Active and Passive Electronic Components

Volume 2018, Article ID 3408480, 14 pages

https://doi.org/10.1155/2018/3408480

## On the Memristances, Parameters, and Analysis of the Fractional Order Memristor

Faculty of Engineering and Graduated School of Information Technology, Siam University, Bangkok, Thailand

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

Received 26 July 2018; Revised 5 September 2018; Accepted 24 September 2018; Published 1 November 2018

Academic Editor: Stephan Gift

Copyright © 2018 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 work, the analytical expressions of memristances, related parameters, and time domain behavioral analysis of the fractional order memristor have been proposed. Both DC with arbitrary delay and many AC waveforms including arbitrary phase sinusoidal and cosinusoidal waveform along with arbitrary periodic waveform have been taken into account. Unlike the previous works, the formerly ignored dimensional consistency has been taken into account and the analytical modelling of the boundary effect has been performed. Moreover, both transient and asymptotic behaviors of the fractional order memristor excited by AC waveform have been distinguished and analyzed. The effect of phase of AC waveform has also been studied. The influence of the fractional order to the areas of voltage-current hysteresis loop and memristance-current lissajous curve has also been clearly discussed and the usage of fractional order memristor in the memristor based circuit has also been demonstrated.

#### 1. Introduction

Recently, a state-of-the-art electrical circuit element, namely, fractional order memristor, is often cited. This circuit element can be obtained from the generalization of the 4th electrical circuit element, namely, memristor, that has been theoretically found by Leon Chua since 1971 [1], by using the concept of fractional calculus which have been adopted in various disciplines, e.g., biomedical engineering [2, 3], control system [4–6], and electronic engineering [7–9]. For decades after Chua proposed his original work, the memristor has been practically realized by a research group in Hewlett Packard (HP) labs [10] in 2008. As a result, the mathematical modelling and analysis attempts of the memristor have been proposed (e.g., [11–15]).

For the fractional order memristor on the other hand, there also exists such modelling and analysis attempts [16–21]. Some of them generalize the memristor by applying concept of the fractional calculus to the voltage-current relationship [16, 17] and termed such generalized memristor as the fracmemristor [17]. On the other hand, others do so by applying the fractional calculus to the memristor’s state equation where the often cited HP memristor has been adopted as the basis [18–21]. However, only the analytical expression of the area of voltage-current hysteresis loop has been proposed in [20] and those of the memristances proposed in [18, 19, 21] are in terms of the input voltage despite the fact that fractional order memristor of interested is the generalization of the HP memristor which is actually of a charge/current controlled type. Moreover, these previous works also neglected the dimensional consistency [22, 23] related issues and the boundary effect, which is an important characteristic of the HP memristor [10], has not been analytically modelled.

By this motivation, we generalize the HP memristor in the fractional order domain by also concerning the formerly ignored dimensional consistency and formulate the analytical expression of memristance in term of the input current where boundary effect has also been modelled. We also derive the expressions of those related parameters of the fractional memristor excited by various exciting waveforms including DC with arbitrary delay and sinusoidal and cosinusoidal with arbitrary phase and arbitrary periodic which are the AC waveforms. With these expressions, parameters, and numerical simulations with MATHEMATICA, the behaviors of the fractional order memristor have been thoroughly explored. Unlike [18–21], both transient and asymptotic behaviors of the fractional order memristor excited by AC waveform have been distinguished and analyzed. The effect of phase of AC waveform has also been studied. Moreover, the influence of the fractional order to the areas of voltage-current hysteresis loop and memristance-current lissajous curve has been clearly discussed and the usage of fractional order memristor in the memristor based circuit has also been demonstrated.

In the following section, the overview of memristor will be briefly given followed by the memristor’s generalization and derivation of our expressions in Section 3 where the behavioral analysis of the fractional order memristor will also be given. The DC waveform will be firstly treated followed by the AC waveforms where the sinusoidal waveform has been emphasized as it is the most fundamental. This is because the memristances and parameters of the memristor excited by the cosinusoidal and arbitrary periodic waveform can be obtained by using those due to the sinusoidal waveform as the basis as will be shown in Section 3 as well. The usage of fractional order memristor in the memristor based circuit will be presented in Section 4 and the conclusion will be finally drawn in Section 5.

#### 2. The Overview of Memristor

Memristor is a nonlinear electrical circuit element. This circuit element relates the instantaneous flux, , and charge, , through the following relationship:

where denotes the memristance.

According to [10], of the HP memristor can be given in terms of the minimum and maximum values of* M*(*t*) denoted by and and the state variable, , as

where , which is dimensionless, can be given in terms of the memristor’s current, , by

Note that , where and , respectively, stand for the ion mobility and semiconductor film of thickness. Therefore, the dimension of is (Asec)^{−1}.

As can be seen from (3) and also according to [10], can be simply given as follows:

Therefore, it can be seen that the HP memristor is charge controlled. Since is a time integration of , it can be stated that the HP memristor is of a current controlled type. Note also that ; thus, as long as the memristor is unsaturated. Otherwise, will be bounded at either 0 or 1 so will be equal to either or according to the boundary effect of the device. Traditionally, such boundary effect can be mathematically modelled by multiplying the RHS of (3) with the window function [24].

#### 3. The Fractional Order Domain Generalization of the Memristor and the Memristances, Related Parameters, and Analysis of the Fractional Order Memristor

By generalizing the memristor in the fractional order domain with the fractional calculus, the fractional order memristor can be obtained. Similarly to [18–21], we perform such generalization by applying the fractional calculus to the memristor’s state equation, i.e., (3). In these previous works, , where stands for the order of the fractional order memristor which can be arbitrary real value and has been obtained from such generalization. However, as is dimensionless; the dimension of the LHS of this previous generalized equation is given by sec^{-α} where that of the RHS is sec^{−1} which means that a dimensional inconsistency has always existed.

Therefore, the fractional time component [22], , which has the dimension of sec, has been introduced for handling this issue. As a result, unlike [18–21], the following generalized state equation has been used instead.

Similarly to that of the RHS, the dimension of the LHS of (5) is sec^{−1}; thus the dimensional inconsistency issue has been resolved. Note also that (5) is reduced to (3) when despite the presence of as become 1 with such value of .

Unlike [18, 19, 21], we derive of the fractional order memristor as a function as it has been assumed that the of fractional order memristor is a generalization of the HP memristor which is of a current controlled type as aforementioned. Therefore we directly determine from (5) by using the Riemann-Liouville fractional order integral [25] as follows:

where and denote the initial value of and the Gamma function [26], respectively.

Since it can be seen from (2) that

where , the initial memristance value, i.e., , can be immediately given by

Thus by substituting (6) into (7) and keeping (8) in mind, of the fractional order memristor can be obtained as follows:

which shows that is current-controlled.

If we let , (9) will be reduced to

Since the integer order integration of gives , we obtain

By using (8), (11) can be simplified under the assumption that and as follows:

which is similar to the original simplified model of the HP memristor [10]. Such correspondence cannot be found in [18, 19, 21] as the integer order integration of the memristor’s voltage, , yields .

For traditionally including the boundary effect, the window function must be introduced to the state equation as mentioned above. In [18], the linear window function given by has been adopted for simplicity as the usage of more accurate yet more complicated window function; e.g., those Joglekar, Biolek, and Prodomakis [24] can be mathematically cumbersome. Unfortunately, using such linear window function is mathematically equivalent to multiplying the RHS of the state equation by 1. As a result, no modification has been made on the state equation; thus the boundary effect modelling has not been performed. Moreover, neither the usage of window function nor alternative boundary effect analytical modelling has been made in both [19] and [21].

In order to model the boundary effect in a simplified manner, we apply two mathematical operators, i.e., and , which, respectively, selects the maximum value and minimum value among and , to (9). As a result, our expression of due to arbitrary exciting waveform can be finally given as follows:

where ; thus the dimension of is A^{−1}sec^{-α}.

Since the dimension of fractional integral of is Asec^{α}, that of is given by Ω which is physically measurable, similarly to those of and . Therefore, and , which are at the RHS of (13), can be physically combined as they have the same dimensions and the dimension of , which is the LHS of such equation, has also been found to be such physically measurable Ω; thus it can be seen that our expression of has dimensional consistency. Moreover, due to the operation of nested and , will be equal to if and only if lies within which means that the fractional order memristor remains unsaturated. Otherwise, will be equal to either or if is lower than or higher than which in turn means that the device become saturated at either its on-state or off-state. Therefore it can be seen that the boundary effect has been modelled without any necessity to use the window function and (13) along with its related results is valid to the saturated fractional order memristor. In the following subsections, ’s due to due to various exciting waveforms and the behavioral analysis of fractional order memristor will be presented.

##### 3.1. DC Waveform

Mathematically, the DC waveform with arbitrary delay (), which is more generic than the undelay waveform assumed in the previous works [18, 19], can be defined as , where and denote the magnitude of the waveform and the unit step function. Therefore, the resulting can be straightforwardly obtained by using (13) as

By using (14) with = 0 sec, = 100000 A^{−1}sec^{-α}, = 1 kΩ, = 100 kΩ, and = 80 kΩ, ’s of the fractional order memristor with various ’s excited by the DC waveform can be numerically simulated as depicted in Figures 1 and 2 where = 110 *μ*A and = -110 *μ*A have been, respectively, assumed and ’s of the HP memristor simulated by using its SPICE model [27] have also been included.