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Analyzing results of impedance spectroscopy using novel evolutionary programming techniques

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Abstract

This paper discusses the application of evolutionary programming methods to the problem of analyzing impedance spectroscopy results. The basic approach is a “direct-problem” one, i.e., to find a time constant distribution function that would create similar impedance results as the measured ones, within experimental error. Two complementary methods have been applied and are discussed here: Genetic Algorithm (GA) and Genetic Programming (GP). A GA can be applied when a known (or desired) model exists, whereas GP can be used to create new models where the only a-priori knowledge is their smoothness and their non-negativity. GP is tuned to prefer relatively non-complex models through penalization of unnecessary complexity.

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Acknowledgements

Partial funding of the Technion's Fund for Promotion of Research and the I. Goldberg Fund for Electronic Research are gratefully acknowledged. S.B. would like to acknowledge the support of The Center for Absorption in Science, Israeli Ministry of Immigrant Adsorption. We would also like to thank an anonymous reviewer for very thorough reading of the manuscript and useful suggestions.

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Authors and Affiliations

  1. Department of Chemical Engineering, Technion, IIT, Haifa, 32000, Israel

    A. B. Tesler, D. R. Lewin, S. Baltianski & Y. Tsur

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  1. A. B. Tesler
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  2. D. R. Lewin
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  3. S. Baltianski
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  4. Y. Tsur
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Correspondence to Y. Tsur.

Appendix: Calculations of average error

Appendix: Calculations of average error

The finite frequency bandwidth of the experiment causes a major difficulty to the implementation of K–K transforms and produces an inherent inaccuracy in the calculation algorithm. Additional inaccuracy of K–K transforms is originated in numerical integration scheme. In order to determine the validity of the measured impedance data, Average Error (AE) was defined as follows:

$${\text{AE}} = 100 \cdot \frac{{\sum\limits_{i = 1}^N {\left| {Z_{{\text{meas}}}^i \left( \omega \right) - Z_{{\text{KKT}}}^i \left( \omega \right)} \right|} }}{{N \cdot Z_{{\text{meas}},\max } }}$$
(15)

where Z meas and Z KKT are the values observed experimentally and calculated by K–K transform. Z meas,max is the maximum value of the experimental relevant data set (real or imaginary) and N is the total number of measured points. Normalization of Eq. 15 by Z meas,max enables the comparison of the different data sets that can differ by orders of magnitude.

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Tesler, A.B., Lewin, D.R., Baltianski, S. et al. Analyzing results of impedance spectroscopy using novel evolutionary programming techniques. J Electroceram 24, 245–260 (2010). https://doi.org/10.1007/s10832-009-9565-z

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