Michel S. Nakhla

Simulation of high-speed interconnects

With the rapid developments in very large-scale integration (VLSI) technology, design and computer-aided design (CAD) techniques, at both the chip and package level, the operating frequencies are fast reaching the vicinity of gigahertz and switching times are getting to the subnanosecond levels. The ever increasing quest for high-speed applications is placing higher demands on interconnect performance and highlighted the previously negligible effects of interconnects such as ringing, signal delay, distortion, reflections, and crosstalk. In this review paper various high-speed interconnect effects are briefly discussed. In addition, recent advances in transmission line macromodeling techniques are presented. Also, simulation of high-speed interconnects using model-reduction-based algorithms is discussed in detail.

Analysis of interconnect networks using complex frequency hopping (CFH)

With increasing miniaturization and operating speeds, loss of signal integrity due to physical interconnects represents a major performance limiting factor of chip-, board- or system-level design. Moment-matching techniques using Pade approximations have recently been applied to simulating modelled interconnect networks that include lossy coupled transmission lines and nonlinear terminations, giving a marked increase in efficiency over traditional simulation techniques. Nevertheless, moment-matching can be inaccurate in high-speed circuits due to critical properties of Pade approximations. Further, moment-generation for transmission line networks can be shown to have increasing numerical truncation error with higher order moments. These inaccuracies are reflected in both the frequency and transient response and there is no criterion for determining the limits of the error. In this paper, a multipoint moment-matching, or complex frequency hopping (CFH) technique is introduced which extracts accurate dominant poles of a linear subnetwork up to any predefined maximum frequency. The method generates a single transfer function for a large linear subnetwork and provides for a CPU/accuracy tradeoff. A new algorithm is also introduced for generating higher-order moments for transmission lines without incurring increasing truncation error. Several interconnect examples are considered which demonstrate the accuracy and efficiency in both the time and frequency domains of the new method. >

A neural network modeling approach to circuit optimization and statistical design

The trend of using accurate models such as physics-based FET models, coupled with the demand for yield optimization results in a computationally challenging task. This paper presents a new approach to microwave circuit optimization and statistical design featuring neural network models at either device or circuit levels. At the device level, the neural network represents a physics-oriented FET model yet without the need to solve device physics equations repeatedly during optimization. At the circuit level, the neural network speeds up optimization by replacing repeated circuit simulations. This method is faster than direct optimization of original device and circuit models. Compared to existing polynomial or table look-up models used in analysis and optimization, the proposed approach has the capability to handle high-dimensional and highly nonlinear problems. >

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.

Analysis of high-speed VLSI interconnects using the asymptotic waveform evaluation technique

The asymptotic waveform evaluation (AWE) technique provides a generalized approach to lumped RLC circuit response approximations. Two results are described: (1) generalization of the AWE method to handle interconnect models which contain distributed elements; and (2) application of the generalized AWE technique to the case where the distributed elements can be modeled as lossy coupled transmission lines. The generalized AWE technique is useful for delay and crosstalk estimation and can be used to evaluate transient responses of high-speed interconnect circuits with negligible error compared with conventional circuit simulators, while being two to three orders of magnitude faster. >

Time-domain analysis of lossy coupled transmission lines

A novel method based on numerical inversion of the Laplace transform is presented for the analysis of lossy coupled transmission lines with arbitrary linear terminal and interconnecting networks. The formulation of the network equations is based on a Laplace-domain admittance stamp for the transmission line. The transmission line stamp can be used to formulate equations representing arbitrarily complex networks of transmission lines and interconnects. These equations can be solved to get the frequency-domain response of the network. Numerical inversion of the Laplace transform allows the time-domain response to be calculated directly from Laplace-domain equations. This method is an alternative to calculating the frequency-domain response and using the fast Fourier transform to obtain the time-domain response. The inversion technique is equivalent to high-order, numerically stable integration methods. Numerical examples showing the general application of the method are presented. It is shown that the inverse Laplace technique is able to calculate the step response of a network. The time-domain independence of the solution is exploited by an efficient calculation of the propagation delay of the network. >

A fast algorithm and practical considerations for passive macromodeling of measured/simulated data

Passive macromodeling of high-speed package and interconnect modules characterized by simulated/measured data has generated immense interest during the recent years. This paper presents a fast algorithm for generating passive macro-models for simulated/measured data, based on linear formulation. New constraints are proposed to quickly generate passive macromodels. Examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

Global passivity enforcement algorithm for macromodels of interconnect subnetworks characterized by tabulated data

With the continually increasing operating frequencies, complex high-speed interconnect and package modules require characterization based on measured/simulated data. Several algorithms were recently suggested for macromodeling such types of data to enable unified transient analysis in the presence of external network elements. One of the critical issues involved here is the passivity violations associated with the computed macromodel. To address this issue, a new passivity enforcement algorithm is presented in this paper. The proposed method adopts a global approach for passivity enforcement by ensuring that the passivity correction at a certain region does not introduce new passivity violations at other parts of the frequency spectrum. It also provides an error estimate for the response of the passivity corrected macromodel.

A general class of passive macromodels for lossy multiconductor transmission lines

This paper presents a general class of passive macromodeling algorithm for multiport distributed interconnects. A new theorem is described that specifies sufficient conditions for matrix-rational approximation of exponential functions in order to generate a passive macromodel. A proof is given showing that the currently existing passive matrix-rational approximation of exponential functions is a subclass of the generic approach presented in this paper. In addition, a technique to obtain a compact passive macromodel with predetermined coefficients, based on near-optimal approximation, is presented. The proposed model can be easily incorporated with recently developed passive model-reduction techniques.

Efficient passive circuit models for distributed networks with frequency-dependent parameters

This paper presents an efficient method for the analysis of multiconductor transmission lines with frequency-dependent parameters. The proposed technique generates positive-real representations for the frequency dependency of transmission line parameters as well as closed-form expressions based on exponential Pade approximants. The new model is suitable for inclusion in general purpose circuit simulators and overcomes the difficulty of mixed frequency/time simulation encountered during transient analysis. In addition, the proposed model can be easily incorporated with the recently developed passive model-reduction techniques. Numerical examples are presented to demonstrate the validity and efficiency of the proposed method.