# A topological approach to linear network analysis

Brownell, Robert Alan (1973) A topological approach to linear network analysis. PhD thesis, University of Tasmania.

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## Abstract

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—their
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.

Item Type: Thesis (PhD) Electric networks The Author Copyright 1973 the Author - The University is continuing to endeavour to trace the copyrightowner(s) and in the meantime this item has been reproduced here in good faith. Wewould be pleased to hear from the copyright owner(s). Thesis (Ph.D.) -- University of Tasmania, 1974. Includes bibliography 25 Nov 2014 00:39 21 Jun 2016 02:21 View statistics for this item