^{1}

^{2}

^{1}

^{2}

^{1}

^{1}

^{3}

^{1}

^{3}

^{1}

^{2}

^{3}

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.

The hysteresis loop pinched at the

In [

On the other hand, the above thesis can be a source of misunderstanding, particularly when insisting on its literal sense. The paper [

Other examples of non-memristive systems providing pinched hysteresis phenomena are given in [

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 [

The question of the so-called new circuit elements, which can be considered as fundamental, is related to Chua’s concept of predictive modelling [

It is well known that the hysteresis loops of memristive systems [

This study suggests a methodology of identifying the devices exhibiting

As follows from a comparison of the original and current classification of memristors and memristive systems in Figures

(a) Original classification of memristors as a special case of memristive systems according to [

Figure

(a) Fragment of Chua’s table of fundamental Higher-Order Elements (HOEs) with

It is important to define the memristor as a fundamental circuit element from Chua’s table, which contains the elements from the well-known Chua’s quadrangle and also from Wang’s triangular table [

The ideal generic memristors as fundamental elements in Figure

Current-controlled ideal generic memristor:

For

Current-controlled ideal memristor:

Current-controlled ideal or ideal generic memristor:

All memristive systems from Figure

Current-controlled extended memristor:

Definitions (

This procedure is definite and thus more conclusive than deducing the type of the element from some limited number of its fingerprints that should hold simultaneously [

It is worth noting that the element is surely not a memristor as fundamental element if its model can be built up from models of the fundamental “no- (-1,-1)” elements in Chua’s table or the storeyed structure. For example, it is obvious from Figure

Notwithstanding that a different notation is used below than in [

It is obvious that the models (

It is shown in [

It is well known that the inductor with a linear current-flux constitutive relation, driven by a sinusoidal voltage or current, generates the (

Consider the current-controlled inductor as fundamental (-1,0) element from Chua’s table with a nonlinear flux (

The terminal voltage of the inductor is

Let the inductor be driven via a sinusoidal current source with the amplitude

It can be derived from (

The impact of the symmetry (

Nonlinear inductor with zero differential inductance for

An analysis of the type of the loop, i.e., whether it is the (non) crossing type and what is the order of touching the loop arms at the origin, can be done via a methodology from [

In order to mimic the definition (

Graphical composition of the

As follows from Figure

Figure

Pinched hysteresis loops generated by nonlinear inductor with asymmetric characteristic

The series connection of a resistor R and the inductor with the nonlinear constitutive relation (

It is essential to emphasize that the mechanism of generating the pinched hysteresis loops in Figure

A similar analysis to that for the inductor from Section

Consider a voltage-controlled capacitor with nonlinear voltage (

When driving the capacitor by a sinusoidal voltage

Consider the pinched hysteresis loop generated by nonlinear inductor with differential inductance^{d}(

It is evident that the loop areas (

Similar computations can be done for the loops generated by a nonlinear capacitor with differential capacitance

Figure

Analysis of the frequency dependence of the area of pinched hysteresis loop of the so-called inverse memristor driven by (a) current (the predictive model is a nonlinear inductor) and (b) voltage (the predictive model is a nonlinear capacitor). A similar analysis can be done for the loop area of classical ideal memristor [

The variables ^{1} of the increasing frequency. As a result, if the frequency of the driving current and thus also the flux frequency increase, then the amplitude of the voltage response increases proportionally, and the lobe area of the

A similar analysis can be done for the nonlinear capacitor, which is a part of the voltage-controlled inverse memristor, with the nonlinear constitutive relation between voltage and charge, the latter being the time integral of the current. This element occupies the diagonal between the nodes

Observations from this study can be summarized into the following paragraphs. (Only conclusions concerning the current-controlled inductors and voltage-controlled capacitors are given below; dual propositions hold for flux–controlled inductors and charge-controlled capacitors.)

Nonlinear inductors and capacitors with unambiguous constitutive relations (

When driving the inductor and capacitor from Paragraph

As a consequence of the symmetry of the responses from Paragraph

The loop symmetry from Paragraph

The hysteresis loop from Paragraph

The hysteresis loops from Paragraphs

The existence of the pinched hysteresis loop of nonlinear inductor and capacitor from Paragraph

The last paragraph in particular expresses an essential difference in the hysteretic effects of nonlinear inductors and capacitors (the loop is pinched only for concrete initial states) and memristors (the pinched loop is their general fingerprint).

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work has been supported by the Czech Science Foundation under grant no. 18-21608S. For research, the infrastructure of K217 Department, UD Brno, was used.