  Citation: Tsouvalas, E.; Kapoulea, S.; Psychalinos, C.; Elwakil, A.S.; Juriši´ c, D. Electronically Controlled Power-Law Filters Realizations. Fractal Fract. 2022, 6, 111. https://doi.org/ 10.3390/fractalfract6020111 Academic Editor: Riccardo Caponetto Received: 12 January 2022 Accepted: 11 February 2022 Published: 14 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article Electronically Controlled Power-Law Filters Realizations Errikos Tsouvalas 1 , Stavroula Kapoulea 1 , Costas Psychalinos 1, * , Ahmed S. Elwakil 2,3,4 and Dražen Juriši´ c 5 1 Department of Physics, Electronics Laboratory, University of Patras, GR-26504 Rio Patras, Greece; up1055602@upnet.gr (E.T.); skapoulea@upnet.gr (S.K.) 2 Department of Electrical and Computer Engineering, University of Sharjah, Sharjah P.O. Box 27272, United Arab Emirates; elwakil@ieee.org 3 Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza 12677, Egypt 4 Department of Electrical and Software Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada 5 Faculty of Electrical Engineering and Computing, University of Zagreb, HR-10000 Zagreb, Croatia; drazen.jurisic@fer.hr * Correspondence: cpsychal@upatras.gr Abstract: A generalized structure that is capable of implementing power-law filters derived from 1st and 2nd-order mother filter functions is presented in this work. This is achieved thanks to the employment of Operational Transconductance Amplifiers (OTAs) as active elements, because of the electronic tuning capability of their transconductance parameter. Appropriate design examples are provided and the performance of the introduced structure is evaluated through simulation results using the Cadence Integrated Circuits (IC) design suite and Metal Oxide Semiconductor (MOS) transistors models available from the Austria Mikro Systeme (AMS) 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. Keywords: non-integer order filters; power-law filters; curve-fitting approximation technique; opera- tional transconductance amplifiers; tunable filters; CMOS analog integrated circuits 1. Introduction The replacement of the Laplacian operator by its fractional-order counterpart (i.e., s → s α , where 0 < α < 1) has been broadly utilized for transposing integer-order transfer functions into the fractional-order domain [1]. Therefore, the rational approximation of the resulted fractional-order filter functions has gained a significant research interest [2–6]. The power-law filters constitute an alternative way for deriving fractional-order trans- fer functions, without employing the approximation of the fractional-order Laplacian operator. In particular, power-law filters are based on the employment of transfer func- tions, which are derived from their integer-order counterparts, raised to a non-integer exponent [7]. Accordingly, starting from an integer-order transfer function H m (s), which will be denoted as the mother function hereinafter, the resulting power-law transfer function can be expressed by (1): H(s)=[ H m (s)] α , (1) where the magnitude and phase responses are related to those of the mother function as: | H(ω)| =[| H m (ω)|] α , (2a) ∠H(ω)= α · ∠H m (ω) , (2b) The insertion of the non-integer exponent α offers an additional degree of freedom, that allows precise adjustment of the filter’s characteristics, including the cutoff frequency and the slope of the transition from the passband to stopband. In particular, the derived frequency responses exhibit an attenuation gradient scaled by a factor equal to the order of the filter, compared to the corresponding response of the mother filter function [8–10]. Fractal Fract. 2022, 6, 111. https://doi.org/10.3390/fractalfract6020111 https://www.mdpi.com/journal/fractalfract