pFFT in FastMaxwell: A Fast Impedance Extraction Solver for 3D Conductor Structures over Substrate

Abstract

Abstract- I. INTRODUCTIONThe integration of RF, analog and digital circuitry on a single integrated-circuit substrate, or system-on-chip (SoC), is associated with many challenges. For instance, substrate noise may lead to altered circuit performance. In addition, conductor skin and proximity effects may impact current return paths in a network of closely-spaced conductors, as well as quality factor in RF inductors. Moreover, the occurrence of radiated electromagnetic interferences (EMI) may transmit disturbances across different SoC systems. In order to avoid all the above problems, electromagnetic simulation tools are needed that can accurately and efficiently extract wide-band full-wave conductor impedances in the presence of semi-conducting substrates.In the last decade, electromagnetic simulators based on integral formulations have been the subject of extensive research due to the development of fast algorithms i.e. O (N log N) In this paper we present a 3D full-wave simulator based on the mixed potential integral equation formulation (MPIE) [7], [8], and accelerated using p-FFT. The solver will be released on public domain starting from December 2006 [9].

Analytical Expressions for High-Frequency VLSI Interconnect Impedance Extraction in the Presence of a Multilayer Conductive Substrate

“…In all cases, the analytical expression for the substrate Green's function shown in (21) provides an excellent agreement with the numerical calculation. Similar results are obtained for (22).…”

“…The relevant regimes of distances and frequencies are covered. The resulting equations (21), (22), (26), (30), (32)- (34), and (36), as well as those shown in the Appendix, are simple analytical expressions depending on parameters that can easily be extracted from real exponential least square fits to known formulae. The suitability of this approach for massive extraction problems is self-evident.…”

“…However, finding a set of complex images (b j , c j ) that result in an accurate approximation to the Green's function remains a challenge [20], [21]. The DCIM has been applied to VLSI interconnect impedance extraction in FastMaxwell [22] for single-layer substrate configurations.…”

Fast Simulation of Complicated 3-D Structures Above Lossy Magnetic Media

“…Where the conductor geometries are complicated, for example in microstrip circuits and VLSI interconnects applications, the Green's functions tend to be simple. The usual practice is to analytically derive and compute the Green's function, and to focus efforts on reducing complexity in the conductors [3], [12], [39], [40]. By contrast, for antennas and scattering applications, the Green's functions themselves contain a great deal of complexity, including resonant and propagation modes, surface waves, and branch cuts in the complex domain, and most of the emphasis is placed upon evaluating the Green's function to high accuracy using sophisticated techniques [13]- [15].…”

“…Firstly, the method avoids discretizing the surrounding freespace and multilayer media, by the use of the multilayer Green's function [11]. This technique is widely used in microelectronic and microwave applications [12]- [15], but normally introduces a complicated Green's function that must be carefully evaluated with a Sommerfeld integral. In the quasistatic limit, we show in Section III that the Green's function is analytic smooth, and harmonic within free-space.…”