Generalized Darlington synthesis

Abstract

Abstract-The existence of a "Darlington embedding" 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.

Minimal symmetric Darlington synthesis

“…whose solvability is ensured by the contractivity of S. Clearly the extension is symmetric if and only if −e iθ 1 p * 2 = e iθ 2 p 2 , which is compatible with (15) if, and only if, all zeros of the polynomial dq(dq) * − dp 1 (dp 1 …”

“…We refer the reader to the nice surveys [11,15] for further references and generalizations (e.g., to the non-stationary case). An imbedding of the form (1) will be called a Darlington synthesis or inner extension, or even sometimes a lossless extension of S. A Darlington synthesis exists provided that S(iω) has constant rank a.e., that the determinant of I p − S(iω)S * (iω) (viewed as an operator on its range) satisfies the Szegö condition (see e.g., [23]), and that S is pseudo-continuable across the imaginary axis meaning that there is a meromorphic function in the left half-plane whose nontangential limits on iR agree with S(iω) a.e.…”

“…When S is rational the above conditions are fulfilled so that a Darlington synthesis always exists. In addition, it can be carried-out without augmenting the McMillan degree (i.e., we may require in (1) that degS = degS) and choosing m = p; this follows easily from Fuhrmann's realization theory [19] and the arguments in [15] or else from more direct computations carried out in [2,24]. In particular S can be chosen rational, and also to have real coefficients if S does.…”

“…This problem was first studied by Darlington in the case of a scalar rational S [D1], was generalized to the matrix case [B1], and finally carried over to non-rational S [A2, D2]. We refer the reader to the nice surveys [BM,D3] for further references and generalizations (e.g. to the non-stationary case).…”

Extended Darlington Synthesis of Fractional Order Immittance Function With Two Element Orders

“…Amarit and Sanjit [25] extended Darlington synthesis to the two-variable case and changed an extracted unit resistor into an impedance function in 1975, but this method has restrictions on the function form and only can be used to realize reciprocity network. After that, Dewilde [26] conducted a detailed analysis of Darlington synthesis and Carlin Herbert [27] summarized Darlington Synthesis. Then Belevitch [28] proposed Darlington synthesis to arbitrary integer order 2n-port in 2011.…”