Addressing Computational Complexity of High Speed Distributed Circuits Using Model Order Reduction

Advanced in the fabrication technology of integrated circuits (ICs) over the last couple of years has resulted in an unparalleled expansion of the functionality of microelectronic systems. Today’s ICs feature complex deep-submicron mixed-signal designs and have found numerous applications in industry due to their lower manufacturing costs and higher performance levels. The tendency towards smaller feature sizes and increasing clock rates is placing higher demands on signal integrity design by highlighting previously negligible interconnect effects such as distortion, reflection, ringing, delay, and crosstalk. These effects if not predicted in the early stages of the design cycle can severely degrade circuit performance and reliability. The objective of this thesis is to develop new model order reduction (MOR) techniques to minimize the computational complexity of non-linear circuits and electronic systems that have delay elements. MOR techniques provide a mechanism to generate reduced order models from the detailed description of the original modified nodal analysis (MNA) formulation. The following contributions are made in this thesis: 1. The first project presents a methodology for reduction of Partial Element Equivalent Circuit (PEEC) models. PEEC method is widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. The PEEC model with retardation has been applied to 3-D analysis but often result in large and dense matrices, which are computationally expensive to solve. In this thesis, a new moment matching technique based on Multi-order Arnoldi is described to model PEEC networks with retardation. 2. The second project deals with developing an efficient model order reduction algorithm for simulating large interconnect networks with nonlinear elements. The proposed methodology is based on a multidimensional subspace method and uses constraint equations to link the nonlinear elements and biasing sources to the reduced order model. This approach significantly improves the simulation time of distributed nonlinear systems, since additional ports are not required to link the nonlinear elements to the reduced order model, yielding appreciable savings in the size of the reduced order model and computational time. 3. A parameterized reduction technique for nonlinear systems is presented. The proposed method uses multidimensional subspace and variational analysis to capture the variances of design parameters and approximates the weakly nonlinear functions as a Taylor series. An SVD approach is presented to address the efficiency of reduced order model. The proposed methodology significantly improves the simulation time of weakly nonlinear systems since the size of the reduced system is smaller than the original system and a new reduced model is not required each time a design parameter is changed

Transient Analysis of High-Speed Channels via Newton-GMRES Waveform Relaxation

This paper presents a technique for the numerical simulation of coupled high-speed channels terminated by arbitrary nonlinear drivers and receivers. The method builds on a number of existing techniques. A Delayed-Rational Macromodel is used to describe the channel in compact form, and a general Waveform Relaxation framework is used to cast the solution as an iterative process that refines initial estimates of transient scattering waves at the channel ports. Since a plain Waveform Relaxation approach is not able to guarantee convergence, we turn to a more general class of nonlinear algebraic solvers based on a combination of the Newton method with a Generalized Minimal Residual iteration, where the Waveform Relaxation equations act as a preconditioner. The convergence of this scheme can be proved in the general case. Numerical examples show that very few iterations are indeed required even for strongly nonlinear termination

Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach

Guaranteed passive parameterized model order reduction of the partial element equivalent circuit (PEEC) method

The decrease of IC feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the system under study as a function of design parameters, such as geometrical and substrate features, in addition to frequency (or time). Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters. We propose an innovative PMOR technique applicable to PEEC analysis, which combines traditional passivity-preserving model order reduction methods and positive interpolation schemes. It is able to provide parametric reduced-order models, stable, and passive by construction over a user-defined range of design parameter values. Numerical examples validate the proposed approach

Addressing Computational Complexity of Electromagnetic Systems Using Parameterized Model Order Reduction

As operating frequencies increase, full wave numerical techniques such as the finite element method (FEM) become necessary for the analysis of high-frequency and microwave circuit structures. However, the FEM formulation of microwave circuits often results in very large systems of equations which are computationally expensive to solve. The objective of this thesis is to develop new parameterized model order eduction (MOR) techniques to minimize the computational complexity of microwave circuits. MOR techniques provide a mechanism to generate reduced order models from the detailed description of the original FEM formulation. The following contributions are made in this thesis: 1. The first project deals with developing a parameterized model order reduction to solve eigenvalue equations of electromagnetic structures that are discretized by using FEM. The proposed algorithm uses a multidimensional subspace method based on modified perturbation theory and singular-value decomposition to perform reduction directly on the finite element eigenvalue equations. This procedure generates parametric reduced order models that are valid over the desired parameter range without the need to redo the reduction when design parameters are changed. This provides significant computational savings when compared to previous eigenvalue MOR techniques, since a new reduced order model is not required each time a design parameter is changed. 2. Implicit moment match techniques such as the Arnoldi algorithm are often used to improve the accuracy of the reduced order model. However, the traditional Arnoldi algorithm is only applicable to first order linear systems and can not directly include arbitrary functions of frequency due to material and boundary conditions. In this work, an efficient algorithm to create parametric reduced order models of distributed electromagnetic systems that have arbitrary functions of frequency (due to material properties, boundary conditions, and delay elements) and design parameters. The proposed method is based on a multi-order Arnoldi algorithm used to implicitly calculate the moments with respect to frequency and design parameters, as well as the cross-moments. This procedure generates parametric reduced order models that are valid over the desired parameter range without the need to redo the reduction when design parameters are changed and provides more accurate reduced order systems when compared with traditional approaches such as Modified Gram Schmidt. 3. This project develops an efficient technique to calculate sensitivities of microwave structures with respect to network design parameters. The proposed algorithm uses a parametric reduced order model to solve the original network and an adjoint variable method to calculate sensitivities. Important features of the proposed method are 1) that the solution of the original network as well as sensitivities with respect to any parameter is obtained from the solution of the reduced order model, and 2) a new reduced order model is not required each time design parameters are varied

Fast methods for full-wave electromagnetic simulations of integrated circuit package modules

Fast methods for the electromagnetic simulation of integrated circuit (IC) package modules through model order reduction are demonstrated. The 3D integration of multiple functional IC chip/package modules on a single platform gives rise to geometrically complex structures with strong electromagnetic phenomena. This motivates our work on a fast full-wave solution for the analysis of such modules, thus contributing to the reduction in design cycle time without loss of accuracy. Traditionally, fast design approaches consider only approximate electromagnetic effects, giving rise to lumped-circuit models, and therefore may fail to accurately capture the signal integrity, power integrity, and electromagnetic interference effects. As part of this research, a second order frequency domain full-wave susceptance element equivalent circuit (SEEC) model will be extracted from a given structural layout. The model so obtained is suitably reduced using model order reduction techniques. As part of this effort, algorithms are developed to produce stable and passive reduced models of the original system, enabling fast frequency sweep analysis. Two distinct projection-based second order model reduction approaches will be considered: 1) matching moments, and 2) matching Laguerre coefficients, of the original system's transfer function. Further, the selection of multiple frequency shifts in these schemes to produce a globally representative model is also studied. Use of a second level preconditioned Krylov subspace process allows for a memory-efficient way to address large size problems.Ph.D.Committee Chair: Swaminathan Madhavan; Committee Member: Papapolymerou John; Committee Member: Chatterjee Abhijit; Committee Member: Peterson Andrew; Committee Member: Sitaraman Sures

Transient simulation of complex electronic circuits and systems operating at ultra high frequencies

The electronics industry worldwide faces increasingly difficult challenges in a bid to produce ultra-fast, reliable and inexpensive electronic devices. Electronic manufacturers rely on the Electronic Design Automation (EDA) industry to produce consistent Computer A id e d Design (CAD) simulation tools that w ill enable the design of new high-performance integrated circuits (IC), the key component of a modem electronic device. However, the continuing trend towards increasing operational frequencies and shrinking device sizes raises the question of the capability of existing circuit simulators to accurately and efficiently estimate circuit behaviour. The principle objective of this thesis is to advance the state-of-art in the transient simulation of complex electronic circuits and systems operating at ultra high frequencies. Given a set of excitations and initial conditions, the research problem involves the determination of the transient response o f a high-frequency complex electronic system consisting of linear (interconnects) and non-linear (discrete elements) parts with greatly improved efficien cy compared to existing methods and with the potential for very high accuracy in a way that permits an effective trade-off between accuracy and computational complexity. High-frequency interconnect effects are a major cause of the signal degradation encountered b y a signal propagating through linear interconnect networks in the modem IC. Therefore, the development of an interconnect model that can accurately and efficiently take into account frequency-dependent parameters of modem non-uniform interconnect is of paramount importance for state-of-art circuit simulators. Analytical models and models based on a set of tabulated data are investigated in this thesis. Two novel, h igh ly accurate and efficient interconnect simulation techniques are developed. These techniques combine model order reduction methods with either an analytical resonant model or an interconnect model generated from frequency-dependent sparameters derived from measurements or rigorous full-wave simulation. The latter part o f the thesis is concerned with envelope simulation. The complex mixture of profoundly different analog/digital parts in a modern IC gives rise to multitime signals, where a fast changing signal arising from the digital section is modulated by a slower-changing envelope signal related to the analog part. A transient analysis of such a circuit is in general very time-consuming. Therefore, specialised methods that take into account the multi-time nature o f the signal are required. To address this issue, a novel envelope simulation technique is developed. This technique combines a wavelet-based collocation method with a multi-time approach to result in a novel simulation technique that enables the desired trade-off between the required accuracy and computational efficiency in a simple and intuitive way. Furthermore, this new technique has the potential to greatly reduce the overall design cycle

Combining Krylov subspace methods and identification-based methods for model order reduction

Many different techniques to reduce the dimensions of a model have been proposed in the near past. Krylov subspace methods are relatively cheap, but generate non-optimal models. In this paper a combination of Krylov subspace methods and orthonormal vector fitting (OVF) is proposed. In that way a compact model for a large model can be generated. In the first step, a Krylov subspace method reduces the large model to a model of medium size, then a compact model is derived with OVF as a second step