Interconnect has become the dominating factor in determining circuit performance and reliability in deep submicron designs. In this embedded tutorial, we first discuss the trends and challenges of interconnect design as the technology feature size rapidly decreases towards below 0.1 micron. Then, we present commonly used interconnect models and a set of interconnect design and optimization techniques for improving interconnect performance and reliability. Finally, we present comparisons of different optimization techniques in terms of their efficiency and optimization results, and show the impact of these optimization techniques on interconnect performance in each technology generation from the 0.35µm to 0.07µm projected in the National Technology Roadmap for Semiconductors.

Since the first papers on asymptotic waveform evaluation (AWE), Padé-based reduced-order models have become standard for improving coupled circuit-interconnect simulation efficiency. Such models can be accurately computed using bi-orthogonalization algorithms like Padé via Lanczos (PVL), but the resulting Padé approximates can still be unstable even when generatedfrom stable RLC circuits. For certain classes of RC circuits it has been shown that congruence transforms, like the Arnoldi algorithm, can generate guaranteed stable and passive reduced-order models. In this paper we present a computationally efficient model-order reduction technique, the coordinate-transformed Arnoldi algorithm, and show that this method generates arbitrarily accurate and guaranteed stable reduced-order models for RLC circuits. Examples are presented which demonstrates the enhanced stability and efficiency of the new method.

A method for the e�cient computation of accu� rate reduced�order models of large linear circuits is de� scribed. The method � called MPVL � employs a novel block Lanczos algorithm to compute matrix Pad�e ap� proximations of matrix�valued network transfer func� tions. The reduced�order models � computed to the re� quired level of accuracy � are used tospeed up the anal� ysis of circuits containing large linear blocks. The lin� ear blocks are replaced by their reduced�order models� and the resulting smaller circuit can be analyzed with general�purpose simulators � with signi�cant savings in simulation time and � practically � no loss of accuracy. 1

Reduced-order modeling techniques are now commonly used to efficiently simulate circuits combined with interconnect, but generating reduced-order models from realistic 3-D structures has received less attention. In this paper we describe a Krylov-subspace based method for deriving reduced-order models directly from the 3-D magnetoquasistatic analysis program FastHenry. This new approach is no more expensive than computing an impedance matrix at a single frequency.

In recent years, reduced-order modeling techniques based on Krylov-subspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the large-scale time-invariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reduced-order modeling techniques based on Krylov subspaces and describes the use of reduced-order modeling in circuit simulation. 1 Introduction Krylov-subspace methods, most notably the Lanczos algorithm [81, 82] and the Arnoldi process [5], have long been recognized as powerful tools for large-scale matrix computations. Matrices that occur in large-scale computations usually have some special structures that allow to compute matrix-vector products with such a matrix (or its transpose) much more efficiently than for a dense, unstructured matrix. The most common structure is sparsity, i.e., only few of the matrix entries are nonzero. Computing a matrix-vector pr...

This paper discusses the analysis of large linear electrical networks consisting of passive components, such as resistors, capacitors, inductors, and trans-formers. Such networks admit a symmetric formula-tion of their circuit equations. We introduce SyPVL, an eficient and numerically stable algorithm for the computation of reduced-order models of large, linear, passive networks. SyPVL represents the specializa-tion of the more general PVL algorithm, to symmetric problems. Besides the gain in eficiency over PVL, SyPVL also preserves the symmetry of the problem, and, as a consequence, can often guarantee the sta-bility of the resulting reduced-order models. Moreover, these reduced-order models can be synthesized as actual physical circuits, thus facilitating compatibility with existing analysis tools. The application of SyPVL is illustrated with two interconnect-analysis examples. 1

An overview of model reduction methods and a comparison of the resulting algorithms is presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better. 1 Introduction and problem statement Direct numerical simulation of dynamical systems has been an extremely successful means for studying complex physical phenomena. However, as more detail is included, the dimensionality of such simulations may increase to unmanageable levels of storage and computational requirements. One approach to overcoming this is through model reduction. The goal is to produce a low dimensional system that has

this paper, we discuss some of the features of the algorithms in the package, with emphasis on the issues related to using the codes. We describe in some detail two routines from the package, one for the solution of linear systems, and the other for the computation of eigenvalue approximations. We present some numerical examples from applications where QMRPACK was used. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems---computations on matrices; G.1.3 [Numerical Analysis]: Numerical Linear Algebra---

Abstract — RF circuits exhibit several distinguishing characteristics that make them difficult to simulate

This paper reviews the main ideas of reduced-order modeling techniques based on Krylov subspaces and describes some applications of reduced-order modeling in circuit simulation