A Parameterization Scheme for Lossy Transmission Line Macromodels with Application to High Speed Interconnects in Mobile Devices

We introduce a novel parameterization scheme based on the generalized method of characteristics (MoC) formacromodels of transmission-line structures having a cross section depending on several free geometrical and material parameters. This situation is common in early design stages, when the physical structures still have to be finalized and optimized under signal integrity and electromagnetic compatibility constraints. The topology of the adopted line macromodels has been demonstrated to guarantee excellent accuracy and efficiency. The key factors are propagation delay extraction and rational approximations, which intrinsically lead to a SPICE-compatible macromodel stamp. We introduce a scheme that parameterizes this stamp as a function of geometrical and material parameters such as conductor-width and separation, dielectric thickness, and permettivity. The parameterization is performed via multidimensional interpolation of the residue matrices in the rational approximation of characteristic admittance and propagation operators. A significant advantage of this approach consists of the possibility of efficiently utilizing the MoC methodology in an optimization scheme and eventually helping the design of interconnects.We apply the proposed scheme to flexible printed interconnects that are typically found in portable devices having moving parts. Several validations demonstrate the effectiveness of the approac

Parametric macromodeling of lossy and dispersive multiconductor transmission lines

We propose an innovative parametric macromodeling technique for lossy and dispersive multiconductor transmission lines (MTLs) that can be used for interconnect modeling. It is based on a recently developed method for the analysis of lossy and dispersive MTLs extended by using the multivariate orthonormal vector fitting (MOVF) technique to build parametric macromodels in a rational form. They take into account design parameters, such as geometrical layout or substrate features, in addition to frequency. The presented technique is suited to generate state-space models and synthesize equivalent circuits, which can be easily embedded into conventional SPICE-like solvers. Parametric macromodels allow to perform design space exploration, design optimization, and sensitivity analysis efficiently. Numerical examples validate the proposed approach in both frequency and time domain

Comprehensive and modular stochastic modeling framework for the variability-aware assessment of Signal Integrity in high-speed links

This paper presents a comprehensive and modular modeling framework for stochastic signal integrity analysis of complex high-speed links. Such systems are typically composed of passive linear networks and nonlinear, usually active, devices. The key idea of the proposed contribution is to express the signals at the ports of each of such system elements or subnetworks as a polynomial chaos expansion. This allows one to compute, for each block, equivalent deterministic models describing the stochastic variations of the network voltages and currents. Such models are synthesized into SPICE-compatible circuit equivalents, which are readily connected together and simulated in standard circuit simulators. Only a single circuit simulation of such an equivalent network is required to compute the pertinent statistical information of the entire system, without the need of running a large number of time-consuming electromagnetic circuit co-simulations. The accuracy and efficiency of the proposed approach, which is applicable to a large class of complex circuits, are verified by performing signal integrity investigations of two interconnect examples

Improving the Convergence of Vector Fitting for Equivalent Circuit Extraction From Noisy Frequency Responses

The vector fitting (VF) algorithm has become a common tool in electromagnetic compatibility and signal integrity studies. This algorithm allows the derivation of a rational approximation to the transfer matrix of a given linear structure starting from measured or simulated frequency responses. This paper addresses the convergence properties of a VF when the frequency samples are affected by noise.We show that small amounts of noise can seriously impair or destroy convergence. This is due to the presence of spurious poles that appear during the iterations. To overcome this problem we suggest a simple modification of the basic VF algorithm, based on the identification and removal of the spurious poles. Also, an incremental pole addition and relocation process is proposed in order to provide automatic order estimation even in the presence of significant noise.We denote the resulting algorithm as vector fitting with adding and skimming (VF-AS). A thorough validation of the VF-AS algorithm is presented using a Monte Carlo analysis on synthetic noisy frequency responses. The results show excellent convergence and significant improvements with respect to the basic VF iteration scheme. Finally, we apply the new VF-AS algorithm to measured scattering responses of interconnect structures and networks typical of high-speed digital systems

Equalization of multi-Gb/s chip-to-chip interconnects affected by manufacturing tolerances

Electrical chip-to-chip interconnects suffer from considerable intersymbol interference at multi-Gb/s data rates, due to the frequency-dependent attenuation. Hence, reliable communication at high data rates requires equalization, to compensate for the channel response. As these interconnects are prone to manufacturing tolerances, the equalizer must be adjusted to each specific channel realization to perform optimally. We adopt a reduced-complexity equalization scheme where (part of) the equalizer is fixed, by involving the channel statistics into the equalizer derivation. For a 10 cm on-board microstrip interconnect with a 10% tolerance on its parameters, we point out that 2-PAM transmission using a fixed prefilter and an adjustable feedback filter, both with few taps, yields only a moderate bit error rate degradation, compared to the all-adjustable equalizer; at a bit error rate of 1e-12 these degradations are about 1.1  dB and 3  dB, when operating at 20 Gb/s and 80 Gb/s, respectively

Wavelet-Based High-Order Adaptive Modeling of Lossy Interconnects

Abstract—This paper presents a numerical-modeling strategy for simulation of fast transients in lossy electrical interconnects. The proposed algorithm makes use of wavelet representations of voltages and currents along the structure, with the aim of reducing the computational complexity of standard time-domain solvers. A special weak procedure for the implementation of possibly dynamic and nonlinear boundary conditions allows to preserve stability as well as a high approximation order, thus leading to very accurate schemes. On the other hand, the wavelet expansion allows the computation of the solution by using few significant coefficients which are automatically determined at each time step. A dynamically refinable mesh is then used to perform a sparse time-stepping. Several numerical results illustrate the high efficiency of the proposed algorithm, which has been tuned and optimized for best performance in fast digital applications typically found on modern PCB structures. Index Terms—Finite difference methods, time-domain analysis, transmission lines, wavelet transforms. I