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].