### 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: Z_{f·FLW}(ω) = Z_{0} · tanh(ωτ_{0})^{n} · (ωτ_{0})^{−n}. τ_{0} 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 τ_{k} = τ_{0}/[π^{2}(k − ½)^{2}] with k = 1, 2, 3, … ∞. The mathematical surface areas of these δ-functions are proportional to τ_{k}. It is found that the FLW impedance can be simulated by an infinite series combination of parallel (R_{k}C_{0})-circuits, with R_{k} = C_{0}×τ_{k} ^{−1} and τ_{k} as defined above. R_{k} = 2τ_{k}×Z_{0} and C_{0} = 0.5×Z_{0} ^{−1}. Z_{0} 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 language | English |
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Pages (from-to) | 154-163 |

Number of pages | 10 |

Journal | Electrochimica acta |

Volume | 252 |

DOIs | |

Publication status | Published - 20 Oct 2017 |

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### Keywords

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

### Cite this

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**Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.** / Boukamp, Bernard A.

Research output: Contribution to journal › Article › Academic › peer-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

Y1 - 2017/10/20

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

KW - Finite Length Warburg

KW - Impedance Spectroscopy

KW - Simulation

UR - http://www.scopus.com/inward/record.url?scp=85028894571&partnerID=8YFLogxK

U2 - 10.1016/j.electacta.2017.08.154

DO - 10.1016/j.electacta.2017.08.154

M3 - Article

VL - 252

SP - 154

EP - 163

JO - Electrochimica acta

JF - Electrochimica acta

SN - 0013-4686

ER -