A topological approach to linear network analysis

The work reported in this thesis was motivated by a desire to develop better practical methods for linear network analysis. The practical aspects of existing methods, together with the new methods arising from this work, are discussed in part II of the thesis. Part I is devoted to a theoretical foundation for the new methods. The analysis method centres on network polynomials‚ÄövÑvÆtheir relationship with network behaviour and with each other. Until recently there has been no satisfactory formal treatment of network polynomials; they tend to be regarded as numerical conveniences arising in various analysis methods. For example, ratios of polynomials may express network 'transfer functions; they characterise linear dynamic systems; and their , roots determine the natural frequencies of networks. In particular, when we analyse a network by inverting the nodal admittance matrix whose elements have been expressed as ratios of polynomials, the polynomials proliferate. It is from this background that most of the theory described here was developed. In 1968, Dr. D.B. Pike, who had been working independently, submitted his Ph.D. thesis on \Linkage Polynomials\" to the University of Sydney. That work which this writer considers to be definitive in its treatment of many aspects of the subject was motivated by problems in the realisation of multiport networks and defines the polynomials by their occurrence as minor determinants of hybrid matrices of multiport networks. This definition relates them directly to network behaviour and their relationships with each other are obtained from Laplace expansions of minor determinants. The most important contribution of Pike's thesis is concerned with the interconnection of two multiport networks; it enunciates the relationships . between the polynomials of the complete network and the polynomials of its two constituent networks. In that work the relationships are obtained with Laplace expansions of the minor determinants of the sum of the two appropriate hybrid matrices of the constituent networks. It is a different enunciation of these same relationships which is Considered ta be the most significant contribution of part I of this thesis. But in this work the subject of network polynomials is approached from an altogether different point of view. Both the point of view and the alternative statement of the main results have an important bearing on the practical implementation of the analysis methods and it is the intended application of the theory which dictates the form of its presentation in part I. The evolution of this approach may be traced from the analysis of networks by the solution of simultaneous linear equations. The conventional elimination techniques are satisfactorily proficient in solving equations with numerical coefficients but are quite clumsy when handling coefficients represented symbolically. In the latter case however application of Cramer's rule leads to a suitable expression of the solution in the form of ratios of determinants and it is left to the numerical analyst to find suitable means for expanding the appropriate determinants. For large determinants containing symbolic entries this task is . cumbersome and for determinants derived from physical structures such as electrical networks concludes with the cancellation of large numbers of terms. It.is to this task that the network topologist with a different point of view of the analysis problem makes a significant contribution. Each term in the expansion is related to a unique set of branches of the network graph and its value is the product of the admittances of those branches. The sets of branches associated with a particular determinant constitute k-trees *of the network graph and the analysis task is therefore one of generating without duplication all the k-trees of a graph. Unfortunately this approach even with the aid of a digital computer is impractical for moderately-sized networks because of the prohibitively large numbers of trees associated with them. * A k-tree of a graph is a tree of a subgraph which although it includes all the nodes of the graph is in k separate parts."