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This paper presents a comprehensive analysis of stability/causality/passivity (SCP) for improper rational transfer functions (TFs) with  complex-conjugate  pair of poles/residues, as an extension of  real  poles/residues case in our previous work. We derive and validate the frequency-domain constraints of SCP for  improper  rational TFs with  complex-conjugate  pair of poles/residues, equivalent to an R-L-R-C circuit branch in parallel with R shunt  and C shunt  elements. We present a novel algorithm for SCP assessment of single-input single-output (SISO) systems at the cellular circuit level, including detailed tables to establish the relationship between the TF parameters (pole, residue, zero), the equivalent circuit parameters (R, L, C), and the SCP conditions, for an improper system of R-L-R-C||R shunt ||C shunt  circuit branches. The aforementioned circuit-block structure serves as an important canonical topology for macromodeling of electrical systems across numerous applications. The presented methodology may be used for synthesis of guaranteed-stable SPICE equivalent circuits of electrical systems represented by network parameters (e.g., S/Y/Z-parameters), including power systems operating at tens of Hertz (Hz), high-speed &#x03BC;wave/mm-wave electronics operating at hundreds of MHz/GHz, and ultra high-bandwidth opto-electronic and optical systems operating at hundreds of THz.

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Stability, Causality, and Passivity of Canonical Equivalent Circuits for Improper Rational Transfer Functions with Complex-Conjugate Poles and Residues