald, Diﬀusion of neutral and charged species under small-signal ac con- ditions, J. Electroanal. Chem. Interfacial Electrochem. 101, 307 (1979)

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