Zdeněk Biolek

Computation of the Area of Memristor Pinched Hysteresis Loop

It is well known that the memristor driven by a periodical voltage or current exhibits pinched v - i hysteresis loop. A novel finding is published in this brief, namely, that the area within the loop is directly related to the value of action potential, which was introduced by Leon Chua in his original work from 1971.

The Art of Finding Accurate Memristor Model Solutions

One of the main issues preventing a large-scale exploration of the full potential of memristors in electrical circuits lies in the convergence issues and numerical errors encountered in the computer-aided integration of the differential algebraic equation set governing the peculiar dynamical behavior of these nonlinear two-terminal electrical components. In most cases the complexity of this equation set prevents an analytical derivation of closed-form state solutions. Therefore the investigation of the nonlinear dynamics of memristors and circuits based upon them relies on software-based integration of the mathematical equations. In this paper, we highlight solution accuracy issues which may arise from an improper numerical integration of the equations, and then propose techniques addressing the problems properly. These guidelines represent a useful guide to engineers interested in the numerical analysis of memristor models.

Some fingerprints of ideal memristors

Since there are many possible types of nonlinear charge-flux constitutive relations of ideal memristors, such elements can manifest various behavior, and the identification of typical memristor fingerprints from the measured data or simulator outputs can be difficult in some cases. The aim of this paper is to reveal several fingerprints of ideal memristors, which extend the repertoire of hitherto published and currently well-known memristor fingerprints. These results can be useful for a clear and fast identification of a system behavior that violates the principles of the operation of ideal memristors.

Analytical Solution of Circuits Employing Voltage- and Current-Excited Memristors

The analysis of complicated circuits containing nonlinear electrical elements is commonly performed via numerical algorithms of simulation programs. However, an analytical solution can be preferable to the numerical approach, particularly for the needs of basic research. This paper introduces a methodology of the analytical solution of voltage/current response of the memristor to its excitation from ideal current/voltage source, with the memristor being characterized by the memristance-charge or memductance-flux relationships. The procedure is explained on examples of a TiO2 memristor with linear dopant drift and of a hydraulic memristor.

Variation of a classical fingerprint of ideal memristor

Gradual disappearance of hysteresis with increasing frequency of the exciting signal is considered a classical fingerprint of general memristive systems. The paper analyzes a special version of this fingerprint, when the value of the charge delivered within the half-period remains constant while the frequency of the sinusoidal current is increasing. Under these conditions, the area of the pinched hysteresis loop of the ideal memristor increases with the square of the frequency. Breaking the rules of this fingerprint indicates reliably that the element analyzed is not an ideal memristor. Copyright

Co)content in Circuits With Memristive Elements

The paper deals with the extension of the concept of content and cocontent functions, known from the theory of nonlinear resistive circuits, to circuits with memristive elements. The connection between the (co)content and the area of the hysteresis loop of a memristive element is indicated. It is proved and demonstrated on examples that the area of the hysteresis loop of a one-port, composed entirely of memristive elements, equals the sum of the areas of the loops of individual memristive elements. It is also proved that the law of conservation of (co)content applies also to circuits with memristive elements.

How Can the Hysteresis Loop of the Ideal Memristor Be Pinched

The hysteresis loop pinched at the origin of the v-i characteristic is the well-known fingerprint of the memristor excited by sinusoidal signal. This brief generalizes the present knowledge of the parameters of the pinched hysteresis loop for a periodical zero-dc driving signal described by an odd function of time. This brief concurrently brings new relationships between the parameter versus state map characteristics, the type of the excitation, and the type of loop pinching.

Analysis of memristors with nonlinear memristance versus state maps

Summary According to the axiomatic definition of the memristor from 1971, its properties are unambiguously determined by the memristance versus charge (or flux) map. The original model of the ‘HP memristor’ introduces this map via a linear function that represents this memristor as a variable resistor whose resistance is linearly dependent on the amount of charge flowing through. However, some analog applications require nonlinear, frequently exponential or logarithmic dependence of the resistance on an external controlling variable. The memristor with nonlinear memristance versus charge map is analyzed in the paper. The results are specified for the exponential type of this nonlinearity, which may be useful for future applications. Analytic formulae of the area of the pinched hysteresis loop of such a memristor are derived for harmonic excitation. It is also shown that the current flowing through such a memristor, which is driven by a voltage of arbitrary waveform, conforms to the Abel differential equation, and its closed-form solution is found. Copyright

Virtual dynamics of DC circuits

It is shown in this contribution that besides the classical dynamics caused by reactance elements, each DC resistive circuit has its special hidden so-called virtual dynamics which contains information about the DC operating point stability. Virtual eigenvalues can be attached to each linear model of resistive circuit linearized around its DC operating point. Corresponding virtual trajectories describe motion of operating point in the virtual space after being deflected from the equilibrium state and they help to decide whether the operating point is stable.

Potential stability of DC operating points in the circuits with controlled sources

A paradox is described related to the fact that in spite of evident circuit instability due to strong positive feedback, all poles of active dynamical circuits lie in the left-side complex plane. It is shown that the key problem is the incomplete circuit model, because the pole location can strongly depend on the inclusion of parasitic reactances. This phenomenon is investigated together with the so-called potential stability of the DC operating point.