Fractional Calculus Background
The design of analog electronic circuits is based on implementing
mathematical operations using the components, like ideal resistor,
inductor and capacitor as linear elements. These elements correspond to
well-known operations in calculus and therefore are used to realize
differential equations of any integer order [9-11]. Real electronic
components depart from the ideal behavior and the extent of this
departure is expressed in terms of the Q (quality) factor which accounts
for these issues. The ideal network components provide a fixed phase
shift to all frequencies which pass through them, but all real
components show parasitic impedances which cause the phase difference to
change with the frequency of the signal passing through them. A
fractional impedance or fractance is an impedance whose Laplace
transform has a fractional order of the form ZF FRAC (s, α) = 1
F.sα. Fractional order elements provide a fixed phase
shift over a wide range of frequencies and a slope of -20α dB per decade
on Bode plot. By manipulating the order of the fractional element, a
component which provides a custom phase shift over a wide frequency
range can be created. This new found freedom to control the magnitude
and phase of a network independently has already proven useful in
modelling systems where cumbersome integer order models are required to
reap the benefit provided by simpler fractional order models. Several
complex systems also lend themselves to fractional order analysis for
better understanding [11-14]. An LC circuit containing a fractional
order circuit exhibits a low roll-off of phase response while the
magnitude rolls off at a much gentler rate. A fractional order LC
circuit can therefore reject out of band noise, leading to lower jitter
noise than non-harmonic oscillators. This gives the impression of a much
lower Q factor in the phase response than what is apparent in the
magnitude response. Fractional calculus (FC) has been there since the
time of well-known integral calculus (IC). However, fractional calculus
progressed slowly due to the lack of tools, definitions and complexity
[1-5]. Fractional Order differential equations were first postulated
by Leibniz as a curiosity which might lead to greater insight later
[32-34]. Examples of fractional order impedances occur in biological
samples whenever a fractal structure exists and power law diffusion
occurs. Fractional order capacitors have been artificially made by using
super-capacitors, active impedance generators; fractal design patches
deposited on silicon and carbon nanotube epoxy devices [9],
[17], [11], [18].