: 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).Beom-Taek Lee, Emre Tuncer, and Dean Neikirk
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 N2 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.
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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.
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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.
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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 105
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.
