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Series RC
E del / E total
electrochemical
capacitor
0.1
0.001 0.01 0.1 1
Pave/ P max
Figure 34 Energy and power relationship for series RC and actual electrochemical capacitor.
The dissipated energy E dis
in a series-RC circuit can also be derived for a series-RC circuit
during a constant-current discharge from V 0
to V 0
/2. This equation is:
Edis
Edel
=
9 ⎡ 8 Pave ⎤
⎢1
− 1−
⎥
4 ⎣ 9 P max ⎦
⎛ Pave ⎞
⎜ ⎟
⎝ P max ⎠
2
The dotted line in Figure 34 shows this as E dis
/E del
. In the case of an electrochemical
capacitor the figure shows that a relatively high percentage of the total energy may not be
available at high power discharge rates.
Distribution of Cell Voltages
It is useful to consider the hypothetical distribution of voltage among cells in a seriesconnected
string as shown in Figure 35. The equations describe the relationship between
the average cell voltage, V a
, and the critical voltage, V c
, above which a cell will fail.
Since no cell in the string can experience a voltage greater than V c
, the average voltage
must be below V c
by an amount that depends on the width of the voltage distribution. So,
without active or passive cell voltage balancing, symmetric capacitor cells must be derated
such that the average cell voltage is V a
< V c
/(1+T/100) where T is the tolerance of
the batch of cells in percent. Such de-rating should be kept to the absolute minimum
because it decreases the stored energy in the device. For example, energy density is
reduced by the factor of (1 + T/100) -2 compared to the energy density of a single cell
operating at its rated voltage. Thus, when the average cell voltage is de-rated by 10%,
the energy density is reduced by over 17%.
Electrochemical Capacitors 64