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 itsel

Efficient Macromodeling and Fast Transient Simulation of High Speed Distributed Interconnects

In the first part of the thesis, an efficient macromodeling technique based on Loewner Matrix (LM) approach has been presented to model multi-port distributed systems using tabulated noisy data. In the proposed method, Loewner Model data from previous rational approximation are used to create less noisy eigenvectors in an iterative manner. As a result, the biasing effect of the LM model approximated by the noisy data is reduced. It is illustrated that this method improves the accuracy of the Loewner Matrix modeling for noisy frequency data. In the second part, a fast and robust algorithm is introduced for time-domain simulation of interconnects with few nonlinear elements based on Large Change Sensitivity approach. After macromodeling interconnects, linear parts of the system construct very large matrix. Large linear matrix with nonlinear components makes time domain simulation a Central Processing Unit (CPU) intensive task where inversion (one Lower/Upper (LU) decomposition and one forward/backward substitution) of this large matrix is done at each step of the Newton-Raphson iteration. Using the proposed method, large system matrix is partitioned into linear and nonlinear parts and LU decomposition of linear matrix is done only once in the entire simulation. Nonlinear elements construct a very small matrix compared to large linear matrix. In this proposed method, small matrix is inverted at each Newton iteration. Cost of inverting a small matrix is much cheaper than inverting a very large matrix. Therefore, this approach is faster than the conventional matrix inversion method. Numerical examples are presented illustrating validity and efficiency of the above method

Time-Domain Macromodeling of High Speed Distributed Networks

With the rapid growth in density, operating speeds and complexity of modern very-large-scale integration (VLSI) circuits, there is a growing demand on efficient and accurate modeling and simulation of high speed interconnects and packages in order to ensure the signal integrity, reliability and performance of electronic systems. Such models can be derived from the knowledge of the physical characteristics of the structure or based on the measured port-to-port response.In the first part of this thesis, a passive macromodeling technique based on Method of Characteristics (referred as Passive Method of Characteristics or PMoC) is described which is applicable for modeling of electrically long high-speed interconnect networks. This algorithm is based on extracting the propagation delay of the interconnect followed by a low order rational approximation to capture the attenuation effects. The key advantage of the algorithm is that the curve fitting to realize the macromodel depends only on per-unit-length (p.u.l.) parameters and not on the length of the transmission line. In this work, the PMoC is developed to model multiconductor transmission lines.Next, an efficient approach for time domain sensitivity analysis of lossy high speed interconnects in the presence of nonlinear terminations is presented based on PMoC. An important feature of the proposed method is that the sensitivities are obtained from the solution of the original network, leading to significant computational advantages. The sensitivity analysis is also used to optimize the physical parameters of the network to satisfy the required design constraints. A time-domain macromodel for lossy multiconductor transmission lines exposed to electromag¬netic interference is also described in this thesis based on PMoC. The algorithm provides an efficient mechanism to ensure the passivity of the macromodel for different line lengths. Numerical examples illustrate that when compared to other passive incident field coupling algorithms, the proposed method is efficient in modeling electrically long interconnects since delay extraction without segmentation is used to capture the frequency response.In addition, this thesis discusses macromodeling techniques for complex packaging structures based on the frequency-domain behavior of the system obtained from measurements or electromagnetic simulators. Such techniques approximate the transfer function of the interconnect network as a rational function which can be embedded with modern circuit simulators with integrated circuit emphasis (SPICE). One of the most popular tools for rational approximations of measured or simulated data is based on vector fitting (VF) algorithms. Nonetheless, the vector fitting algorithms usually suffer convergence issues and lack of accuracy when dealing with noisy measured data. As a part of this thesis, a methodology is presented to improve the convergence and accuracy issues of vector fitting algorithm based on instrumental variable technique. This methodology is based on obtaining the “instruments” in an iterative manner and do not increase the complexity of vector fitting to capture the frequency response and minimize the biasing