90
Electrochemical Supercapacitors for Energy Storage and Delivery
The overall charge (Q S(stack) ) density accumulated over the stack can be calculated as:
n
n
Q ( ) = C ( ) V S stack ) =
C i
S stack
S stack
(
∑ ∑ V i
(2.84)
1
1
If all individual supercapacitors are identical, the total charge can be
expressed as:
C i
Q ( ) = nV i = C V i
(2.85)
S stack
i
n
Therefore, the overall charge in a series stack is equal to that of a single cell’s
charge. If all individual cells are not identical, the cells having small capacitances would weigh more in total capacitance and charge. As to the specific
energy and power densities of a stack, the following equations can be written
according to the individual cell’s expressions if all individual cells are identical:
1
2
( )
E
(
= ( )
C n V
(2.86)
m
sp
1
S stack )
i
2
1 nV i
2
P m (
=
(Note that (R esr ) P(stack ) = nR esr )
(2.87)
( )
s
S stack )
4m R esr
Equations (2.86) and (2.87) indicate that the energy and power densities of a
stack containing n identical single cells are the sums of all individual cells’
energy and power densities.
2.7.2 Stacking in Parallel
If a number of single supercapacitors are connected in parallel, all single
cells should be the same, equal to the stack’s voltage (V P(stack) ). The capacitance
of the entire stack (C P(stack) ) can be expressed as:
n
C P stack ) = C 1 + C 2 + C 3 + ..... = ∑ C i
(2.88)
(
1
If all individual cells are identical, the stack capacitance should be:
C P(stack) = nC i
(2.89)