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

This paper presents a comprehensive analysis of stability/causality/passivity (SCP) for improper rational transfer functions 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 transfer functions 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 also develop detailed tables to establish a relationship between the transfer function 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 multi-port 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 micro-wave/mm-wave electronics operating at hundreds of MHz/GHz, and ultra high-bandwidth opto-electronic and optical systems operating at hundreds of THz.