Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.

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Abstract

An analytic Distribution Function of Relaxation Times (DFRT) is derived for the fractal Finite Length Warburg (f-FLW, also called ‘Generalized FLW’) with impedance expression: Zf·FLW(ω) = Z0 · tanh(ωτ0)n · (ωτ0)−n. τ0 is the characteristic time constant of the f-FLW. Analysis shows that for n → 0.5 (i.e. the ideal FLW) the DFRT transforms into an infinite series of δ-functions that appear in the τ-domain at positions given by τk = τ0/[π2(k − ½)2] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τk. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with Rk = C0×τk −1 and τk as defined above. Rk = 2τk×Z0 and C0 = 0.5×Z0 −1. Z0 is the dc-resistance value of the FLW. A full analysis of these DFRT expressions is presented and compared with impedance inversion techniques based on Tikhonov regularization and multi-(RQ) CNLS-fits (m(RQ)fit). Transformation of simple m(RQ)fits provide a reasonably close presentation in τ-space of the f-FLW, clearly showing the first two major peaks. Impedance reconstructions from both the Tikhonov and m(RQ)fit derived DFRT's show a close match to the original data.

Original languageEnglish
Pages (from-to)154-163
Number of pages10
JournalElectrochimica acta
Volume252
DOIs
Publication statusPublished - 20 Oct 2017

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Fractals
Relaxation time
Distribution functions
Networks (circuits)

Keywords

  • Distribution Function of Relaxation Times
  • Finite Length Warburg
  • Impedance Spectroscopy
  • Simulation

Cite this

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title = "Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.",
abstract = "An analytic Distribution Function of Relaxation Times (DFRT) is derived for the fractal Finite Length Warburg (f-FLW, also called ‘Generalized FLW’) with impedance expression: Zf·FLW(ω) = Z0 · tanh(ωτ0)n · (ωτ0)−n. τ0 is the characteristic time constant of the f-FLW. Analysis shows that for n → 0.5 (i.e. the ideal FLW) the DFRT transforms into an infinite series of δ-functions that appear in the τ-domain at positions given by τk = τ0/[π2(k − ½)2] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τk. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with Rk = C0×τk −1 and τk as defined above. Rk = 2τk×Z0 and C0 = 0.5×Z0 −1. Z0 is the dc-resistance value of the FLW. A full analysis of these DFRT expressions is presented and compared with impedance inversion techniques based on Tikhonov regularization and multi-(RQ) CNLS-fits (m(RQ)fit). Transformation of simple m(RQ)fits provide a reasonably close presentation in τ-space of the f-FLW, clearly showing the first two major peaks. Impedance reconstructions from both the Tikhonov and m(RQ)fit derived DFRT's show a close match to the original data.",
keywords = "Distribution Function of Relaxation Times, Finite Length Warburg, Impedance Spectroscopy, Simulation",
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Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg. / Boukamp, Bernard A.

In: Electrochimica acta, Vol. 252, 20.10.2017, p. 154-163.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.

AU - Boukamp, Bernard A.

PY - 2017/10/20

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N2 - An analytic Distribution Function of Relaxation Times (DFRT) is derived for the fractal Finite Length Warburg (f-FLW, also called ‘Generalized FLW’) with impedance expression: Zf·FLW(ω) = Z0 · tanh(ωτ0)n · (ωτ0)−n. τ0 is the characteristic time constant of the f-FLW. Analysis shows that for n → 0.5 (i.e. the ideal FLW) the DFRT transforms into an infinite series of δ-functions that appear in the τ-domain at positions given by τk = τ0/[π2(k − ½)2] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τk. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with Rk = C0×τk −1 and τk as defined above. Rk = 2τk×Z0 and C0 = 0.5×Z0 −1. Z0 is the dc-resistance value of the FLW. A full analysis of these DFRT expressions is presented and compared with impedance inversion techniques based on Tikhonov regularization and multi-(RQ) CNLS-fits (m(RQ)fit). Transformation of simple m(RQ)fits provide a reasonably close presentation in τ-space of the f-FLW, clearly showing the first two major peaks. Impedance reconstructions from both the Tikhonov and m(RQ)fit derived DFRT's show a close match to the original data.

AB - An analytic Distribution Function of Relaxation Times (DFRT) is derived for the fractal Finite Length Warburg (f-FLW, also called ‘Generalized FLW’) with impedance expression: Zf·FLW(ω) = Z0 · tanh(ωτ0)n · (ωτ0)−n. τ0 is the characteristic time constant of the f-FLW. Analysis shows that for n → 0.5 (i.e. the ideal FLW) the DFRT transforms into an infinite series of δ-functions that appear in the τ-domain at positions given by τk = τ0/[π2(k − ½)2] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τk. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (RkC0)-circuits, with Rk = C0×τk −1 and τk as defined above. Rk = 2τk×Z0 and C0 = 0.5×Z0 −1. Z0 is the dc-resistance value of the FLW. A full analysis of these DFRT expressions is presented and compared with impedance inversion techniques based on Tikhonov regularization and multi-(RQ) CNLS-fits (m(RQ)fit). Transformation of simple m(RQ)fits provide a reasonably close presentation in τ-space of the f-FLW, clearly showing the first two major peaks. Impedance reconstructions from both the Tikhonov and m(RQ)fit derived DFRT's show a close match to the original data.

KW - Distribution Function of Relaxation Times

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KW - Impedance Spectroscopy

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U2 - 10.1016/j.electacta.2017.08.154

DO - 10.1016/j.electacta.2017.08.154

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JO - Electrochimica acta

JF - Electrochimica acta

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