PhD Thesis - Cranfield University

Chapter 3
where,
C0
1
b ≡ , c ≡ , 2
2αR
αR
C = C − αV
=
0
V is the voltage across the ultracapacitor, C o is the static capacitance at zero voltage, α is an
empirically determined constant and q is the stored charge
Currently, more complex models are being investigated to capture other long-term effects
such as charge leakage and temperature parameters. However, if the required observation of
the model and the actual practical usage of the ultracapacitor is only for relatively of short
cyclic periods, a more simplified equivalent circuit is sufficient as leakage resistance and
other long-term steady state losses may be negligible.
3.19 Ultracapacitor Power and Energy
Sizing of an ultracapacitor system requires the specification of the power and energy
requirements. For a fixed ultracapacitor bank, these quantities dictate the number of
ultracapacitors needed but do not represent the same constraints [108]. The minimum
number of ultracapacitors needed is determined by the energy profile that the
supercapacitive bank has to assume. However, due to the voltage decay property of
ultracapacitors, not all the stored energy can be utilised. Therefore the sizing is based on the
usable energy that the ultracapacitor bank can transfer. Following this, the usable energy can
either be consumed very fast or throughout a long period. How fast the energy can be
extracted determines the power constraint.
The fundamental electrical equations defining an ultracapacitor are:
dv
i = C
( 3-34)
dt
1
Cv
2
2
E = ( 3-35)
87
q
V