Reevaluation of Performance of Electric Double-layer Capacitors from Constant-current Charge/Discharge and Cyclic Voltammetry

TLDR: This paper revisits the calculation of capacitance, power and energy of EDLCs from the time domain constant-current step response and linear voltage waveform, under the assumption that the device behaves as an equivalent fractional-order circuit consisting of a resistance Rs in series with a constant phase element (CPE(Q, α).

Abstract: The electric characteristics of electric-double layer capacitors (EDLCs) are determined by their capacitance which is usually measured in the time domain from constant-current charging/discharging and cyclic voltammetry tests, and from the frequency domain using nonlinear least-squares fitting of spectral impedance. The time-voltage and current-voltage profiles from the first two techniques are commonly treated by assuming ideal RsC behavior in spite of the nonlinear response of the device, which in turn provides inaccurate values for its characteristic metrics [corrected]. In this paper we revisit the calculation of capacitance, power and energy of EDLCs from the time domain constant-current step response and linear voltage waveform, under the assumption that the device behaves as an equivalent fractional-order circuit consisting of a resistance Rs in series with a constant phase element (CPE(Q, α), with Q being a pseudocapacitance and α a dispersion coefficient). In particular, we show with the derived (Rs, Q, α)-based expressions, that the corresponding nonlinear effects in voltage-time and current-voltage can be encompassed through nonlinear terms function of the coefficient α, which is not possible with the classical RsC model. We validate our formulae with the experimental measurements of different EDLCs.

Full text

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Scientific REPORTS | 6:38568 | DOI: 10.1038/srep38568
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Reevaluation of Performance of
Electric Double-layer Capacitors
from Constant-current Charge/
Discharge and Cyclic Voltammetry
Anis Allagui
1,2
, Todd J. Freeborn
3
, Ahmed S. Elwakil
4,5
& Brent J. Maundy
6
The electric characteristics of electric-double layer capacitors (EDLCs) are determined by their
capacitance which is usually measured in the time domain from constant-current charging/discharging
and cyclic voltammetry tests, and from the frequency domain using nonlinear least-squares tting of
spectral impedance. The time-voltage and current-voltage proles from the rst two techniques are
commonly treated by assuming ideal R
s
C behavior in spite of the nonlinear response of the device,
which in turn provides inaccurate values for its characteristic metrics. In this paper we revisit the
calculation of capacitance, power and energy of EDLCs from the time domain constant-current step
response and linear voltage waveform, under the assumption that the device behaves as an equivalent
fractional-order circuit consisting of a resistance R
s
in series with a constant phase element (CPE(Q, α),
with Q being a pseudocapacitance and α a dispersion coecient). In particular, we show with the
derived (R
s
, Q, α)-based expressions, that the corresponding nonlinear eects in voltage-time and
current-voltage can be encompassed through nonlinear terms function of the coecient α, which is not
possible with the classical R
s
C model. We validate our formulae with the experimental measurements of
dierent EDLCs.
e soaring demand for portable consumer electronic products and alternative energy vehicles created a unique
market place for electrochemical energy storage in double-layer capacitors (EDLC). EDLCs are known for their
high power density and high degree of reversibility, an energy density that bridges the gap between conventional
electrostatic/electrolytic capacitors and rechargeable batteries, and long-term self-discharge, while remaining low
cost and environmentally compatible devices
1
. Unlike batteries and fuel cells that harvest the energy stored in
chemical bonds through faradic reactions, the outstanding properties of EDLCs are principally the result of the
nanometer-sized electrostatic charge separation at the interface between the large surface area porous electrode
material and the electrolyte
2
. However, Chmiola et al. showed that sub-nanometric pore size smaller than the sol-
vated ions (such as carbide-derived carbon) can drastically increase the energy being stored in the device
3
, which
challenges the widely accepted traditional charge storage mechanism
4,5
. It has also been proven that the hybrid
conguration, in which the electrodes are made out of porous carbon material for surface-based double-layer
capacitance combined with battery material for volume-based pseudocapacitance, is an eective approach to
enhance the energy density of the storage devices
6–11
. us, because the energy and power performance of EDLCs
are determined by the capacitance of its electrodes, and the ionic and electronic charge transports in the cell
12
,
most of the research studies are focused on the rational design and optimization of nanostructured materials,
electrolytes, and auxiliary components
1,3,11,13,14
.
Notwithstanding, in contrast with conventional capacitors that have been available for over a century, the
measurement methods for determining the main metrics of EDLCs, i.e. capacitance, internal resistance, stored
energy and power, are still not properly standardized
15,16
. As a result, we see that the estimation of such parameters
1
Dept. of Sustainable and Renewable Energy Engineering, University of Sharjah, PO Box 27272, Sharjah, UAE.
2
Center for Advanced Materials Research, University of Sharjah, PO Box 27272, Sharjah, UAE.
3
Dept. of Electrical and
Computer Engineering, University of Alabama, PO Box 870286, Tuscaloosa, USA.
4
Dept. of Electrical and Computer
Engineering, University of Sharjah, PO Box 27272, Sharjah, UAE.
5
Nanoelectronics Integrated Systems Center
(NISC), Nile University, Cairo, Egypt.
6
Dept. of Electrical and Computer Engineering, University of Calgary, Alberta,
Canada. Correspondence and requests for materials should be addressed to A.A. (email: aallagui@sharjah.ac.ae)
received: 02 September 2016
Accepted: 10 November 2016
Published: 09 December 2016
OPEN

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Scientific REPORTS | 6:38568 | DOI: 10.1038/srep38568
from the commonly employed steady-state and impulse electroanalytical techniques
15–18
are most of the time
adapted from the formulae used for ideal capacitors. For instance, with galvanostatic charge/discharge, which
consists of studying the transient voltage response of the device when a stepping current I
cc
is applied, the average
capacitance is usually calculated from ref.15:
=C
dV dt
I
/
(1)
cc
with dV/dt being the slope of the time-voltage curve. e capacitance is typically measured from the response of
the device under dierent values of I
cc
. With cyclic voltammetry (CV) experiment, the current is recorded vs. a
linearly changing cell voltage between the two terminals of the device, giving qualitative and quantitative infor-
mation about the electrode processes. e integral capacitance C of the target electrode can be calculated from
the CV curves as
18
:
∫
=
∆
C
VdVdt
IdV
1
(/)(2)
with the numerical integration of the current being over the half-cycle potential window (∆ V), and dV/dt is
the voltage scan rate
15
. Since C depends on the sweep rates in the CV test, the device is usually charged and dis-
charged at dierent rates and thus at dierent powers
19
. e stored energy is estimated from the dc capacitance
and the voltage window as E = C∆ V
2
/2, and the power from the ohmic drop as p = V
2
/4R
s
, by assuming an ideal
capacitive behavior
11
.
However, from the very specic name of EDLC devices (lossy capacitors, leaking capacitors, or pseudocapaci-
tive devices), it is actually misleading to assume their ideality which is what is being done when using equations1
and 2. In the frequency domain, EDLC devices exhibit constant phase or fractional power characteristics dierent
from the response of ideal capacitors
20
. In Fig.1(a) we show the Nyquist plot representation of impedance spec-
troscopy (EIS) for two EDLCs subject of this study, i.e. a Cooper Bussmann PowerStor supercapacitor (denoted
PS, rated as 2.7 V with 3 F nominal capacitance and 0.060  maximum equivalent series resistance (ESR) at 1 kHz)
and a NEC/TOKIN supercapacitor (denoted NEC, rated as 5.5 V with 1 F nominal capacitance and 65  maxi-
mum ESR at 1 kHz). e impedance response of the PS device is typical for an equivalent series resistance (R
s
)
in series with a constant-phase element (CPE) behavior
21–26
, as it consists of a straight line of slope 13.17 /
(i.e. 85.6°) with R
s
= 40 m (Im(Z) = 0) at 5.3 kHz. Complex nonlinear least-squares tting of the impedance
response using the model R
s
–CPE (Z
CPE
= 1/Qs
α
in which the pseudocapacitance Q is in units of F s
α−1
, s = jω,
and the dispersion coecient α can take on values between 1, for an element acting as an ideal capacitor, to 0, for
a resistor
24
) resulted in (R
s
, Q, α) = (50 m, 2.04 F s
α−1
, 0.95). Although α is very close to one, the device cannot
be considered ideal and energy dissipation is expected. e impedance response of the NEC EDLC, on the other
hand, shows a nonlinear behavior with more deviation from ideality, including a depressed semi-circle of 3.44 
diameter representing, a 43.0°-inclined pseudo-Warburg region (824 Hz to 28 mHz), and a quasi-vertical line
of 85.4° inclination vs. the real axis in the low frequency region 28 mHz to 5 mHz. e R
s
-CPE tting plot to the
experimental data from 824 Hz to 5 mHz, and its parameters are shown in Fig.1, although a double-dispersion
model with two CPEs would be a better t
11
. e dispersion coecient α is found to be 0.70, further away from
ideal capacitor. e Bode diagrams, i.e. phase angle shi of impedance vs. frequency, of both PS and NEC EDLCs
are plotted in Fig.1(b), and from which one can deduct more detail on the electric behaviors of the devices. e
capacitive behavior manifests itself only close to the dc limit where the phase angle vs. log(|f|) tends to − 90°. e
PS EDLC shows a close-to-ideal capacitance over a wider frequency range. In an intermediary frequency region
that spans a few decades, the devices show a tendency towards a resistive behavior with loss of capacitance, as the
porous electrode material is not allowed to be fully charged.
Now giving that in the frequency domain EDLCs exhibit constant phase behavior, it is incorrect to revert
to R
s
C modeling for the analysis of their behavior in the time domain
20
. Furthermore, although EIS is a pow-
erful characterization tool to evaluate the equivalent electric circuits of dynamic processes, frequency response
analysis remains quite expensive from hardware and soware viewpoints. In this study, we show how to accu-
rately characterize an EDLC device by extracting its equivalent circuit element parameters from the time domain
constant-current charge/discharge responses and cyclic voltammetry using fractional-order calculus. We adopted
Figure 1. (a) Nyquist representation of impedance and (b) Bode diagrams for the NEC and PS EDLCs. Plots
and parameters using complex nonlinear least-squares tting to R
s
–CPE(Q, α) model are also shown.

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Scientific REPORTS | 6:38568 | DOI: 10.1038/srep38568
the R
s
−CPE(Q, α) model which oers one extra degree of freedom when compared to the commonly used R
s
C
model to better accommodate the response of the device. Our results on the time-domain responses of EDLCs
have the merit to (i) provide additional and inexpensive alternatives to the standard EIS technique, while (ii)
redening the metrics of these devices (capacitance, power and energy) that are commonly and wrongly adopted
from R
s
C models.
Constant-Current Charging/Discharging
When the impedance Z(s) = R
s
+ 1/Qs
α
of an equivalent electric model R
s
–CPE(Q, α) is excited by an input cur-
rent step of amplitude I
cc
> 0 (note that the same analysis below can be applied for a constant-current discharging
step), the output voltage is given by:
=+





+






α
Vs
ss
R
Qs
()
VI
1
(3)
s
0cc
where V
0
is the initial voltage. V(s) can be re-arranged to:
=+






+






α
−
Vs
s
R
sQ
s
s
()
V
I
1
(4)
s0
cc
1
We now apply the inverse Laplace transform to equation4 using the formula
27
:














=±
αβ
α
αβ
αβ
α−
−
+
+−

ks
sa
tEat
!
()
()
(5)
k
kk1
1
1
,
()

where E is the Mittag-Leffler function. Noting that k = 0, a = 0, and β = α + 1, this yields the time domain
equation:
=+





+






α
αα+
Vt R
t
Q
E() VI (0)
(6)
s0cc,1
ii
in which
Γ=+α
αα+
E (0)1/(1)
,1
ii
, which reduces to the voltage-time characteristics of an R
s
–CPE equivalent
model as:
=+






+
Γ+α






α
Vt R
t
Q
() VI
(1 )
(7)
s0cc
We verify that for an ideal capacitor (i.e. α = 1 and Q = C in Farads) we have the linear relationship:
=+




+





Vt R
t
C
() VI
(8)
s0cc
which leads to the capacitance expression in equation1 by setting R
s
= 0. Note that an eective capacitance
28
in
units of Farad can be dened for an EDLC by equating equations7 and 8 giving:
=Γ +α
α−
CQ t(1 )
(9)
eff
1
Figure2(a)and(d) show in solid lines the h cycles of voltage-time response (the rst few cycles are usually
disregarded because of undened initial conditions) collected at dierent dc charge/discharge currents for the PS
and NEC EDLCs, respectively. All plots are practically symmetric showing rst a steep increase of voltage due
to dissipation in the equivalent series resistance R
s
, followed by a second quasi-linear stage corresponding to the
charging of the pseudocapacitive material of the device. e discharge curves are also nonlinear by exhibiting
rst a quick voltage drop that increases with the increase of current rate, followed by a nonlinear capacitive region
until zero voltage. ese nonlinearities in voltage-time curves are characteristic features of the electric behavior of
porous electrode capacitors
29
, that can not be properly captured by an R
s
C model. e deviation from ideality can
also be demonstrated from the discrete Fourier transform analysis of harmonics obtained from the decomposi-
tion of current waveforms, as shown in Fig.S1. In particular, for NEC (see Fig.S1(c)and(d) showing the spectral
amplitude of measured charging and discharging current signals respectively), the non-fundamental components
extending to a few tens of mHz contribute with relatively important weights (vs. the magnitude of the fundamen-
tal harmonic) to the overall signals. is is most noticeable with the increase of current charge/discharge rates (see
frequency response from ± 25 mA waveforms). e eect of these frequency components of the power system
harmonics can be demonstrated in connection with the Nyquist plot of impedance shown in Fig.1(a). Around
the knee frequency of ca. 28 mHz the impedance of NEC changes from the pseudo-Warburg impedance ((R
s
, Q,
α) = (8.06 , 0.15 F s
α−1
, 0.47) over the frequency range 28 mHz–824 Hz) to a close-to-ideal capacitive behavior
((R
s
, Q, α) = (16.6 , 0.56 F s
α−1
, 0.93) over the frequency range 5 mHz–28 mHz). is knee frequency is one of the
harmonics resulting from the Fourier transform analysis shown in Fig.S1(c)and(d), which means that the electric
characteristics of the device extracted from galvanostatic charge/discharge measurements are averaged integral
values over the frequency domain that take into account the contributions of both capacitive and pseudo-Warburg
regions. When the knee frequency is within the interval set for tting in the Nyquist plot, we clearly see that
both Q and α decrease consequently, e.g. (R
s
, Q, α) is equal to (16.6 , 0.56 F s
α−1
, 0.93) over the frequency

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Scientific REPORTS | 6:38568 | DOI: 10.1038/srep38568
range 5 mHz–28 mHz, (9.62 , 0.29 F s
α−1
, 0.74) over the range 5 mHz–824 Hz, and (8.18 , 0.25 F s
α−1
, 0.70)
over the range 5 mHz–3 MHz (note that the device is not well tted with the model of a series resistance associ-
ated with a CPE over extended ranges of frequency). is eect is not taken into account when the simple R
s
C
model is used to analyze the voltage-time proles of charging/discharging EDLCs.
We show in Fig.2(a)and(d) using dash-dot lines the tted data with equation8 (the average capacitance
values are shown in the second column of TablesS1andS2 for PS and NEC EDLCs respectively, which were
calculated from the slopes of the charge and discharge time-voltage curves using equation1 and in which it
is implicitly assumed that R
s
= 0). e deviation from the experimental data is more pronounced for the NEC
device characterized by a lower dispersion coecient. In dashed lines we show the time-voltage plots of charging
and discharging using equation7, in which the parameters R
s
, Q, and α (summarized in TablesS1andS2) were
extracted from the experimental data using nonlinear least-squares tting with optimization search for global
minimum. It is clear that the R
s
–CPE (Q, α)-modeled data for both PS and NEC are in excellent agreement with
the nonlinear behavior of the experimental charging/discharging of the devices. In contrast with the R
s
C model,
the R
s
–CPE model takes into account the nonlinear behavior of an EDLC through the dispersion coecient α
which cannot be one.
From equation7, we derive the following expressions for the power of an EDLC:
=++
Γ+α
α
pt
Q
t() VI RI
I
(1 )
(10)
0ccscc
2
cc
2
=+ +
C
tVI RI
I
(11)
0ccscc
2
cc
2
eff
e energy is therefore given by:
∫
αα
==++
Γ+ +
α+
Et Vtdt tt
Q
t() I()VIRI
I
(1 )( 1) (12)
t
0
cc 0ccscc
2
cc
2
1
α
=++
+
tt
C
tVI RI
I
(1)
(13)
0ccscc
2
cc
2
eff
2
For V
0
= 0, R
s
= 0 and α = 1, we verify that E(t) = q
2
/2C = CV
2
/2 (q is the stored charge in Coulomb). e accu-
mulated energy under dierent constant-current charge/discharge rates for both PS and NEC devices are plotted
in Fig.2(b)and(e), respectively. e experimental and modeled data using R
s
C and R
s
–CPE are shown in solid,
dash-dot, and dashed lines, respectively. In contrast with the R
s
C-modeled data, the R
s
–CPE model shows excel-
lent agreement with the experiment as a consequence of the proper calculation of the eective capacitance of
the devices given by C
e
= QΓ(1 + α)t
1−α
, and by taking into account their dispersion coecients (see computed
values in TablesS1andS2). In particular, the dependence of maximum power vs. maximum energy (i.e. Ragone
plot, using the expression of C
e
with t = t
ss
the time needed for full charge (or full discharge)) which is plotted
in Fig.2(c)and(d) for PS and NEC respectively, shows a much better agreement with the measurements for the
R
s
–CPE-model when compared to the commonly used R
s
C model.
Figure 2. Constant-current time-voltage (a) and (d), time-energy (b) and (e), and power-energy (c) and
(f) proles of PS and NEC EDLCs respectively. e solid lines represent the experimental data, whereas the
dash-dot lines and the dashed lines represent the tted data using the R
s
C model and the R
s
–CPE(Q, α) model,
respectively.

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Scientific REPORTS | 6:38568 | DOI: 10.1038/srep38568
Cyclic Voltammetry
In this section we revisit the calculation of the metrics of an EDLC from its charging with constant positive dV/dt.
e same procedure can be followed for discharging to simulate the EDLC dynamics in CV experiment. Consider
rst the general case of an applied voltage v(t) across an R
s
–CPE(Q, α) model. e voltage v(t) can be written as:
=+
α
α
vt RQ
dv t
dt
vt()
()
()
(14)
s
Q
Q
in which v
Q
(t) is the voltage across the pseudocapacitance Q. Applying the Laplace transform yields:
=+
α
Vs RQsVs() (1 )()
(15)
sQ
us, for v(t) = V
cc
t/t
ss
;
⩽t t
ss
(V
cc
is the steady-state upper limit of the linear voltage scan that will be reached at
time t = t
ss
), we obtain:
=
+
=
+
α
α
Vs
Vs
RQs
RQss
()
()
(1 )
V
t
1
(1 )
(16)
Q
s
s
cc
ss
2
With a = 1/R
s
Q, we can write:
=




+




α
α
α
sV s
s
as
sa s
()
V
t
1
()
(17)
Q
cc
ss
Using the inverse Laplace transform we get:
∫
=−
α
α
α
α
dv t
dt
a
Eatdt
()
V
t
()
(18)
Q
cc
ss
where E is again the Mittag-Leer function. Now if we dene a = b
α
(i.e. b = (1/R
s
Q)
1/α
), then for signicantly
small R
s
such that b → ∞ , it is possible to write the following expansion
30
:
∫
∫
ΓαΓα
=−
=




−
−
−
+




α
α
α
α
α
α
αα−−

Q
dv t
dt
Q
b
Ebtdt
Q
b
bt bt
dt
()
V
t
[
()
]
V
t
()
(1 )
()
(1 2)
(19)
Q
cc
ss
cc
ss
2
where Γ( · ) is Euler’s gamma function. e le-hand side of equation19 is equal to the current i(t) = dq(t)/dt ,
which aer integration with respect to time must be equal to the charge on a CPE alone if R
s
→ 0 (i.e. b → ∞ ):
Γα
=
−
α−
qt Q
t
()
V
t(3)
(20)
cc
ss
2
Note that at steady-state the charge
Γα
=−
=
α−
qQ C(t )Vt/(3 )V
ss cc ss
(1 )
effcc
, where C
e
is an eective capaci-
tance in units of Farad dened as:
Γα
=
−
α−
C
Q t
(3 )(21)
eff
ss
(1 )
en the current can be found to be:
Γα Γα
=




−
−
−
+




αα
α
−−
it Q
tt
b
()
V
t(2) (2 2)
(22)
cc
ss
112
Γα
Γα Γα
=
−




−
−
−
+




α
αα
α
−
−−

tt
b
C
(3 )
t
(2 )(22)
V
(23)
ss
(2 )
112
effcc
We verify that for an ideal capacitor, i.e. R
s
= 0, α = 1 and Q = C , the current-time relationship given by equa-
tion23 reduces to i(t) = CV
cc
/t
ss
. It is worth mentioning here that the eective capacitance is a quantity that
depends on the way the EDLC device has been excited (compare equations9 and 21) and eventually on the elec-
tric model being used.
In Fig.3(a)and(d), we show in solid lines the h half-cycles of current-voltage responses measured at
dierent positive voltage scan rates (i.e. 2, 5, 10, 20, and 50 mV s
−1
) for PS and NEC EDLCs respectively. e
voltammograms are close, but not ideally rectangular in shape, which is characteristic of non-ideal capacitors (for
pure voltage-independent capacitive behavior, the current is linear with the voltage sweep rate and the voltam-
mograms are ideal rectangles with mirror-image symmetry with respect to the zero current axis
31
). e increase
of distortion, i.e. increase of positive slope of voltammetric current responses, is more noticeable for NEC which
exhibits further deviation from ideality (Fig.3(d)), and increases with the increase of voltage sweep rate for both