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

Analysis of crosstalk and field coupling to lossy MTL's in a SPICE environment

This paper proposes a circuit model for lossy multiconductor transmission lines (MTLs) suitable for implementation in modern SPICE simulators, as well as in any simulator supporting differential operators. The model includes the effects of a uniform or nonuniform disturbing field illuminating the line and is especially devised for the transient simulation of electrically long wideband interconnects with frequency dependent per-unit-length parameters. The MTL is characterized by its transient matched scattering responses, which are computed including both dc and skin losses by means of a specific algorithm for the inversion of the Laplace transform. The line characteristics are then represented in terms of differential operators and ideal delays to improve the numerical efficiency and to simplify the coding of the model in existing simulators. The model can be successfully applied to many kinds of interconnects ranging from micrometric high-resistivity metallizations to low-loss PCBs and cables, and can be considered a practical extension of the widely appreciated lossless MTL SPICE model, which maintains the simplicity and efficienc

Adaptive transient solution of nonuniform multiconductor transmission lines using wavelets

Abstract—This paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations. The discretization of the governing equations is obtained through a weak formula-tion using biorthogonal wavelet bases as trial and test functions. It is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes. A simple yet effective time–space adaptive al-gorithm capable of selecting the minimal number of unknowns at each time iteration is described. Numerical results show the high degree of adaptivity of the proposed scheme. Index Terms—Electromagnetic (EM) transient analysis, multi-conductor transmission lines (TLs), wavelet transforms. I

Impact on signal integrity of interconnect variabilities

In this paper, literature results on the statistical simulation of lossy and dispersive interconnect networks with uncertain physical properties are extended to general nonlinear circuits. The approach is based on the expansion of circuit voltages and currents into polynomial chaos approximations. The derivation of deterministic circuit equivalents for nonlinear components allows to retrieve the unknown expansion coefficients with a single circuit simulation, that can be carried out via standard SPICE-type solvers. These coefficients provide direct statistical information. The methodology allows the inclusion of arbitrary nonlinear elements and is validated via transmission-line networks terminated by diodes and driven by inverter

Analysis of nonuniform transmission lines with an iterative and adaptive perturbation technique

This paper presents an iterative and adaptive perturbation technique for the analysis of nonuniform transmission lines. Place-dependent variations of the per-unit-length parameters are interpreted as perturbations with respect to their average values along the line. This allows casting the governing equations for the corresponding perturbations of the voltages and currents as those of a uniform transmission line with distributed sources. Therefore, standard transmission line theory is used to calculate these perturbation terms. Specifically, perturbations of increasing order are computed iteratively starting from the solution of the unperturbed line. The accuracy is adaptively adjusted by setting a threshold on the convergence of the solution. The algorithm turns out to be simple to implement and very accurate, yet faster than traditional approaches based on the discretization of the line into uniform sections. The technique is validated through the analysis of several nonuniform transmission line structures of relevance in EMC applications, namely uniformly and nonuniformly twisted wire pairs as well as a cable bundle with lacing cords

Application of Taylor models to the worst-case analysis of stripline interconnects

This paper outlines a preliminary application of Taylor models to the worst-case analysis of transmission lines with bounded uncertain parameters. Taylor models are an algebraic technique that represents uncertain quantities in terms of a Taylor expansion complemented by an interval remainder encompassing approximation and truncation errors. The Taylor model formulation is propagated from input uncertainties to output responses through a suitable redef nition of the algebraic operations involved in their calculation. While the Taylor expansion def nes an analytical and parametric model of the response, the remainder provides a conservative bound inside which the true value is guaranteed to lie. The approach is validated against the SPICE simulation of a coupled stripline and shows promising accuracy and eff ciency

Wavelet-Based Adaptive Solution for the Nonuniform Multiconductor Transmission Lines

Abstract—A time-domain technique for the solution of arbi-trary nonuniform multiconductor transmission lines (NMTL’s) is presented. The technique is based on a weak formulation of the NMTL equations obtained through spatial expansion of the voltage and current vectors into biorthogonal wavelet func-tions. Wavelets allow adaptive representations of the solution by using few expansion coefficients, with any fixed approximation order. The set of significant expansion coefficients is determined automatically from the solution, which can be computed very efficiently. A numerical example illustrates the high adaptivity of the method. Index Terms—Distributed parameter circuits, multiconductor transmission lines, time domain analysis, wavelet transforms. I