Figure 2 : (a) Single stage RC tree. (b) Equivalent RC tree
for fractional order capacitor of any order [8, 16].
Figure 2(a) represents a single stage RC ladder structure for designing
a pseudo-capacitance of a particular fractional order(α) and figure 2(b)
shows an equivalent RC tree formed by cascading many such individual
single stage RC networks as in figure 2(a) for better approximations of
frequency and phase over a particular frequency range.
RC Ladder component values
generation.
Fractional order capacitors are generated for various orders; both
fractance and their frequency response curves are studied herewith. This
is done to ensure whether the designed pseudo-capacitances have
fractional behavior or not in a desired frequency and phase spectrum.
Figure 3 depicts a RC ladder of n equivalent stages. The values
of resistors and capacitors of the progressive ladder structure vary
based on the fractional order, α, for which the fractance has been
designed (keeping the pseudo-capacitance value same). Keeping the
fractance value same, magnitude curves have been plotted for different
fractional orders, α ranging from 0.2 to 0.9, as can be seen in Figure
4. The simulated R and C values for the ladder network have also been
taken for various fractional orders ranging from, α=0.2 to α=0.9. Table
1 depicts the ladder values for same fractance, F=10n, at different
order, α varying from 0.2 to 0.9. Although, the pseudo-capacitance is
same, that is F=10n, the design using different fractional order α
offers different properties to the fractance device. FOCs offer a unique
phase and frequency control by changing α, keeping the magnitude fixed.