LETTER Analysis and realization of a switched fractional-order-capacitor integrator Costas Psychalinos 1 , Ahmed Elwakil 2,3 * ,† , Brent Maundy 4 and Anis Allagui 5 1 Electronics Laboratory, Department of Physics, University of Patras, Rio Patras GR 26504, Greece 2 Department of Electrical and Computer Engineering, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates 3 Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt 4 Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta Canada 5 Department of Sustainable and Renewable Energy, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates SUMMARY Using fractional calculus, we analyze a classical switched-capacitor integrator when a fractional-order capacitor is employed in the feed-forward path. We show that using of a fractional-order capacitor, signif- icantly large time constants can be realized with capacitances in the feedback path much smaller in value when compared with a conventional switched-capacitor integrator. Simulations and experimental results using a commercial super-capacitor with fractional-order characteristics confirmed via impedance spectros- copy are provided. Copyright © 2016 John Wiley & Sons, Ltd. Received 29 October 2015; Revised 5 January 2016; Accepted 18 January 2016 KEY WORDS: circuit theory; super-capacitors; switched-capacitor circuits; fractional-order circuits 1. INTRODUCTION A constant phase element (CPE) [1] or a fractional-order capacitor (FC) is essentially a lossy capacitor with less than unity dispersion coefficient, resulting in a current–voltage phase angle less than (π/2) [2, 3]. In this device current, charge and voltage are related by it ðÞ¼ dq=dt ¼ C ˆ d a vt ðÞ=dt a (1) where Ĉ is known as the pseudocapacitance with units Farad sec (a À 1) and a is the dispersion coefficient (0 < a < 1). For ideal capacitive behavior, a ≃ 1 and Ĉ has the exact units of Farads. In general, the value of the capacitance in a FC can only be found at a fixed frequency in the form C = Ĉ/ω (1 À a) . Several recent publications have proposed promising techniques for the realization of fractional capacitors [4, 5] and particularly using nano-materials [6]. Meanwhile, switched-capacitor (SC) circuits are standard blocks important for many applications. It is assumed in such circuits that ideal capacitors are employed with dispersion coefficient a = 1. However, by employing FCs, it is possible to design precise fractional-step SC filters [7]. In [7], a second-order approximation of the FC was used in the design, and all analyses were carried out in the frequency domain. In this work, we use fractional calculus to derive exact time-domain *Correspondence to: Ahmed Elwakil, Department of Electrical and Computer Engineering, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates. † E-mail: elwakil@ieee.org Copyright © 2016 John Wiley & Sons, Ltd. INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. (2016) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cta.2197