Passive synthesis of compact frequency-dependent interconnect models via quadrature spectral rules

In this paper, we present a reduced order modeling methodology, based on the utilization of optimal non-uniform grids generated by Gaussian spectral rules, for the direct passive synthesis of SPICE-compatible modeling of multi-conductor interconnect structures. The algorithm is based on a Pade-Chebyshev approximation of the frequency-dependent input impedance matrix of the passive interconnect system. The synthesized circuit is represented as the concatenation of a number of non-uniform sections of passive lumped coupled circuits. However, contrary to the popular uniform segmentation-based distributed circuit models for interconnects, where 10 to 15 segments per minimum wavelength are needed for multi-GHz accuracy, the proposed model is "optimal" in the sense that highly-accurate responses can be obtained with a number of segments per minimum wavelength barely exceeding the Nyquist limit of 2. This high accuracy stems from the super-exponential convergence of the Pade-Chebyshev approximation of the input impedance of the transmission-line model of the interconnect, and results in the synthesis of MNA stamps for the interconnect structure with five to ten times reduction in the number of state variables compared to uniform grids. Moreover, the passivity of the generated SPICE-compatible multi-port models is guaranteed through the use of passive equivalent circuits for the representation of the frequency-dependent, per-unit-length series impedance and shunt admittance matrices of the interconnect.