### 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 |

### Keywords

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