Piero Triverio

Stability, Causality, and Passivity in Electrical Interconnect Models

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.

A Parameterized Macromodeling Strategy With Uniform Stability Test

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.

Delay-Based Macromodeling of Long Interconnects From Frequency-Domain Terminal Responses

We present a robust and efficient scheme for the generation of delay-based macromodels from frequency-domain tabulated responses. These responses can come from both simulation or direct measurement. The main algorithm is based on an iterative weighted least-squares process for the identification of delayed rational approximations. In case that pole relocation is performed during the iterations, the scheme can be interpreted as a generalization of the well-known vector fitting algorithm to delayed systems. Therefore, we denote this algorithm as delayed vector fitting (DVF). In case no pole relocation is performed, the scheme generalizes the so-called Sanathanan-Koerner iteration, calling for the denomination of delayed Sanathanan-Koerner algorithm. These techniques produce compact macromodels that are readily synthesized in SPICE-compatible equivalent circuits including delayed sources or ideal transmission line elements. These macromodels outperform classical rational macromodels in terms of simulation time. Several examples illustrate the advantages of proposed methodology.

A robust causality verification tool for tabulated frequency data

We introduce a robust algorithm for the qualification of tabulated frequency data representing the port responses of a linear subsystem. This algorithm is aimed at the verification that frequency responses obtained via full-wave electromagnetic simulation or from direct measurement of some interconnect structure are self-consistent and causal before they are used for the generation of macromodels for system-level analysis and design purposes. The technique is based on an advanced formulation of the Hilbert transform, leading to a direct check for causality of the raw data. One of the main advantages of the proposed method is the possibility to provide accurate estimates for error bounds due to various sources, including sampling frequency and data availability over a limited bandwidth

Robust Causality Characterization via Generalized Dispersion Relations

The self-consistency of frequency responses obtained via numerical simulations or measurements is of paramount importance in the analysis and design of linear systems. In particular, tabulated responses with flaws and causality violations have been demonstrated to be the root cause for numerical problems and unreliability in modeling and simulation tasks. In this work, we present the generalized dispersion relations as a robust and reliable tool for the causality characterization of frequency responses. Several applications are presented, including causality and passivity verification for tabulated data and causality-controlled interpolation schemes. Practical examples illustrate the excellent performance of the proposed techniques.

Passive parametric macromodeling from sampled frequency data

We present an efficient algorithm for the generation of passive parametric macromodels from impedance, admittance, or scattering frequency samples. The model formulation includes design variables in symbolic form and guarantees stability, causality, and passivity by construction. Model accuracy and efficiency are superior to previous solutions, because of a generalized formulation based on parameter-dependent poles. Two application examples demonstrate the excellent algorithm performance in modeling linear passive devices and interconnects.

An Equivalent Surface Current Approach for the Computation of the Series Impedance of Power Cables with Inclusion of Skin and Proximity Effects

We present a fast numerical technique for calculating the series impedance matrix of systems with round conductors. The method is based on a surface admittance operator in combination with the method of moments and it accurately predicts both skin and proximity effects. Application to a three-phase armored cable with wire screens demonstrates a speedup by a factor of about 100 compared to a finite elements computation. The inclusion of proximity effect in combination with the high efficiency makes the new method very attractive for cable modeling within EMTPtype simulation tools. Currently, these tools can only take skin effect into account.We present an efficient numerical technique for calculating the series impedance matrix of systems with round conductors. The method is based on a surface admittance operator in combination with the method of moments and it accurately predicts skin and proximity effects. The application to a three-phase armored cable with wire screens demonstrates a speedup by a factor of about 100 compared to a finite-elements computation. The inclusion of proximity effect, in combination with the high efficiency, makes the new method very attractive for cable modeling within Electromagnetic Transients Program-type simulation tools. Currently, these tools can only take skin effect into account.

Identification of Highly Efficient Delay-Rational Macromodels of Long Interconnects From Tabulated Frequency Data

We introduce a novel formulation of black-box models for long multiconductor interconnects, together with an identification algorithm from tabulated scattering parameters. The fundamental assumption requires a modal decomposition matrix that does not depend on frequency. The model structure includes low-order rational coefficients with suitable delay operators. The latter are included in a feedback loop; therefore, all infinite signal reflections are automatically accounted for. This feature may lead to significant speed up factors during circuit-based transient simulation with respect to other state-of-the-art solutions. The model is cast in a delayed descriptor form, leading to a straightforward conversion to an equivalent SPICE-compatible netlist. Finally, a purely algebraic stability test is presented based on a linear matrix inequality. The very high efficiency of the proposed models is demonstrated through several application examples.

Parametric Macromodeling of Multiport Networks from Tabulated Data

We propose a numerical technique to compute, from tabulated frequency data, compact macromodels parameterized by design variables, to be used for efficient optimization, Monte Carlo analysis and design centering of complex systems. Important theoretical results on the stability of parameter-dependent models are also presented.

An improved fitting algorithm for parametric macromodeling from tabulated data

This paper introduces a new scheme for the identification of multivariate behavioral macromodels from tabulated frequency-domain data. The method produces closed-form parametric expressions that reproduce with excellent accuracy the external port behavior of the structure, both as function of frequency and one or more external parameters. The numerical robustness of the main algorithm is demonstrated on two significant examples.