Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters

Abstract

Summary Approximations of the fractional‐order differentiator and integrator operators s±r are proposed in this work. These approximations target the realization of these operators using standard active filter transfer functions. Hence, circuit implementations in integrated circuit form or in discrete component form are significantly facilitated. Complementary metal‐oxide‐semiconductor (CMOS) realizations of the proposed approximations are given and validated via simulations using the AMS 0.35 μm CMOS technology, while experimental results using operational amplifier circuits are tested and confirm the proposed theory.

Reconnection–less reconfigurable low–pass filtering topology suitable for higher–order fractional–order design

“…As mentioned before, the proposed filter was implemented with RC networks to substitute the function of fractional–order capacitors due to rather high complexity of the constructed structure, unless it is all designed on one chip, and the complexity of the resulting control necessary for proper function. The electronic control of the fractional order and thus, the reconnection–less reconfiguration of both the integer and fractional order can be achieved by the replacement of the RC ladder structures by electronically adjustable emulators of a fractional–order capacitor/inductor introduced in [57] , [29] , [30] , [31] , for example. The other possible solution is to replace the building blocks (OTA + C) of the designed structure i.e.…”

“…In recent years, the matter of the fractional–order (FO) calculus received an increased attention of many scientists due to its possible utilization in various spectrums of industry branches including medicine [1] , [2] , [3] , agriculture [1] , [4] and, of course, electrical engineering [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] giving many new potential applications a chance to arise. In comparison to the integer–order circuits, FO circuits provide an increased degree of freedom, due to the presence of the non–integer–order parameter ( α ).…”

“…There have been some reports of a fabrication of fractional–order capacitors [34] , [35] , [36] nonetheless, these elements are expensive and difficult to implement and they are not commercially available. Therefore, they are usually substituted by passive elements (RC ladder networks) [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] or electronic emulators [29] , [30] , [31] approximating the FO capacitor (or inductor). The other approach involves the approximation of Laplacian operator of fractional–order s α by an integer–order function [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] .…”

Fracmemristor chaotic oscillator with multistable and antimonotonicity properties

“…Fractional-order differential equations are in the group of nonlinear and complex systems [12] , [13] , [14] . These systems have shown different complex properties such as hyperchaos [15] , self-producing attractors, and strange maps [16] , which enabled them to be used in modeling of biological phenomena, electrical components, controllers, and filters [17] . Multistability and antimonotonicity are two features that have been reported in fractional-order systems [18] .…”

Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters

“…Fractional-order filters, implemented using various types of active elements such as Operational Amplifiers (op-amps), Operational Transconductance Amplifiers (OTAs), second generation Current Conveyors (CCIIs), Current-Feedback Operational Amplifiers (CFOAs) and current-mirrors (CMs), have been already proposed in the literature [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. All of them suffer from the increased number of MOS transistors which are required for implementing the active elements.…”