The tractability index of memristive circuits: branch-oriented and tree-based models

The memory-resistor or memristor is a new electrical element characterized by a nonlinear charge-flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper we address the index analysis of certain differential-algebraic models of memristive circuits; specifically, our attention is focused on so-called branch-oriented models, which include in particular tree-based formulations of the circuit equations. Our approach combines results coming from DAE theory, matrix analysis and the theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits

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

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.Comment: Published in 2013. Journal reference included as a footnote in the first pag

Cyclic matrices of weighted digraphs

We address in this paper several properties of so-called augmented cyclic matrices of weighted digraphs. These matrices arise in different applications of digraph theory to electrical circuit analysis, and can be seen as an enlargement of basic cyclic matrices of the form B W \rsp B^T, where $B$ is a cycle matrix and $W$ is a diagonal matrix of weights. By using certain matrix factorizations and some properties of cycle bases, we characterize the determinant of augmented cyclic matrices in terms of Cauchy-Binet expansions and, eventually, in terms of so-called proper cotrees. In the simpler context defined by basic cyclic matrices, we obtain a dual result of Maxwell's determinantal expansion for weighted Laplacian (nodal) matrices. Additional relations with nodal matrices are also discussed. Finally, we apply this framework to the characterization of the differential-algebraic circuit models arising from loop analysis, and also to the analysis of branch-oriented models of circuits including charge-controlled memristors

Structural characterization of classical and memristive circuits with purely imaginary eigenvalues

The hyperbolicity problem in circuit theory concerns the existence of purely imaginary eigenvalues (PIEs) in the linearization of the time-domain description of the circuit dynamics. In this paper we characterize the circuit configurations which, in a strictly passive setting, yield purely imaginary eigenvalues for all values of the capacitances and inductances. Our framework is based on branch-oriented, semistate (differential-algebraic) circuit models which capture explicitly the circuit topology, and uses several notions and results from digraph theory. So-called P-structures arising in the analysis turn out to be the key element supporting our results. The analysis is shown to hold not only for classical (RLC) circuits but also for nonlinear circuits including memristors and other mem-devices

An Adaptive Sampling Process for Automated Multivariate Macromodeling Based on Hamiltonian-Based Passivity Metrics

This paper introduces a fully automated greedy algorithm for the construction of parameterized behavioral models of electromagnetic structures, targeting at the same time uniform model stability and passivity. The proposed algorithm is able to determine a small set of parameter configurations for which an external solver provides on the fly the sampled scattering parameters of the structure over a predetermined frequency band. These samples are subjected to a multivariate rational/polynomial fitting process, which iteratively leads to a parameterized descriptor realization of the model. The main novel contribution in this work is the adoption of a model-based approach for the adaptive augmentation of an initially small set of frequency responses, each corresponding to a randomly-selected parameter configuration. In particular, the locations of the in-band passivity violations of intermediate macromodels constructed at each iteration are used as a proxy for the model-data error in those regions where input data are not available. This physics-based consistency check, which is enabled by recent developments in multivariate passivity characterization based on Skew-Hamiltonian-Hamiltonian (SHH) spectra, is combined with standard space exploration metrics to obtain a small-size and automatically-determined distribution of points in the parameter space, leading to the construction of an accurate macromodel with a very limited number of external field solver runs. The embedded passivity check and enforcement process guarantees that either the final model is passive throughout the parameter space, or the residual violations, if present, are negligible for practical purposes. Several examples validate the proposed approach for up to three concurrent parameters

Hybrid analysis of nonlinear circuits: DAE models with indices zero and one

We extend in this paper some previous results concerning the differential-algebraic index of hybrid models of electrical and electronic circuits. Specifically, we present a comprehensive index characterization which holds without passivity requirements, in contrast to previous approaches, and which applies to nonlinear circuits composed of uncoupled, one-port devices. The index conditions, which are stated in terms of the forest structure of certain digraph minors, do not depend on the specific tree chosen in the formulation of the hybrid equations. Additionally, we show how to include memristors in hybrid circuit models; in this direction, we extend the index analysis to circuits including active memristors, which have been recently used in the design of nonlinear oscillators and chaotic circuits. We also discuss the extension of these results to circuits with controlled sources, making our framework of interest in the analysis of circuits with transistors, amplifiers, and other multiterminal devices

Passivity-preserving balanced truncation for electrical circuits

We present a passivity-preserving balanced truncation model reduction method for differential-algebraic equations arising in circuit simulation. This method is based on balancing the solutions of projected Lur'e equations. By making use of the special structure of circuit equations, we can reduce the numerical effort for balanced truncation significantly. It is shown that the property of reciprocity is also preserved in the reduced-order model. Network topological interpretations of certain circuit effects are given. The presented theory is illustrated by a numerical example

5 Post-processing methods for passivity enforcement

Many physical systems are passive (or dissipative): they are unable to generate energy on their own, but they can store energy in some form while exchanging power with the surrounding environment. This chapter describes the most prominent approaches for ensuring that Reduced Order Models are passive, so that their math- ematical representation satisfies an appropriate dissipativity condition. The main focus is on Linear and Time-Invariant (LTI) systems in state-space form. Different conditions for testing passivity of a given LTI model are discussed, including Linear Matrix Inequalities (LMIs), Frequency-Domain Inequalities, and spectral conditions on associated Hamiltonian matrices. Then we describe common approaches for perturbing a given non-passive system to enforce its passivity. Various examples from electronic applications are used to demonstrate both theory and algorithm performance