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Abstract The tests and calculation of the key performance metrics of supercapacitors including capacitance, power and energy stored are commonly reported by the academia and the industry based on formulæ valid only for ideal capacitors. This is inconsistent with the fact that supercapacitors exhibit electrical behaviors that are different from those of ideal capacitors whether they are looked at in the time domain or in the frequency domain. This results inevitably in errors in their characterization, design and system integration. Based on recent literature, this review article is an attempt to present and discuss the main differences between ideal capacitors and supercapacitors, and especially how the performance metrics of the latter depend on the operating frequency, the charging/discharging waveform type as well as their deviation from ideality. We present a set of calculation methods for supercapacitor metrics using fractional-order calculus when such devices are operated under the commonly used (i) sinusoidal excitation, (ii) step current input and (iii) linear voltage input. We hope to bring such analysis, which is proven to be much more reliable and effective than the standard integer-order-based analysis of ideal capacitors, to the attention of researchers, manufacturers, and end-users of these devices particularly as their range of application is surging. Highlights • Time & frequency domain, fractional-order characterization of super-capacitors. • Performance metrics of supercapacitors depend on frequency & (dis)charging waveforms. • Fractional-order expressions for capacitance, power & energy stored in supercapacitors. [ABSTRACT FROM AUTHOR]

The response of a commercial super‐capacitor to an applied periodic current excitation in the form of a triangular waveform is investigated in this study. This waveform has a linear‐with‐time variation which enables linear charging and discharging of the device. A model consisting of a linear resistance Rs and a constant phase element is used to describe the super‐capacitor impedance and expressions for the voltage across the device, the power, and stored energy are derived using concepts from fractional calculus. Experimental results are shown and an application of the study to super‐capacitor parameter extraction is described. [ABSTRACT FROM AUTHOR]

The porous nature of the electrode material in supercapacitors and the apparent conductivity of the electrolyte cause their impedance to show a complex frequency-dependent behavior, which in turn makes it incorrect to treat them as ideal capacitors, even at a frequency of a few millihertz. This is particularly crucial if the intended application requires a configuration that uses stacked supercapacitor banks, in which errors in defining the metrics of the individual components accumulate. Although manufacturers provide supercapacitor ratings under DC only, by using a detailed impedance spectroscopy study of all possible series and parallel combinations of two different commercial 1 Farad carbon-carbon supercapacitors, we show that these nominal DC capacitances are not suitable to evaluate the equivalent capacitance. Instead, by using a model that consists of a series resistance and a constant-phase element, we employed a real, effective capacitance (in proper Farad units) suitable for direct application. This effective capacitance can be used to define the frequency-dependent quality factor of a supercapacitor and enables the easy calculation of series and parallel associations of identical or different devices. [ABSTRACT FROM AUTHOR]

The focus in supercapacitor research typically falls into one of two categories: (i) the rational design and engineering of electrode materials and electrolyte formulation to achieve high performance devices at competitive costs, and (ii) the modeling of their resulting behavior in response to constant-current charging/discharging, cyclic voltammetry or impedance spectroscopy. However, less work has been dedicated to new ways for charging these devices. In this work we show that charging a supercapacitor, modeled as a constant phase element with a series resistor, using a linear voltage ramp results in higher stored charge and higher effective capacitance value than when using a linear current ramp. This is despite the reciprocity of the device, as we proved analytically. The theoretical analysis and numerical simulations are in excellent agreement with the experimental results carried out on a commercial supercapacitor. The findings can be viewed as a step towards finding the optimum charging waveforms for these devices that would maximize their effective capacitance. [ABSTRACT FROM AUTHOR]