# How to Measure and Calculate Equivalent Series Resistance of Electric Double-Layer Capacitors

^{1}

^{2}

^{*}

## Abstract

**:**

_{max}) of these devices uses the relation P

_{max}= U

^{2}/4R

_{ESR}, where U stands for the cell voltage and R

_{ESR}for the equivalent series resistance. Despite the relevance of R

_{ESR}, one can observe a lack of consensus in the literature regarding the determination of this parameter from the galvanostatic charge-discharge findings. In addition, a literature survey revealed that roughly half of the scientific papers have calculated the R

_{ESR}values using the electrochemical impedance spectroscopy (EIS) technique, while the other half used the galvanostatic charge discharge (GCD) method. R

_{ESR}values extracted from EIS at high frequencies (>10 kHz) do not depend on the particular equivalent circuit model. However, the conventional GCD method better resembles the real situation of the device operation, and thus its use is of paramount importance for practical purposes. In the latter case, the voltage drop (ΔU) verified at the charge-discharge transition for a given applied current (I) is used in conjunction with Ohm’s law to obtain the R

_{ESR}(e.g., R

_{ESR}= ΔU/ΔI). However, several papers have caused a great confusion in the literature considering only applied current (I). In order to shed light on this important subject, we report in this work a rational analysis regarding the GCD method in order to prove that to obtain reliable R

_{ESR}values the voltage drop must be normalized by a factor of two (e.g., R

_{ESR}= ΔU/2I).

## 1. Introduction

^{−1}) and power (W kg

^{−1}) normalized per weight of the device (or the electrode material). In this sense, the maximum output energy (E

_{max}) and power (P

_{max}) are determined using the relations E

_{max}= CU

^{2}/2 and P

_{max}= U

^{2}/4R

_{ESR}, respectively, where U stands for cell voltage, C for specific capacitance, and R

_{ESR}for equivalent series resistance (ESR) [5].

_{ESR}(e.g., R

_{ESR}= ΔU/ΔI and not R

_{ESR}= ΔU/I) [5,10,11,12,13,14,15,16]. However, the ‘ad hoc’ adoption in the literature of non-standard normalizing factors for ΔU have led to a great confusion when findings present in different papers obtained for supercapacitor devices were compared.

_{ESR}values requires the voltage drop to be normalized by a factor of two (e.g., R

_{ESR}= ΔU/2I and not R

_{ESR}= ΔU/I). Simulations using canonic circuit models were carried out to emphasize the theoretical aspects inherent to the present work. Furthermore, a comparison of the theoretical electrochemical behaviors of these circuits in the frequency domain is presented using the EIS method.

## 2. Fundamentals of the EIS and GCD Methods

_{ESR}. It is worth mentioning that the impedance response verified at very high frequencies does not depend on the particular equivalent circuit model used in the simulation process [7,8,17,18,19,20,21,22]. In addition, for practical purposes, since the EIS is a steady-state technique obeying the linear theory of systems, one has that highly accurate values of resistances and capacitances can be obtained for EDLCs using a low amplitude sinusoidal voltage (e.g., δU = 10 mV (peak-to-peak)) and scanning the frequency for various orders of magnitude (e.g., Δf = 100 kHz to 10 mHz).

_{ESR}) is connected to a branch containing a capacitor (C

_{EDL}), representing the charge storage process on the electrical double-layer (EDL) formed at the electrode/electrolyte interface, which stands in parallel with a leakage resistance (R

_{L}). Despite the use of a canonic model, it is important noting that the R

_{ESR}-value obtained at high-frequencies always appear as a generalized resistance connected in series with the other circuit elements of the particular circuit model [4].

_{ESR}-values and using an R

_{L}(impedance to leak current)-value of 1 MΩ. Obviously, an ideal EDLC device has a very high leakage resistance (R

_{L}→ ∞) and the phenomenon of frequency dispersion is absent, i.e., the complex-plane plot is characterized by a perfect vertical line (see Figure 1a). Thus, the extrapolation of this line on the real axis (Z

_{real}) at very high frequencies (e.g., ω → ∞) yields the desired value of the R

_{ESR}. Therefore, in real systems the true value of R

_{ESR}is commonly determined by extrapolation using a high-frequency value of ≈1.0 kHz.

_{drop}) at the inversion of polarity. From the theoretical viewpoint, in this case the determination of the R

_{ESR}-value involves the application of a square wave current function with the inversion in polarity (e.g., I

_{(+)}↔ I

_{(−)}and |I

_{(+)}| = |I

_{(−)}|).A voltage drop is observed at the reversal of polarization with a voltage increase after the sign of the current was reversed. During a continuous repetition of the charge–discharge processes, the positive (anode) and negative (cathode) electrodes were constantly charged and discharged, respectively, for equal times by applying positive (I

_{(+)}) and negative (I

_{(−)}) currents of the same magnitude (|I

_{(+)}| = |I

_{(−)}|). Therefore, for an ideal case where only a capacitive behavior exists (R

_{ESR}= 0) one would obtain as the response a symmetric triangular voltage wave since the capacitive voltage (U

_{c}) increases linearly with the stored charge (Q) for a given capacitance (C), i.e., δU

_{c}= δQ/C. However, in real cases where R

_{ESR}> 0 the anodic branch (e.g., the straight line with a positive slope) of the voltage wave referring to the charging process (e.g., δU

_{c(+)}= δQ

_{(+)}/C) is displaced to more positive values by a constant value dictated by U

_{ESR(+)}= R

_{ESR}× I

_{(+)}= constant. Therefore, the instantaneous values of the overall voltage are given by U

_{i}= U

_{c(+)}+ U

_{ESR(+)}. Accordingly, the cathodic branch (e.g., the straight line with a negative slope) of the voltage wave associated with the discharging process (e.g., δU

_{c(−)}= δQ

_{(−)}/C) is displaced to more negative voltages due to the reversal in polarity of the applied current where δI = I

_{(+)}− I

_{(−)}= 2I, since |I

_{(+)}| = |I

_{(−)}|.

_{i}= U

_{c}− U

_{ESR}= U

_{c}− R

_{ESR}× 2I. In this sense, the overall voltage drop during reversal of the polarization is U

_{ESR}= − R

_{ESR}× 2I since ΔU

_{ESR}< 0, i.e., R

_{ESR}= ΔU

_{ESR}/2I and, therefore, the voltage drop must be normalized by a factor of 2 [5].

## 3. Theoretical Electric Response of the GCD Curves Using the Canonic Equivalent Circuit Model

#### 3.1. Deriving the Theoretical Formula for the Equivalent Series Resistance

_{ESR}, the canonic circuit presented in Figure 2 representing the charge–discharge processes was employed to obtain the pertinent equations. In short, key equations for the charge and discharge processes and a combination of them are presented to obtain the desired theoretical model.

_{cell(t)}describing the transient response after the switcher K is closed is given by the following equation:

_{EDL}, respectively, were obtained at constant current from numerical simulation (e.g., Simulink of PSIM software) using different R

_{ESR}-values (e.g., 0, 0.01, and 0.1 Ω) keeping the R

_{L}= 1.0 MΩ and C

_{EDL}= 0.1 F (see Figure 3). The anodic (positive) and cathodic (negative) currents were alternated between +1 A and −1 A, respectively. In this case, a virtual controlled current source was applied referring to a charging time of ~1 s. Then, the current direction was reversed, thus characterizing the discharge of the C

_{EDL}. The magnitude of the charge and discharge currents was maintained so that the desired characteristics of the circuit could be graphically observed.

_{ESR}-value can be determined by the voltage drop during the reversal of the polarity, i.e., when the charging process is discontinued to obtain the discharging curves. In the case of the cell voltage during the charging process (see Equation (1)), when t → ∞ the capacitor is fully charged. Therefore, one has for this particular condition that:

_{L}I

_{cell}is equal to the voltage in the capacitor (U

_{SC}

^{0}), the following equation can be obtained:

#### 3.2. Validation of the Theoretical Expression Obtained for the Equivalent Series Resistance Using a Commercial Supercapacitor of 200 F

_{d.c.}= 0 V, and Δf = 10 kHz to 10 mHz). The impedance data of a real electric double-layer capacitor (EDLC) were quite different from the impedance response of a conventional (passive) capacitor due to the presence of the frequency dispersion phenomenon [4,17]. In fact, as discussed earlier by Conway [4], the EDLCs composed of high surface area porous carbon materials cannot be represented by a simple capacitance or even by a simple RC circuit due to the influence of the ions in conjunction with the porous behavior exhibited by the electrode material, i.e., the high-frequency voltage hardly penetrates inside the narrow pores while the low-frequency voltage penetrates inside the porous electrode structure.

_{ESR}-value is obtained at very high frequencies from extrapolation on the real axis of the Nyquist plot. Thus, it was verified in the present case that R

_{ESR}= 22 mΩ, which is in good agreement with that of 25 mΩ obtained in this work using the classical GCD method.

_{CPE}= 1/Y

_{o}(jω)

^{n}and n ≅ 1. In this case, R* contains information about the electrolyte resistance inside the pores while the parameter Y

_{o}represents the non-ideal capacitance.

_{ESR}-values obtained from the GCD method as a function of the applied current. As it can be seen, a stationary value of 25 mΩ was obtained for applied currents higher than 1 A. These findings are in agreement with the experimental value of 22 mΩ obtained using the EIS technique. It must be emphasized that for an applied current lower than 1 A the R

_{ESR}values were overestimated in comparison with those obtained with the EIS technique. Therefore, the present study suggests that reliable values of the R

_{ESR}must be obtained in the case of the GCD method using different currents in order to verify a stationary (constant) value. To the best of our knowledge, this important issue regarding the influence of the applied current on the R

_{ESR}values has not been addressed in the literature.

## 4. Conclusions

_{ESR}) using the galvanostatic charge–discharge method. It was verified that the voltage drop must be normalized by a factor of two in order to obtain meaningful findings. The present work compared the applications of the electrochemical impedance spectroscopy (EIS) technique and galvanostatic charge/discharge (GCD) method in order to determine the value of the equivalent series resistance (R

_{ESR}) present in electric double-layer capacitors (EDLCs). The derived equation was applied to obtain the R

_{ESR}of a 200 F commercial supercapacitor.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of (

**a**) the complex-plane plots and (

**b**) the galvanostatic charge-discharge curves evidencing the voltage drop (U

_{drop}). The inset in Figure 1a shows the canonic circuit model. Simulation was carried out considering different values of the R

_{ESR}and R

_{L}= 1 MΩ.

**Figure 2.**Electric circuits containing the canonic equivalent circuit model representing the electrochemical behavior of electric double layer capacitors during the charging and discharging processes carried out at constant current. Definitions: I

_{cell}is the constant current applied; R

_{ESR}is the equivalent series resistance; R

_{L}is the leakage resistance; C

_{EDL}represents the equivalent capacitance of the symmetric coin cell; U

_{cell}is the overall cell voltage and U

_{SC}

^{0}is the voltage across the C

_{EDL}when the capacitor is fully charged.

**Figure 3.**Simulation of the galvanostatic charge–discharge curves using different values of R

_{ESR}. Conditions: R

_{ESR}= 0, 0.01, and 0.1 Ω; R

_{L}= 1.0 MΩ; and C

_{EDL}= 0.1 F. The anodic (positive) and cathodic (negative) currents were alternated between +1 A and −1 A, respectively.

**Figure 4.**Experimental GCD curves obtained for a 200F commercial supercapacitor (2.7V D35H62 PTH S0016) as a function of the applied current.

**Figure 5.**(

**a**) Nyquist plot obtained for the commercial supercapacitor of 200 F. (

**b**) Equivalent series resistance obtained from the GCD method as a function of the applied current.

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**MDPI and ACS Style**

Vicentini, R.; Da Silva, L.M.; Cecilio Junior, E.P.; Alves, T.A.; Nunes, W.G.; Zanin, H.
How to Measure and Calculate Equivalent Series Resistance of Electric Double-Layer Capacitors. *Molecules* **2019**, *24*, 1452.
https://doi.org/10.3390/molecules24081452

**AMA Style**

Vicentini R, Da Silva LM, Cecilio Junior EP, Alves TA, Nunes WG, Zanin H.
How to Measure and Calculate Equivalent Series Resistance of Electric Double-Layer Capacitors. *Molecules*. 2019; 24(8):1452.
https://doi.org/10.3390/molecules24081452

**Chicago/Turabian Style**

Vicentini, Rafael, Leonardo Morais Da Silva, Edson Pedro Cecilio Junior, Thayane Almeida Alves, Willian Gonçalves Nunes, and Hudson Zanin.
2019. "How to Measure and Calculate Equivalent Series Resistance of Electric Double-Layer Capacitors" *Molecules* 24, no. 8: 1452.
https://doi.org/10.3390/molecules24081452