The Physical Meaning of Compactness

Abstract

A lossless twoport N, whose impedances z_{11}, z_{12} , and z_{22} possess poles at the complex frequencies s = \pm j\omega_p , is considered. It is shown that the algebraic alternatives of compactness, or lack of compactness, for the z_{ij} at s = \pm j\omega_p correspond directly to the physical alternatives of whether or not a pair of N's natural oscillations at the frequency \omega = \omega_p are scaled replicas of one another throughout N. A secondary result of the paper is called the Energy Theorem. This theorem assumes that the natural oscillations of a lossless oneport have been excited by a unit impulse of current, and states that each impedance residue of the oneport is proportional to the energy stored in the corresponding one of these oscillations.