Analytical expressions for electrochemical supercapacitor with potential dependent capacitance
Introduction
The potential or voltage dependency of the capacitance is well known and reported on the literature of electrochemical super or ultracapacitors [1], [2], [3], [4]. This capacitance is usually referred as the differential capacitance of a supercapacitor and several models have been proposed in the literature for explaining this and other supercapacitors related phenomena [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. The RC series model is the simplest one and considers only the equivalent series resistance [11,13]. The Two-Branch model considers the differential capacitance, ESR and the equivalent parallel resistance [11,17]. The Zubieta model is a sophistication of the previous model by the inclusion of a variable capacitor [7]. The Zubieta model is more complex, as it has three branches with capacitors in parallel, which means working with equations formed by non-linear third-order ordinary differential equations (ODE) that have no trivial solution. Long, medium and short term diffusion on the charging process, have also been considered in the Zubieta model. Moreover, there is also the Multibranch model that considers the electrochemical supercapacitor as a transmission line [1,2,9,14]. Due to their relevance, extensive research has been concentrated on the study of distinct models for the electrochemical supercapacitor, as shown in recent reviews on the matter and thorough investigations of high-performance supercapacitors [18], [19], [20], [21], [22], [23]. This paper addresses the Three-Branch model and gives the analytical equations governing this theoretical equivalent circuit of a electrochemical symmetric supercapacitor. Experimental data have also been introduced in the general expressions to be compared with cyclic voltammetry (CV) experimental curves of a carbon-based supercapacitor with organic electrolyte [5].
Section snippets
The equivalent circuits and related equations
In this paper two electric circuits are considered, the first treated by 2RC is shown in Fig. 1 and the second treated by 2R(C+kUC(t))) is presented in Fig. 2. The former is applied to electrochemical supercapacitors without considering the potential dependent capacitance. The latter is employed for potential dependent capacitance, represented by kUC(t) in the 2R(C+kUC(t))) circuit, where k is an index obtained experimentally and indicates how much the potential dependent capacitance, kUC(t),
Experimental
Commercially available symmetric carbon-based electrochemical supercapacitors nominally rated at 10 F, hereafter referred as SC10F, with maximum operational potential of 2.7 V (with organic electrolyte) were used in this investigation. Cyclic voltammetry measurements were carried out using a computer analyzer Arbin BT4 with MITSPRO program employing a scan rate of 20 and 50 mVs−1, and a potential window of 2.5 V. The average value of capacitance (disregarding kUC(t)) of this supercapacitor was
Results and discussion
Fig. 3 shows the cyclic voltammetry curve for a commercial carbon-based organic electrolyte supercapacitor with a nominal capacitance of 10 F measured using a potential scan rate of 20 mVs−1 and 50 mVs−1, with a potential window of 2.5 V. The shape of this CV curve nearly resembles that rectangular curve of an ideal electrochemical symmetric supercapacitor. The angle between the base of the curve and the x-axis is caused by the EPR. The calculated capacitance using the CV curve area in Eq. (43)
Conclusions
Mathematical expressions for an electrochemical symmetric supercapacitor with and without potential-dependent capacitance have been deduced in the present study. The equations were deduced using two circuits, one for the case in which k is equal to zero, represented by 2RC, and other for the case in which k is greater than zero, represented by 2R(C+kUC(t)). Theoretical CV curves similar to the experimental CV curves of a commercial electrochemical symmetric carbon-based supercapacitor with
Declaration of Competing Interest
The authors declare that there is no conflict of interest.
Acknowledgment
The work of A. P. R. Fernandez was supported by the Faculty of Technology SENAI Theobaldo De Nigris, São Paulo, Brazil.
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