Analysis and investigation of CDBA based fractional-order filters
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Analysis and investigation of CDBA based fractional-order filters Gagandeep Kaur1
•
Abdul Quaiyum Ansari1 • M. S. Hashmi2
Received: 1 November 2019 / Revised: 30 May 2020 / Accepted: 22 June 2020 Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, a new design of continuous time current differencing buffered amplifier based low-pass, high-pass, band-pass, all-pass and notch fractional-order filters is reported. The design of proposed filters is based on the approximation of fractional-order filters by using an appropriate integer order transfer function. Signal flow graph approach is used for the realization of fractional-order filters of order (1 ? a). The frequency responses of the proposed circuits are verified using MATLAB in conjunction with SPICE. The evaluation of the realized fractional-order filters are also performed through the AC analysis and corner analysis. Furthermore, stability and sensitivity investigations are also investigated. It is observed from the simulation results that the fractional-order filters are appropriate for IC implementation. Keywords Fractional-order filters Fractional theory CDBA Signal flow graph
1 Introduction Fractional calculus is the branch of mathematics that deals with the derivative and integrals of non integer order [1–3]. Due to the intrinsic characteristic of infinite memory of fractional calculus,the conception of fractional hypothesis has been explored in several engineering applications corresponding to fluid mechanics [4], bioengineering [5], electrical networks [6], electromagnetic theory [7, 8], mechanics [9], biomedical [6, 10], signal processing and automatic control [11–18, 43–49]. Several definitions of fractional derivative such as the Caputo definitions, the Gru¨nwald-Letnikov and the Riemann–Liouville are considered within the literature [1–3]. The Gru¨nwald–Letnikov definition of order a is expressed in (1), where Dt is the
& Gagandeep Kaur [email protected] Abdul Quaiyum Ansari [email protected] M. S. Hashmi [email protected] 1
Department of Electrical Engineering, Jamia Millia Islamia, New Delhi, Delhi, India
2
Department of Electrical and Computer Engineering, Nazarbayev University, Nur-Sultan, Kazakhstan
integration step, while the Riemann–Liouville definition of the fractional derivative is expressed in (2). n X Cðj aÞ Da f ðtÞ,ðDtÞa f ððn jÞDtÞ ð1Þ C ð a ÞCðj þ 1Þ j¼0 da f ðt Þ : dta 1 dn t f ðsÞ r ds n 1\a\n; aþ1n n ¼ Cðn n aÞ dt 0 ðt sÞ d f ðt Þ a ¼ n: dtn
D a f ðt Þ
ð2Þ L 0 dta f ðtÞ ¼ sa F ðsÞ
ð3Þ
. Now, the Laplace transformation of (2) with zero initial conditions yields (3). Therefore the impedance of fractional element is proportional to sa, where a is non integer. The phase difference of such devices is ap/2 if the order is 1, 0 and - 1 they corresponds to inductor, resistor and capacitor respectively. In the planning of FO systems, the realization of FO capacitors (FCs) and FO inductors (FOIs) are the key challenges. It ought to be mentioned at
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