Theory of Distributed Systems

Abstract

This paper concerns the wide class of physical networks made up of lumped components with transfer matrices whose elements are rational functions of the Laplace variables and hydraulic, pneumatic, electric, thermal, and elastic lines. It is advisable to lump on the basis of mathematical considerations after first obtaining the system equations with the lines unlumped. It is proved here that in a first approximation, the output vector of a system with hydraulic, pneumatic, electric, and elastic lines is related to the input vector by a transfer matrix whose elements are quotients of linear combinations of hyperbolic sines and cosines of Ki s for constants {Ki } with coefficients which are polynomials in s. First approximations are often if not generally sufficient. The numerators and denominators of the elements of the transfer matrix may be expanded into infinite products with the aid of computers, and a finite number of dominant terms kept so that the numerators and denominators are now polynomials in s. If the elements of the input vector are rational functions of s, the inverse Laplace transforms of the elements of the output vector are readily obtained by known techniques. Thus lumping on the basis of mathematical considerations is accomplished. When thermal lines are included rational functions of s and hyperbolic functions of constants times s also arise.