Hierarchical Nanostructures: Application to Supercapacitors - Hierarchical Nanostructures for Energy Devices - page 208

In principle, the capacitance of EDLC can be expressed by the following
basic relation:

C ¼ e r e 0 A
d

(9 : 1)

d n 3 r 4 n g | 4

where e r is the dielectric constant of the electrolyte, e 0 is that of the vacuum
(e 0 ¼ 1), A is the surface area of the electrode, and d is the effective thickness
of the double layer developed in the electrolyte. Given a capacitance C, the
amount of charge stored in the electrode can be simply calculated by:

Q ¼ CV (9.2)

where Q is the stored charge and V is the differential potential applied to the
electrodes. While the capacitance of EDLC can be estimated based on eqn
(9.1), the basic parameters lack sucient accuracy because of non-ideal
behavior. Instead, practical evaluation can be made by directly measuring
the apparent capacitance of EDLC.
The capacitance of EDLC can be obtained from galvanostatic charging-
discharging curves (CDCs). In CDC, a potential response to charging-
discharging processes with a constant current is recorded as a V-t curve.
From eqn (9.2), the capacitance can be expressed by the following relation:

I
dV = dt

C ¼ (9 : 3)

According to eqn (9.3), the slope of the CDC (dV/dt) is directly correlated with
the capacitance. In general, discharging curves are used for this purpose.
Alternatively, the capacitance can be calculated from cyclic voltammetry
(CV). In CV curves, the rate of potential change (dV/dt) is kept constant, and
the current response is recorded with respect to the transient voltage (I-V
curves). In contrast to an ideal capacitor that shows a rectangular response
in CV characterization, general EDLCs show curved close-loops. In this case,
the capacitance value is obtained by integrating and then normalizing the
current response.
As discussed in section 9.1, specific energy and specific output power are
most important figures of merit for supercapacitors. In conjunction with the
above equations, the total energy (E) and the power (P) of EDLC can be
expressed as:

.

E ¼ 1

2 CV 2

(9 : 4)

V 2
4R ESR

P ¼ (9 : 5)

where R ESR indicates equivalent series resistance (ESR). ESR originates from
(1) the contact resistance of the electrode and current collector; (2) the in-
ternal resistance of the electrode itself; (3) the resistance of electrolyte
(bulk þ pore); 8

(4) the resistance of the external lead contact. ESR can be