Synthesis of lossy-lumped-distributed cascaded networks

Abstract

The synthesis of lossy lumped-distributed networks is important for many applications; for example, in the analysis of large systems such as the interconnections of the circuits on an LSI or VLSI silicon chip, such networks have been used as models, and a solution of the synthesis problem will thus aid in the design of these chips. In this paper singlevariable realizability conditions and synthesis procedures are established for the class of lossy and/or lossless lumped-distributed cascade networks described by an input-impedance expression of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_0 = \frac{\sum_{i=0}^{6n}a_i(s)e^{sT(2i-n)}}{\sum_{i=0}^{n}b_i(s)e^{sT(2i-n)}}</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a_i(s), b_i(s)</tex> real polynomials in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . The cascade networks consist of commensurate, uniform, lossless transmission lines interconnected by passive, lumped (lossless and/or lossy) two-ports and terminated in a passive load which can be prescribed as part of the specifications. Moreover, the results of this paper are also applicable to lumpeddistributed cascade networks which contain noncommensurate, tapered and/or lossy transmission lines (e.g., RC lines, distortionless lines) and to nonelectrical systems which can be modeled as distributed or lumpeddistributed cascades of types similar to the ones described above (e.g., acoustic filters).