Environmental Engineering Reference
In-Depth Information
The capacitor, by virtue of its characteristic energy density, has available an
amount of energy dictated by its voltage swing and active mass. The charge/dis-
charge efficiency of the capacitor system is taken as
h
c
¼
0.9. The voltage swing in
a direct parallel connection is limited to the battery minimum voltage to nominal
voltage ratio,
s ¼
0.78, for an NiMH pack. The available capacitor energy is then
E
c
-
avail
¼ h
c
ð
1
s
2
Þg
E
c
m
c
(J)
ð
10
:
35
Þ
It should be evident from (10.35) that a direct parallel (tandem) connection has
severely restricted available capacitor energy due to the battery chemistry limited
voltage swing (e.g. ultra-capacitor voltage is clamped). If an independent power
processor interfaces the capacitor to the voltage bus, then its voltage swing
is limited only by the minimum input voltage of its power converter, which
is typically one-third the operating voltage. This means
s¼
0.33 and virtually
all the energy of the capacitor system is available (89% can be discharged in
this case).
In the parallel connection, Figure 10.27(a), the capacitor must deliver the full
peak power since the battery (1) cannot respond as quickly as the ultra-capacitor
and (2) the ultra-capacitor has much higher pulse power capability. This scenario
represents the power limited case for which the ultra-capacitor mass is given as
P
pk
h
c
g
P
c
m
c
¼
ð
kg
Þ
ð
10
:
36
Þ
Now, if the capacitor energy available, as given in (10.35), is less than or equal
to the energy required, as given in (10.33), the system will be energy limited.
Visualize the distinction between power limited and energy limited as follows. If,
in the system described in Figure 10.28, ultra-capacitor has excess power beyond
the system target but it just meets the energy demand, then it will be energy limited.
If, however, the capacitor just meets the power target but has excess energy still
available, then it is power limited. Neither case is optimal since in both instances
the capacitor energy storage component is more capable in power needs or has
excess mass and the system will be heavier than necessary. Setting (10.35) equal to
(10.34), we obtain
P
pk
t
m
ð
1
=
2
Þ
t
m
g
P
b
m
b
3,600
h
c
ð
1
s
2
Þg
E
c
m
c
ð
kWh
Þ
ð
10
:
37
Þ
Equation (10.37) states that for a given mass of ultra-capacitor, the system is
energy limited because the specification demand exceeds the combined energy of
both the ultra-capacitor and battery. Rewriting (10.37) to clarify and using the
relations in (10.34), we obtain
1
2
t
m
P
b
P
pk
t
m
3,600
h
c
ð
1
s
2
Þ
E
c
avail
þ
ð
kWh
Þ
ð
10
:
38
Þ
