(4)Beom-Taek Lee, Emre Tuncer, and Dean Neikirk
For other examples of the use of the conformal mapping method combined with the Effective Internal Impedance (EEI), please see:
In addition to a model for the internal behavior of the conductors (as represented by the EII approximations discussed above), a method of finding the external inductive interaction between conductors must specified. One of the approaches we have developed is based on the use of conformal mapping [8-10]. To apply the conformal mapping technique to finite conductivity interconnects, the EII, ZS, must first be found in the real-space domain (the x-y -plane) for each isolated conductor. In general, ZS will depend on x and y (and hence, on u and v, the coordinates in the conformally mapped space). Once this is found using the approaches discussed above, a conformal map is used to transform the full transmission line into a parallel plate configuration. Current crowding effects can then be accounted for in the mapped domain, as discussed below. Also note a fundamental assumption in quasi-statics: the series impedance (i.e., the resistance and inductance) of a transmission line does not depend on the dielectrics used. Thus, the conformal map appropriate for a homogeneous dielectric problem is all that is required. All experimental data we have measured are in excellent agreement with this approximation.
To illustrate, assume a conformal map f(z) has been found for a particular transmission line structure. The map is such that it produces parallel plates in the w-plane (see Fig. 3), with the plates parallel to the u -axis, extending from 0 to uo, the top plate located at vt, and the bottom plate at vb. A point (x, jy) on the surface of the real space conductor has corresponding EII, ZS(x, y). For simplicity, assume this point maps onto a point in the w-plane at (u, jvt), on the top parallel plate. The scaled impedance in the w-plane is given by
(4)
where 
is the inverse of the mapping function, and
(5)
The differential series impedance per unit length dZtop due to a differential width du of the top plate is then
(6)
The bottom plate also contributes to the series impedance
(7)
To find the total differential series impedance per unit length dZtot due to this portion of the transmission line we assume the magnetic fields between the plates are uniform, giving rise to an inductance
(8)
where uo is the permeability of free space. Finally, the total differential series impedance per unit length dZtot is
(9)
The total (quasi-static) series impedance per unit length Z([[omega]]) for the transmission line is due to the parallel combination of each differential impedance, so
(10)
where for simplicity in notation we have used
(11)
The complex propagation constant [[gamma]] for the transmission
line is now given by 
, where 
is the shunt admittance per unit length
for the transmission line, and Z([[omega]]) is found from eq. 10.
Previous attempts to use conformal mapping to evaluate conductor loss have
used "effective" (average) values for the scale factor [11,12],
and generally have not placed M and the surface impedance within the integral
for the series impedance Z([[omega]]). This can lead to significant
inaccuracy, since the frequency dependence of current crowding is not properly
accounted for. As an example of the application of this technique, Fig. 4 shows a comparison between the measured and calculated
series impedance of two coplanar strips in close proximity to one another.
The agreement is excellent. We have also made measurements for closely space
circular wires, square bars, and coplanar waveguide; in all cases agreement
has been excellent [8]. Example calculations using this technique, along
with those based on another approach, are also discussed in the next section.
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.