Complexity

Volume 2018, Article ID 2806976, 10 pages

https://doi.org/10.1155/2018/2806976

## Fractional-Order Memristor Emulator Circuits

^{1}UATx, Mexico^{2}ITESM, Mexico

Correspondence should be addressed to C. Sánchez-López; xm.moc.oohay@xmnaslrac

Received 16 March 2018; Revised 24 April 2018; Accepted 24 April 2018; Published 28 May 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 C. Sánchez-López et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This brief leads the synthesis of fractional-order memristor (FOM) emulator circuits. To do so, a novel fractional-order integrator (FOI) topology based on current-feedback operational amplifier and integer-order capacitors is proposed. Then, the FOI is substituting the integer-order integrator inside flux- or charge-controlled memristor emulator circuits previously reported in the literature and in both versions: floating and grounded. This demonstrates that FOM emulator circuits can also be configured at incremental or decremental mode and the main fingerprints of an integer-order memristor are also holding up for FOMs. Theoretical results are validated through HSPICE simulations and the synthesized FOM emulator circuits can easily be reproducible. Moreover, the FOM emulator circuits can be used for improving future applications such as cellular neural networks, modulators, sensors, chaotic systems, relaxation oscillators, nonvolatile memory devices, and programmable analog circuits.

#### 1. Introduction

Resistors, inductors, capacitors, and memristors are basic network elements and the real behavior of each of them is time-varying and nonlinear [1–3]. For the last three cases, the real behavior of each element has always been modeled from integer-order differential equations. However, it is well known that this kind of modeling is only a narrow subset of fractional calculus, which is a generalization of arbitrary order differentiation and integration, and this last approach can be used to better model the description of natural phenomena [4–8]. In this context, fractional calculus is beginning to be used for describing the behavior of memristive elements and systems, i.e., memristors, memcapacitors, meminductors, and any combination of them. Particularly, few studies have been realized on fractional-order memristors (FOM). Thus, [9] analyzes the FOM state equation behavior when a step signal is applied and demonstrates that by controlling fractional parameters associated with the FOM, the saturation time of the resistance can be controlled. In [10], fractional calculus is used to generalize the memristor and higher-order elements, although without any physical meaning. From a mathematical point of view, [11] reports the memfractance concept and according to the fractional-order, it shows the interpolated characteristics between different memristive elements. In [12], the relationship between fracmemristance and fractance is discussed. By combining capacitors together with memristors, net-grid-type structures were also described to approximate the capacitive and inductive fracmemristor. In [13], the no ideal fractional interaction between flux and charge of a memristor is described. However, a piecewise nonlinear model of the memristor is considered and as a consequence, the fractional-order dynamic system is approached but again without any deep physical understanding. More recently, [14] reports the use of Valsa-algorithm for approximating a fractional-order capacitor. Afterwards, this element is substituting the integer-order capacitor into a memristor emulator circuit, obtaining the FOM behavior. However, the main disadvantage of [14] is that not only large RC-circuits are obtained, but the numerical value of each resistive and capacitive element is not commercially available and hence, parallel-series networks must again be used. Despite the FOM concept has been mathematically studied and ideal numerical results were shown, neither physical solid-state device nor emulator circuit has been developed until today. In this scope, this paper addresses the synthesis of FOM emulator circuits from integer-order memristor emulator circuits previously reported in the literature [15–18]. The rest of the paper is organized as follows. In Section 2, a novel fractional-order integrator (FOI) topology based on current-feedback operational amplifier (CFOA) and integer-order capacitors is discussed. In Section 3, the FOI previously designed is replacing the integer-order integrator (IOI) inside flux- or charge-controlled memristor emulator circuits, at their floating and grounded versions, and the FOMs can also be configured for operating at incremental or decremental mode [2]. Section 4 shows HSPICE simulation results, showing that the fingerprints of an integer-order memristor are holding up for their fractional versions. Finally, some conclusions are summarized in Section 5.

#### 2. Fractional-Order Integrator

A challenge at fractional calculus is the building or in best of cases, the approximation of fractances [19, 20]. In this sense, several mathematical approximations were researched and by its quickly convergence, continuous fractional expansion approach is the most adequate. Thus, the first-order approximation of an FOI is given bywhere is the fractional-order. It is important to mention that high-order fractance approximations can also be obtained; however, the synthesis of them leads to complex and bulky circuits [21, 22]. A simple circuit able to synthesize (1) is given in Figure 1, whose transfer function iswhere and are the voltage and current gains of the voltage and current followers associated with* X-Y*,* W-Z,* and* Z-X* terminals of the CFOA, respectively. To design the FOI, we propose the following design guide: (1)Given , use (1) to compute* B*.(2)Choose* C* = 0.1 mF and evaluate kΩ.(3)Using the numerical value of obtained in the first step, evaluate* BC* and of Figure 1. Resistances with noncommercial values are adjusted with precision potentiometers and capacitances with series and parallel connections.(4)Frequency denormalization is done for , where* kf* is the denormalization constant.