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

Abstract

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.

System Characterization of Equivalent Circuits of Improper Rational Transfer Functions with Complex-Conjugate Poles and Residues

“…Previous work in [19]. for the admittance transfer function (ATF) with complexconjugate zeros, or equivalently the impedance transfer function (ITF) with complex-conjugate poles, and the immittance transfer functions (ATF and ITF) with complexconjugate poles/residues, including expressions for transformation of shunt elements in equivalent circuit form to/from the pole/residue form, (2) we perform frequency-domain SCP analysis of the admittance and impedance transfer functions with complex-conjugate poles/residues, and derive the SCP conditions for ATF and ITF across parameters in the equivalent circuit form and the pole/residue form, summarized in concise and comprehensive tables, and (3) we derive time-domain expressions of the impulse-response for the ATF and the ITF (i.e., y(t) and z(t)), and the instantaneous power due to an ideal voltage source (IVS) for an ATF or an ideal current source (ICS) for an ITF.…”

“…This section briefly reviews the fundamental definitions (e.g., the forms of rational transfer function, stability definitions, Routh-Hurwitz stability criterion, and excitation source) from previous works such as [9], [19]- [26].…”

“…The transfer function (TF) of a system in the Laplace domain may be represented as a ratio of two polynomials as follows [9], [19]- [21].…”

“…where p n = p nr + jp ni , c n = c nr + jc ni , x nr and x ni denotes the real and imaginary part of x n ; respectively, * is complex conjugate operator, {c nr , c ni , p nr , p ni } ∈ Reals, and all poles are assumed to be distinct. The expression for TF with real pole/zero is provided in [19]. As we mentioned, (3) is the partial fraction form of a strictly proper TF; however, if a rational TF is not strictly proper (r ≤ 0), we may decompose (1) by dividing polynomials of numerator to denominator, as…”

“…The work in [9] uses a semi-analytic and cellular approach to characterize the SCP of a strictly proper rational transfer function (TF) comprised of either real pole/residue and/or pair of complex-conjugate poles/residues, through the R-L equivalent circuit branch and/or the R-L-R-C equivalent circuit branch, respectively. Considering that certain fitting algorithms [11] may achieve improved model of a network's response by using additional coefficients equivalent to shunt elements (e.g., R shunt and C shunt ), our previous work in [19] analyzed the SCP of improper rational TFs with real pole/residue (or real pole/zeros), equivalent to R-L circuit branch parallel to R shunt and C shunt elements. The R-L circuit branch in parallel with R shunt and C shunt elements (i.e., R-L∥R∥C) forms an improper rational TF with real pole/residue; however, elucidation of the improper rational TF and the canonical form would be incomplete without adding a rigorous treatment of the complexconjugate poles/residues.…”