Abstract
Abstract Supercapacitors are an emerging technology with applications in pulse power, motive power, and energy storage. However, their carbon electrodes show a variety of non-ideal behaviours that have so far eluded explanation. These include Voltage Decay after charging, Voltage Rebound after discharging, and Dispersed Kinetics at long times. In the present work, we establish that a vertical ladder network of RC components can reproduce all these puzzling phenomena. Both software and hardware realizations of the network are described. In general, porous carbon electrodes contain random distributions of resistance R and capacitance C, with a wider spread of log R values than log C values. To understand what this implies, a simplified model is developed in which log R is treated as a Gaussian random variable while log C is treated as a constant. From this model, a new family of equivalent circuits is developed in which the continuous distribution of log R values is replaced by a discrete set of log R values drawn from a geometric series. We call these Pascal Equivalent Circuits. Their behaviour is shown to resemble closely that of real supercapacitors. The results confirm that distributions of RC time constants dominate the behaviour of real supercapacitors.
Highlights
In order to explore some of the properties of Model II, we introduce a new family of equivalent circuits that can mimic the wide distributions of resistances found in carbon supercapacitors
This illustrates the effect of charging the “Row 5” Pascal Equivalent Circuit shown in Fig. 2B for different periods of time at 2:0 V
The successful implementation of Pascal Equivalent Circuits in silico prompted us to explore the possibility of implementing them in operando Before embarking on this task, it was first necessary to ensure that any commercial resistors and capacitors that we used would respond as closely as possible to their mathematically idealized counterparts
Summary
Supercapacitors differ from ideal capacitors in a number of important ways. They show voltage decay at open circuit, capacitance loss at high frequency, and voltammetric distortions at high scan rate. The first model that we consider (denoted Model I) assumes that the micro-capacitors are all different while the resistors are all the same. N1⁄41 where NL (dimensionless) is the total number of rungs in the ladder, and Rn and Cn are discrete random variables. If C is a random variable while R is a constant, the total admittance YL of the ladder network as a function of frequency u takes the much simpler form: YL ð∞Þ. In the case of a real electrode, the function gðtÞ will be essentially continuous
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