Problem In Network Analysis Research Articles

Hydraulic network analysis using (generalized) geometric programming

AbstractThe problem of determining the flows (and pressures) in the pipes of a general hydraulic network, for given input and output flows and/or given input and output pressure heads, is shown to be equivalent to either of a pair of convex programming problems with linear constraints. This is accomplished via the theory of (generalized) Geometric Programming. The equivalence of these problems is exploited to prove existence and uniqueness of the flow solution under certain conditions as well as to derive an algorithm which calculates this solution. Computational aspects of implementing the algorithm are considered in some detail and results obtained for a general example problem are presented. A brief discussion of the application of the methods of the paper to problems in electrical network analysis, transportation network analysis and the elastic analysis of structural trusses, is also given.

Generalized networks: Networks embedded on a matroid, part I

AbstractThe definition of ordinary electric networks is based on graphs. Since such a formulation is inadequate in many respects, we define a new network concept, namely, a generalized network, which is a network based on a matroid. To make the paper self‐contained an introduction to basic matroid theory is given in the first part of the paper. We then define a generalized network and formulate the network analysis and network synthesis problems for generalized networks. A number of new results are obtained for both analysis and synthesis and some old results for networks on graphs are generalized to networks on matroids. It is shown that the principle of duality, which does not hold for networks on graphs but is valid for generalized networks, adds power and insight for solving analysis and synthesis problems. The analysis problem, though formulated in terms of resistance networks, holds for networks containing all types of elements. The synthesis problem that is treated is the crucial one of the realization of a p‐port resistance network.

Some aspects of the theory of nonlinear networks

In the development of network theory over the years, the primary focus of attention has been in the area of linear systems. Several reasons for this emphasis can easily be cited, but perhaps the foremost reason is that it has long been thought that, except in certain very special cases, little progress toward a rigorous definitive theory could be expected once the hypothesis of linearity is discarded. The recent success in the use of numerical methods for computing solutions of the equations for specific nonlinear networks (the importance of which is not to be minimized) has, furthermore, resulted in a certain complacency on the part of many engineers who occasionally need to solve network problems. One senses their outlook as being, basically, that whenever a particular nonlinear problem arises, one need only then run, data in hand, to the computer. Somewhat ironically, however, the development of computer-aided network analysis techniques has also been a prime impetus for many of the recent theoretical investigations in the field of nonlinear networks, and although much remains to be done, a rather comprehensive body of knowledge in this area has begun to take form. A number of related recent contributions to the theory of non-linear networks are reviewed here. As distinct from the computational aspects of the network analysis problem, we discuss work whose primary purpose is to yield an understanding of the nature of the equations that describe the behavior of nonlinear networks, and to identify and relate certain properties of the network elements, and the manner of their interconnection, to properties of the equations and their solutions. In addition, we do frequently touch on the problem of computation since, as has already been implied, it is indeed one of the purposes of the work discussed here to provide more of a theoretical foundation on which to base the numerical analyses.

On Monte Carlo algebra

A Monte Carlo method is proposed and demonstrated for obtaining an approximate algebraic solution to linear equations with algebraic coefficients arising from first order master equations at steady state. Exact solutions are hypothetically obtainable from the spanning trees of an associated graph, each tree contributing an algebraic term. The number of trees can be enormous. However, because of a high degeneracy, many trees yield the same algebraic term. Thus an approximate algebraic solution may be obtained by taking a Monte Carlo sampling of the trees, which yields an estimate of the frequency of each algebraic term. The accuracy of such solutions is discussed and algorithms are given for picking spanning trees of a graph with uniform probability. The argument is developed in terms of a lattice model for membrane transport, but should be generally applicable to problems in unimolecular kinetics and network analysis. The solution of partition functions and multivariable problems by analogous methods is discussed.

On Monte Carlo algebra

A Monte Carlo method is proposed and demonstrated for obtaining an approximate algebraic solution to linear equations with algebraic coefficients arising from first order master equations at steady state. Exact solutions are hypothetically obtainable from the spanning trees of an associated graph, each tree contributing an algebraic term. The number of trees can be enormous. However, because of a high degeneracy, many trees yield the same algebraic term. Thus an approximate algebraic solution may be obtained by taking a Monte Carlo sampling of the trees, which yields an estimate of the frequency of each algebraic term. The accuracy of such solutions is discussed and algorithms are given for picking spanning trees of a graph with uniform probability. The argument is developed in terms of a lattice model for membrane transport, but should be generally applicable to problems in unimolecular kinetics and network analysis. The solution of partition functions and multivariable problems by analogous methods is discussed.

The Generalized Inverse in Network Analysis and Quadratic Error-Minimization Problems

The Generalized Inverse in Network Analysis and Quadratic Error-Minimization Problems

A novel method for solving the uniform distributed network analysis problem

An application of the Laplace transform method of Ogata to the solution of the uniform distributed network analysis problem is described.

Network Synthesis with Negative Resistors

The development of new solid-state active elements, such as variable-capacitor diodes and tunnel diodes, has stimulated the network theorist to consider the negative resistor as an additional basic circuit element to be included in problems of linear network analysis and synthesis. In this paper it is first shown that if the negative resistor is added to the usual set of lumped passive building blocks, then it is possible to represent as a network any linear relation between n-port voltages and currents prescribed in terms of real, rational functions of a complex-frequency variable. This leads to the synthesis of some novel pathologic circuits which have neither immittance nor scattering representations, such as a one-port, which is simultaneously an open circuit and a short circuit (v=i=0, the "nullator"), and the linear network in which voltages and currents at the ports are completely arbitrary (the "norator," the unique, linear nonreciprocal, one-port). These elements are shown to be basic linear circuit building blocks. The second part of the paper considers the synthesis in the frequency domain of a real, rational n×n immitance matrix in which pole locations and pole multiplicities are completely arbitrary. It is shown that such a matrix can always be realized with lossless elements and at most n positive and n negative resistors.

A general method of digital network analysis particularly suitable for use with low-speed computers

The concept of digital solutions of many of the network analysis problems associated with the design and operation of electrical power systems is now generally appreciated. There is, however, little indication of an increasing routine use of such methods in the United Kingdom, although many types of digital computer are now available. It is suggested that this is partly due to the fact that most of the existing programmes in this field employ techniques which have been prepared by skilled programmers for use with large high-speed computers, and digital analyses consequently tend to be restricted to the realm of special studies at computing centres.The paper describes the network theory, programme details and application of a general method of digital network analysis particularly suitable for use with low-speed computers. The method has been proved using both high- and lower-speed computers and the results of sample load-flow and short-circuit studies are discussed.

Some aspects of the network analysis of sequence transducers

An algebraic formulation is presented for use in the analysis of networks of the synchronous type of sequence transducer discussed by Mealy (1).2 With the network analysis problem in mind, the operation of a sequence transducer is characterized by two matrices and the formulation is given for elementary cases of cascade, parallel, and feedback connections. This formulation involves matrix-type operations and derives the characterizing matrices for the over-all network from those of the component sequence transducers. The formulations are intended to serve as a framework for the general network analysis problem. The analysis problem for feedback connections is also discussed.

The Design of Wide-Band Phase Splitting Networks

A number of articles and patents dealing with the properties and design of phase splitting networks, particularly in conjunction with single sideband modulators, have been published in the last few years. However, all of them have been restricted either to particular methods of design or to a particular number of design parameters. The present paper gives the results of a general investigation of phase splitting networks, dealing separately with network analysis, network synthesis and performance curve approximation problems. For the most important types of curve approximation, Taylor and Tchebycheff approximations, explicit formulas for any number of design parameters and for any required closeness of approximation are stated. Alternatives to the classical all-pass lattice network are given, and dissipation compensated phase shift networks are developed. In this way, clear and comparatively simple design instructions for simple as well as for difficult specifications for phase splitting networks are obtained. Furthermore, it is believed that some of the theoretical results obtained and methods developed, e.g., the Taylor and Tchebycheff approximations, the method of obtaining dissipation compensation, one of the methods of network synthesis, and the representation of approximating curves as iterated functions of two variables, with fractional index of iteration, are novel and of general theoretical interest.