844 research outputs found
Dynamical properties of electrical circuits with fully nonlinear memristors
The recent design of a nanoscale device with a memristive characteristic has
had a great impact in nonlinear circuit theory. Such a device, whose existence
was predicted by Leon Chua in 1971, is governed by a charge-dependent
voltage-current relation of the form . In this paper we show that
allowing for a fully nonlinear characteristic in memristive
devices provides a general framework for modeling and analyzing a very broad
family of electrical and electronic circuits; Chua's memristors are particular
instances in which is linear in . We examine several dynamical
features of circuits with fully nonlinear memristors, accommodating not only
charge-controlled but also flux-controlled ones, with a characteristic of the
form . Our results apply in particular to Chua's
memristive circuits; certain properties of these can be seen as a consequence
of the special form of the elastance and reluctance matrices displayed by
Chua's memristors.Comment: 19 page
Twin subgraphs and core-semiperiphery-periphery structures
A standard approach to reduce the complexity of very large networks is to
group together sets of nodes into clusters according to some criterion which
reflects certain structural properties of the network. Beyond the well-known
modularity measures defining communities, there are criteria based on the
existence of similar or identical connection patterns of a node or sets of
nodes to the remainder of the network. A key notion in this context is that of
structurally equivalent or twin nodes, displaying exactly the same connection
pattern to the remainder of the network.
The first goal of this paper is to extend this idea to subgraphs of arbitrary
order of a given network, by means of the notions of T-twin and F-twin
subgraphs. This is motivated by the need to provide a systematic approach to
the analysis of core-semiperiphery-periphery (CSP) structures, a notion which
somehow lacks a formal treatment in the literature. The goal is to provide an
analytical framework accommodating and extending the idea that the unique
(ideal) core-periphery (CP) structure is a 2-partitioned K2. We provide a
formal definition of CSP structures in terms of core eccentricities and
periphery degrees, with semiperiphery vertices acting as intermediaries. The
T-twin and F-twin notions then make it possible to reduce the large number of
resulting structures by identifying isomorphic substructures which share the
connection pattern to the remainder of the graph, paving the way for the
decomposition and enumeration of CSP structures. We compute the resulting CSP
structures up to order six.
We illustrate the scope of our results by analyzing a subnetwork of the
network of 1994 metal manufactures trade. Our approach can be further applied
in complex network theory and seems to have many potential extensions
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 is a cycle matrix and 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
Circuit theory in projective space and homogeneous circuit models
This paper presents a general framework for linear circuit analysis based on
elementary aspects of projective geometry. We use a flexible approach in which
no a priori assignment of an electrical nature to the circuit branches is
necessary. Such an assignment is eventually done just by setting certain model
parameters, in a way which avoids the need for a distinction between voltage
and current sources and, additionally, makes it possible to get rid of voltage-
or current-control assumptions on the impedances. This paves the way for a
completely general -dimensional reduction of any circuit defined by
two-terminal, uncoupled linear elements, contrary to most classical methods
which at one step or another impose certain restrictions on the allowed
devices. The reduction has the form Here, and capture
the graph topology, whereas , , comprise homogeneous
descriptions of all the circuit elements; the unknown is an -dimensional
vector of (say) ``seed'' variables from which currents and voltages are
obtained as , . Computational implementations
are straightforward. These models allow for a general characterization of
non-degenerate configurations in terms of the multihomogeneous Kirchhoff
polynomial, and in this direction we present some results of independent
interest involving the matrix-tree theorem. Our approach can be easily combined
with classical methods by using homogeneous descriptions only for certain
branches, yielding partially homogeneous models. We also indicate how to
accommodate controlled sources and coupled devices in the homogeneous
framework. Several examples illustrate the results.Comment: Updated versio
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