Infinite Electrical Networks Research Articles

Resistances of Infinite Electrical Networks

It is interesting to find the equivalent resistance between two nodes of an infinite electrical network. In this paper, we consider an infinite electrical network that can be described as a series of squares whose edges are resistors with resistance $R$ and whose corresponding vertices are joined successively by resistors with resistance $R$ as well. Our major work is to find the equivalent resistance between the diagonal vertices of the base square of this infinite network. First, we apply the techniques of balanced bridges and symmetry of voltages to convert each iteration of the network to a parallel circuit that includes the previous iteration. Then, we evaluate the equivalent resistance of each iteration of the network and derive a recursive sequence of equivalent resistances with iterations. After that, we prove that the recursive sequence is convergent using the contraction theorem in real analysis. Finally, we claim that the limit of the recursive sequence is the equivalent resistance of the infinite network.

Formula omitted]-Potentials in an infinite network

This paper is a unified study of real-valued functions on an infinite network, with results generalizing some of those proved in the case of random walks, finite and infinite electrical networks and Schrödinger operators.

Estimation From Relative Measurements: Electrical Analogy and Large Graphs

We examine the problem of estimating vector-valued variables from noisy measurements of the difference between certain pairs of them. This problem, which is naturally posed in terms of a measurement graph, arises in applications such as sensor network localization, time synchronization, and motion consensus. We obtain a characterization on the minimum possible covariance of the estimation error when an arbitrarily large number of measurements are available. This covariance is shown to be equal to a matrix-valued effective resistance in an infinite electrical network. Covariance in large finite graphs converges to this effective resistance as the size of the graphs increases. This convergence result provides the formal justification for regarding large finite graphs as infinite graphs, which can be exploited to determine scaling laws for the estimation error in large finite graphs. Furthermore, these results indicate that in large networks, estimation algorithms that use small subsets of all the available measurements can still obtain accurate estimates.

Infinite regular electrical networks

Infinite regular electrical networks

Nonstandard electrical networks and the resurrection of Kirchhoff's laws

Kirchhoff's laws fail to hold in general for infinite electrical networks. Standard calculus is simply incapable of resolving this paradox because it cannot provide the infinitesimals and more generally the hyperreal currents and voltages that such networks often require. However, nonstandard analysis can do precisely this. The idea of a nonstandard electrical network is introduced in this paper and is used to reestablish Kirchhoff's laws for a fairly broad class of infinite electrical networks. The second section herein presents a fairly brief tutorial on infinitesimals, hyperreal numbers, and the key ideas of nonstandard analysis needed for a comprehension of this paper.

A maximum principle for node voltages in finitely structured, transfinite, electrical networks

Transfinite electrical networks have unique finite-powered voltage-current regimes given in terms of branch voltages and branch currents, but they do not in general possess unique node voltages. However, if their structures are sufficiently restricted, those node voltages will exist and will satisfy a maximum principle much like that which holds for ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalized the idea of local-finiteness. Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's current laws for nodes of any ranks and the permissibility of connecting pure voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a subsequent work.

Uncountable transfinite graphs and electrical networks

AbstractUp to now the theory of infinite and transfinite graphs and electrical networks has been established only for the case where the branch sets are countable. This work extends that prior theory to graphs and networks with uncountable branch sets, a structure that arises quite naturally in the theory of transfinite graphs. Moreover, conditions on the local structure of the transfinite graph are given which insure the countability of the branch set. Finally, it is shown that only countably many of the branches in an uncountable electrical network can have non‐zero voltages and currents, and therefore the restriction to countability in the prior theory is not essentially restrictive.

Morphisms and currents in infinite nonlinear resistive networks

In this paper we develop a theory of nonlinear resistive infinite electrical networks in the framework of modular sequence spaces. After introducing the notion of network morphism, we show that currents in infinite networks can be approximated by currents in suitable finite networks. Rayleigh monotonicity law for nonlinear networks is studied in detail. Parabolic and hyperbolic networks are introduced and characterized in analogy to the linear case.

Transfinite graphs and electrical networks

All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen’s equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff’s laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson’s least power principle to transfinite networks.

Resistances and currents in infinite electrical networks

We regard a graph as an electrical network where every edge is a resistance of 1 ohm. We show that, for an infinite graph of linear growth, Kirchhoff's laws combined with the finite power condition ensures the existence and uniqueness of an electric current when a current generator is added. We also show that the effective resistance between any two adjacent vertices in an infinite edge-transitive d-regular graph of moderate growth is 2 d , a result which has been expected by electrical engineers, but has previously been proved rigorously only in special cases.

A Thomson’s principle for infinite, nonlinear resistive networks

Suppose Γ \Gamma is an infinite resistive electrical network with resistors of the form (2). We prove an existence and uniqueness theorem for the current generated by external current sources by establishing the analogue of Thomson’s principle in suitable modular sequence spaces.

Finite-difference analysis of borehole flows involving domain contractions around three-dimensional anomalies

A finite-difference analysis in cylindrical coordinates of borehole flows governed by Poisson's equation yields an infinite cylindrical electrical grid as a model. The customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks. When the medium around the borehole is either uniform or cylindrically layered except for a three-dimensional anomalous region near the borehole, infinite electrical network theory allows one to represent the medium outside any cylinder S containing the anomaly and centered on the borehole by an infinite continued fraction of circulant-Laurent operators. The continued fraction in turn can be represented by scalar conductances connected to the grid's nodes on S. By choosing S just large enough to contain the anomaly, one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on S. An operational calculus for the circulant-Laurent operators and the fast Fourier transform expedites the computations, and moreover allows a rapidly computed determination of the solution outside S.

Infinite electrical networks: a reprise

A tutorial on resistive infinite electrical networks at an undergraduate level is presented. Rather than presenting a comprehensive survey on the wide variety of results existing in this subject, the author introduces the basic ideas with several examples and puzzles, examines how the theory branches into two separate avenues of investigation, points out how infinite networks differ in their behavior from finite networks, and ends with a brief survey of the literature. >

Infinite electrical networks with finite sources at infinity

Various physical situations leading to the exterior problem for partial differential equations are modeled by infinite electrical networks having finite-valued sources connected to extremities of the network at infinity. Up to now, there was no theory for such a network. The present work establishes one. An existence and uniqueness theorem is proven for a countably infinite resistive network that has an infinity of voltage and current sources, some of which connect to different "parts of infinity," as well as certain finite and infinite nodes that are shorted to different "parts of infinity" as well. The phrase in quotes is made precise by introducing the new concepts of extended nodes, extended branches, pathlike extremities, and extremities for the network. Moreover, short circuits between different pathlike extremities are allowed. The hypothesis of the theorem requires that the maximum power available from all the sources in the network be finite. Some examples are given, relating to petroleum flow in an oil well and to well-logging for geophysical exploration, to show how the considered infinite electrical networks can arise from practical applications.

Operator-valued transmission lines as models of a horizontally layered earth under transient two-dimensional electromagnetic excitation

A new method of computing the transient behavior of a horizontally layered earth under two-dimensional electromagnetic excitation is presented. The earth's conductivity, magnetic permeability, and electric permittivity are assumed to vary only with depth, but those variations are allowed to be quite arbitrary. Displacement current is not neglected. After discretization of Maxwell's equations, the earth is modeled by infinite RLC electrical networks. No truncations of those networks are invoked during the computational procedure. Instead, those networks are viewed as infinite ladder networks of Hilbert ports, that is, as operator-valued lumped transmission lines, and the computations are based upon the recently devised theory of such systems. This analysis yields a new surface impedance operator, which provides a considerably more comprehensive representation of the earth's behavior than does the Tikhonov-Cagniard surface impedance. For example, the Tikhonov-Cagniard surface impedance is applicable only when the electric or magnetic fields vary slowly or linearly with respect to horizontal displacements. The surface-impedance operator derived herein does not require those restrictions; only quadratic summability and Laplace transformability are needed.

A solution for an infinite electrical network arising from various physical phenomena

AbstractA procedure for determining the currents and voltages in a square or cubic grounded grid, wherein the grounding conductances can differ in quite arbitrary ways, is presented. Such a semi‐infinite electrical network is of practical importance, for it arises naturally from the discretization of a number of physical phenomena. There are very few classes of infinite electrical networks for which computational methods of solution exist; the present work adds one more class to that brief list.

A matroid related to finitely chainlike, countably infinite networks

AbstractThe determination of the currents and voltages in a countably infinite electrical network containing infinite energy requires not only the imposition of Kirchoff's laws and Ohm's laws but also the arbitrary assignment of the current‐voltage pairs in certain sets of branches, called joints. For finitely chainlike, countably infinite networks the maximal joint sets satisfy the base axioms of a matroid.

Nonuniform Semi-Infinite Grounded Grids

Semi-infinite resistive grounded grids are countably infinite electrical networks that arise from the discretization of the partial differential equation governing the minority-carrier density in a doped semiconductor. If the doping varies with depth from the surface of the semiconductor, the grid’s resistances also vary with distance from the inputs to the grid. This nonuniformity prevents the use of the characteristic-resistance method for determining currents and voltages. A computational method for making such a determination is presented herein. It is based upon the theory of infinite continued fractions whose entries are positive operators on a Hilbert space. It is also know that the solution given by the method is precisely that solution for which the power dissipated in the network is finite. Finally, the method is extended to RLC networks, and this allows the computation of transient responses in semi-infinite grounded grids of positive-real impedances.

The Characteristic-Resistance Method for Grounded Semi-Infinite Grids

A variety of existence and uniqueness theorems for the current flows in infinite electrical networks have been previously established, but there is virtually no information on how to compute those current flows under the practical requirement that the power dissipated and energy stored in the network be finite. This problem is addressed herein. It is shown that the characteristic-impedance method of analyzing lumped transmission lines can be adapted to semi-infinite half-plane resistive grids and to three-dimensional half-space resistive grids. The method yields the one and only current flow within the grid for which the total power dissipation is finite. It also yields a practical procedure for computing the currents and voltages in the grid. That procedure is remarkably efficient and uses very little computer time. Moreover, semi-infinite grids whose branch impedances are positive real functions can also be analyzed in a similar way. This allows the computation of transient behavior in the presence of energy-storage elements, but the required computer time is now considerably longer. Nonlinear grids can also be encompassed by an approximation technique. These results appear to provide a basis for the numerical analysis of certain boundary value problems of practical importance in engineering and the physical science.

Countably Infinite Time-Varying Electrical Networks

An existence and uniqueness theorem is established for the currents and voltages in a nonlinear time-varying countably infinite electrical network whose parameters are restricted to inductors and capacitors except possibly in certain branches, called joints, which are allowed to have any kind of electrical parameters. The LC form of the network allows the basic determining equation of the network to be obtained in normal form, and this in turn leads to substantially stronger results than those obtained in prior works. Similar results are obtained for linear time-varying RLC networks as well.