On the physical and circuit-theoretic significance of the Memristor

It is noticed that the inductive and capacitive features of the memristor reflect (and are a quintessence of) such features of any resistor. The very presence in the resistive characteristic v = f(i) of the voltage and current state variables, associated by their electrodynamics sense with electrical and magnetic fields, forces any resister to cause to accumulate some magnetic and electrostatic fields and energies around itself. The present version is strongly extended in the sense of the circuit theory discussion.

💡 Research Summary

The paper by Emanuel Gluskin revisits the physical and circuit‑theoretic role of the memristor, arguing that its significance lies not merely in being a fourth fundamental element but in exposing the hidden inductive (L) and capacitive (C) aspects that are inherently present in any resistive component. The author begins by recalling Chua’s original proposal of the memristor as a device linking electric charge and magnetic flux, and notes that practical realizations have appeared as two‑state switches. However, Gluskin stresses that a true distinction between a conventional resistor and a memristor must be grounded in physics rather than in abstract circuit symbols.

In the second section, the conventional definition of a resistor as a one‑port with a one‑to‑one voltage‑current relation v = f(i) is examined. Gluskin points out that whenever a current i flows through a conductor, a magnetic field H surrounds the wire, implying an associated magnetic energy proportional to H²—i.e., an inductance. Likewise, a voltage drop creates an electric field E that extends outside the conductor, storing electrostatic energy proportional to E²—i.e., a capacitance. By invoking the Poynting vector S = E × H, he shows that power flows from the surrounding space into the resistor, where it is dissipated as heat (p = vi). A simple cylindrical geometry yields the familiar power expression, confirming that the presence of L and C is independent of the linearity or non‑linearity of the v‑i characteristic.

Section III raises a relativistic objection: applying Ohm’s law instantaneously along a multi‑kilometer resistive chain would imply superluminal signal propagation. Gluskin resolves this by emphasizing that the establishment of a steady dc current always involves an initial electromagnetic wave that propagates at finite speed, after which the dc state still contains steady energy flow through the implicit L and C surrounding the resistor.

The core of the paper is the mathematical description of a memristive one‑port (equations (3)). Instead of a direct v(i) relation, the model uses a state vector x (or flux ψ, charge q) with dynamics ẋ = f(x,i) and a “memristance” R(x,i) that multiplies the instantaneous current. This formulation inherently incorporates memory because solutions depend on initial conditions. Gluskin explores under what circumstances R can be reduced to a function of a single internal variable, thereby linking charge q and flux ψ directly. He also notes that the model may become energetically problematic if R explicitly contains time derivatives, prompting a discussion of alternative formulations where voltage is expressed as a function of flux and current as a function of charge.

To illustrate the practical relevance, the author examines fluorescent lamp circuits, which exhibit pronounced hysteresis loops in their v‑i characteristics. The lamp’s voltage can be written as v = A·sign(i) + L·di/dt, where the inductive term captures the observed phase lead of voltage over current, yet the stored energy is of electrostatic origin (charge separation in the gas) rather than magnetic flux. By starting from a simple hard‑limiter model v = A·sign(i) and adding the inductive term, Gluskin derives equations (8)–(9) that reproduce the measured hysteresis. He further shows that the external ballast inductance L and the internal hysteresis parameter A′ are not independent; dimensional analysis yields a relation A′ = A + k·L, with k≈2. This demonstrates that the lamp cannot be modeled in isolation; its behavior is inseparable from the surrounding circuit.

Section VI introduces the concept of “zero‑crossing nonlinearity.” Here, the times at which a waveform crosses zero (t_k) are treated as analytical parameters that cannot be prescribed a priori; they depend on the system’s state. This viewpoint classifies nonlinearity into three categories: constant coefficients, known time‑varying coefficients, and coefficients that depend on the state variables themselves. Zero‑crossing nonlinearity belongs to the third class and provides a natural language for describing memristive switched circuits where the traditional v‑i characteristic is deliberately avoided.

In the concluding remarks, Gluskin emphasizes three main messages: (1) every resistor intrinsically possesses inductive and capacitive energy storage due to its surrounding electromagnetic fields; (2) the memristor makes these hidden L and C elements explicit, thereby eliminating the need for a direct v‑i characteristic and providing a unified framework for memory and nonlinearity; (3) existing textbooks neglect the sophisticated models required for real‑world nonlinear devices such as fluorescent lamps, leading engineers to rely on oversimplified linear R‑L approximations. The paper calls for broader adoption of memristor‑based modeling and for incorporating zero‑crossing nonlinearity into circuit theory curricula.