The Brune Synthesis in State-space Terms*

The development of synthesis procedures for passing from a prescribed rational positive real function or matrix to a linear lumped network of passive components possessing the prescribed quantity as its impedance has been one of the major problems confronting network theorists in the past. One of the earliest such synthesis procedures is the Brune synthesis, see Reference 1 and for more recent treatments, References 2 and 3. As presented in References 1-3, the synthesis is of positive real functions, rather than positive real matrices. Mnitiport generalizations may also be fo~nd."~ Reference 9 by Newcomb, besides referencing earlier technical reports by the same author, provides comparisons of the various multiport approaches. Our goal here is twofold. First, we aim to present the Brune cycle in state-space terms. As we show here, carrying out a Brune cycle is equivalent to the problem of finding a state-space description of an impedance in a special co-ordinate basis. In this special co-ordinate basis, a certain matrix appearing in the so-called Positive Real Lemma10-'3-the fundamental result on positive realness in state-space terms-takes on a special form which enables its part identification. Since, as argued in Reference 10, synthesis in some ways is equivalent to the complete identification of this matrix, it becomes reasonable that part identification corresponds to part synthesis. Our second aim is to illustrate how the multiport case is a natural extension of the single port case when viewed in state-space terms-perhaps more so than when a classical viewpoint is taken. Further, and as one would hope, in synthesizing a symmetric impedance matrix, gyrators are automatically excluded. The layout of the paper is as follows. In section 2 we analyse, as opposed to synthesize, the structure resulting from a Brune cycle, to exhibit the sort of state-space equations one needs in order to execute a synthesis step. Section 3 contains our fundamental lemma, explaining how one can change the co-ordinate basis to get the right equations (The proof has common roots with that used by Yakubovic in a proof of the Positive Real Lemma".) The section 'One-Port Brune Synthesis' explains the one-port synthesis, and the section 'Multiport Brune Synthesis' with the aid of a generalization of the fundamental lemma, discusses the multiport problem. We caution the reader that in the course of the paper, some familiarity with the Brune synthesis is expected, including some familiarity with the properties of positive real functions and matrices. References 2 , 3 , 9 will provide adequate background. Further, some familiarity with state-variable descriptions of networks is also expected, see e.g. Reference 10.