Abstract
Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order differentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order differentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup’s approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of differentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy offered by the Continued Fraction Expansion and the Oustaloup’s approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational amplifiers, operational transconductance amplifiers, current conveyors, and current feedback operational amplifiers realized in commercially available discrete-component IC form, are compared in terms of the most important performance characteristics. The most suitable of them are further compared using the OrCAD PSpice software.
Highlights
Owing to the interdisciplinary nature of the fractional calculus,[1] there is a growing research interest in the development of fractional-order circuits includinglters,[2,3,4,5,6,7,8,9] oscillators,[10,11,12,13] biological tissues emulators,[14] and energy storage devices.[15]
The fractional-order integrator/di®erentiator is approximated by an appropriate integer-order transfer function and the achieved accuracy depends on the order of approximation which re°ects into the circuit complexity required for implementing the corresponding transfer function
According to the provided results, the conclusions are the following: (i) Comparing the accuracy of the 3rd-order Continued Fraction Expansion (CFE) and Oustaloup's approximations, it is obvious that CFE is more e±cient
Summary
Owing to the interdisciplinary nature of the fractional calculus,[1] there is a growing research interest in the development of fractional-order circuits includinglters,[2,3,4,5,6,7,8,9] oscillators,[10,11,12,13] biological tissues emulators,[14] and energy storage devices.[15]. The value of the conventional inductance (L) in Henry equivalent in its impedance at specic frequency (!) to the fractional-order inductor can be obtained by the formula in (4). Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators emulator and a Generalized Impedance Converter (GIC).[23,6] Another solution, offering more design °exibility than that o®ered by the previous one, is the employment of fractional-order integrator/di®erentiation stage and, an appropriate voltage-to-current (V =I) converter.[24,25] The fractional-order integrator/di®erentiator is approximated by an appropriate integer-order transfer function and the achieved accuracy depends on the order of approximation which re°ects into the circuit complexity required for implementing the corresponding transfer function.
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