First order devices, hybrid memristors, and the frontiers of nonlinear circuit theory

TL;DR

This paper classifies nonlinear circuit devices with memory effects, introduces hybrid memristors as a frontier of first order circuit theory, and explores the taxonomy and mathematical properties of such mem-devices.

Contribution

It proposes a new classification framework for mem-devices based on their variables and differential order, defining hybrid memristors that relate all four fundamental circuit variables.

Findings

Hybrid memristors accommodate all four fundamental variables.

A taxonomy of C^1-devices with nonlinear characteristics is developed.

The differential-algebraic index of circuits with mem-devices is characterized.

Abstract

Several devices exhibiting memory effects have shown up in nonlinear circuit theory in recent years. Among others, these circuit elements include Chua's memristors, as well as memcapacitors and meminductors. These and other related devices seem to be beyond the, say, classical scope of circuit theory, which is formulated in terms of resistors, capacitors, inductors, and voltage and current sources. We explore in this paper the potential extent of nonlinear circuit theory by classifying such mem-devices in terms of the variables involved in their constitutive relations and the notions of the differential- and the state-order of a device. Within this framework, the frontier of first order circuit theory is defined by so-called hybrid memristors, which are proposed here to accommodate a characteristic relating all four fundamental circuit variables. Devices with differential order two and mem-systems are discussed in less detail. We allow for fully nonlinear characteristics in all circuit elements, arriving at a rather exhaustive taxonomy of C^1-devices. Additionally, we extend the notion of a topologically degenerate configuration to circuits with memcapacitors, meminductors and all types of memristors, and characterize the differential-algebraic index of nodal models of such circuits.