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In this paper we show how to compute port behaviour of multiports which have port equations of the general form $B v_P - Q i_P = s,$ which cannot even be put into the hybrid form, indeed may have number of equations ranging from $0$ to $2n,$ where $n$ is the number of ports. We do this through repeatedly solving with different source inputs, a larger network obtained by terminating the multiport by its adjoint through a gyrator. The method works for linear multiports which are consistent for arbitrary internal source values and further have the property that the port conditions uniquely determine internal conditions. We also present the most general version of maximum power transfer theorem possible. This version of the theorem states that `stationarity' (derivative zero condition) of power transfer occurs when the multiport is terminated by its adjoint, provided the resulting network has a solution. If this network does not have a solution there is no port condition for which stationarity holds. This theorem does not require that the multiport has a hybrid immittance matrix., Comment: {keyword} keywords: Basic circuits, Implicit duality, Thevenin, Maximum power. arXiv admin note: substantial text overlap with arXiv:2005.00838

In this paper, we define the notion of rigidity for linear electrical multiports and for matroid pairs. We show the parallel between the two and study the consequences of this parallel. We present applications to testing, using purely matroidal methods, whether a connection of rigid multiports yields a linear network with unique solution. We also indicate that rigidity can be regarded as the closest notion to duality that can be hoped for, when the spaces correspond to different physical constraints, such as topological and device characteristic. A multiport is an ordered pair $(\V^1_{AB},\A^2_{B}),$ where $\V^1_{AB}$ is the solution space on $A\uplus B$ of the Kirchhoff current and voltage equations of the graph of the multiport and $\A^2_{B}\equivd \alpha_B+\V^2_B$ is the device characteristic of the multiport, with $A$ corresponding to port voltages and currents and $B$ corresponding to internal voltages and currents. The pair $\{\V^1_{AB},\alpha_B+\V^2_{B}\}$ is said to be rigid iff it has a solution $(x_A,x_B)$ for every vector $\alpha_B$ and given a restriction $x_A$ of the solution, $x_B$ is unique. A matroid $\M_S$ on $S,$ is a family of `independent' sets with the property that maximal independent sets contained in any given subset of $S$ have the same cardinality. The pair $\{\M^1_{AB},\M^2_{B}\}$ is said to be rigid iff the two matroids have disjoint bases which cover $B.$ We show that the properties of rigid pairs of matroids closely parallel those of rigid multiports. We use the methods developed in the paper to show that a multiport with independent and controlled sources and positive or negative resistors, whose parameters can be taken to be algebraically independent over $\Q,$ is rigid, if certain simple topological conditions are satisfied by the device edges., Comment: keywords: Rigidity, multiports, matroids, implicit duality

In this paper we derive some basic results of circuit theory using `Implicit Linear Algebra' (ILA). This approach has the advantage of simplicity and generality. Implicit linear algebra is outlined in [1]. We denote the space of all vectors on $S$ by $\mathcal{F}_S$ and the space containing only the zero vector on $S$ by $\mathbf{0}_S.$ The dual $\mathcal{V}_S^{\perp}$ of a vector space $\mathcal{V}_S$ is the collection of all vectors whose dot product with vectors in $\mathcal{V}_S$ is zero. The basic operation of ILA is a linking operation ('matched composition`) between vector spaces $\mathcal{V}_{SP},\mathcal{V}_{PQ}$ (regarded as collections of row vectors on column sets $S\cup P, P\cup Q,$ respectively with $S,P,Q$ disjoint) defined by $\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (g_P,h_Q) \in \mathcal{V}_{PQ}\},$ and another ('skewed composition`) defined by $\mathcal{V}_{SP}\rightleftharpoons \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (-g_P,h_Q) \in \mathcal{V}_{PQ}\}.$ The basic results of ILA are the Implicit Inversion Theorem (which states that $\mathcal{V}_{SP}\leftrightarrow(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_S)= \mathcal{V}_S,$ iff $\mathcal{V}_{SP}\leftrightarrow \mathbf{0}_P\subseteq \mathcal{V}_S\subseteq \mathcal{V}_{SP}\leftrightarrow\mathcal{F}_S$) and Implicit Duality Theorem (which states that $(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ})^{\perp}= (\mathcal{V}_{SP}^{\perp}\rightleftharpoons \mathcal{V}_{PQ}^{\perp}$). We show that the operations and results of ILA are useful in understanding basic circuit theory. We illustrate this by using ILA to present a generalization of Thevenin-Norton theorem where we compute multiport behaviour using adjoint multiport termination through a gyrator and a very general version of maximum power transfer theorem, which states that the port conditions that appear, during adjoint multiport termination through an ideal

Electric circuits are usually described by charge- and flux-oriented modified nodal analysis. In this paper, we derive models as port-Hamiltonian systems on several levels: overall systems, multiply coupled systems and systems within dynamic iteration procedures. To this end, we introduce new classes of port-Hamiltonian differential-algebraic equations. Thereby, we additionally allow for nonlinear dissipation on a subspace of the state space. Both, each subsystem and the overall system, possess a port-Hamiltonian structure. A structural analysis is performed for the new setups. Dynamic iteration schemes are investigated and we show that the Jacobi approach as well as an adapted Gauss-Seidel approach lead to port-Hamiltonian differential-algebraic equations., Comment: 33 pages, 1 figure

Neuromorphic computing describes the use of VLSI systems to mimic neuro-biological architectures and is also looked at as a promising alternative to the traditional von Neumann architecture. Any new computing architecture would need a system that can perform floating-point arithmetic. In this paper, we describe a neuromorphic system that performs IEEE 754-compliant floating-point multiplication. The complex process of multiplication is divided into smaller sub-tasks performed by components Exponent Adder, Bias Subtractor, Mantissa Multiplier and Sign OF/UF. We study the effect of the number of neurons per bit on accuracy and bit error rate, and estimate the optimal number of neurons needed for each component.

The structural analysis, i.e., the investigation of the differential-algebraic nature, of circuits containing simple elements, i.e., resistances, inductances and capacitances is well established. However, nowadays circuits contain all sorts of elements, e.g. behavioral models or partial differential equations stemming from refined device modelling. This paper proposes the definition of generalized circuit elements which may for example contain additional internal degrees of freedom, such that those elements still behave structurally like resistances, inductances and capacitances. Several complex examples demonstrate the relevance of those definitions.

We introduce in this paper an equivalence notion for submersions $U \to \R$, $U$ open in $\R^2$, which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as {\em classes of equivalent functions}. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced.

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The present paper investigates state space analysis of memristor based series and parallel RLCM circuits. The stability analysis is carried out with the help of eigenvalues formulation method, pole-zero plot and transient response of system. The state space analysis is successfully applied and eigenvalues of the two circuits are calculated. It is found that the, system follows negative real part of eigenvalues. The result clearly shows that addition of memristor in circuits will not alter the stability of system. It is found that systems poles located at left hand side of the S plane, which indicates stable performance of system. It clearly evident that eigenvalues has negative real part hence two systems are internally stable., Comment: 9 pages, 4 figures and 1 table

In this paper, we investigate few memristor-based analog circuits namely the phase shift oscillator, integrator, and differentiator which have been explored numerously using the traditional lumped components. We use LTspice-IV platform for simulation of the above-said circuits. The investigation resorts to the nonlinear dopant drift model of memristor and the window function portrayed in the literature for nonlinearity realization. The results of our investigations depict good agreement with the conventional lumped component based phase shift oscillator, integrator, and differentiator circuits. The results are evident to showcase the potential of the memristor as a promising candidate for the next generation analog circuits., Comment: 11 Pages, 9 Figures

The use of reverse time chaos allows the realization of hardware chaotic systems that can operate at speeds equivalent to existing state of the art while requiring significantly less complex circuitry. Matched filter decoding is possible for the reverse time system since it exhibits a closed form solution formed partially by a linear basis pulse. Coefficients have been calculated and are used to realize the matched filter digitally as a finite impulse response filter. Numerical simulations confirm that this correctly implements a matched filter that can be used for detection of the chaotic signal. In addition, the direct form of the filter has been implemented in hardware description language and demonstrates performance in agreement with numerical results., Comment: 9 pages, 17 figures

To analyze the failure risk of asynchronous digital circuits the time-parameter is introduced into the Boolean algebra replacing the arithmetic operations by logical operations. There considered an example of construction of signals passing through the logical elements, using the described below mathematical apparatus., Comment: In this work there was considered the method of failure risk analysis of digital circuits using an analytic representation of signals within the circuit. At that logical operations had to by replaced by arithmetic operations. The transitions from one signal state to another is described by the Heaviside function

We here use notions from the theory linear shift-invariant dynamical systems to provide an easy-to-compute characterization of all rational wavelet filters. For a given N bigger or equql to 2, the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree $N((N-1)/2+m). Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parameterization is exploited for a step-by-step construction (where in each the index m is increased by one) of state space model of wavelet filters., Comment: The present version is better coordinated with another work of us on the same subject, entitled Extending wavelet filters. Infinite dimensions, the non-rational case and indefinite-inner product spaces, and also available on arxiv at: http://arxiv.org/abs/1106.2303

Reversible computing is a concept reflecting physical reversibility. Until now several reversible systems have been investigated. In a series of papers Kenichi Morita defines the rotary element RE, that is a reversible logic element. By reversibility, he understands that 'every computation process can be traced backward uniquely from the end to the start. In other words, they are backward deterministic systems'. He shows that any reversible Turing machine can be realized as a circuit composed of RE's only. Our purpose in this paper is to use the asynchronous systems theory and the real time for the modeling of the ideal rotary element, Comment: presented at the 12-th Symposium of Mathematics and its Applications, "Politehnica" University of Timisoara, Timisoara, 2009

We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, we partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix.

In this paper, by way of three examples - a fourth order low pass active RC filter, a rudimentary BJT amplifier, and an LC ladder - we show, how the algebraic capabilities of modern computer algebra systems can, or in the last example, might be brought to use in the task of designing analog circuits., Comment: V1: documentclass IEEEtran, 7 pages, 6 figures. Re-release of the printed version, with some minor typographical errors corrected

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way., Comment: V1: documentclass IEEEtran, 6 pages, 1 figure. Re-release of the printed version, with some minor typographical errors corrected

The paper is concerned with defining the electrical signals and their models. The delays are discussed, the asynchronous automata - which are the models of the asynchronous circuits - and the examples of the clock generator and of the R-S latch are given. We write the equations of the asynchronous automata, which combine the pure delay model and the inertial delay model; the simple gate model and the complex gate model; the fixed, bounded and unbounded delay model. We give the solutions of these equations, which are written on R->{0,1} functions, where R is the time set. The connection between the real time and the discrete time is discussed. The stability, the fundamental mode of operation, the combinational automata, the semi-modularity are defined and characterized. Some connections are suggested with the linear time and the branching time temporal logic of the propositions.

The paper studies some important properties of the asynchronous (=timed) automata: the delay-insensitivity, the hazard-freedom, the semi-modularity and the technical condition of good running. Time is discrete.

The paper presents the differential equations that characterize an asynchronous automaton and gives their solution x:R->{0,1}x...x{0,1}. Remarks are made on the connection between the continuous time and the discrete time of the approach. The continuous and the discrete time, the linear and the branching temporal logics have the semantics depending on x and their formulas give the properties of the automaton.

We write the relations that characterize the simpliest timed automaton, the inertial delay buffer, in two versions: the non-deterministic and the deterministic one, by making use of the derivatives of the R->{0,1} functions.