Journal of Electrical and Computer Engineering

Volume 2019, Article ID 4190641, 13 pages

https://doi.org/10.1155/2019/4190641

## Effects of Parasitic Fractional Elements to the Dynamics of Memristor

Correspondence should be addressed to Rawid Banchuin; moc.oohay@b_diwar

Received 25 March 2019; Revised 27 May 2019; Accepted 4 June 2019; Published 27 June 2019

Academic Editor: Muhammad Taher Abuelma’atti

Copyright © 2019 Rawid Banchuin. 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

In this research, we study the effects of the parasitic fractional elements to the dynamic of the memristor where both flux- and charge-controlled memristors have been considered. For doing so, the fractional differential equation-based approach has been used for modeling the memristor and the memristor-based circuits under the effects of the parasitic fractional elements where the resulting equations have been solved both analytically and numerically. From the obtained solutions and simulations, the effects of the parasitic fractional elements to the dynamic of the memristor have been studied. We have found that the parasitic fractional elements cause the charge and flux decay of the memristor similarly to the conventional parasitic elements. Moreover, the impasse points of the phase portraits between flux and charge of the memristor-based circuits can also be broken by the parasitic fractional elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been studied.

#### 1. Introduction

Memristor is the 4th electrical circuit element that has been theoretically found by Leon Chua since 1971 [1]. The memristor can be simply thought of as the resistor with memory. For decades after Chua proposed his original work, the memristor has been practically realized by a research group in Hewlett Packard (HP) labs [2] in 2008. The memristor has been applied in various applications, e.g., parametric oscillator, DRAM, signal processing, neural networks, programmable logic, cross bar switch, passive switch, and control systems. [3–6]. According to the importance of the memristor, the studies in its circuit theoretical aspects have been proposed [7–10]. In [9], the effects of the parasitic element to the memristor have been studied where it has been found that such parasitic element exhibits significant effects to the memristor and memristor-based circuits. However, it was assumed that the orders of the parasitic elements are strictly integers despite that they can be fractional in practice as those parasitic element can also be the fractional elements or fractances [11–13].

Therefore, we study the dynamic of the memristor and the memristor-based circuits under the effects of parasitic elements, which are the fractional elements, in this work where comparison with [9] has also been made. For simplicity, we appeal the parasitic elements, which are the fractional elements, as the parasitic fractional elements. Here, both flux- and charge-controlled memristors have been considered. Since the order of the parasitic fractional element can be fractional, the fractional differential equation (FDE), which its order can be fractional, must be used for modeling the memristor and the memristor-based circuits under the effects of such parasitic fractional element instead of the ordinary differential equation (ODE) which its order is strictly integer. The FDE is the extension of the ODE in the domain of fractional calculus which plays a fundamental role in various engineering fields, e.g., bioengineering, control theory, electronics, robotics, and signal processing [14–18]. The analytical solutions of the formulated FDEs, which are nonlinear, have been determined by using the Adomian decomposition method [19]. On the other hand, the numerical solutions have been obtained by using the integer derivative-based approximation of the fractional derivative with the moments of a function [20]. From the obtained solutions and simulations with MATHEMATICA, the effects of the parasitic fractional element to the dynamic of the memristor and the memristor-based circuits have been studied. We have found that the parasitic fractional element also causes the charge and flux decay of the memristor and breaks the impasse points of the phase portraits between flux and charge of the memristor-based circuits. Moreover, the effects of the order and the nonlinearity of the parasitic fractional element have also been studied.

In the following section, some background theory on the memristor and the fractional element will be briefly given. The flux and charge decaying of the memristor due to the parasitic fractional element will be illustrated in Section 3 followed by the impasse points breaking of the phase portraits between flux and charge of the memristor-based circuits in Section 4 where the effects of the order of the parasitic fractional element will be studied in these sections. The nonlinear effect of the parasitic fractional element will be explored in Section 5, and the conclusion will be finally drawn in Section 6.

#### 2. Background Theory

##### 2.1. Memristor

Memristor is a nonlinear electrical circuit element. This circuit element relates the flux (*φ*) and charge (*q*) through one of the following constitutive relations [9]:

Noted that both *f*(·) and must be differentiable and scalar-valued. Moreover, the memristor with its constitutive relation given by (1a) and (1b) can be referred to as the *q*-controlled memristor and the *φ*-controlled memristor, respectively. According to [21], the ideal memristor is nonvolatile. Therefore, it can retain the values of *φ* and *q* at the moment that it is switched off, i.e., *φ* and *q* are always conserved without any change as time goes by. This property of the ideal memristor corresponds to the fundamental circuit law on *φ* and *q*, namely, the principle of the conservation of charge and flux [22].

From (1a) and (1b), the following mathematical expression of small-signal memristance, *M*(*q*), and memductance, *W*(*φ*), of the memristor can be obtained [9]:

##### 2.2. Fractional Element

Fractional element is a circuit element with the impedance function of fractional order, *α*, where 0 < *α* < 1. According to [23], the fractional element which the phase of its voltage always leads that of its current is referred to as the fractional inductor. On the other hand, such element which the phase of its voltage always lags that of its current is referred to as fractional capacitor. For the fractional inductor, its s-domain impedance function can be given bywhere *L*_{α} denotes the inductivity of the fractional inductor [24]. On the other hand, we have the following impedance function for the fractional capacitor:where *C*_{α} stands for the pseudocapacitance of the fractional capacitor [25].

By using (4) and (5), the following flux-charge relationships can be, respectively, obtained for the fractional inductor and fractional capacitor:where *D*^{α}[ ] denotes the fractional derivative operator. Moreover, () and () stand for the flux and charge of the fractional capacitor (fractional inductor).

#### 3. Flux and Charge Decay by the Parasitic Fractional Elements

Despite the conservation of *φ* and *q* in the ideal phenomena, the memristor with parasitic elements exhibits the decaying of both *φ* and *q* after being switched off as proposed in [9]. In this work, we will show that such decaying of both *φ* and *q* can also be occurred by the parasitic fractional elements.

Firstly, the parasitic fractional capacitor will be considered. The memristor with parasitic fractional capacitor can be modeled as an ideal memristor connected in parallel with a fractional capacitor as depicted in Figure 1; thus, we have . If we assume that the memristor is *φ*-controlled, the following relationship can be obtained by using the conventional circuit analysis after the memristor is switched off: