Letter pubs.acs.org/JPCL
Determination of Peukert’s Constant Using Impedance Spectroscopy: Application to Supercapacitors Edmund Martin Mills and Sangtae Kim* Department of Materials Science and Engineering, University of California, Davis, California 95616, United States ABSTRACT: Peukert’s equation is widely used to model the rate dependence of battery capacity, and has recently attracted attention for application to supercapacitors. Here we present a newly developed method to readily determine Peukert’s constant using impedance spectroscopy. Impedance spectroscopy is ideal for this purpose as it has the capability of probing electrical performance of a device over a wide range of time-scales within a single measurement. We demonstrate that the new method yields consistent results with conventional galvanostatic measurements through applying it to commercially available supercapacitors. Additionally, the novel method is much simpler and more precise, making it an attractive alternative for the determination of Peukert’s constant.
T
We will demonstrate the determination of Peukert’s constant using IS, by first presenting a description of the approach, then applying it to commercially available energy storage devices. The results are compared to those obtained from traditional galvanostatic methods. The approach will be applied to commercial supercapacitors. Supercapacitors (SCs) are an emerging energy storage device with relatively high power density and low energy density in comparison with batteries.9,10 Peukert’s equation has recently attracted interest for the evaluation of the rate-dependent capacity of supercapacitors, expanding its use beyond batteries alone.11 Peukert’s equation relates the discharge current of a battery and its available capacity. At elevated discharge currents, the losses associated with rate-dependent internal resistance are larger. The capacity (C) depends on the discharge current (I) and discharge time (t) according to
he capacity of energy storage devices such as batteries and supercapacitors depends on their discharge rate. One of the most widely used relationships that represent the ratedependent capacity of a battery is Peukert’s equation, an empirical power-law equation that relates the discharge rate and battery capacity, discovered in 1897.1,2 Due to its simplicity and accuracy under a wide range of conditions, Peukert’s equation is still in use, for example, in battery state-of-charge estimation.3,4 Within this relationship, Peukert’s constant k is a parameter that accounts for losses associated with discharging batteries at higher currents, with values near unity representing ideal, efficient operation and higher values indicating loss. This parameter can be used to situate novel energy storage devices, operating principles, or materials in the context of established device designs, compare their rate-performance, and even shine light on fundamental processes.5 In this work, we develop a novel method to determine Peukert’s constant with the use of impedance spectroscopy (IS), which is simpler and more precise than conventional determination with galvanostatic charge/discharge measurements. IS measures electrical behavior over a broad range of frequencies/time-scales, which makes it ideal for evaluating the rate-dependent performance of energy storage devices. In addition to providing insight into physical processes within a device, we will show that this aspect of IS gives it the capability of easily determining Peukert’s constant. Furthermore, the small currents used in IS minimize the device heating during discharge that complicates conventional dc determination of Peukert’s constant.6 Simple determination of Peukert’s constant by IS facilitates the evaluation of the rate-dependent properties, as IS is already widely used to characterize energy storage devices. This may also be of use battery monitoring systems and state-of-charge estimation, where IS has already been integrated.7,8 © XXXX American Chemical Society
C = I kt
(1)
where k is Peukert’s constant, which is dimensionless and empirical. A value of 1 indicates a current-independent capacity. Values of k larger than 1 indicate that the capacitance will decrease as the current is increased. As k increases, this effect becomes more pronounced. In typical batteries, k is between 1.1 and 1.5. Peukert’s law can be written using reference values for the capacitance and discharge time and current (Co, to, Io): ⎛ C ⎞k − 1 It = Co⎜ o ⎟ ⎝ Ito ⎠
(2)
Received: October 20, 2016 Accepted: November 30, 2016 Published: November 30, 2016 5101
DOI: 10.1021/acs.jpclett.6b02441 J. Phys. Chem. Lett. 2016, 7, 5101−5104
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The Journal of Physical Chemistry Letters
Figure 1. Discharge curves (a−c) and capacitance plotted versus discharge time (d−f) from the galvanostatic measurements. Peukert’s constant kdc was determined from the indicated fit.
When written in terms of discharge time, this becomes
⎛ t ⎞k − 1/ k C =⎜ ⎟ Co ⎝ to ⎠
and C im = (3)
(4)
To generate an expression for Peukert’s constant that allows it to be easily determined using IS, we convert it to terms of frequency. During IS measurements, the energy storage device is charged and discharged within a small voltage range by the ac signal. The discharge time is then proportional to the inverse of the frequency, t ∝ 1/f. Combining this with eq 3 gives ⎛ f ⎞1 − k / k C = ⎜⎜ ⎟⎟ Co ⎝ fo ⎠
(5)
It should be noted that the dc and ac cases are not identical: in the case of dc, the current follows a square wave over a single charge−discharge cycle, while in the case of ac the current is sinusoidal. Despite this small difference, we show experimentally that the results from impedance spectroscopy and the galvanostatic measurements are consistent. To determine the capacitance from the impedance to use with eq 5, one can use the complex capacitance
C = Cre + jC im
2πf |Z(f )|2
Table 1. Values of Peukert’s Constant Determined by Galvanostatic Measurements and Impedance Spectroscopy
(6)
The real and imaginary components are Cre =
Maxwell 1F Nesscap 1.5F Kemet 1F
−Z im(f ) 2πf |Z(f )|2
(8)
where Z is the complex impedance, Z = Zre + jZim. Equation 5 predicts that a plot of Cre versus f on a logarithmic scale will have a linear relationship, and that k can be easily determined from the slope of this plot. Therefore, it is evident that Peukert’s constant can be straightforwardly determined through the use of impedance spectroscopy. Three commercial SCs were evaluated with both galvanostatic charge−discharge measurements and IS to determine Peukert’s constant k. The results of the galvanostatic measurements are shown in Figure 1(a−c), alongside the associated capacity versus current plots, Figure 1d−f. All SCs show typical charge/discharge curves. Each of the discharge curves appears linear, indicating that there is no appreciable contribution of pseudocapacitance. The capacitance decreases with increasing current, as expected. The data were fit with eq 4 to find kdc from the slope, reported in Table 1. The Maxwell 1F and Kemet 1F each appear to have two distinct slopes; all data points were used in the fitting to find a single, average k applicable for this range of currents. The trend in ratedependence is expressed in the trend in k, increasing from near the ideal value of one (1.01) for the Nesscap 1.5 SC to quite high (1.29) for the Kemet 1F SC. The R2 values for the fits are
Or, in terms of the current, ⎛ I ⎞1 − k C =⎜ ⎟ Co ⎝ Io ⎠
Zre(f )
(7) 5102
kdc (R2)
kis (R2)
1.06 ± 0.022 (0.56) 1.01 ± 0.008 (0.32) 1.29 ± 0.200 (0.67)
1.06 ± 0.004 (0.96) 1.04 ± 0.001 (0.99) 1.25 ± 0.025 (0.99)
DOI: 10.1021/acs.jpclett.6b02441 J. Phys. Chem. Lett. 2016, 7, 5101−5104
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The Journal of Physical Chemistry Letters quite poor, leading to a possibility of doubt in the accuracy of this method. The impedance spectra (i.e., Nyquist plots) of the SCs are plotted in Figure 2. The typical impedance spectrum of a
Figure 2. Impedance spectra of the three supercapacitors, with the high frequency behavior magnified, inset.
supercapacitor contains three regions: a nearly vertical spike at low frequencies representing the SC’s capacitance, a semicircular arc at high frequencies representing the electrolyte’s impedance response, and an intermediate sloped region deviating from ideal capacitive behavior, often resulting from porosity and local dispersion in the capacitance. All three studied SCs have typical behavior. In the Nesscap 1.5F SC, no high frequency arc is apparent due to the minimal impedance of the electrolyte. Figure 3a−c shows the plot of Cre versus frequency for each SC, determined from the impedance using eq 7. Equation 5 indicates that Cre should be linear if it follows Peukert’s law, with a slope of (1 − k)/k. For typical SCs, Cre is nearly constant at low frequencies and drops off as the frequency increases; there are distinct linear regions at both low and intermediate frequencies. We are primarily interested in the low frequency region: this is the region of device operation, where the majority of the SC’s capacitance is available. k is determined from the slope of Cre in the low frequency region. The fits are shown in Figure 3 and the values are reported in Table 1. In comparing the results from the two methods, it can be seen that the values of kdc and kis are consistent for all SCs measured. In contrast with the galvanostatic measurements, the R2 values for the fitting of the IS data is excellent. Additionally, the uncertainty in k is smaller in the IS measurements, indicating that the IS method is more precise. These results demonstrate the use of IS as a viable and attractive alternative to conventional dc measurements for the determination of Peukert’s constant. In summary, a novel determination of Peukert’s constant using impedance spectroscopy was presented. This method was compared to galvanostatic charge/discharge measurements through application to three commercial SCs. The results confirmed the consistency of the two approaches, while also demonstrating that the IS method provides greater precision. As Peukert’s equation attracts attention for application to supercapacitors, this work clearly offers a simple method for the determination of Peukert’s constant, with the advantage of greater precision and simplicity than the conventional method.
Figure 3. Real capacitance Cre versus frequency for each supercapacitor, with the fit to find Peukert’s constant kis shown in red.
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EXPERIMENTAL METHODS Maxwell 1F, Nesscap 1.5F, and Kemet 1F commercial supercapacitors were characterized with IS and galvanostatic charge/discharge measurements, at room temperature. A Novocontrol Apha-A impedance analyzer was used for IS over the frequency range of 10−2 to 105 Hz, with 30 mV applied ac bias. Galvanostatic measurements were conducted using an Arbin Instruments MSTAT. Identical charge and discharge currents were used, ranging from 10 mA to 2A, with no hold time. The capacitance was calculated using C =
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tdischarge·I V − VIR
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Address: Department of Materials Science and Engineering, University of California, Davis, CA, 95616, USA. ORCID
Edmund Martin Mills: 0000-0003-4888-9139 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported by GO! KRICT Project of Korea Research Institute of Chemical Technology, Korea, by the R&D Convergence Program of NST (National Research Council of Science & Technology) of Korea, and by a grant (2011-0031636) from the Center for Advanced Soft Electronics under the Global Frontier Research Program of the Ministry of Science, ICT and Future Planning, Korea. 5103
DOI: 10.1021/acs.jpclett.6b02441 J. Phys. Chem. Lett. 2016, 7, 5101−5104
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The Journal of Physical Chemistry Letters
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REFERENCES
(1) Peukert, W. Elektrotechnische Zeitschrift 1897, 20, 20−21. (2) Hausmann, A.; Depcik, C. Expanding the Peukert Equation for Battery Capacity Modeling Through Inclusion of a Temperature Dependency. J. Power Sources 2013, 235, 148−158. (3) Cugnet, M.; Dubarry, M.; Liaw, B. Y. Peukert’s Law of a LeadAcid Battery Simulated by a Mathematical Model. ECS Trans. 2009, 25, 223−233. (4) Omar, N.; Van den Bossche, P.; Coosemans, T.; Van Mierlo, J. Peukert RevisitedCritical Appraisal and Need for Modification for Lithium-Ion Batteries. Energies 2013, 6, 5625−5641. (5) Griffith, L. D.; Sleightholme, A. E. S.; Mansfield, J. F.; Siegel, D. J.; Monroe, C. W. Correlating Li/O2 Cell Capacity and Product Morphology with Discharge Current. ACS Appl. Mater. Interfaces 2015, 7, 7670−7678. (6) Doerffel, D.; Sharkh, S. A. A Critical Review of Using the Peukert Equation for Determining the Remaining Capacity of Lead-Acid and Lithium-Ion Batteries. J. Power Sources 2006, 155, 395−400. (7) Rodrigues, S.; Munichandraiah, N.; Shukla, A. K. A Review of State-of-Charge Indication of Batteries by Means of A.C. Impedance Measurements. J. Power Sources 2000, 87, 12−20. (8) Waag, W.; Fleischer, C.; Sauer, D. U. Critical Review of the Methods for Monitoring of Lithium-Ion Batteries in Electric and Hybrid Vehicles. J. Power Sources 2014, 258, 321−339. (9) Simon, P.; Gogotsi, Y. Capacitive Energy Storage in Nanostructured Carbon-Electrolyte Systems. Acc. Chem. Res. 2013, 46, 1094−1103. (10) Yan, J.; Wang, Q.; Wei, T.; Fan, Z. Recent Advances in Design and Fabrication of Electrochemical Supercapacitors with High Energy Densities. Adv. Energy Mater. 2014, 4, 1300816. (11) Zhang, S.; Pan, N. Supercapacitors Performance Evaluation. Adv. Energy Mater. 2015, 5, 1401401.
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DOI: 10.1021/acs.jpclett.6b02441 J. Phys. Chem. Lett. 2016, 7, 5101−5104
