Environmental Engineering Reference
In-Depth Information
(10.69) expands into a power series that represents an
RC
ladder network. The time
domain equivalent is
k
2
1
n
p
j
w
k
1
k
2
þ
2
k
1
k
1
j
w
k
2
k
1
e
ð
n
2
p
2
k
1
=
k
2
Þ
t
p
coth
¼)
¼
1
p
t
ð
10
:
70
Þ
k
1
¼
C
t
C
k
2
¼
Equation (10.70) in the time domain represents a fixed capacitor in series with an
infinite number of
RC
parallel networks. This is reminiscent of the advanced battery
distributed model shown in Figure 10.46, which was obtained from DIS. In the case of
the ultra-capacitor, the time domain model (10.70) expands into Figure 10.56.
R
p
1
R
p
N
R
i
L
C
C
/2
C
/2
Z
uc
(
t
)
n
= 1, ...
n
= N
Figure 10.56 Ultra-capacitor time domain model (after Reference 40)
The various values for the parallel network are obtained from (10.70) by set-
ting
n
¼
1, 2, 3, . . . ,
N
. From Reference 40 the values for
C
are defined as
k
2
k
1
C
¼
ð
10
:
71
Þ
k
2
2
k
1
¼
C
2
C
n
¼
The distributed parallel resistances are each defined in terms of the frequency
domain time constant and capacitance value, but with each successively higher
term divided by
n
2
. This is a very promising model for describing the dynamic
behaviour of ultra-capacitors. Global coordination of ultra-capacitor standards,
regulatory matters and education and outreach fall under the auspices of the Kilo-
Farad International organization consisting of manufacturers and users [41].
It is instructive to view the ultra-capacitor model from a network synthesis
vantage point [42]. In the context of a network, the MIT short term model [43] can
be slightly modified into a Foster II network having three time constants. Schindall
et al
. [43] describe the ultra-capacitor in terms of short term (
<
3,000 s) behaviour
using a slow, medium and fast time constant approach. The Foster II equivalent
circuit model can be converted to a Cauer I network by taking the continued
