Modelling supercapacitors using a dynamic equivalent circuit with a distribution of relaxation times

L.E Helseth

doi:10.1016/j.est.2019.100912

Abstract

• Supercapacitors can be modelled precisely using a dynamic equivalent circuit with a distribution of relaxation times. • Distribution of relaxation times provides an indicator of charge dynamics at the electrodes. • Both time dynamics (charging and self-discharging) and impedance spectroscopy can be studied within the model. Supercapacitors are often modelled using electrical equivalent circuits with a limited number of branches. However, the limited number of branches often cannot explain long-term dynamics, and one therefore has to resort to more computationally challenging basic models governing diffusion and drift of ions. Here, it is shown that consistent modelling of a supercapacitor can be done in a straightforward manner by introducing a dynamic equivalent circuit model that naturally allows a large number or a continuous distribution of time constants, both in time and frequency domains. Such a model can be used to explain the most common features of a supercapacitor in a consistent manner. In the time domain, it is shown that the time-dependent charging rate and the self-discharge of a supercapacitor can both be interpreted in this model with either a few or a continuous distribution of relaxation times. In the frequency domain, the impedance spectrum allows one to extract a distribution of relaxation times. The unified model presented here may help visualizing how the distribution of relaxation times or frequencies govern the behaviour of a supercapacitor under varying circumstances.

Highlights

Electrochemical double layer capacitors, often called supercapacitors, were patented in 1957 and later commercialized [1,2]
It is shown that the time-dependent charging rate and the self-discharge of a supercapacitor can both be interpreted in this model with either a few or a continuous distribution of relaxation times
The aim of this study was to demonstrate that the dynamic equivalent circuit can be used to model the behavior of supercapacitors if one allows for an interpretation in terms of a distribution of relaxation times

Summary

Introduction

Electrochemical double layer capacitors, often called supercapacitors, were patented in 1957 and later commercialized [1,2]. Supercapacitors are often modelled using equivalent circuits composed of resistors and capacitors, including inductive elements which may become important at higher frequencies, both in the time and frequency domains [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Representing a supercapacitor in terms of an equivalent circuit composed of passive elements is a convenient and computationally efficient method to describe the supercapacitor, it is based on sound physics. Comparisons have been made between the equivalent circuits to better understand their performances [24,25,26], and multiscale models have been shown to represent impedance spectra well [27]. Methods for accurate fitting of model parameters are needed [28], and recursive algorithms have been used to obtain better fits to experimental data characterizing charge-discharge cycles [29]

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