Environmental Engineering Reference
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R p
R i
L
C dl
Z uc ( jw )
Figure 10.52 EIS frequency domain ultra-capacitor model (Aachen
University model)
those of the advanced battery model in which the distributed RC nature of porous
electrodes is modelled. However, in the Aachen model a total of four parameters
need to be characterized in the frequency domain to obtain excellent agreement
with physical behaviour. Series resistance of the contacts and electrolyte is mod-
elled as R i along with inductance of the terminals and electrodes L . The interesting
part of this model is the parallel impedance representing the complex pore beha-
viour. The distributed ion transfer resistance and double layer capacitance are
modelled as a parallel network. Note that the Aachen model does not include a self-
discharge term, which would help in accuracy over long operating durations, nor
does it include any voltage non-linearity of the double layer capacitor. The fre-
quency domain mathematical model is
p
j wt
t coth ð
Þ
Z uc ð j wÞ¼ R i þ j w L þ
C
p
ðWÞ
ð 10 : 69 Þ
j wt
Equation (10.69) can be solved for a representative ultra-capacitor that was
laboratory characterized at the Aachen Technical Institute (the Aachen model)
using EIS to obtain the following set of four parameters:
R i ¼ 1.883 m W
L ¼ 50 nH
1.67 s
C ¼ 1,130 F
When these parameters are substituted into (10.69) and the complex impe-
dance solved for frequencies ranging from 200 mHz to 2.5 Hz, the following results
are obtained. In Figure 10.53 the lowest frequency is at the upper left.
In Figure 10.54 the lowest frequency is at the upper right. At low frequencies
the ultra-capacitor exhibits nearly pure capacitive effects. As frequency increases
towards 1 Hz, the real component becomes more evident and the complex impe-
dance tends towards the series resistance value of 1.883 m W at high frequency
(2.5 Hz in this plot). The real and imaginary components of Z uc ( j w ) are plotted in
Figure 10.55(a) and (b).
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