Stability, Causality, and Passivity Analysis of Canonical Equivalent Circuits of Improper Rational Transfer Functions With Real Poles and Residues

This paper extends a previous work on synthesis of equivalent circuits using strictly proper canonical R-L and R
_a
-L-R
_b
-C circuit branches, and presents a thorough time-domain and frequency-domain analysis of stability/causality/passivity (SCP) of the R-L circuit branch (based on real pole/residue) and two additional shunt elements R
_shunt
and C
_shunt
parallel to the R-L branch, leading to an improper rational transfer function. We develop a rigorous and comprehensive table of sign-relationships including pole/residue, pole/zero, R/L/C elements, and SCP conditions to describe the interaction of R
_shunt
and C
_shunt
elements with an R-L branch. We also examine the effects on SCP due to negative gain coefficient of the transfer functions. Because such a topology can commonly occur as a result of applying fitting algorithms (e.g., Vector Fitting) on the electrical response of multi-port networks (e.g., impedance, admittance, or scattering parameters), it is important to understand the above SCP conditions for synthesis of practical SPICE models for stable time-domain simulations.