Abstract—In this paper, we describe the algorithms used in FastImp, a program for accurate analysis of wide-band electromagnetic effects in very complicated geometries of conductors. The program is based on a recently developed surface integral formulation and a precorrected fast Fourier transform (FFT) accelerated iterative method, but includes a new piecewise quadrature panel integration scheme, a new scaling and preconditioning technique as well as a generalized grid interpolation and projection strategy. Computational results are given on a variety of integrated circuit interconnect structures to demonstrate that FastImp is robust and can accurately analyze very complicated geometries of conductors. Index Terms—Fast integral equation solver, panel integration, parasitic extraction, preconditioner, surface integral formulation, wide-band analysis. I.

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Shrinking feature sizes and increasing speeds of operation make interconnect-related effects very relevant for current circuit verification methodologies. Reliable and accurate system verification requires the full analysis of circuits together with the environment that surrounds them, including the common substrate, the packaging structures, and perhaps even board information. In this paper we discuss circuit-level simulation algorithms that enable the analysis of the impact of strongly coupled interconnect structures on nonlinear circuit operation, so as to allow reliable and accurate system verification. 1

Abstract—In this paper, a closed-form integral representation for the eddy-current losses over a conductive substrate is presented. The results are applicable to monolithic inductors and transformers, especially when such structures are realized over an epitaxial CMOS substrate. The technique is verified against measured results from 100 MHz to 14 GHz for spiral inductors. Index Terms—CMOS substrate losses, eddy currents, monolithic inductors, monolithic transformers, spiral inductors, spiral transformers. I.

We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a so-called Fourier integral operator (FIO) of the form ∫ e2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) isan amplitude, and Φ(x, ξ) is a phase function, which is typically as large as |ξ|; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size N by N would require O(N 4) operations. This paper develops a new numerical algorithm which requires O(N 2.5 log N) operations and as low as O ( √ N) in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e2πiΦ(x,ξ) a(x, ξ) into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.

Integral equation based approaches are popular for extracting the capacitance of integrated circuit structures. Typically, first-order collocation or Galerkin methods are used. The resulting dense system of equations is e#ciently solved by combining matrix sparsification with an iterative solver. While the speed-up over direct factorization is substantial, the first-order methods still lead to large systems even for simple problems. In this paper we introduce a high-order Nystrom scheme. For the same level of discretization, the high-order schemes can be an order of magnitude more accurate than the first-order approaches at the same computational cost. As a consequence, we obtain the same level of accuracy with a much smaller matrix. 1 Introduction With decreasing feature sizes and increasing frequencies, accurate and e#cient capacitance extraction has become critical for design. In recent years, capacitance extractors based on integral equations [3, 7] have become popular. Integral ...

With the increase in circuit performance (higher speeds) and density (smaller feature size) in deep submicrometer (DSM) designs, interconnect parasitic effects are increasingly becoming more important. This paper first surveys the state of the art in parasitic extraction for resistance, capacitance, and inductance. The paper then covers other related issues such as interconnect modeling, model order reduction, delay calculation, and signal integrity issues such as crosstalk. Some future trends on parasitic extraction, model reduction and interconnect modeling are discussed and a fairly complete list of references is given.

This paper proposes a novel wideband modeling technique for high-performance RF passives and linear(ized) analog circuits. The new method is based on a recently proposed sdomain hierarchical modeling and analysis method [27]. Theoretically, we show that the s-domain hierarchical reduction is equivalent to implicit moment matching around s = 0, and that the existing hierarchical reduction method by one-point expansion is numerically stable for general tree-structured circuits. Practically, we propose a hierarchical multi-point reduction scheme for high-fidelity, wideband modeling of general passive or active linear circuits. A novel explicit waveform matching algorithm is proposed for searching the dominant poles and residues from different expansion points based on the unique hierarchical reduction framework. Experimental results with large analog circuits, on-chip spiral inductors are presented to validate the proposed method. I.

Electrostatic analysis of complicated molecular surfaces arises in a number of nanotechnology applications including: biomolecule design, carbon nanotube simulation, and molecular electron transport. Molecular surfaces are typically smooth, without the corners common in electrical interconnect problems, and are candidates for methods with higher order convergence than that of the commonly used flat panel methods. In this paper we describe and demonstrate a spectrally accurate approach for analyzing molecular surfaces described by a collection of surface points. The method is a synthesis of several techniques, and starts by using least squares to fit a high order spherical harmonic surface representation to the given points. Then this analytic representation is used to construct a differentiable map from the molecular suface to a cube, an orthogonal basis is generated on the rectangular cube surfaces, and a change of variables is used to desingularize the required integrals of products of basis functions and Green’s function. Finally, an efficient method for solving the discretized system using a matrix-implicit scheme is described. The combined method is demonstrated on an analytically solvable sphere problem, capacitance calculation of complicated molecular surface, and a coupled Poisson/Poisson-Boltzmann problem associated with a biomolecule. The results demonstrate that for a tolerance of 10 −3 this new approach requires one to two orders of magnitude fewer unknowns than a flat panel method. 1.

We describe powerful new techniques for the analysis of RF circuits. Next-generation CAD tools based on such techniques should enable RF designers to obtain a more accurate picture of how their circuits will operate. These new simulation capabilities will be essential in order to reduce the number of design iterations needed to produce complex RF ICs. 1 Introduction Design methodology and superior computer-aided design tools are key to success in the integrated circuit (IC) business. They are particularly important in the case of radio-frequency (RF) IC applications, where the digital IC divide-and-conquerdesign style, based on partitioning by functional blocks and abstraction levels, does not apply. The goal of an RF designer is to get a manufacturable design that meets the specifications with minimum cost, under severe time-to-market constraints. Unlike traditional discretecomponent RF design, prototyping is practically impossible, and the validation of a design can only be done b...