Distributed networks with small parasitic elements: Input-output stability

This paper considers a linear time-invariant network
\cal N
, made of lumped and distributed elements (
R, L, C, M,
transformers, gyrators, controlled and independent sources, transmission lines). The positive number
\epsilon
is a small number, proportional to the size of the stray lumped energy-storing elements: as
\epsilon \rightarrow 0,
the stray elements disappear: stray capacitors (inductors) become open (short, respectively) circuits. The problem is to find what additional condition is required to guarantee that if
{\cal N}_0
--the network
\cal N_{\epsilon}
, with
\epsilon
set to zero-is input-output stable, then for any
\epsilon
\epsilon
sufficiently small
{\cal N}_{\epsilon}
, is also input-output stable. The additional condition requires that some approximate "high-frequency" network be also input-output stable. It is also shown that if either of these conditions fail so does the input-output stability of
{\cal N}_{\epsilon}
for any
\epsilon
sufficiently small.