Analytic Mapping between Porous Electrode Theory and Equivalent Circuit
Authors: Zelič Klemen , Pačnik Ivo , Gaberscek Miran , Katrašnik Tomaž 
Published: December 2020

Battery equivalent circuit models (ECMs) are frequently used in battery management systems of high end applications due to their low computational complexity, robustness and simplicity. Nevertheless, empirical nature of such models and lack of physiochemical background calls for excessive experimental parametrization of the ECMs to ensure their usability. Experimentally parameterized empirical models, in general, lack of prediction capability outside of the region of their parameterization. Due to these deficiencies of the ECMs, there is an increasing demand to develop the bridging methodology interlinking ECMs and continuum electrochemical models. Such a methodology introduces physiochemical significance to the ECMs by providing guidance to their topological structure and physiochemical meaning of their elements resultantly increasing their prediction capability and reducing the amount of data needed to parameterize the ECM.
Constituent parts for the ECM with electrochemical basis already exist. Jamnik et.al [1] proposed the so called transmission lines (equivalent circuit representations) for several different electrochemical processes. Coupling constituent parts proposed by Jamnk et. al [1], one can design equivalent circuit model of the battery where every single element in ECM represent a distinctive part of the electrochemical system. With the goal to describe the electric impedance of the battery, several such models were proposed lately, i.e. [2, 3, 4]. Even though these models function well only for small harmonic perturbation (electric impedance simulation), their insightful results first clearly show that mapping between electrochemical battery model and equivalent circuit approach is possible and second they demonstrate the usefulness of electrochemically based ECMs. Nevertheless, none of the authors unambiguously shows the connection between physiochemical equations of the electrochemical model and the topology of the proposed circuit. Circuits are rather constructed based on intuition and justified by very plausible results that they generate. Here, we present the complete mapping between equivalent circuit model of one electrode and equations of porous electrode theory [5], which is one of the most commonly used continuum electrochemical model.
The methodology that enables complete mapping between equations of porous electrode theory and equations of equivalent circuit follows finite volume discretization. The whole derivation is demonstrated for a one dimensional case but it can simply be generalized to higher dimensionality. First porous electrode was divided in finite number of control volumes along the thickness of the electrode. Selected number of control volumes determines the final resolution and computational complexity of the equivalent circuit. Four partial differential equations of porous electrode theory were discretized in the direction of electrode thickness by volume integration across all control volumes. Thereby, the system of partial differential equations is transformed to a large set of ordinary differential equations (ODE). Reformulation of the derived ODE system yields equations that can be interpreted as Kirchhoff’s laws of the electric circuit. Thereby identities between parameters of equivalent circuit and parameters of electrochemical model are established and thus complete mapping between both.
Obtained general equivalent circuit model was first tested in the limit of small harmonic perturbation, since many already existing references reports the same case. The insightful results obtained by the model were compared to the analytical solution of the porous electrode theory equations for small harmonic perturbation calculated by Huang and Zhang [6]. Excellent matching between both shows credibility of the mapping that we propose. Further simulation was performed not only for the small harmonic perturbation but also for the high amplitude signals. Validity and plausibility of the proposed ECM is strengthened by the non-linear response of the proposed circuit to the high amplitude signals.
References:
[1] Jamnik, J. and Maier, J., 2001. Generalised equivalent circuits for mass and charge transport: chemical capacitance and its implications. Physical Chemistry Chemical Physics, 3(9), pp.1668-1678.
[2] Lai, W. and Ciucci, F., 2011. Mathematical modeling of porous battery electrodes—Revisit of Newman's model. Electrochimica Acta, 56(11), pp.4369-4377.
[3] von Srbik, M.T., Marinescu, M., Martinez-Botas, R.F. and Offer, G.J., 2016. A physically meaningful equivalent circuit network model of a lithium-ion battery accounting for local electrochemical and thermal behaviour, variable double layer capacitance and degradation. Journal of Power Sources, 325, pp.171-184.
[4] Moškon, J., Drvarič Talian, S., Dominko, R. and Gaberšček, M., 2020. Advances in understanding Li battery mechanisms using impedance spectroscopy. Journal of Electrochemical Science and Engineering, 10(2), pp.79-93.
[5] Newman, J. and Tiedemann, W., 1975. Porous‐electrode theory with battery applications. AIChE Journal, 21(1), pp.25-41.
[6] Huang, J. and Zhang, J., 2016. Theory of Impedance Response of Porous Electrodes: Simplifications, Inhomogeneities, Non-Stationarities and Applications. Journal of The Electrochemical Society, 163(2) pp.A1983-A2000