arXiv subjects

Ahmed Elwakil

Publications and source records attributed to Ahmed Elwakil.

7 recordsLinked to original sources

arXiv2026

Inversion of Electrochemical Immittance Spectra based on the Mellin Transform

In this work, we show that the Fredholm integral equations underlying the distribution of relaxation times (DRT), the distribution of capacitive times (DCT), and related frameworks share a common mathematical structure, namely that of a Mellin convolution. This comes from the fact that all standard immittance (impedance or admittance) kernels depend on the product $ωτ$ rather than on $ω$ and $τ$ independently. Exploiting this structure, we derive an exact algebraic inversion formula in Mellin space that converts the deconvolution problem into a closed-form relation between the Mellin transform of the measured immittance and that of the unknown distribution function. The framework is validated analytically on a set of examples including the constant phase element (CPE), the Davidson-Cole (DC) model, and the finite-length Warburg model with blocking boundary conditions. It is also validated numerically using the fast Mellin transform via the fast Fourier transform algorithm for both the CPE and the DC model, including their DRT and DCT recovery under clean and noisy conditions. The approach unifies the impedance- and admittance-based inversions under a single spectral framework, and provides a new approach for the characterization of electrochemical systems from immittance data.

physics.app-ph↗

arXiv2025

Energy-based interpretation of the dispersion coefficient of the constant phase element

The dispersion coefficient of the constant phase element (CPE) is typically treated as an empirical fitting parameter in the analysis of impedance spectroscopy data, with no clear physical meaning. Here we seek to establish a energy-based interpretation for this coefficient by linking it to the ratio of the dissipated or stored energy in the CPE relative to that supplied by the input source. Using the $RC$ network equivalency of a CPE, we decompose the total input energy into a contribution stored in the capacitive modes and another dissipated in the resistive modes. Analytical expressions are derived for three test examples: (i) a constant voltage, (ii) a voltage ramp, and (iii) a quadratic input of the form $v(t)=λt^2$. In all cases we found that the ratios of any two of these energy quantities reduce to pure functions of the dispersion coefficient of the CPE, independent of excitation amplitude or material parameters. This result provides a new perspective of the CPE's dispersion coefficient from a thermodynamic/energetic basis, with direct implications for supercapacitor characterization, battery modeling, as well as for the analysis of other electrochemical systems and devices exhibiting the CPE behavior.

physics.app-ph↗

arXiv2024

Procedure for Obtaining the Analytical Distribution Function of Relaxation Times for the Analysis of Impedance Spectra using the Fox $H$-function

The interpretation of electrochemical impedance spectroscopy data by fitting it to equivalent circuit models has been a standard method of analysis in electrochemistry. However, the inversion of the data from the frequency domain to a distribution function of relaxation times (DFRT) has gained considerable attention for impedance data analysis, as it can reveal more detailed information about the underlying electrochemical processes without requiring a priori knowledge. The focus of this paper is to provide a general procedure for obtaining analytically the DFRT from an impedance model, assuming an elemental Debye relaxation model as the kernel. The procedure consists of first representing the impedance function in terms of the Fox $H$-function, which possesses many useful properties particularly that its Laplace transform is again an $H$-function. From there the DFRT is obtained by two successive iterations of inverse Laplace transforms. In the passage, one can easily obtain an expression for the response function to a step excitation. The procedure is tested and verified on some known impedance models.

physics.chem-ph↗

arXiv2024

Generalized Distribution Function of Relaxation Times with the Davidson-Cole Model as a Kernel

In this paper we propose a generalized distribution function of relaxation times (DFRT) considering the Davidson-Cole model as an elementary process instead of the standard Debye model. The distribution function is retrieved from the inverse of the generalized Stieltjes transform expressed in terms of iterated Laplace transforms. We derive computable analytical expressions of the generalized DFRT for some of the most known normalized impedance (or admittance) models including the constant phase element, the Davidson-Cole, Havriliak-Negami and the Kohlrausch-Williams-Watts models.

physics.chem-ph↗

arXiv2023

Tikhonov regularization for the deconvolution of capacitance from voltage-charge response of electrochemical capacitors

The capacitance of capacitive energy storage devices can not be directly measured, but can be estimated from the input and output signals expressed in the time or frequency domains. Here the time-domain voltage-charge relationship in non-ideal electrochemical capacitors is treated as an ill-conditioned convolution integral equation where the unknown capacitance kernel function is to be found. This comes from assuming \emph{a priori} that in the frequency domain the charge is equal to the product of capacitance by voltage. The computation of a stable solution to this problem particularly when dealing with experimental data is highly sensitive to noise as it may lead to an oscillating output even in the presence of small errors in the measurements. In this work, the problem is treated using Tikhonov's regularization method, where a degree of damping is added to each singular value decomposition (SVD) component of the solution, thus effectively filtering out the components corresponding to the small singular values.

physics.app-ph↗

arXiv2023

Further Experimental Evidence of the Dead Matter Has Memory Conjecture in Capacitive Devices

This study provides new sets of experimental results supporting Westerlund's conjecture that Dead Matter Has Memory. Memory effects in the dynamic response of electric double-layer capacitors (EDLCs) that integrate its prior history of stimulation and state have been experimentally observed and reported in a few recent studies. The different excitation signals used to quantify such effects in these studies aimed at charging a device to the same voltage value and the exact same accumulated charge level but in different manners. Having reached the same unique voltage-charge point, it was observed that different yet repeatable discharge patterns occur, proving the existence of memory. The aim of this work is to provide further experimental evidence of the inherent memory effect in EDLCs in response to time-varying stationary input excitations with different statistical properties. In particular, different sets of charging voltage waveforms composed of fixed dc values with superimposed uniformly-distributed random fluctuations of different amplitudes were created and used to charge the same EDLC device to a unique voltage-charge point. The duration of these signals was the same but with different values of variance around the mean value. We observed different time-charge responses depending on the extent of the noise level in these charging waveforms. This is interpreted and discussed in the context of inherent memory using fractional-order voltage-charge equations of non-ideal capacitors.

physics.app-ph↗

arXiv2023

Time-domain Response of Supercapacitors using their Impedance Parameters and Fourier Series Decomposition of the Excitation Signal

Supercapacitors are mostly recognized for their high power density capabilities and fast response time when compared to secondary batteries. However, computing their power in response to a given excitation using the standard formulæ of capacitors is misleading and erroneous because supercapacitors are actually non-ideal capacitive devices that cannot be characterized with a single constant capacitance. In this study we show how to estimate accurately the time-domain power and energy of supercapacitors in response to any excitation signal represented in terms of its Fourier series coefficients with the sole knowledge of the frequency-domain impedance parameters of device. The presented theory is first verified and validated with simulations conducted on an equivalent fifth-order RC circuit emulating the behavior of a fractional circuit consisting of a resistance (Rs) in series with a constant phase element (CPE) of fractional impedance ZCPE = 1/Cαsα. Then we do the same for a commercial supercapacitor modeled as an Rs-CPE circuit, and subjected to both a periodic triangular voltage waveform and a random voltage excitation. The results are conclusive and very promising for adopting the proposed procedure to estimate the power and energy performance of supercapacitors in response to real-world charging and discharging signals.

physics.app-ph↗