Beom-Taek Lee, Emre Tuncer, and Dean P. Neikirk*
Department of Electrical and Computer Engineering
University of Texas at Austin
Austin, Texas 78712
This paper is abstracted from "Efficient 3-D Series Impedance Extraction using Effective Internal Impedance," by B.-T. Lee, E. Tuncer, and D. P. Neikirk, IEEE 4th Topical Meeting on Electrical Performance of Electronic Packaging (EPEP), Portland, OR, Oct. 2-4, 1995, pp. 220-222. Also see our talk from EPEP '95.
The effective internal impedance approach, combined with the current-filament technique, has been shown to be a very efficient way to extract frequency dependent resistance and inductance of uniform interconnects. In this study, we show that this approach can be extended to find the series impedance of three dimensional structures. A microstrip bend is studied as a simple example.
Integrated circuits, IC packages, and PWBs contain many structures (such as bends or vias) that should be treated as three dimensional discontinuities in what may otherwise be uniform two-dimensional interconnects. The efficient extraction of an equivalent series impedance is especially difficult if the conductor geometries are comparable to the skin depth. This can easily occur in the high frequency analog sections of a mixed signal package where lead frames and interconnect thicknesses are typically in excess of 25 um. The frequency dependence of the series impedance, however, can still be found using magneto-quasi-static treatments. For uniform lines, various methods have been proposed, such as Weeks' current filament method  and conformal mapping techniques . The volume current filament technique is based on the method of partial inductances [3-5] and has been used as the basis for a number of inductance simulators. For skin depth limited problems the volume filament method divides the conductor into rectangular filaments that fill the entire cross section of the conductor. For accurate results the side of each filament must be no longer than about 1/3 of the skin depth, leading to very large matrix sizes for even simple calculations.
Efficient application of such techniques as conformal mapping to series impedance extraction  depends fundamentally upon the use of an effective internal impedance (EII) that represents the "internal" behavior of the conductors at their surfaces; conformal mapping then solves the problem outside the conductors. We have recently shown that a surface effective internal impedance can also be used in conjunction with the filament method . Using this approach the filament method can be applied to surface "ribbons" instead of volume filaments. This leads to a substantial reduction in problem size, with a corresponding increase in computational efficiency. The reduction in the size is from N2 for the conventional approach, to 4N for the ribbon approach, where N represents the number of segments per side used. For example, for a single metal bar 5 um x 25 um, at 10 GHz accurate simulation using a volume filament method required a 35 x 7 mesh, generating a 245 x 245 partial inductance matrix, and took 30 seconds of CPU time on a SPARC 10 to find the resistance and inductance of the bar; using this same mesh to find the frequency dependence between dc and 10 GHz took 200 sec for ten frequency points. Using our surface ribbon method we needed only an 8 x 2 mesh, and found the full frequency dependent resistance and inductance from dc to 10 GHz in 1 sec.
Electromagnetic analysis of 3-D structures is particularly demanding, especially when finite conductivity effects are included; even dc resistance calculation is nontrivial for three dimensional structures. The finite difference time domain (FDTD) technique and integral equation techniques using the methods of moments have been developed for high frequency modeling of various discontinuities such as vias and bends . For the frequencies of interest the quasi-TEM assumption is not valid for such discontinuities, (although it is valid for long, uniform interconnect sections) but the quasi-static assumption should still valid. Full-wave solvers are therefor unnecessarily expensive for the extraction of the discontinuities, even though they give S-parameters, radiation effects and higher mode effects at very high frequencies. To address this, several quasi-static methods have also been developed. Cangellaris et al. developed a hybrid finite element and moment method for arbitrary three dimensional structures . Ruelhi's partial element equivalent circuit (PEEC) approach has also been adapted for three dimensional problems . Rubin has developed a three dimensional solver, but an eigenvalue search is necessary, and the application is limited . All of these methods require fine discretization of conductors, so memory and CPU time burden is high.
The objective of this work is the extension of the surface ribbon technique to 3-D bends and discontinuities. To apply the surface effective internal impedance to parameter extraction of discontinuities, modeling of the effective internal impedance and modeling of the interaction between patches must be done. In three dimensional structures, current exist in all directions (Figure 1). The effective internal impedance must then be defined in three orthogonal directions. The inside of the conductors must be properly divided into patches for each of the three directions. To find the resistance and inductance of discontinuities the interaction between patches can be considered using external solvers and connections between patches.
To test this approach, we have first considered the simplest case of a single right angle bend. The circuit representation and connections between patches are shown in Figure 2a. The calculation of frequency dependent resistance of the discontinuities is then simply evaluated by solving the small network. To verify this simple approximation the result has been compared to a full volume element technique, the so-called Partial Element Equivalent Circuit (PEEC) technique . In Figure 2b, our simple approximation is fairly close to the PEEC solution over the entire frequency range. To identify the effect of the discontinuity the frequency dependent resistance of a uniform line of equivalent length (i.e., the length is chosen to produce the same dc resistance as the bend) was found using a volume current filament approach. The difference between uniform lines and the discontinuity is large at high frequency, for example, 45 % at 1 GHz, and over 100 % at 10 GHz. The CPU time for our approximation is less than a second for sixty frequencies, for the PEEC over 15,000 seconds for ten frequencies (valid only to about 1 GHz). This illustrates the significant computational savings that can be achieved using a 3-D EII approach.
More complex examples are illustrated in our talk fron EPEP '95.
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(a) (b) (c)
Figure 1: Discretization of right-angle bends and definition of effective internal impedance for (a) y-axis, (b) z-axis, (c) x-axis; arrows indicate direction of current flow.
Figure 2: (a) Circuital representation of the surface approximation. (b) Resistance vs.. frequency for a right angle bend; crosses: PEEC, accurate to 1 GHz; : our approximation; : conventional volume current filament technique for uniform lines with length chosen to give the same dc resistance as the bend.