B.-T. Lee and D. P. Neikirk, "Minimum Segmentation in the Surface Ribbon Method for Series Impedance Calculations of Microstrip Lines," IEEE 5th Topical Meeting on Electrical Performance of Electronic Packaging, Napa, CA, October 28-30, 1996, pp. 233-235.
Minimum Segmentation in the Surface Ribbon Method for Series Impedance Calculations of Microstrip Lines
Beom-Taek Lee, and Dean P. Neikirk*
Department of Electrical Engineering and Computer Engineering
University of Texas at Austin
Austin, Texas 78712
Tel:(512) 471-4669
Fax:(512) 471-5445
*e-mail: neikirk@mail.utexas.edu
Abstract
The efficiency and accuracy of the surface ribbon method has been shown in calculating frequency-dependent resistance and inductance. In this paper, the further numerical efficiency is obtained by deriving a minimum segmentation scheme for microstrip lines, without loss in accuracy.
I. Introduction
For performance evaluation of integrated circuits accurate characterization of the electrical parameters of various interconnects and packages is essential. As operating speeds and wiring densities increase skin and proximity effects become more important, and cause significant loss and coupling. Various electromagnetic field solvers have been derived to accurately calculate frequency-dependent resistance and inductance, such as the volume filament method (VFM) [1], the finite element method (FEM), the boundary element method (BEM) [2]. However, for complex two and three dimensional geometries more efficient field solution approaches are desirable. To reduce the number of unknowns and the computational load, non-uniform segmentation schemes, efficient matrix manipulation, and iterative matrix solvers have been developed for the volume filament method [3]. The surface ribbon method (SRM) [4,5] has also been developed to reduce the problem size using the effective internal impedance (EII) and surface segmentation instead of volume segmentation. In SRM, greater numerical efficiency can be obtained by using minimum number of segments. In this paper, a minimum segmentation scheme for SRM is described, and efficiency and accuracy are examined in case of a microstrip line over a ground plane and a microstrip line over a meshed ground plane.
II. Surface Ribbon Method and Minimum Segmentation
For accurate results at high frequencies, the thickness of a skin depth must be divided into several segments in VFM. Hence, matrix sizes become large, especially at high frequency. To avoid discretizing the conductor interior, the effective internal impedance (EII) is combined with the current integral equation assuming current flows only at the conductor surface. The method of moments is then applied to this surface current integral equation. The internal behavior of conductor is represented at the conductor surface using an EII and the conductor interior is replaced by the exterior material. Hence, this SRM replaces volume discretization by surface discretization, reducing the number of unknowns, giving fast and accurate results.
Unlike VFM, SRM does not require the use of several segments for the width of a skin depth and, therefore, can significantly reduce the number of unknowns. Figure 1(a) shows a minimum segmentation scheme for the case of a microstrip line over a ground plane. For a signal line with comparable width to thickness ratio, the line can be represented using four segments, one segment for each side of the conductor. In the case of a lossy ground plane, the upper surface of the ground plane must also be discretized. Here we show that only five segments are necessary for accurate results: one segment directly below the signal line with width of w (the width of the signal line), adjacent segments on each side with width of 3h (three times the dielectric thickness), and finally one more segment on each side for the remaining ground plane. Therefore, the total number of unknowns becomes nine. This minimum segmentation scheme can be extended to structures having multi-conductors and multi-ground planes.
III. A Microstrip Line over a Ground Plane or a Meshed Ground Plane
For comparison, we have also considered two other minimum segmentation schemes in SRM. Firstly, a single segment has been assigned to the ground plane. Secondly, the ground plane is discretized into three segments: one segment directly below the signal line with the width of w , and one more segment on each side for the remaining ground plane. Both use four segments for the signal line. For two different microstrip line geometries, resistance and inductance have been calculated using the different segmentation schemes of SRM, as well as with full VFM and a finely divided SRM; example 1 has a signal line 10 um wide (w = 10 um) and 10 um thick (t = 10 um), and a ground plane 500 um wide and 10 um thick. Example 2 has a signal line of 10 um wide and 1 um thick, and a ground plane 500 um wide and 1 um thick. Figure 1(b), (c), and (d) compare resistance and inductance as a function of strip height above the ground plane h, from a height equal to 0.1w to a height equal to 10w , at low frequency (where the skin depth delta equals 10t ) and at high frequency (where delta = 0.1t ). With five segments for the ground plane SRM gives accuracy within 10% for resistance and inductance for both examples. With three segments for the ground SRM deviates by as much as 30% for resistance and inductance. Table 1 shows speed gains of several thousands times compared to VFM and of several ten times compared to SRM using fine segments.
Similarly, the minimum segmentation scheme can be applied to the three-dimensional surface ribbon method (3DSRM). As an example, the series impedance of a microstrip line obliquely oriented over a meshed ground plane has been calculated, where the signal line is 12 um wide, 2.5 um thick and 12 um over the meshed ground plane of 100x100 um period with 50x50 um holes (Fig. 2). In 3DSRM using minimum segments, the signal line is segmented into 3 x 1 segments with width ratio of 2.8, and an arm of the meshed ground is divided into three segments. For comparison, the partial element equivalent circuit method (PEEC) [6] is also applied, where three different segmentation schemes are used. In PEEC1 the meshed ground is approximated by cascaded straight rectangular bars and divided into 6 x 3 segments; in PEEC2 an arm of the meshed ground is divided into four segments for wide side and one segment for the thickness; and in PEEC3 an arm of the meshed ground is divided into four segments for wide side and two segment for the thickness. In PEECs the signal line is divided into non-uniform 12 x 4 segments. Figure 2 shows the structure and compares resistance and inductance calculated using different schemes. For inductance, PEEC1 is considerably off about 23% from others and 3DSRM gives the result with 5% accuracy. For resistance, PEECs do not capture the skin effect of the ground plane due to coarse segments, and the results deviates more than 20% from the result of 3DSRM. Table 2 compares run time and the number of unknowns of 3DSRM and PEECs.
In summary, a minimum segmentation scheme has been derived for SRM, and through several examples accuracy and efficiency of minimum segmentation has been examined. In three dimensional structure as well as two dimensional structures, SRM using minimum segments reduces considerably the amount of computational time required.
References
[1] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, "Resistive and inductive skin effect in rectangular conductors," IBM J. Res. Develop., vol. 23, pp. 652-660, November 1979.
[2] M. J. Tsuk, and J. A. Kong, "A Hybrid method for the calculation of the resistance and inductance of transmission lines with arbitrary cross sections," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1338-1347, August 1991.
[3] M. Kamon, M. J. Tsuk, and J. K. White, "FASTHENRY: A multipole-accelerated 3-D inductance extraction program," IEEE Trans. Microwave Theory Tech., vol. 42, pp. 1750-1758, September 1994.
[4] E. Tuncer, B.-T. Lee, and D. P. Neikirk, "Interconnect series impedance determination using a surface ribbon method," IEEE 3rd Topical Meetings on Electrical Performance of Electronic Packaging, Monterey, CA, November 2-4, 1994, pp. 249-252.
[5] B.-T. Lee, E. Tuncer, and D. P. Neikirk, "3-D series impedance extraction using effective internal impedance," IEEE 4th Topical Meeting on Electrical Performance on Electronic Packaging, Portland, OR, October 2-4, 1995, pp. 220-222.
[6] A. E. Ruehli, "Equivalent circuit models for three-dimensional multiconductor systems," IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 216-221, March 1974.
(a) Minimum segmentation scheme for a microstrip line
Segmentation scheme for coupled lines
(b) Comparison of high frequency resistance
(c) Comparison of low frequency inductance
(d) Comparison of high frequency inductance
Fig. 1: Minimum segmentation scheme for a microstrip line, and comparison of resistance and inductance calculated using different segmentations in SRM. Resistance and inductance are normalized by the results of VFM. A(solid line): fine segments; B(dot-and-dashed line): one segment for the ground plane; C(dashed line): three segments for the ground plane; D(dotted line): five segments for the ground plane. Minimum segmentations use four segments for the signal line.
Method |
Number of unknowns |
CPU time (sec) | |
Assembling |
Solve per frequency | ||
VFM |
660 |
55 |
101 |
SRM1 |
120 |
0.3 |
0.89 |
SRM2 |
9 |
- |
0.012 |
Table 1: Comparison of run time and the number of unknowns on an IBM RISC 6000 for VFM, SRM, and SRM using minimum segments. Matrix solutions done using simple gaussian elimination.
(a) A microstrip line over a meshed ground plane
(b) Comparison of resistance and inductance
Fig. 2: A microstrip line 45 degrees obliquely oriented over a meshed ground plane, and comparison of resistance and inductance between PEECs with three segmentation schemes and 3DSRM. A(dashed line): PEEC with straight line approximation; B(dotted line): PEEC with one layer segment for the ground; C: PEEC with two layer segments for the ground; D(solid line) 3DSRM. Meshed ground is considered by 5 apertures perpendicular to the signal line and 9 apertures along the signal line to obtain constant per unit length inductance.
Method |
Number of unknowns |
CPU time (sec) | |
Assembling |
Solve per frequency | ||
PEEC1 |
398 |
1360 |
18.6 |
PEEC2 |
283 |
1788 |
8 |
PEEC3 |
743 |
4745 |
147 |
3D SRM |
148 |
37 |
3.1 |
Table 2: Comparison of run time and number of unknowns on an IBM RISC 6000 for PEECs and 3DSRM. * is in case of 9 x 5 apertures of the meshed ground. Matrix solutions done using simple gaussian elimination.