64
Electrochemical Supercapacitors for Energy Storage and Delivery
⎛ C C
⎞
C ⎜= ⎜
H , n diff , n
dl , n
⎟ ⎟ ,
⎝ C H , n + C diff ,n ⎠
the total capacitance of the supercapacitor ( C
T
dl ) is:
C C
C
T
dl =
dl, P dl ,n
(2.33)
C dl , P + C dl ,n
For electrochemical double-layer supercapacitors, the two electrodes are
identical, meaning that C dl,P = C dl,n . Therefore, Equation (2.33) can be converted into:
T
1
1
C dl = C , = C
(2.34)
2
dl P
2
dl ,n
The total capacitance of a double-layer supercapacitor is equal to half of the
individual electrode capacitance. Note that Equation (2.34) is applicable only
to symmetric supercapacitors whose two electrodes are identical. It is not
applicable to asymmetric supercapacitors in which the two electrodes are
not identical. However, Equation (2.33) is applicable for both symmetric and
asymmetric supercapacitors. In an asymmetric supercapacitor, the electrode
with the smaller capacitance will dominate the total capacitance.
2.5.2 Equivalent Series Resistance (ESR)
As discussed in Chapter 1, if a sinusoidal alternative current is applied on
an ideal capacitor, the output voltage should be out of phase by 90°, independent of the frequency. However, in a supercapacitor, the output voltage
is normally out of phase fewer than 90°, suggesting that an equivalent series
ohmic resistor is coupled. This ohmic component is defined as the equivalent
series resistance (ESR).
In electrochemical supercapacitors, the ESR is a real series resistance
that involves: the contact resistance between the current collector and
the electrode layer, the resistance of the electrode layer interparticles due
to the porous and particulate nature of the electrode matrix, the resistance of the external lead contact, the resistance of the electrolyte, and
the resistance caused by the dielectric loss of the interphasal solvent and
ions when the AC frequency is higher than hundreds of megahertz (MHz).
Figure 2.13 displays a simple equivalent circuit of a supercapacitor in the
presence of ESR.
As described in Chapter 1, the complex AC impedance of this equivalent circuit ( Z
j
cell ) in Figure 2.13, which is the complex AC impedance of the
Electrochemical Supercapacitors for Energy Storage and Delivery
⎛ C C
⎞
C ⎜= ⎜
H , n diff , n
dl , n
⎟ ⎟ ,
⎝ C H , n + C diff ,n ⎠
the total capacitance of the supercapacitor ( C
T
dl ) is:
C C
C
T
dl =
dl, P dl ,n
(2.33)
C dl , P + C dl ,n
For electrochemical double-layer supercapacitors, the two electrodes are
identical, meaning that C dl,P = C dl,n . Therefore, Equation (2.33) can be converted into:
T
1
1
C dl = C , = C
(2.34)
2
dl P
2
dl ,n
The total capacitance of a double-layer supercapacitor is equal to half of the
individual electrode capacitance. Note that Equation (2.34) is applicable only
to symmetric supercapacitors whose two electrodes are identical. It is not
applicable to asymmetric supercapacitors in which the two electrodes are
not identical. However, Equation (2.33) is applicable for both symmetric and
asymmetric supercapacitors. In an asymmetric supercapacitor, the electrode
with the smaller capacitance will dominate the total capacitance.
2.5.2 Equivalent Series Resistance (ESR)
As discussed in Chapter 1, if a sinusoidal alternative current is applied on
an ideal capacitor, the output voltage should be out of phase by 90°, independent of the frequency. However, in a supercapacitor, the output voltage
is normally out of phase fewer than 90°, suggesting that an equivalent series
ohmic resistor is coupled. This ohmic component is defined as the equivalent
series resistance (ESR).
In electrochemical supercapacitors, the ESR is a real series resistance
that involves: the contact resistance between the current collector and
the electrode layer, the resistance of the electrode layer interparticles due
to the porous and particulate nature of the electrode matrix, the resistance of the external lead contact, the resistance of the electrolyte, and
the resistance caused by the dielectric loss of the interphasal solvent and
ions when the AC frequency is higher than hundreds of megahertz (MHz).
Figure 2.13 displays a simple equivalent circuit of a supercapacitor in the
presence of ESR.
As described in Chapter 1, the complex AC impedance of this equivalent circuit ( Z
j
cell ) in Figure 2.13, which is the complex AC impedance of the