No abstract

Modern packaging design requires extensive signal integrity simulations in order to assess the electrical performance of the system. The feasibility of such simulations is granted only when accurate and efficient models are available for all system parts and components having a significant influence on the signals. Unfortunately, model derivation is still a challenging task, despite the extensive research that has been devoted to this topic. In fact, it is a common experience that modeling or simulation tasks sometimes fail, often without a clear understanding of the main reason. This paper presents the fundamental properties of causality, stability, and passivity that electrical interconnect models must satisfy in order to be physically consistent. All basic definitions are reviewed in time domain, Laplace domain, and frequency domain, and all significant interrelations between these properties are outlined. This background material is used to interpret several common situations where either model derivation or model use in a computer-aided design environment fails dramatically. We show that the root cause for these difficulties can always be traced back to the lack of stability, causality, or passivity in the data providing the structure characterization and/or in the model itself.

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed techniques.

This paper is devoted to transient analysis of lossy transmission lines characterized by frequency-dependent parameters. A public dataset of parameters for three line examples (a module, a board, and a cable) is used, and a new example of on-chip interconnect is introduced. This dataset provides a well established and realistic benchmark for accuracy and timing analysis of interconnect analysis tools. Particular attention is devoted to the intrinsic consistency and causality of these parameters. Several implementations based on generalizations of the well-known method-of-characteristics are presented. The key feature of such techniques is the extraction of the line modal delays. Therefore, the method is highly optimized for long interconnects characterized by significant propagation delay. Nonetheless, the method is also successfully applied here to a short high/loss on-chip line, for which other approaches based on lumped matrix rational approximations can also be used with high efficiency. This paper shows that the efficiency of delay extraction techniques is strongly dependent on the particular circuit implementation and several practical issues including generation of rational approximations and time step control are discussed in detail.

This paper presents a strategy for the construction of parameterized linear macromodels from tabulated port responses. These macromodels are able to reproduce the input-output behavior of the structure of interest both in terms of frequency and one or more design variables such as geometry and material parameters. A highly efficient combination of rational identification and piecewise linear interpolation leads to a macromodel form which can be cast as a polytopic descriptor form. This in turns enables the construction of a numerically robust testing procedure, based on linear matrix inequalities, for the assessment of uniform model stability within any prescribed region of the parameters space. Several numerical examples are used to illustrate the theory on practical application cases.