Mathematical Problems in Engineering

Volume 2018, Article ID 1747865, 10 pages

https://doi.org/10.1155/2018/1747865

## About v-i Pinched Hysteresis of Some Non-Memristive Systems

^{1}Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic^{2}Faculty of Military Technologies, University of Defence, Brno, Czech Republic^{3}Faculty of Electrical and Computer Engineering, Technical University Dresden, Germany

Correspondence should be addressed to Zdenek Biolek; moc.liamg@keloib.kenedz

Received 17 June 2018; Accepted 7 August 2018; Published 24 September 2018

Academic Editor: Isabel S. Jesus

Copyright © 2018 Dalibor Biolek 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

A special subset of two-terminal elements providing pinched hysteresis loops in the voltage-current plane with the lobe area increasing with the frequency is analysed. These devices are identified as non-memristive systems and the sufficient condition for their hysteresis loop to be pinched at the origin is derived. It turns out that the analysed behaviour can be observed only for just one concrete initial state of the device. This knowledge is conclusive for understanding why such devices cannot be regarded as memristors.

#### 1. Introduction

The hysteresis loop pinched at the* v*–*i* (voltage-current) origin is the most widely known fingerprint of the ideal memristor introduced by L. Chua in 1971 [1] and also of the more general memristive systems [2], today referred to as the extended memristors [3].

In [4], L. Chua introduces a frequently cited terse thesis “If it’s pinched, it’s a memristor”. Such a hyperbole reflects the fact that many devices, behaving as a resistor whose resistance depends on the state of associated dynamical networks, comply with the classical definition of the memristive system [2]. Examples are given in [5–7], where the circuits containing conventional nonlinear devices and linear accumulating elements are described which generate pinched hysteresis loops.

On the other hand, the above thesis can be a source of misunderstanding, particularly when insisting on its literal sense. The paper [8] presents a number of models of systems that, even if they are not of memristive nature, generate pinched hysteresis loops. The results from [8] are analysed in [9], claiming that the hysteresis loops of the memristive system must provide pinching at the* v*–*i* origin regardless of the parameters of the driving signal and the initial state of the system. It is argued in [9] that since the systems from [8] do not conform to all these conditions, their hysteretic behaviour cannot be interpreted as a manifestation of the memristive effect.

Other examples of non-memristive systems providing pinched hysteresis phenomena are given in [10], namely, the nonlinear inductor with a linear resistor in series and the nonlinear capacitor with a parallel linear resistor. These reactive elements are studied with quadratic nonlinearities and under sinusoidal excitation. It is demonstrated that the hysteresis loops are pinched at the* v*–*i* origin for all amplitudes and all frequencies of the driving signal and that their areas increase proportionally with the frequency, which violates the well-known fingerprint of diminishing hysteresis when the frequency tends to infinity [2]. That is why this phenomenon is referred to in [10] as “pinched hysteresis with inverse-memristor frequency characteristics”. It is concluded therein that the existence of the pinched hysteresis loop is not the sufficient condition for identifying a memristor and that it is important to clearly identify what exactly the memristor is and in what sense it differs from other nonlinear systems. In the paper [11] the corresponding elements are already called the inverse memristors.

To illustrate the inconsistency of today’s understanding of what is/is not a memristor, let us mention that, in contrast to memristors and memristive systems introduced in 1971 [1] and 1974 [2], the current classification of memristors recognizes ideal, ideal generic, generic, and extended memristors [3]. In addition, the so-called second- or higher-order memristors [15] or the above inverse memristors [11] are also discussed. The possible confusion can be amplified by the frequently used notation of the memristor as the fourth fundamental circuit element [1]. Does it mean that all the above devices called memristors are the fundamental elements?

The question of the so-called new circuit elements, which can be considered as fundamental, is related to Chua’s concept of predictive modelling [12]. The term predictive means the model’s ability to predict the behaviour of the modelled subject in various modes of its operation. Chua showed that such models can be built from the predictive models of fundamental circuit elements organized in Chua’s table [12]. Their models are in the form of unambiguous constitutive relations, which do not depend on the way the element interacts with the surroundings and on the initial state of the element. Each candidate for a “new element” should be put to the test whether it can or cannot be replaced with a combination of existing elements from Chua’s table. If yes, then it is surely NOT a new element.

It is well known that the hysteresis loops of memristive systems [2] are also governed by other regularities, which should be taken into account when determining the type of the two-terminal device producing the hysteresis. For example, the ideal memristor driven by a signal modelled by an odd function of time must generate the odd-symmetric loops [16] whereas the loops of the extended memristors can be of both type I and type II (crossing type, CT, and non–crossing type, NCT) with a general order of touching at the* v*–*i* origin [17]. All memristive systems without any exceptions exhibit the fingerprint of a gradual [2] or sudden [18] disappearance of the hysteresis if the frequency of applied voltage or current increases above a certain limit, whereas the regularity of this disappearance depends on the concrete characteristics of the memristor [19]. The rule of homothety for ideal memristors, which is published in [20, 21], implies the rule of the quadratic increase in the loop area with increasing frequency if the memristor is driven by a charge or flux waveform of a fixed level.

This study suggests a methodology of identifying the devices exhibiting* v*-*i* pinched hysteresis loops that cannot be classified as memristors or memristive systems, or, more generally, as new fundamental circuit elements. In the first step, the classical definitions of the memristor and memristive system are confronted with the current classification of these devices from the point of view of Chua’s concept of predictive modelling [12]. This approach will help in clarifying whether the analysed device is a fundamental circuit element. As a demonstration, the devices with the inverse-memristor frequency characteristics are analysed. A more general class of nonlinear inductors and capacitors with the “inverse-memristor pinched hysteresis” than those in [10] is identified. The differences with regard to the loops of the memristive systems are highlighted, and the mechanism explaining why the loop area increases with increasing frequency is revealed. All these new pieces of knowledge can be used for a correct identification of the so-called inverse memristors.

#### 2. Fundamental Elements, Memristors, and Memristive Systems

As follows from a comparison of the original and current classification of memristors and memristive systems in Figures 1(a) and 1(b), the latter is governed by the thesis “If it’s pinched, it’s a memristor”.