Fractional calculus definitions, approximations, and engineering applications

The basic idea behind fractional calculus is that it considers derivatives and integrals of non-integer orders giving extra degrees of freedom and tuning knobs for modeling complex and memory dependent systems with compact descriptions. This paper reviews fractional calculus history, theory, and its applications in electrical engineering. The basic definitions of fractional calculus are presented together with some examples. Integer order transfer function approximations and constant phase elements (CPEs) emulators are overviewed due to their importance in implementing fractional-order circuits and controllers. The stability theory of fractional-order linear systems is outlined and discussed. Four common electrical engineering applications are surveyed. Fractional-order oscillators allow controlling the phase difference, as well as achieving high oscillation frequency independently. Fractional order electronic filters are used to provide non-integer order slopes eliminate the need to round up the filter order and achieve the exact required time and frequency domain specifications. Studying fractional-order bioimpedance models provides better fitting to the measured data from fruits and vegetables. Fractional order DC-DC converter models provide a better estimation of the power conversion efficiency by incorporating frequency-dependent losses. © 2020, Cairo University. All rights reserved.