Topology Synthesis Research Articles

Some applications of graph theory to network analysis and synthesis

This paper describes applications of graph theory to some of the problems of network analysis and synthesis, starting with the early days of network theory. The first section is devoted to the classical results related to the topological analysis of linear, passive, and transfornerless networks. Then it goes over to contributions dealing with generalization of these methods to networks with mutual couplings and active elements. Appearance of integrated circuits and digital computers provided both motivation and means for studying large-scale systems. A part of the paper is devoted to publications on large-scale network analysis dealing with diacoptics and graph theoretic methods for defining efficient Gaussian elimination algorithms for solution of sparse sets of linear algebraic equations. The part on topological synthesis starts with the early results dealing with relations between the structure and the properties of passive, reciprocal networks without mutual inductances. The last section is devoted to contributions in the field of topological synthesis of pure resistive networks dealing with such topics as conditions of realizability, minimal realizations and bounds on the number of port-terminals.

Computer Network Optimization for Air Τrasportation System Control

Methodology of topological synthesis of computer network for air transportation system control is presented. The problem of synthesis is described by a complex of interconnected models. Procedures for evaluation topological paramètres of the network is suggested.

Some Properties of Matrices with 1, -1 and 0 Entries

A class of matrices with 1, -1 and 0 entries, called the H-matrices, is introduced and properties of such matrices are investigated. It is shown that for an H-matrix K the matrix G = KDK T, where D is a diagonal matrix with nonnegative diagonal entries, has certain properties applicable to topological synthesis of networks. A technique for decomposing a particular symmetrical matrix G, called the Q-matrix, into KDK T, where K is an H-matrix and D is a diagonal matrix with positive diagonal entries, is also developed.

Summary of the History of Circuit Theory

After a brief survey of the state of circuit theory before World War I, the various directions of its development in the last 50 years are discussed, mainly in relation with applications to communication engineering. The early period of network design (1920-1925) was followed by the beginning of synthesis (1926-1935). The next steps were the development of feedback amplifier theory and insertion loss filter theory. The numerous new directions of research started during and after the second war are briefly mentioned. Finally recent progress is reported in formal realizability theory and in topological synthesis. A last section deals with nonlinear and linear variable circuits.

Two Theorems in Multi-Weighted Sums

Systems of linear relationships have received an increasing amount of attention in recent years, partly in their own right as mathematical entities, but probably more so because of their frequent occurrence in physical situations. Multilinear systems, however, involving linear combinations of linear expressions, seem to have been almost completely neglected. Yet many physical situations are more adequately described by multilinear than by linear equations. Examples arise in the topological synthesis of electrical networks, and in sensitivity and stability studies of active networks. There is therefore a need for mathematical development in this area. To the end of initiating this, two theorems, based on the representation of a multilinear system as a multi-weighted sum of vectors belonging to some linear space, are presented in this paper, together with an example of their application in the stability analysis of linear systems.

Topological synthesis of non-reciprocal resistance networks

The paper deals with the topological synthesis of non-reciprocal resistance networks specified by the so-called driving-conductance matrix. Synthesis of non-reciprocal networks has already been treated by H. J. Carlin and Y. Oono, and their networks are specified by scattering matrices and consequently n-port impedance matrices. One of the merits of this paper, which differs from their paper, is in the determination of the connection and ideal transformers. Factors which determine the network are the connection and element resistances, and they therefore make possible the determination of the node-branch incidence matrix and resistances so that the driving-conductance matrix may equal the given matrix. When we denote the branch e.m.f. matrix by V and the branch-current matrix by I, I = GV, where G is called the driving-conductance matrix, which is assumed to be given, and, in general, G′ ≠ G. If a maximal set of solutions for Gx = 0 and x′G = 0 is denoted by X as the matrix expression, X is called the zero factor of G. Provided that the zero factor X satisfies the character of a node-branch incidence matrix, the connection can be fixed directly. If not, the connection is determined by the part of X which satisfies the character of a node-branch incidence matrix, and the ideal transformers needed are determined by the other part. After the connection and the ideal transformers are determined, the branch resistances are determined by comparing the given matrix with the matrix calculated from the newly determined connection and the unknown branch resistances.

On Coefficients of Polynomials in Network Functions

This paper presents a study of the relationships between the missing powers of polynomials in network functions and the network geometry. The elementary transformation of trees and the 2-trees of a network are introduced to obtain the necessary and sufficient conditions for polynomials in network functions to have missing powers. It is shown that the polynomial in the numerator of the transfer function of a grounded two-terminal-pair RLC network cannot have two successive missing powers unless some common factors of the numerator and the denominator are cancelled. This result is useful in topological synthesis where one must usually restore all the necessary surplus factors before deciding on the minimum number of vertices and the geometry of the network.

Topological Synthesis of Non-Reciprocal Linear-Networks

Topological Synthesis of Non-Reciprocal Linear-Networks

Topological Synthesis of Transfer-Admittance Matrices

Combinatorial topology is useful for handling not only problems in analysis, but also problems in synthesis. In this paper, the author shows topological foundations and one of the typical methods for network synthesis. This paper deals with topological synthesis of an electrical network specified by a given transfer-admittance matrix. Factors that determine the network are the connection and branch impedances which then make it possible to determine the node-branch incidence matrix and branch impedances so that the transfer-admittance matrices of the network may equal the given. matrix. This node-branch incidence matrix is determined by a zero factor of the given transfer-admittance matrix. Therefore, whether ideal transformers are necessary or not depends on whether this zero factor satisfies the character of a node-branch incidence matrix. After the connection and ideal transformers are determined, we can determine branch impedances by comparing the given matrix with the matrix calculated from the newly determined connection and the unknown branch impedances.

Topological Synthesis of Transfer Admittance?Matrix

Topological Synthesis of Transfer Admittance?Matrix