Environmental Engineering Reference
In-Depth Information
Equation (10.68) describes the relation between model parameters in light of
the definition of porosity. It can be seen from the model that, during stand time
after a charge, double layer charge will continue to redistribute between the indi-
vidual branches, which indeed is the case with a physical ultra-capacitor. Model
parameters are extracted from pulse testing by application of constant current
pulses of short duration to charge the first branch of the model. After the current
source is disconnected, the charge redistributes to the delayed and long term
branches. During this redistribution the delayed branch parameters are extracted.
Finally, the long term branch with time constant on the order of hours is calculated
by noting the open circuit voltage decay representing the phenomena.
A perhaps less refined model, but one that is more behavioural, is illustrated
in Figure 10.51 and consists of a series resistance, a parallel impedance and a
main series capacitance. This model is derived from EIS measurements, where R s
models the electrolyte, R p and C dl the charge transfer resistance and C a the bulk
capacitance [16].
R p
R s
C dl
C a
Figure 10.51 EIS derived behavioural model of ultra-capacitor (University of
Toronto model)
Recent work on ultra-capacitor modelling involving EIS at the Aachen Uni-
versity of Technology was reported by S. Buller et al . [40]. In Buller's work the
complex plane representation of measured impedance is taken for four different
voltages and six different temperatures at frequencies ranging from 10 m Hz to
6 kHz. The premise of this work, and the model resulting from it, is that simpler
models do not accurately portray the dynamic effects of ultra-capacitors nor their
energy efficiency during dynamic current profiles such as that exhibited in a hybrid
powertrain. A new approach is described in which the ultra-capacitor is char-
acterized in the frequency domain, and the resulting four experimental model
parameters are adequate to then derive a distributed time constant time domain
model with ten or more RC time constants. In the frequency domain the model is
very simple and illustrated in Figure 10.52.
The Aachen model in the frequency domain gives very good results for ultra-
capacitor behaviour in dynamic applications. The model parameters are similar to
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