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+
+
+
+ 20250128195518-b6aeff7ac968ea5f435222522e1a1a2dde544f97
+ 20250128195518
+
+ JOSS Admin
+ admin@theoj.org
+
+ The Open Journal
+
+
+
+
+ Journal of Open Source Software
+ JOSS
+ 2475-9066
+
+ 10.21105/joss
+ https://joss.theoj.org
+
+
+
+
+ 01
+ 2025
+
+
+ 10
+
+ 105
+
+
+
+ nleis.py: A Nonlinear Electrochemical Impedance Analysis Toolbox
+
+
+
+ Yuefan
+ Ji
+
+ Department of Chemical Engineering & Clean Energy Institute, University of Washington, Seattle, WA, United States of America
+
+ https://orcid.org/0000-0003-1912-767X
+
+
+ Matthew D.
+ Murbach
+
+ Independent Researcher, United States of America
+
+ https://orcid.org/0000-0002-6583-5995
+
+
+ Daniel T.
+ Schwartz
+
+ Department of Chemical Engineering & Clean Energy Institute, University of Washington, Seattle, WA, United States of America
+
+ https://orcid.org/0000-0003-1173-5611
+
+
+
+ 01
+ 28
+ 2025
+
+
+ 7424
+
+
+ 10.21105/joss.07424
+
+
+ http://creativecommons.org/licenses/by/4.0/
+ http://creativecommons.org/licenses/by/4.0/
+ http://creativecommons.org/licenses/by/4.0/
+
+
+
+ Software archive
+ 10.5281/zenodo.14606367
+
+
+ GitHub review issue
+ https://github.com/openjournals/joss-reviews/issues/7424
+
+
+
+ 10.21105/joss.07424
+ https://joss.theoj.org/papers/10.21105/joss.07424
+
+
+ https://joss.theoj.org/papers/10.21105/joss.07424.pdf
+
+
+
+
+
+ Electrochemical impedance spectroscopy
+ Orazem
+ 978-0-470-04140-6
+ 2008
+ Orazem, M. E., & Tribollet, B. (2008). Electrochemical impedance spectroscopy. Wiley. ISBN: 978-0-470-04140-6
+
+
+ Nonlinear electrochemical impedance spectroscopy of lithium-ion batteries: Experimental approach, analysis, and initial findings
+ Murbach
+ Journal of The Electrochemical Society
+ 11
+ 165
+ 10.1149/2.0711811jes
+ 0013-4651
+ 2018
+ Murbach, M. D., Hu, V. W., & Schwartz, D. T. (2018). Nonlinear electrochemical impedance spectroscopy of lithium-ion batteries: Experimental approach, analysis, and initial findings. Journal of The Electrochemical Society, 165(11), A2758–A2765. https://doi.org/10.1149/2.0711811jes
+
+
+ Impedance.py: A python package for electrochemical impedance analysis
+ Murbach
+ Journal of Open Source Software
+ 52
+ 5
+ 10.21105/joss.02349
+ 2020
+ Murbach, M. D., Gerwe, B., Dawson-Elli, N., & Tsui, L. (2020). Impedance.py: A python package for electrochemical impedance analysis. Journal of Open Source Software, 5(52), 2349. https://doi.org/10.21105/joss.02349
+
+
+ AC impedance technique in PEM fuel cell diagnosis—a review
+ Yuan
+ International Journal of Hydrogen Energy
+ 32
+ 10.1016/j.ijhydene.2007.05.036
+ 2007
+ Yuan, X., Wang, H., Sun, J. C., & Zhang, J. (2007). AC impedance technique in PEM fuel cell diagnosis—a review. International Journal of Hydrogen Energy, 32, 4365–4380. https://doi.org/10.1016/j.ijhydene.2007.05.036
+
+
+ Application of electrochemical impedance spectroscopy to commercial li-ion cells: A review
+ Meddings
+ Journal of Power Sources
+ 480
+ 10.1016/j.jpowsour.2020.228742
+ 2020
+ Meddings, N., Heinrich, M., Overney, F., Lee, J.-S., Ruiz, V., Napolitano, E., Seitz, S., Hinds, G., Raccichini, R., Gaberšček, M., & Park, J. (2020). Application of electrochemical impedance spectroscopy to commercial li-ion cells: A review. Journal of Power Sources, 480, 228742. https://doi.org/10.1016/j.jpowsour.2020.228742
+
+
+ Tables of degenerate electrical networks for use in the equivalent‐circuit analysis of electrochemical systems
+ Fletcher
+ Journal of The Electrochemical Society
+ 141
+ 10.1149/1.2055011
+ 1994
+ Fletcher, S. (1994). Tables of degenerate electrical networks for use in the equivalent‐circuit analysis of electrochemical systems. Journal of The Electrochemical Society, 141, 1823–1826. https://doi.org/10.1149/1.2055011
+
+
+ Extending newman’s pseudo-two-dimensional lithium-ion battery impedance simulation approach to include the nonlinear harmonic response
+ Murbach
+ Journal of The Electrochemical Society
+ 164
+ 10.1149/2.0301711jes
+ 2017
+ Murbach, M. D., & Schwartz, D. T. (2017). Extending newman’s pseudo-two-dimensional lithium-ion battery impedance simulation approach to include the nonlinear harmonic response. Journal of The Electrochemical Society, 164, E3311–E3320. https://doi.org/10.1149/2.0301711jes
+
+
+ Nonlinear electrochemical impedance spectroscopy for lithium-ion battery model parameterization
+ Kirk
+ Journal of The Electrochemical Society
+ 170
+ 10.1149/1945-7111/acada7
+ 2023
+ Kirk, T. L., Lewis-Douglas, A., Howey, D., Please, C. P., & Chapman, S. J. (2023). Nonlinear electrochemical impedance spectroscopy for lithium-ion battery model parameterization. Journal of The Electrochemical Society, 170, 010514. https://doi.org/10.1149/1945-7111/acada7
+
+
+ Second-harmonic nonlinear electrochemical impedance spectroscopy: Part I. Analytical theory and equivalent circuit representations for planar and porous electrodes
+ Ji
+ Journal of The Electrochemical Society
+ 170
+ 10.1149/1945-7111/ad15ca
+ 2023
+ Ji, Y., & Schwartz, D. T. (2023). Second-harmonic nonlinear electrochemical impedance spectroscopy: Part I. Analytical theory and equivalent circuit representations for planar and porous electrodes. Journal of The Electrochemical Society, 170, 123511. https://doi.org/10.1149/1945-7111/ad15ca
+
+
+ Second-harmonic nonlinear electrochemical impedance spectroscopy: Part II. Model-based analysis of lithium-ion battery experiments
+ Ji
+ Journal of The Electrochemical Society
+ 171
+ 10.1149/1945-7111/ad2596
+ 2024
+ Ji, Y., & Schwartz, D. T. (2024). Second-harmonic nonlinear electrochemical impedance spectroscopy: Part II. Model-based analysis of lithium-ion battery experiments. Journal of The Electrochemical Society, 171, 023504. https://doi.org/10.1149/1945-7111/ad2596
+
+
+
+
+
+
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+
+
+
+
+
+
+
+Journal of Open Source Software
+JOSS
+
+2475-9066
+
+Open Journals
+
+
+
+7424
+10.21105/joss.07424
+
+nleis.py: A Nonlinear Electrochemical Impedance Analysis
+Toolbox
+
+
+
+https://orcid.org/0000-0003-1912-767X
+
+Ji
+Yuefan
+
+yuefan@uw.edu
+
+*
+
+
+https://orcid.org/0000-0002-6583-5995
+
+Murbach
+Matthew D.
+
+
+
+
+https://orcid.org/0000-0003-1173-5611
+
+Schwartz
+Daniel T.
+
+
+
+
+
+Department of Chemical Engineering & Clean Energy
+Institute, University of Washington, Seattle, WA, United States of
+America
+
+
+
+
+Independent Researcher, United States of
+America
+
+
+
+
+* E-mail: yuefan@uw.edu
+
+
+20
+10
+2024
+
+10
+105
+7424
+
+Authors of papers retain copyright and release the
+work under a Creative Commons Attribution 4.0 International License (CC
+BY 4.0)
+2025
+The article authors
+
+Authors of papers retain copyright and release the work under
+a Creative Commons Attribution 4.0 International License (CC BY
+4.0)
+
+
+
+Python
+impedance.py
+Electrochemistry
+EIS
+NLEIS
+2nd-NLEIS
+Lithium-ion Batteries
+Fuel Cells
+
+
+
+
+
Building on the growing adoption of impedance.py
+(Murbach
+et al., 2020) as an open-source software tool within the
+electrochemical impedance spectroscopy (EIS) community, nleis.py is a
+toolbox for impedance.py that aims to provide an easily accessible tool
+to perform second harmonic nonlinear EIS (2nd-NLEIS) analysis, with the
+ability to extend to higher harmonic analysis in the future. The toolbox
+is designed with impedance.py in mind to minimize the learning curve for
+users. It inherits the basic functionality of impedance.py, introduces
+paired linear and 2nd-harmonic nonlinear circuit elements, and enables
+the simultaneous analysis of EIS and 2nd-NLEIS. With this toolbox, one
+can choose to individually analyze an EIS or 2nd-NLEIS spectra or
+perform simultaneous parameter estimation of linear and nonlinear
+impedance data using an impedance.py workflow. Ultimately, the nleis.py
+toolbox will be integrated into impedance.py as adoption grows, while
+maintaining the standalone version of nleis.py as a platform to develop
+advanced features as the field matures.
+
+ Background
+
Electrochemical impedance spectroscopy (EIS) is a widely accepted
+ electroanalytical method that is often used to characterize engineered
+ electrochemical systems like fuel cells
+ (Yuan
+ et al., 2007) and lithium-ion batteries (LIBs)
+ (Meddings
+ et al., 2020). EIS experiments and modeling require
+ linearization of system response, leading to unavoidable information
+ loss and model degeneracy challenges in real-world nonlinear
+ electrochemical processes
+ (Fletcher,
+ 1994;
+ Orazem
+ & Tribollet, 2008). Second-harmonic nonlinear
+ electrochemical impedance spectroscopy (2nd-NLEIS) is emerging as a
+ powerful and complementary tool to EIS in lithium-ion battery
+ research. 2nd-NLEIS uses a moderately larger input modulation than
+ conventional EIS to drive the electrochemical system into the weakly
+ nonlinear regime where the fundamental frequency continues to
+ represent the linear system response, and a small additional
+ 2nd-harmonic signal adds key new information about the nonlinear
+ dynamics of the interfaces under study
+ (Murbach
+ et al., 2018). Analyzing a 2nd-NLEIS signal unavoidably
+ complicates the mathematical modeling compared to linear system
+ theory, but it also provides a sensible way to break EIS degeneracy
+ and generate key new insights into charge transfer, transport, and
+ thermodynamic parameters that are inaccessible to linear EIS alone.
+ Early work with the pseudo-two-dimensional (P2D) LIB model provided
+ the first physical insights into the potential value of 2nd-NLEIS
+ signals for battery research
+ (Murbach
+ & Schwartz, 2017), whereas quantitative parameter
+ estimation of 2nd-NLEIS experiments has required adoption of
+ physically-insightful reduced order models, such as Kirk et al.’s work
+ developing a nonlinear single particle model (SPM)
+ (Kirk
+ et al., 2023), and our work defining nonlinear Randles circuit
+ (RC) and porous electrode models (PEM)
+ (Ji
+ & Schwartz, 2023).
+
+
+ Statement of Need
+
As an emerging technique requiring nonlinear dynamic modeling to
+ analyze experimental data, 2nd-NLEIS method adoption is slowed by a
+ lack of commercial or open-source software for parameter estimation
+ from experiments, even though NLEIS experiments can be performed with
+ EIS equipment offered by several vendors
+ (Murbach
+ et al., 2018). Within the linear EIS community, the adoption of
+ an open-source impedance.py equivalent circuit modeling workflow has
+ successfully facilitated reproducible, easy-to-use, and transparent
+ impedance analysis that supports an active community of users. By
+ introducing nonlinear equivalent circuit modeling through nleis.py, we
+ seek to enhance the accessibility of this powerful new technique with
+ a streamlined data analysis pipeline that researchers are already
+ familiar with, hence accelerating the co-development of theory and
+ experiments. Moreover, there is neither a research nor industry
+ standard platform available to measure and analyze 2nd-NLEIS.
+ Consequently, we aim to use nleis.py as a starting point to establish
+ 2nd-NLEIS measurements and analysis best-practices while working in
+ concert with the impedance.py user community.
+
+
+ Naming Conventions and Parameter Assignments
+
The linear and second harmonic nonlinear circuit elements are
+ defined in a pair with an addition of ‘n’ after the nonlinear circuit
+ element to facilitate the simultaneous analysis of linear and
+ nonlinear impedance response. Additionally, because of the nature of
+ the nonlinear response, the simplest possible circuit element is a
+ wrapper Randles circuit (RC) rather than the Resistor (R) and
+ capacitor (C) element defined in impedance.py. For example, the linear
+ and nonlinear Randles circuits are defined in pairs as RC and RCn
+ respectively. Lastly, parameter assignments should follow a convention
+ that first defines the linear parameters of a model
+ (
+
+ p1)
+ then the nonlinear parameters (
+
+ p2).
+ For the RC circuit as an example, RC should only take
+ [
+
+ p1]
+ as parameter inputs, while RCn should take
+ [
+
+ p1,
+
+
+ p2]
+ as parameter inputs, as described in
+ (Ji
+ & Schwartz, 2023).
The 2nd-harmonic nonlinear Equivalent Circuit Fitting is
+ accomplished with NLEISCustomCircuit. It
+ inherits most features from the impedance.py
+ CustomCircuit, but provides an extra level of
+ flexibility for performing NLEIS specific tasks. Overall, the users
+ should expect the same workflow as impedance.py.
+
+
+ Simultaneous Equivalent Circuit Fitting of EIS and
+ 2nd-NLEIS
+
EISandNLEIS is the key feature of nleis.py
+ that enables the simultaneous analysis of EIS and 2nd-NLEIS with
+ equivalent circuit modeling. The visual representation of nonlinear
+ equivalent circuit representation can be found in
+ (Ji
+ & Schwartz, 2023). Everything works like impedance.py,
+ but the users should provide the correct pair of linear and
+ nonlinear circuit strings with a single initial guess that is
+ consistent with both linear and nonlinear circuit features seen in
+ the data. For EIS and 2nd-NLEIS data with known error structure or
+ relative magnitudes, the users can also specify the optimization,
+ weighting, and normalization method for the EIS and 2nd-NLEIS data
+ parameter estimation process, as introduced in
+ (Ji
+ & Schwartz, 2024;
+ Kirk
+ et al., 2023).
+
+
+ Visualization
+
The user can choose to use the plotting function in impedance.py
+ or a customized plotting function for EIS (plot_first) and 2nd-NLEIS
+ (plot_second) to get a correctly labeled Nyquist plot.
+
+
+ Nonlinear Circuit Elements
+
nleis.py supports a variety of linear and 2nd harmonic nonlinear
+ circuit element pairs from simple Randles circuits to analytical
+ porous electrode and numerical transmission line models. These
+ models all rely on the foundation of the analytical theory developed
+ by
+ (Ji
+ & Schwartz, 2023) for Randles and porous electrodes. Just
+ like impedance.py, nleis.py supports manual element definition. If
+ you want your model to be included in future releases,
+ create
+ an issue on GitHub with your models to contribute to the
+ project.
+
+
+
+ Side-by-Side Comparison between impedance.py and nleis.py
+ API
+
nleis.py comes with detailed documentation with examples and
+ concepts for new users to
+ get
+ started. For existing impedance.py users, we expect this
+ side-by-side comparison between impedance.py and nleis.py can reduce
+ barriers for extending your impedance analysis to the weakly nonlinear
+ regime.
+
+
Side-by-side comparison between
+ impedance.py and
+ nleis.py API
There is only a simple data processing function for frequency
+ domain data truncation now. We hope to include data processing and
+ conversion from either time or frequency domain data in the future
+ (i.e. FFT capability for time domain data). If you have equipment
+ that allows you to perform 2nd-NLEIS and wish it to be compatible
+ with nleis.py,
+ create
+ an issue with a sample file to help the advancement of
+ 2nd-NLEIS.
+
+
+ Data Validation
+
Unlike EIS, which already has a set of well-established
+ validation methods available, there is not yet a standard set of
+ tools to quickly validate the causality and stationarity
+ requirements of 2nd-NLEIS data. We are actively working on the
+ development of such a data validation method, which will be
+ incorporated in the future. If you have a new method for 2nd-NLEIS
+ validation, we are also interested in including it in the future
+ after it is peer reviewed and published.
+ Create
+ an issue or
+ submit
+ a pull request to initiate review for inclusion.
+
+
+ Easily Extendable to Higher Harmonics
+
Though the analysis of harmonics greater than the second are not
+ yet implemented here, there is no restriction for higher harmonic
+ circuit element definitions. A general consideration for higher
+ harmonics is to avoid numbers in the definition of raw circuit
+ elements. Higher harmonics definitions are expected to follow the
+ Naming Conventions and Parameter Assignments section above.
+
+
+ Contribute to the Project
+
In general, 2nd-NLEIS is a novel technique for electrochemical
+ science and engineering research and development. Many areas
+ familiar to EIS analysts are not fully developed for nonlinear
+ systems. If you are publishing theoretical or experimental work that
+ is advancing the field and would like to disseminate it as software
+ others can use as part of nleis.py, we encourage you to
+ create
+ an issue on GitHub and become a contributor to
+ nleis.py.
+
+
+
+ Acknowledgments
+
This work has been supported by funding from State of Washington
+ proviso funding for Clean Energy Institute Graduate Fellowships and
+ the Boeing-Sutter endowment for excellence in engineering. We thank
+ An-Hung Shih, Lauren Frank, and Rebecca C. Vincent for their
+ preliminary testing and invaluable feedback. An up-to-date list of
+ contributors can be found on
+ GitHub.
+
+
+
+
+
+
+
+
+ OrazemMark E.
+ TribolletBernard
+
+
+ Wiley
+ Hoboken, N.J
+ 2008
+ 978-0-470-04140-6
+
+
+
+
+
+ MurbachMatthew D.
+ HuVictor W.
+ SchwartzDaniel T.
+
+ Nonlinear electrochemical impedance spectroscopy of lithium-ion batteries: Experimental approach, analysis, and initial findings
+
+ 2018
+ 20181226
+ 165
+ 11
+ 0013-4651
+ https://iopscience.iop.org/article/10.1149/2.0711811jes
+ 10.1149/2.0711811jes
+ A2758
+ A2765
+
+
+
+
+
+ MurbachMatthew D.
+ GerweBrian
+ Dawson-ElliNeal
+ TsuiLok-kun
+
+ Impedance.py: A python package for electrochemical impedance analysis
+
+ The Open Journal
+ 2020
+ 5
+ 52
+ https://doi.org/10.21105/joss.02349
+ 10.21105/joss.02349
+ 2349
+
+
+
+
+
+
+ YuanXiaozi
+ WangHaijiang
+ SunJian Colin
+ ZhangJiujun
+
+ AC impedance technique in PEM fuel cell diagnosis—a review
+
+ Elsevier
+ 2007
+ 32
+ https://www.sciencedirect.com/science/article/pii/S036031990700328X
+ 10.1016/j.ijhydene.2007.05.036
+ 4365
+ 4380
+
+
+
+
+
+ MeddingsNina
+ HeinrichMarco
+ OverneyFrédéric
+ LeeJong-Sook
+ RuizVanesa
+ NapolitanoEmilio
+ SeitzSteffen
+ HindsGareth
+ RaccichiniRinaldo
+ GaberščekMiran
+ ParkJuyeon
+
+ Application of electrochemical impedance spectroscopy to commercial li-ion cells: A review
+
+ Elsevier
+ 2020
+ 480
+ https://www.sciencedirect.com/science/article/pii/S0378775320310466
+ 10.1016/j.jpowsour.2020.228742
+ 228742
+
+
+
+
+
+
+ FletcherStephen
+
+ Tables of degenerate electrical networks for use in the equivalent‐circuit analysis of electrochemical systems
+
+ 1994
+ 141
+ https://iopscience.iop.org/article/10.1149/1.2055011
+ 10.1149/1.2055011
+ 1823
+ 1826
+
+
+
+
+
+ MurbachMatthew D.
+ SchwartzDaniel T.
+
+ Extending newman’s pseudo-two-dimensional lithium-ion battery impedance simulation approach to include the nonlinear harmonic response
+
+ 2017
+ 164
+ https://iopscience.iop.org/article/10.1149/2.0301711jes
+ 10.1149/2.0301711jes
+ E3311
+ E3320
+
+
+
+
+
+ KirkToby L.
+ Lewis-DouglasAdam
+ HoweyDavid
+ PleaseColin P.
+ ChapmanS. Jon
+
+ Nonlinear electrochemical impedance spectroscopy for lithium-ion battery model parameterization
+
+ 2023
+ 170
+ https://iopscience.iop.org/article/10.1149/1945-7111/acada7
+ 10.1149/1945-7111/acada7
+ 010514
+
+
+
+
+
+
+ JiYuefan
+ SchwartzDaniel T.
+
+ Second-harmonic nonlinear electrochemical impedance spectroscopy: Part I. Analytical theory and equivalent circuit representations for planar and porous electrodes
+
+ 2023
+ 170
+ https://iopscience.iop.org/article/10.1149/1945-7111/ad15ca
+ 10.1149/1945-7111/ad15ca
+ 123511
+
+
+
+
+
+
+ JiYuefan
+ SchwartzDaniel T.
+
+ Second-harmonic nonlinear electrochemical impedance spectroscopy: Part II. Model-based analysis of lithium-ion battery experiments
+
+ 2024
+ 171
+ http://iopscience.iop.org/article/10.1149/1945-7111/ad2596
+ 10.1149/1945-7111/ad2596
+ 023504
+
+
+
+
+
+
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