Topological Considerations in the Design of Driving-Point Functions

As contrasted with the conventional discussions of network synthesis which concern themselves with the analytic character of the network functions, this paper presents the algebraic topological considerations in the realisation of the driving point functions. Abstractly a network may be considered as a linear graph with certain "weights" attached to the elements of the graph. These weights may be either impedance or admittance functions of the elements. If the network contains no mutual inductances, the network functions are expressible simply in terms of the topology of this weighted graph. On the other hand if the function is specified in terms of these weights it should be possible to deduce the topology. The paper begins with the assumption that the given driving point admittance function
Y_{d} (s)
is expressed as a ratio of homogeneous polynomials
V(y_i)/W(y_i)
in the weights or the elementary admittance functions of resistors, inductors and capacitors. Just how the function
Y_{d} (s)
is to be expressed as
V(y_{i})/W(y_{i})
is left as an unsolved problem. The properties of the polynomials
V(y_{i}
and
W( y_{i} )
with regard to their realisability are the main results of the paper. A procedure for exhibiting the network from the realisable functions
V(y_{i})
and
W(y_{i })
is also given. Relationships between the vertex and circuit matrices of the network and the polynomials
V(y_{i})
and
W(y_{i})
are derived for this purpose. A generalization of Brune's theorem I and a sufficient criterion for a matrix to be a circuit matrix are two auxiliary results obtained. The value of the paper is primarily academic in providing results which are useful in a discussion of basic ideas and in suggesting problems in the rich and unexplored field of topological methods of network synthesis.