Chapter Eleven - Fractional calculus in electronic circuits: a review

Abstract

Fractional-order (FO) systems offer a unique flexibility of arbitrary slope matching in system gain which is not possible in conventional circuit theory. This has made application of fractional calculus in electronic circuits an emerging research area. This chapter presents a thorough account of the development of FO circuit theory which covers major research topics since the 2000s. The chapter presents a brief discussion on FO circuit elements (fractors), their specifications, and two major lines of realizations of practical fractors. Next, it is highlighted how fractors can be realized in different quadrants of the impedance plane using operational amplifiers (Op-Amps), operational transconductance amplifiers (OTAs), and current feedback Op-Amps (CFOAs). The last two make it a suitable candidate for IC level fabrication. The chapter follows the gradual development of FO filter design theory with comparative studies among different approximation algorithms, realization circuits (Sallen–Key, biquad, and follow-the-leader feedback topologies, among others) and design topologies. Several optimization algorithms which are adopted by different researchers are mentioned in this context. Examples of FO phase-locked loops are discussed as an application of FO filters. Besides, different FO oscillators are presented with a detailed discussion of their design techniques, frequency ranges, and limitations. Special attention has been given to recent trends on FO circuit synthesis which uses OTA, current conveyors, CFOA, and current mirrors, suitable for low-power applications. Overall, this review chapter aims to highlight key features of different FO circuits reported across literature and presents a consolidated analysis on the advantages and limitations.