Beom-Taek Lee, Emre Tuncer, and Dean Neikirk

Other material related to the Surface Ribbon Method:

- Interconnect Series Impedance Calculator
- SIMIAN: A program
for calculating the series impedance of two-dimensional multi-conductor
interconnects and transmission lines using the Surface
Ribbon Method.
- Users Manual (html version; other versions available here).
- Links to the computer code (password required; inquire via e-mail).

- Interconnect
Series Impedance Determination Using a Surface Ribbon Method
- Slide show on the 2-D ribbon method (15 slides,300K total)
- Minimum ribbons for very efficient modeling of microstrips including finite resistance ground planes

- Extraction of Series Impedance of Three Dimensional Discontinuities
- Slide show on 3-D extraction (13 slides, 260K total)

One traditional technique for the calculation of the frequency dependent total series impedance for an interconnect on an MCM is Reuhli's and Weeks' filament method [4,5]. A much more efficient application of this technique can be implemented using the EII approach. Using the EII, one has to solve only the exterior problem, and so the filament method can be applied to surface "ribbons" instead of volume filaments [13]. This leads to a substantial reduction in problem size, with a corresponding increase in computational efficiency.

The volume filament method divides the conductor into rectangular filaments
that fill the entire cross section of the conductor. The size of the problem
can be reduced by considering only the surface of the conductor. Instead
of using "volume filaments" we use "surface ribbons,"
each assigned an impedance based on the EII appropriate for that location
on the conductor surface. The reduction in the size is thus from N^{2}
to 4N, where N represents the number of segments per side used. As a test
case, we have calculated the series impedance of a two conductor transmission
line using Weeks' filament method and by the modified filament method for
surface impedance approaches discussed above. The copper conductors are
coplanar, 10 um thick and 10 um wide, separated by 10 um. For the volume
filament method each conductor is divided into a 20x20 mesh, resulting in
800x800 matrix size. For the surface ribbon method the same mesh is used,
but this time only at the surface. The size of the matrix for the latter
case is 160x160. Figure 5a and 5b compares the series
impedance per unit length for the coplanar bars found with the different
models. For comparison, the conformal mapping result from [8] is also shown.
All techniques are in good agreement for the series resistance of the interconnect.
The main discrepancies are in the low frequency inductance; the conformal
mapping being the highest. Both the volume filament and surface ribbon calculations
are in excellent agreement over the full frequency range.

The main advantage of the surface filament approach is seen when CPU times are compared. For the ten frequency point calculation shown in Fig. 5, the volume filament calculation required 3 hours on a Sparc 10 workstation.; the same curve required only 100 sec to calculate using the surface ribbons (including the calculation of the surface impedance). The conformal mapping calculation required only 48 sec (including both surface impedance and map coefficient calculations), although at the cost of loss of accuracy in inductance at low frequency. Another comparison for coplanar, thick strips is shown in Fig. 6: here the ribbon technqiue is about 200 times faster than the conventional volume approach. Table 1 shows a comparison of run-times for volume and ribbon calculations of the self-inductance of a rectangular metal bar. It should also be noted that most techniques designed to speed up volume filament calculations can also be applied to the surface ribbon approach. To summarize, a major contribution of this work has been the development of a new effective internal impedance model, that when used in combination with a surface ribbon calculation, can accurately find the series impedance per unit length of multi-wire lossy interconnects using significantly less CPU time.

1. S. Y. Kim, E. Tuncer, R. Gupta, B. Krauter, D. P. Neikirk, and L.
T. Pillage, "An Efficient Methodology for Extraction and Simulation
of Transmission Lines for Application Specific Electronic Modules,"
*International Conference on CAD - 93*, Santa Clara, CA, Nov. 7-11,
1993, pp. 58-65.

2. E. Tuncer, S. Y. Kim, L. T. Pillage, and D. P. Neikirk, "A new,
efficient circuit model for microstrip lines including both current crowding
and skin depth effects," *IEEE 2nd Topical Meeting on Electrical
Performance of Electronic Packaging*, Monterey, CA, Oct. 20-22, 1993,
1993, pp. 85-88.

3. 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 Transactions on Microwave Theory and Techniques*,
vol. 39, pp. 1338-1347, 1991.

4. W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, "Resistive
and inductive skin effect in rectangular conductors," *IBM Journal
of Research and Development*, vol. 23, pp. 652-660, 1979.

5. A. E. Ruehli, "Inductance Calculations in a Complex Integrated
Circuit Environment," *IBM Journal of Research and Development*,
pp. 470-481, 1972.

6. W. Jingguo, J. D. Lavers, and Z. Peibai, "Modified surface impedance
boundary condition applied to eddy current problems," *IEEE Transactions
on Magnetics*, vol. MAG-26, pp. 1197-1200, 1992.

7. E. Tuncer and D. P. Neikirk, "Efficient Calculation of Surface
Impedance for Rectangular Conductors," *Electronics Letters*,
vol. 29, pp. 2127-2128, 1993.

8. E. Tuncer, B.-T. Lee, M. S. Islam, and D. P. Neikirk, "Quasi-Static
Conductor Loss Calculations in Transmission Lines using a New Conformal
Mapping Technique," *IEEE Transactions on Microwave Theory and Techniques*,
1994.

9. M. S. Islam, E. Tuncer, and D. P. Neikirk, "Calculation of Conductor
Loss in Coplanar Waveguide using Conformal Mapping," *Electronics
Letters*, vol. 29, pp. 1189-1191, 1993.

10. M. S. Islam, E. Tuncer, and D. P. Neikirk, "Accurate Quasi-Static
Model for Conductor Loss in Coplanar Waveguide," *1993 IEEE MTT-S
International Microwave Symposium Digest*, Atlanta, GA, June 15-18, 1993,
pp. 959-962.

11. R. Schinzinger, "Conformal transformations in the presence of
field components along a third axis - part 2," *International Journal
of Electrical Engineering Education*, vol. 13, pp. 127-131, 1976.

12. R. Schinzinger and P. Laura, *Conformal Mapping: Methods and Applications*.
New York: Elsevier, 1991.

13. E. Tuncer, B.-T. Lee, and D. P. Neikirk, "Interconnect Series
Impedance Determination Using a Surface Ribbon Method," *IEEE 3rd
Topical Meeting on Electrical Performance of Electronic Packaging*, Monterey,
CA, Nov. 2-4, 1994, pp. 249-252.

Figure 1: Regions to calculate Effective Internal Impedance; region A uses simple hyperbolic tangent equation (eqn. (1)), and region C' uses eqn. (2).

Figure 2: Each half corner (region C' in fig. (1)) is subdivided into triangles. The equivalent impedance seen at the bottom surface of triangles is calculated by eq. (2).

Figure 3: Conformal mapping process which
unfolds conductors in real space (the **z**-plane), scaling conductor
conductivity and surface impedance ZS into the mapped plane (the **w**-plane).

Figure 4: Comparison between
measured (solid lines) and simulated series impedance per unit length (real
and imaginary parts) for thin coplanar strips. The strips were supported
by a 0.08 cm thick [[epsilon]]r = 2.55 substrate, and the strip dimensions
were: d = 0.05 cm, w = 1.3 cm, t = 17 um, and *[[sigma]]* = 5.8 x 10^{5}
S^{.}cm^{-1} (copper). ^{ } : conformal
mapping results from eq. 10; cross : imaginary part calculated using Maxwell
reg.; cross : real part calculated using Maxwell reg.; ^{ } : skin effect
resistance only (no current crowding).

Figure 5: Series impedance per unit length for copper, coplanar bars, calculated from the volume filament and surface ribbon methods using various surface impedance models. Solid line: volume filament method (from Weeks, [4]);

surface ribbon method, EII from Jingguo et al. [6];

: EII from [7]; ;

EII from skin depth limited simplification of [7, 8];

Figure 6: Comparison of volume and surface ribbon calculations for thick, coplanar strips.

Table 1: Comparison of run-times for volume and ribbon calculations of the self-inductance of a rectangular metal bar.