Optimization-based Wideband Basis Functions for Efficient Interconnect Extraction

Abstract

Abstract-This paper introduces a technique for the numerical generation of basis functions that are capable of parameterizing the frequency-variant nature of cross-sectional conductor current distributions. Hence skin and proximity effects can be captured utilizing much fewer basis functions in comparison to the prevalently-used piecewise-constant basis functions. One important characteristic of these basis functions is that they only need to be pre-computed once for a frequency range of interest per unique conductor cross-sectional geometry, and they can be stored off-line with a minimal associated cost. In addition, the robustness of these frequency-independent basis functions are enforced using an optimization routine. It has been demonstrated that the cost of solving a complex interconnect system can be reduced by a factor of 170 when compared to the use of piecewise-constant basis functions over a wide range of operating frequencies. [6] that are capable of rapid and accurate resolution of large conductor system impedance, some of which can even account for the effects invoked by the presence of a semi-conductive substrate. However, the efficiency of these solvers is being continuously challenged by the ever increasing operating frequencies which generate skin and proximity effects that need to be carefully modeled in order to provide accurate impedance solutions. I. INTRODUCTIONSkin and proximity effects are troublesome for today's fast solvers due to the fact that most of the mixed potential integral equation (MPIE) solvers rely on piecewise-constant [7], [2], [6] or piecewise-linear basis functions to capture conductor current distributions. As frequency increases, the use of these basis functions has proven to be computationally expensive because a very fine discretization scheme must be applied in order to faithfully capture the exponential variation in crosssectional conductor current, thus generating densely-populated clusters of filaments with highly uneven aspect ratios. One might think this is not problematic if the discretized system were to be solved by a fast technique since, theoretically speaking, the computational cost only scales in a linear fashion with the total number of basis functions. However, in practice, direct computation must be used to account for the numerous near-distanced interactions generated by clusters of long and skinny filaments which takes a quadratic order of complexity to resolve.The explosive cost associated with high-frequency impedance simulations has spurred the development of methods that seek to either represent interior conductor current using surface field quantities [8]

Stochastic High Order Basis Functions for Volume Integral Equation with Surface Roughness

“…Despite the fact that the number of unique SHOBF integrals is K 2 less than the SPCBF integrals (K is defined in (4)), the matrix fill time did not reduce significantly. This is because one single SHOBF integral is exactly K 2 more expensive than one SPCBF integral (5). The reduction in the matrix solution time is however O(R 3 ), where R is the reduction in the number of basis functions, as a consequence of using gaussian elimination to solve the linear system.…”

“…In this paper we use the wideband proximity templates basis functions generation procedure described in [4], [5], and summarized in II-B. However, in our case, the cross sectional area of the conductor is defined by rough surfaces, and is therefore not deterministic.…”

“…Such calculations can be carried out very efficiently since they involve solving (1) on a geometry including only two wires, that can therefore be discretized using for instance simple thin filaments (piecewise constant basis functions). Finally, for wideband basis functions that can generate well-conditioned systems we can, as in [5], generate cross-section current distributions at a large number of frequency points and proximity configurations, assemble them as columns of a matrix, use singular value decomposition to obtain the principal orthogonal vectors, and optimize them to guarantee a physical non-zero current sum.…”

“…However, a very large number of thin cross sectional volumetric filaments may be required to accurately describe the current within each segment and accurately account for roughness effects, since the correlation length of the surface roughness can be very small relative to the cross section. High order basis functions have been proposed and developed in [4], [5] as an efficient way to reduce the number of cross sectional basis functions in deterministic solvers. In this paper we make use of a similar wideband basis function generation technique, and extend it further to account for stochastic variations.…”