Environmental Engineering Reference
In-Depth Information
branches of the short term model are retained, but the simulation must solve for
N
(
M
1)
þ
1 nodes instead of just
M
nodes per equivalent circuit.
The short term ultra-capacitor model can also be viewed as a particular case of
a Foster II electrical network consisting of
R
s and
C
s (alternative representation to
Figure 10.56). In its basic form, the admittance function for the Foster II network is
Y
ð
s
s
¼
ð
s
þ
a
1
Þð
s
þ
a
3
Þ
ð
s
þ a
2
Þð
s
þ a
4
Þ
(S)
ð
10
:
72
Þ
k
2
s
s
þ a
2
þ
k
4
s
s
þ a
4
Y
ð
s
Þ¼
Hs
þ
k
0
þ
where the partial fraction expansion of the admittance can be seen to approximate
the short term model of the ultra-capacitor given in Figure 10.57. The values of the
individual coefficients are taken as the residues at the admittance poles;
Y
(
s
¼
0)
¼
k
0
,
Y
(
s
¼
a
2
)
¼
k
2
etc. The parameter
a
is the reciprocal of each time constant.
Figure 10.59 illustrates the Foster II circuit configuration. In this figure the basic
short term model given in Figure 10.57 is slightly modified to fit the Foster II
equivalent by removing the fast branch resistance and combining it with the
terminal ESR. The Foster II model is then transformed to its Cauer I equivalent
using continued fractions.
1/
k
2
1/
k
4
H
1/
k
0
α
k
2
/
2
k
4
/
α
4
Z
(
s
)
Figure 10.59 Foster II network approximation of an ultra-capacitor
The Cauer I circuit representation gives somewhat more insight into the origins
of the three time constant approximation of an ultra-capacitor model. In this
modified form the ESR represents the combined effect of terminations, metal foil
current collectors and its interfacial resistance to the carbon matte electrodes. The
ESR term is separated from the short term model to facilitate the equivalent circuit
transformation and be included as ESR
0
. The resulting equivalent circuit shown in
Figure 10.60 then approximates the highly distributed nature of carbon matte
resistance, ionic conduction and Helmholtz double layer capacitances existing at
macro, meso and micro pores [43].
Table 10.14 lists the parameter values for both the Foster II and Cauer I
equivalent models derived from network synthesis and illustrated in Figure 10.60.
