79
Fundamentals of Electrochemical Double-Layer Supercapacitors
2.5.4.7 AC Impedance Equivalent Circuit
The complex impedance of the equivalent circuit ( Z
j
cell ) shown in Figure 2.14
can be written as:
π ( )
2
R
2 f R C
T
Z ce
j
= R +
p
− j
d
esr
p
l
ll
(
)
2
2
(2.63)
1 2
+ πfR
T
T
p d
C l
1+ + ( 2πfR p d
C l)
The magnitude of this complex impedance can be expressed as:
⎛
⎞
2 ⎛
⎜
R p
⎟ ⎜ 2π
2
⎞
2
f R
=
( )
⎜ R esr +
p p
C
T
dl ⎟
Z ell
c
(
2 ⎟ + ⎜
63
⎜
2 ⎟
(2. a)
1 2
+ πfR C
T
) ⎟ ⎜ 1 2
+ ( πfR C
T
⎠
) ⎟
⎝
p dl
⎝
p dl
⎠
Using AC impedance spectroscopy, R esr , R p , and C
T
dl can be obtained simultaneously from the complex plane (Nyquist) plot. Chapter 7 will provide a
more detailed discussion of supercapacitor measurements including the AC
impedance spectroscopic method.
2.6 Energy and Power Densities of
Electrochemical Supercapacitors
2.6.1 Energy Densities
Energy density is one of the most important parameters for evaluating an
electrochemical supercapacitor. In a double-layer supercapacitor, the energy
density can be expressed as:
∫
q
q
2
T
2
C V
q
1 q
1
1
=
dl sc
E
V
T
sc
=
( )
dq = ∫ dq
=
= C V
2
dl
T
T
T
sc
(2.64)
C dl
2 C dl 2 C C dl
2
0
0
where q is the total charge quantity stored in the supercapacitor (C.cm –2 ) and the
double-layer capacitance of the cell (C
T
–2
dl ) is expressed as F.cm . In practical applications, the specific energy density is more popular and useful and is defined as:
1 C
1
E m =
m V
2
= C V
2
sc
sp sc
(2.65)
2 m
2