Pinched hysteresis with inverse-memristor frequency characteristics in some nonlinear circuit elements
Introduction
In the past few years, a significant number of publications related to memristors; their modeling and their applications have been published [1], [2], [3], [4], [5], [6], [7], [8]. Due to the fact that memristors are nonlinear devices which exhibit pinched hysteresis in the plane, simple models that can capture this behavior are necessary. One such mathematical model was recently proposed by the authors in [9] where memristance was linked to the existence of a nonlinear term of the form resulting in a pinched-hysteresis that declines with increased frequency [10]. The state variable x(t) may represent the voltage v(t) in a voltage-controlled memristor or the current i(t) in a current-controlled memristor. In [10], the self-crossing (pinched) hysteresis loop was shown to be a necessary characteristic of all memristive devices. However, [11] added two more conditions on memristive devices which are
- (i)
starting from some critical frequency, the hysteresis lobe area should decrease monotonically as the excitation frequency increases and
- (ii)
the pinched hysteresis loop should shrink to a single-valued function when the frequency tends to infinity. This means that the lobe area declines with increased frequency.
In this work, however, we propose a system that represents a nonlinear inductor or a nonlinear capacitor (with quadratic nonlinearity) that exhibits self-crossing pinched hysteresis. The same system may also represent a nonlinear derivative-controlled transconductor; which we verify experimentally and observe inverse-memristor frequency characteristics in the form of widening (rather than shrinking) in the lobe area with increased frequency. These observations lead to the conclusion that self-crossing pinched hysteresis can be obtained from other nonlinear elements. This observation is in line with the observations of other researchers as well [12].
Section snippets
System with pinched hysteresis
Consider the following model:where y is a normalized output, x is a normalized input signal, and are scaling constants. Under sinusoidal excitation where we obtainUsing trigonometric identities, one obtains
This equation has the following properties:
- (i)
There exists a line of symmetry given by the first order equation:Evidently, for a=0, the y-axis is the line of symmetry.
- (ii)
A pinched double-loop hysteresis behavior is
Circuit identification
From an electrical circuit point of view, Eq. (1) can represent different types of circuits based on the nature of x and y. Restricting ourselves to the plane, the possible choices of x(t) and y(t) are either a voltage v(t) or a current . When and then (1) can be translated into series connected components. Alternatively, if and then (1) can be translated into parallel connected components. These components are identified as follows.
Experimental validation
The design principle to validate (12) is shown in Fig. 3(a) where an applied voltage V is differentiated using a floating differentiator circuit and then used to control a voltage-controlled transconductance Gm through its control voltage Vc. Gm is implemented using an LM13700 chip [13] where Gm is proportional to a bias voltage Vc given by [13]where RA and RB are external biasing resistors. If the control voltage Vc is forced to be equal to the derivative of the
Conclusion
In this work, we have demonstrated the fact that the existence of pinched hysteresis is not a sufficient condition to identify a memristor since it can be observed from other nonlinear devices; particularly with quadratic-type nonlinearity. The authors in [12] have also recently demonstrated pinched hysteresis from circuits with standard components. In our proposed model of (1), we were also able to identify an equivalent standard component to each term in the model. In view of recent
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The author is also with the Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt.