, where u and [[sigma]] are permeability and
conductivity of the conductor, respectively. When the thickness of the slab
is finite, the surface impedance can be calculated using transmission line
equations,Beom-Taek Lee, Emre Tuncer, and Dean Neikirk
The impact of finite interconnect conductivity remains one of the more difficult problems in transmission line analysis. Traditional approaches include the incremental inductance rule and integration over surface current distribution. One difficulty with these techniques is their failure to adequately model the transition from low frequency to high frequency behavior. When calculating broad bandwidth time-domain pulse propagation, for instance, the dispersion induced by this transition can be significant. For digital IC interconnect modeling, several other techniques for predicting the frequency dependence of the series impedance per unit length have also been developed [3-5]. There is a continuing need for models which accurately predict the frequency dependent attenuation and phase constants from dc to high frequency for transmission lines using conductors with finite resistance.
Various methods have been used to calculate the total series impedance of a transmission line structures, such as Weeks' filament method, conformal mapping, and the ribbon technique. Efficient application of such "external" field solvers depends fundamentally upon representing the "internal" behavior of the conductors at the periphery. Arduous calculations can be eliminated by dividing the problem into two steps: first, calculate a quantity which will represent the internal behavior at the surface of the conductors; and second, solve the external problem using this quantity. Similar approaches have been used at high frequencies by applying the surface impedance concept. Although past approaches give good results at frequencies where the skin-depth is much smaller than the cross sectional dimensions of conductors, at and below the frequency range where the skin-depth is comparable to the cross sectional dimensions the conventional surface impedance concept is not applicable. Even at high frequencies, the surface impedance concept is erroneous due to the fact that it does not account for sharp edges. We have developed a new modified internal impedance concept, the Effective Internal Impedance (EII), that is more accurate near sharp edges and covers the transition from high frequency to dc behavior.
EII can be viewed as a quantity which represents the current distribution
inside the conductor. At high frequencies, the EII is the same as the surface
impedance. The surface impedance can be defined as the ratio of total tangential
electric field to total tangential magnetic field at the conductor surface
at a given frequency. The surface impedance of an semi-infinite conductor
slab is , where u and [[sigma]] are permeability and
conductivity of the conductor, respectively. When the thickness of the slab
is finite, the surface impedance can be calculated using transmission line
equations,
(1)
where [[gamma]] is the complex propagation constant in the conductor
and d is the thickness. For more useful conductor geometries calculation
becomes much more complex. The surface impedance can also be calculated
from a conventional full electromagnetic solution, such as one based on
the filament method [4]. However, calculating actual surface impedance using
such an approach leads to no improvement in numerical efficiency, since
the full problem must be solved.
By approximating the surface impedance of a thick rectangular bar with an EII significant speed-up can be obtained. Jingguo et al. calculates surface impedance assuming four plane waves incident normal to the surfaces of the conductor [6]. We have recently developed two new methods that use geometrical segmentation to mimic electromagnetic wave distribution inside the conductor with non-uniform transmission lines [7]. In Figure 1, the EII for the middle sections are calculated using eq. (1) with d=t/2. For the corner sections, each square is divided into two triangles and each triangle further divided into smaller slanted triangles as in Figure 2. For each slanted triangle the input impedance at the base is calculated by
(2)
where J0 and J1 are Bessel functions and hn and wn are given
as follows,
(3a)
(3b)
where N is the total number of slanted triangles and n=0,1,2,...,N-1.
For N=4 this model predicts the series resistance of a square bar
with in 1% from dc to intermediate frequencies.
A modified version of this method is obtained by making the conductor a hollow tube, with wall thickness of 3d (where d is the skin-depth) as the frequency increases. For the corner regions the same technique is used as that discussed above, while for the rectangular regions the simple hyperbolic tangent expression is used. However, since the thickness of the wall is always 3d, the argument to the special functions (hyperbolic tangents and the Bessel functions in eqs. 1 & 2) is always the same at each frequency once the frequency is high enough such that 3d is smaller than thickness of the conductor. For lower frequencies the dc resistance is used for that section. The impact on numerical efficiency of the EII approach will be illustrated in the next two sections.
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 105
S.cm-1 (copper). 
: conformal
mapping results from eq. 10; Z : imaginary part calculated using Maxwell
reg.; 8 : 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.