Modeling and Simulations in Symmetrical Supercapacitors Using Time Domain Mathematical Expressions

This study presents the deduction of time domain mathematical equations to simulate the curve of the charging process of a symmetrical electrochemical supercapacitor with activated carbon electrodes fed by a source of constant electric potential in time ε and the curve of the discharge process through two fixed resistors. The first resistor R Co is a control that aims to prevent sudden variations in the intensity of the electric current i 1 (t) present at the terminals of the electrochemical supercapacitor at the beginning of the charging process. The second resistor is the internal resistance R A of the ammeter used in the calculation of the intensity of the electric current i 1 (t) over time in the charging and discharging processes. The mathematical equations generated were based on a 2R(C + kU C (t)) electrical circuit model and allowed to simulate the effects of the potential-dependent capacitance (kU C (t)) on the charge and discharge curves and hence on the calculated values of the fixed capacitance C, the equivalent series resistance (ESR), the equivalent parallel resistance (EPR) and the electrical potential dependent capacitance index k.


Introduction
Recently, it has been shown that it is possible to simulate voltammogram curves of carbon-based with organic electrolyte symmetrical supercapacitors [1]. The inclusion of the potential dependency of the capacitance on the time domain equations was essential to properly assess the experimental curves. This potential dependent capacitance implies that the capacitance changes as a function of the potential applied on the electrochemical supercapacitor and plays a crucial role in the time domain equations for the voltammetry simulations studies. More recently, it has also been shown that the capacitance dependent on the electrical potential becomes a factor that considerably increases the degree of difficulty in obtaining the time domain equations that aim to simulate galvanostatic curves of supercapacitors [2]. A straightforward electrical circuit was used and fed by a source of electric current of constant intensity in order to simulate the galvanostatic curves of the activated carbon-based supercapacitors.
Although modeling studies on electrochemical carbon-based supercapacitors have been carried out, the measurement techniques used dedicated apparatus such as potentiostat/galvanostat [1]- [14]. However, simpler setups can enable the collection of experimental data and assessment of proposed models. The present study uses a standard resistor for the discharge process and gives the analytical equations governing a straightforward theoretical equivalent circuit of an electrochemical symmetric supercapacitor. Charging has also been carried out employing a standard power supply operating at a constant potential mode. As in previous studies, the simulation of the curves for carbon-based supercapacitors with organic electrolytes also considered the effects of capacitance dependent on electric potential and the analysis was carried out in the time domain using a 2R(C + kU C (t)) model electric circuit [1] [2]. Analytical equations have been presented and comparisons between experimental and theoretical curves were made on a quantitative basis. The electric parameters have been determined using the equations and simulations with a straightforward but effective experimental setup.
The organizational structure of this paper has been divided into three main parts, namely: 1) obtaining the equations through which it is possible to simulate the experimental curves by means of the solution of non-linear and non-homogeneous first-order ODEs (obtained from the analysis in the time domain of an electrical circuit fed by a triangular electromotive force source with discharge through a resistor); 2) proof of the functionality of the deduced equations by comparing the experimental curves and the theoretical curves; 3) the conclusions of the work are presented, through which the functionality of the equations deduced in this study are observed. Experimental curves will be shown in black lines and the theoretical curves will be in dotted red.

Equivalent Circuits and Equations
As demonstrated in previous studies, a symmetrical carbon-based supercapacitor can be represented by the electric circuit 2R(C + kU C (t)) (inside the rectangle made with dashed lines), as shown schematically in Figure 1 [1] [2]. The circuit used to charge and control the initial electric current flowing through a supercapacitor is composed of a source of constant potential in time ε, the control resistor R Co used in order to avoid overloads in the ammeter with internal electrical Figure 1. Schematic of the charging process of an electrochemical supercapacitor by a source of constant electric potential ε considering that it can be represented by the electrical circuit 2R(C + kU C (t)) fed by a source ε in series with a control resistance R Co and with the internal resistance R A of an ammeter used to determine the values of i 1 (t). resistance R A and the equivalent 2R(C + kU C (t)) supercapacitor circuit (enclosed in dashed line). The charging process of the capacitor of fixed capacitance C is carried out by the source of constant potential and the discharging process is controlled by resistors. In the circuit, the equivalent series resistance (ESR) is represented by the electrical resistance R 1 , the equivalent parallel resistance (EPR) by the electrical resistance R 2 and the electric potential-dependent capacitance by the electric potential-dependent variable capacitance capacitor kU C (t). By the analysis of the circuit shown in Figure 1, an equation can be obtained so that it is possible to calculate the theoretical values of t as a function of the experimental values of the electric current i 1 (t) at the terminals of the electrochemical supercapacitor with electrodes of activated carbon material and organic electrolyte for its charging process.
In accordance with the circuit 2R(C + kU C (t)) shown in Figure 1 for the charging process of the fixed capacitance capacitor C, it is verified that: as well as: therefore: It turns out that: which implies: which results: To obtain the expression of the constant c 1 , it is considered that the beginning of the charging process of the fixed capacitance capacitor occurs at t = t (iPC) and that at such instant U C (t) = U C (t (iPC) ). In this way, it can be seen that: Substituting the term to the right of Equation (14) in Equation (13) one has that: ln .
Referring to Equation (15), it is not possible to obtain the solution analysis of the same for U C (t) because it is an electrical parameter that cannot be measured directly in an electrochemical supercapacitor, be it symmetrical or not (U C (t) is the electrical potential between the terminals of an ideal capacitor used to build an electrical circuit that represents a real supercapacitor). However, it is possible to obtain the analytical solution for the theoretical values of t taking as reference the experimental values of the current i 1 (t) present in the branch that connects Journal of Applied Mathematics and Physics an electrochemical supercapacitor to the source of constant electric potential in time ε. Thus, considering the charging process of the fixed capacitance capacitor C contained in the electrical circuit 2R(C + kU C (t)), it is verified that: It is also observed for the charging process of the fixed capacitance capacitor C that: as well as: Substituting the expression to the right of Equations (17) and (18) in Equation (15) it is verified that: By using Equation (19), it is possible to calculate the theoretical values of t as a function of the experimental values of i 1 (t) collected during the charging process of a supercapacitor evaluated during its charging carried out through a source of constant electric potential in time ε considering that it can be represented by the electric circuit 2R(C + kU C (t)). Although the value of the electric potential ε is adjusted in the source of constant potential so that the supercapacitor nominal value is not exceeded (causing decomposition of the electrolyte), the electrical control resistor R Co must be used.
Although the value of the electric potential ε can be adjusted in the source of constant potential so that the maximum value of this parameter recommended by the manufacturer of the electrochemical supercapacitor is not exceeded (higher potential will cause decomposition of the electrolyte) the electrical control resistor R Co must be used. Considering that a symmetric supercapacitor can be represented by the electrical circuit 2R(C + kU C (t)), an equation must be obtained by means of which it is possible to calculate the theoretical values of t as a function of the experimental values of i 1 (t) collected during the process of discharging the supercapacitor through the resistors R Co and R A (when its evaluation by the technique of charge at constant electric potential and discharge through resistors). To obtain this equation, Figure 2 is taken as a reference, so that, the process of discharging the fixed capacitance capacitor C of the 2R(C + kU C (t)) electrical circuit through resistors R Co and R A is carried out. Journal of Applied Mathematics and Physics Figure 2. Discharge process of the fixed capacitance capacitor C contained in the electrical circuit 2R(C + kU C (t)) through resistors R A and R Co .
In accordance with the circuit 2R(C + kU C (t)) for the charging process of the fixed capacitance capacitor C, it is verified that: as well as: It occurs for the fixed capacitance capacitor discharge process that: which implies: what results: ln .
Considering that the beginning of the discharge process of the capacitor C contained in the electrical circuit 2R(C + kU C (t)) occurs immediately at the end of the charging process, it results that t (iPD) is equal to t (fPC) , just as U C (t (iPD) ) is equal to U C (t (fPC) ).
In this way it can be seen that: Substituting the term to the right of Equation (33) in Equation (32): ln .
It turns out that it is not possible to evaluate the electrical potential U C (t) of a supercapacitor (whether symmetrical or not), but it is perfectly possible to assess the intensity of the electric current i 1 (t) present in the terminals of the same terminal using an ammeter. Thus, considering the discharge process of the fixed capacitance capacitor C contained in the electrical circuit 2R(C + kU C (t)), it is verified that: which implies: Replacing the term to the right of Equations (35) and (36) in Equation (34) it is verified that: with Equation (37), it is possible to calculate the theoretical values of t as a function of the experimental values of i 1 (t) obtained in the process of discharging a supercapacitor evaluated by the discharge through resistors, considering that the supercapacitor can be represented by the 2R(C + kU C (t)) circuit initially charged by a source of constant potential ε and discharged through resistors R Co and R A. The equation to calculate the initial value of R 1 can be found in Equation (1), which is repeated and deduced as follows: In Equation (38), U C (t (iPC) ) is the electric potential between the terminals of the fixed capacitance capacitor C in the electrical circuit 2R(C + kU C (t)) at the beginning of the charging process.
Given the fact that it is not possible to measure U C (t (iPC) ) in a supercapacitor, it is necessary that the charging process starts with it fully discharged, which implies that U(t (iPC) ), which is the electrical potential existing between the terminals of the supercapacitor at the beginning of the charging process, must be equal to zero, thus imposing that U C (t (iPC) ) is also equal to zero. Thus, Equation (38) can be rewritten as follows: By using Equation (40), it is possible to calculate the ESR a symmetric electrochemical supercapacitor for the case in which it is charged by a source of constant potential in time ε, that is, as long as it is fully discharged at the beginning of the charging process. The ESR can also be obtained in the shortest time interval in which there is a change from the charging process to the discharging process of the supercapacitor. Considering that the electric potential U C (t) at the end of the charging process and at the beginning of the discharging process of the capacitor C in the electrical circuit 2R(C + kU C (t)) are equal, it can be seen that: In Equation (41), U C (t (fPC) ) is the electrical potential between the terminals of the fixed capacitance capacitor C in the electrical circuit 2R(C + kU C (t)) at the end of the process of charge. U C (t (iPD) ) is the electric potential existing between the terminals of that capacitor at the beginning of the discharge process.
Considering that: in the case in which t is equal to t (fPC) and U C (t) is equal to U C (t (fPC) ): so that: for t equal to t (iPD) it is verified that U C (t) is equal to U C (t (iPD) ), which leads to: In this way, it appears that: By means of Equation (46), it is possible to calculate the ESR based on the values of the electric current i 1 (t) at the end of the charging process and at the beginning of the discharging process of a supercapacitor evaluated by means of charge in constant potential and of discharge through resistors.

Experimental
The experimental values of the electric current i 1 (t) and the respective experimental values of t presented in this study were obtained by the analysis of a commercial symmetrical electrochemical activated carbon supercapacitor with 1 F of nominal capacitance using discharge by resistors after charge at a constant potential. Mostly used electrolytes in these supercapacitors with carbon material electrodes are tetraethylammonium tetrafluoroborate salt (NEt4BF4) in acetonitrile (ACN) or propylene carbonate (PC), for the good conductivity [2]. Care was taken for not exceeding the full scale of the ammeter with the experimental electric current i 1 (t) and the supercapacitor maximum nominal potential (5.5 V).
The charging process of the supercapacitor was terminated as soon as the electric current intensity i 1 (t) reached 5% of the value verified at the beginning of the charging process. As soon as the charging process was completed, the discharging process began. It was found that the intensity of the electric current at the beginning of the discharging process was slightly lower than that verified at the end of the charging process. The discharge process ended as soon as the intensity of the electric current i 1 (t) reached a value equal to 5% of the value verified at the beginning of the charging process. The interruption of the charging process of the carbon-based supercapacitor based on the value of 5% of the electric current at the beginning of the charging process was determined by providing experimental values of i 1 (t) and t suitable for the purposes of generating the theoretical curves. It was verified by testing that experimental curves generated with very long discharge times for the supercapacitor (12 hours) presented great similarity to those obtained with one-hour duration (considering their non-asymptotic region). Cyclic voltammetry and galvanostatic cycling were used as a comparison of real-time experimental setup using a Versastat 4 potentiostat. Table 1 shows the values of t and i 1 (t) collected during the charging and discharging processes of the commercial activated carbon supercapacitor with the nominal capacitance of 1 F which was charged to a constant potential of 5.5 V from a source ε with an internal resistance R A of 6.84 Ω in series with resistance R Co of  Figure 3 shows the curve of i 1 (t) as a function of t for the charging and discharging processes of the supercapacitor evaluated with discharge by resistors. Using Equation (40) and considering that ε = 5.5 V, i 1 (t (iPC) ) = 0.00712 A (see Table 1), R Co = 750 Ω and R A = 6.84 Ω, R 1 was calculated as being 15.6 Ω. Identical value (15.6 Ω) was obtained using Equation (46), using ε = 5.5 V, i 1 (t (fPC) ) = 0.00033 A (see Table 1), i 1 (t (iPD) ) = −0.00679 A (see Table 1), R Co = 750 Ω and R A = 6.84 Ω. The value of the capacitance index dependent on the electrical potential k was initially considered null. Using Equations (19) and (37), R 1 = 15.63 Ω, R 2 = 694.00 kΩ (obtained through the self-discharge technique for a total of 80 hours), C = 1.12 F (obtained through the technique of direct current discharge I) and k = 0 FV −1 , it was possible to produce the theoretical inverted charge and discharge curves in accordance with the theoretical values of t and the respective experimental values of i 1 (t). These curves are shown in Figure 3.

Results and Discussion
Taking Figure 4 as a reference, it can be seen that the theoretical curves obtained by means of Equations (19) and (37) are very different from the experimental curves, indicating that the initial parameters obtained by other techniques need to be adjusted. The value of the sum between the modules of the differences of the theoretical and experimental values of t of the curves presented in Figure 4 was approximately 22,498 s. Figure 5 shows the comparison between the experimental curves of i 1 (t) as a function of t and inverted theoretical curves of t as a function of i 1 (t). The inverted theoretical curves were produced by inserting the values R 1 = 15.63 Ω, R 2 = 18 kΩ, C = 1 F and k = 0 FV −1 in equations (19) and (37). As can be seen, there was a significant improvement in terms of similarity between the theoretical and experimental curves of i 1 (t) as a function of t. The value of the sum of the modules A. P. R. Fernandez et al.   of the differences between the theoretical and experimental values of t was approximately 5471 s, demonstrating that the values of R 1 and C were more adequate than the previous ones.     Figure 9 shows the galvanostatic curves for this supercapacitor also using a real time setup potentiostat. An ESR of 25 Ω was obtained at a discharge current of 5 mA and 32 Ω at 14 mA. A lower value (6 Ω) was obtained using the low-cost setup and this can be attributed to the different power sources' signal and the duration of the charge-discharge cycle time of each method. Galvanostatic cycling uses a direct current source, whereas cyclic voltammetry uses a ramp signal. The present setup uses a source of constant potential in time. Furthermore, the capacitance value determined by the voltammogram area depends on the scan rate whereas in the galvanostatic discharge curve on de current density.
The obtained values of the electrical parameters fixed capacitance C and ESR by the cyclic voltammetry differ slightly from the values presented by the resistor technique not only because the source signal is different but also because of the  duration of each cycle of loading and unloading for each of the methods presented. In the galvanostatic cycle a direct current source is used and in the cyclic voltammetry the source generates a ramp signal (at each instant, the potential is increased or reduced in the potentiostat source). In this work, the source is of constant electric potential in time. Test time is preponderant in the case of the EPR parameter, which, when evaluated by means of self-discharge, can take tens, perhaps hundreds of hours to be obtained, while in the method presented in this article, it is obtained in a few hours. In the present technique the supercapacitor must be fully discharged to be tested.
The technique presented here can be used together with conventional techniques in order to provide a comparison of values, as well as for cases in which a potentiostat/ galvanostat is unavailable. It can also be used to provide a comparison between supercapacitors produced in series based on a reference charge and discharge curve. In this case, it should be borne in mind that the technique is valid only for symmetrical electrochemical supercapacitors.
Although this study does not address the main theorem to be studied, it has as an innovation the presentation of equations through which it is possible to simulate charge and discharge curves of electrochemical supercapacitors that are obtained in sophisticated and expensive equipment, as is the case of a potentiostat/galvanostat typically used in the analysis of the electrical parameters equivalent series resistance (ESR), equivalent parallel resistance (EPR) and capacitance C of electrochemical supercapacitors by means of an apparatus of simple assembly and of low cost constituted basically of a potentiometer, a voltmeter and an ammeter. The method, and therefore the equations presented here, also has the Journal of Applied Mathematics and Physics advantage of providing values of the capacitance parameter dependent on the electric potential kU C (t), which cannot be obtained by means of cyclic voltammetry and galvanostatic cycle analysis techniques.
Given the present theoretical results, future studies are being carried out using the 2R(C + kU C (t)) circuit fed by a triangular potential source in order to generate equations in the frequency domain and create a method of analysis of symmetrical supercapacitor electrical parameters considering the potential capacitance dependence in the frequency domain.

Conclusion
The majority of publications on electrochemical supercapacitors circuit analysis are in the frequency domain and this paper is in the time domain. The electrical potential capacitance dependence in the time domain is a complicating factor for obtaining the analytical expressions. Despite that, the mathematical equations that aim to simulate the curves of the electric current intensity in the terminals of an activated carbon symmetrical electrochemical supercapacitor with organic electrolyte were successfully deduced in the time domain. Evaluation has been carried out by the charging technique in constant electric potential and discharge by resistors. The activated carbon material supercapacitor was represented by an equivalent modeled electrical circuit 2R(C + kU C (t)). These