Generalized Darlington synthesis

Abstract

The existence of a has been the topic of vigorous debate since the time of Darlington's original attempts at synthesizing a lossy input impedance through a lossless cascade of sections terminated in a unit resistor. This paper gives a survey of present insights in that existential question. In the first part it considers the multiport, time invariant case, and it gives the necessary and sufficient conditions for the existence of the Darlington embedding, namely that the matrix transfer scattering function considered must satisfy a special property of analyticity known as pseudomeromorphic continuability (of course aside from the contractivity condition which ensures lossiness). As a result, it is reasonably easy to construct passive impedances or scattering functions which do not possess a Darlington embedding, but they will not be rational, i.e. they will have infinite dimensional state spaces. The situation changes dramatically when time-varying systems are concerned. In this case also Darlington synthesis is possible and attractive, but the anomalous case where no synthesis is possible already occurs for systems with finite dimensional state spaces. We give precise conditions for the existence of the Darlington synthesis for the time-varying case as well. It turns out that the main workhorse in modern Darlington theory is the geometry of the so called Hankel map of the scattering transfer function to be embedded. This fact makes Darlington theory of considerably larger importance for the understanding of systems and their properties than the original synthesis question would seem to infer. Although the paper is entirely devoted to the theoretical question of existence of the Darlington embedding and its system theoretic implications, it does introduce the main algorithm used for practical Darlington synthesis, namely the 'square root algorithm' for external or inner-outer factorization, and discusses some of its implications in the final section.