Important references for this material are:

- 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. - 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*, vol. 42, pp. 1807-1815, 1994. - E. Tuncer and D. P. Neikirk, "Efficient Calculation of Surface
Impedance for Rectangular Conductors,"
*Electronics Letters*, vol. 29, pp. 2127-2128, 1993.

Beom-Taek Lee, Emre Tuncer, and Dean Neikirk

In this phase of our work we have concentrated on the development of efficient methods for the modeling of frequency-dependent resistance and inductance in MCM interconnects. Since the geometrical cross-sections of MCM-level (i.e., inter-chip interconnects) interconnects are relatively large (compared to intra-chip interconnects) their behavior is influenced by not only their capacitance and resistance, but also by their inductance. These effects are intimately related to the distribution of current across the cross-section of the interconnects, that in turn, is influenced by interactions with other conductors (external proximity effects) and by the skin effect (internal proximity effects). Conventional "full wave" electromagnetic calculations of the frequency dependent series impedance (i.e., the resistance and inductance of the interconnect) are numerically intensive, making them unsuitable for CAD tools. We have developed several new approaches that provide excellent accuracy, along with a significant improvement in computation speed, and have demonstrated their use for CAD purposes [1,2]. For instance, by combined a new effective internal impedance with the current-filament method for extraction we have achieved excellent accuracy with at least a 100 times reduction in computation time. Comparison between conventional and our new "surface-ribbon" calculations over a wide range of conductor geometries (including high aspect ratio cross-sections, such as square conductors) has shown excellent agreement from dc to 100 GHz.

Conformal Mapping

Current Ribbon Technique for Frequency Dependent Interconnect Modeling

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
(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.