Synthesis of lumped-distributed cascades with lossy transmission lines

In the analysis of large systems such as high-speed digital computing networks and circuits on an LSI or VLSI silicon chip, lossy lumped-distributed networks have been used to model their interconnections. A solution of the synthesis problem for these networks will aid in the design of these circuits. This paper establishes single-variable realizability conditions and synthesis procedures for the class of lossy lumped-distributed cascade networks which contain lossy transmission lines and are desctribed by a driving point impedance expession of the form
Z_{0}=\frac{\sum_{i=0}^{n} a_{i}(s,z_{0})e^{(2i-n)}T_{0}\gamma (s)} {\sum_{i=0}^{n} b_{i}(s, z_{0})e^{(2i-n)}T_{0}\gamma (s)}
where
a_{i}(s, z_{0}), b_{i}(s, z_{0})
are two-variable, real polynomials in
s
and
z_{0}
, with
z_{0}
the characteristic impedance,
\gamma (s)
the progagation constant, and To the total "electrical length" characterizing each of the lossy lines. The cascade networks consist of commensurate, uniform and/or tapered, lossy (except distortionless [3], [4]) transmission lines interconnected by passive, lumped (lossless and/or lossy) two-ports and terminated in a passive load. This class includes general lines, leakage-free lines,
RC
-lines and acoustic filters. The results also apply to cascades with noncommensurate lines and to cascades of mixed transmission-line types.