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Beom-Taek Lee#, Sangwoo Kim, Emre Tuncer*, and Dean P. Neikirk
Electrical Engineering Research Laboratory
Department of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX 78712
# current address: Intel Corporation, 2111 N.E. 25th Avenue, M/S: JF1-56, Hillsboro, OR 97124
* current address: Quad Designs, 1385 Del Norte Road, Camarillo, CA 93010
corresponding author: Dean P. Neikirk, FAX: (512) 47104669, e-mail: email@example.com
A new impedance boundary condition method for calculating the series impedance of lossy quasi-TEM transmission lines has been developed. Previous techniques using a more standard impedance boundary condition are of limited usefulness at low and middle frequency where the skin depth is comparable to or larger than the cross-sectional dimensions of the transmission line conductors since it is quite difficult to accurately approximate, a priori, the surface impedance. The new approach is based on the effective internal impedance (EII) that can be accurately approximated using the surface impedance of an isolated conductor. In this paper twin circular wires are used as an example to illustrate the accuracy and efficiency of the new EII-based method compared to conventional surface impedance boundary condition-based approaches.
Impedance boundary conditions (IBCs) are widely used in scattering problems, eddy current problems, and lossy transmission line problems. An IBC is usually adopted to simplify the problems by eliminating from the domain to be solved, for example, lossy dielectrics, multi-layered coatings on conductors in scattering problems, or the lossy conductors in eddy current and transmission line problems. Solutions can then be obtained by applying the finite element method (FEM) [1,2], the boundary element method (BEM) [3, 4], the electric/magnetic field integral method (MFIE/EFIE) [5, 6], the finite difference time domain method (FDTD) [7, 8], etc.. In each case, the use of an IBC reduces the number of unknowns and can substantially reduce the computation time required. But, in general, the boundary condition must be known (at least approximately) a priori on the surface, and the accuracy of the field solution is determined by the accuracy with which the IBC is approximated.
The most widely used IBC is the standard impedance boundary condition (SIBC), also called the Leontovich boundary condition [9, 10], that was originally developed for use when the skin depth is small relative to other dimensions of the problem. In this paper, an efficient approach for calculating the series impedance of lossy multi-conductor lines from DC to high frequency is presented that use the effective internal impedance (EII) as an impedance boundary condition. The advantage of the EII when compared to the SIBC is that even for strongly coupled transmission lines it is easily approximated from low frequency (i.e., the skin depth is larger than the cross-sectional dimensions of the conductors) to high frequency (i.e., the skin depth is far smaller than the dimensions of the conductors) using the surface impedance of an isolated conductor. As an example, the EII-based method and the SIBC combined with the BEM are compared in the case of twin lead conductors, and the appropriateness of the EII approach is shown for series impedance calculations of lossy quasi-TEM transmission lines from DC to high frequency.
Schelkunoff  first introduced the concept of surface impedance in electromagnetics in 1934 for the analysis of coaxial cables. In the 1940's Leontovich  as well as many other workers studied the basic properties of the surface impedance for a semi-infinite plane of an isotropic linear medium and on a conductor-backed thin lossy dielectric layer where a plane wave is incident. Senior  explained in detail the Leontovich boundary conditions and the requisites to be satisfied. For a semi-infinite thickness plane of an isotropic linear lossy conductor, the surface impedance is given by
where is the frequency in radians per second, the permeability, the conductivity, and the skin depth.
In order to use (2) to approximate the SIBC for multi-conductor problems the relationship between the tangential electric and magnetic fields at any point on the boundary must be (at least approximately) a local one, depending only on the curvature of the surface and the local electromagnetic properties (i.e., u, [epsilon], and [sigma]) of the bodies. Hence, to use the simple expressions given by (2), the surface impedance must not be altered by the global geometry of the conductor or the existence of other conductors in a problem. This constraint will hold if the operating frequency is "high", i.e., when the curvature radius of the surface is larger than the skin depth and when the other dimensions of the problem are smaller than the dielectrics wavelength. For geometries having curvature, Leontovich introduced a first order curvature correction term to the surface impedance for small radii of curvature and Mitzner  later refined this.
At low and middle frequency where the skin depth is comparable to or larger than the dimensions of the conductors, the surface impedance is strictly no longer a local property and depends on the global geometries of the conductors. For "thin-film" transmission lines using metals with thicknesses of less than a few micrometers, such a condition may hold for frequencies well into the GHz range. Under such conditions the surface impedance when other conductors are present may considerably differ from the surface impedance of an isolated conductor. Hence, it is difficult to know or approximate a priori the surface impedance in lossy multi-conductor systems at low to middle frequencies. This limits the usefulness of the SIBC for calculating the series impedance of lossy quasi-TEM transmission lines at low and middle frequencies, and necessitates complicated models for the IBC.
To examine the use of various conductor boundary conditions in multi-conductor transmission line analysis, we first formulate the problem using the boundary element method for finite conductivity conductors, and then identify the relationships giving the transmission line series impedance parameters (resistance and inductance).
From Maxwell's equations and Green's theorem, the coupled boundary integral equations are set up at the surface of the conductor [14, 15]. In two dimensions, for an m conductor transmission line, from the Poynting theorem  with the assumptions of non-radiation and quasi-static fields, the following equation is derived
where V is a volume enclosing all the conductors, is the applied potential on the qth conductor, J is the current density, and A is the magnetic vector potential. The left-hand side of the equation is the power applied, the first term of right-hand side is the power dissipated, and the second term corresponds to the magnetic energy stored. This term can be further decomposed into the magnetic energy stored interior and exterior to the conductor .
Using Green's theorem and the power dissipation term in (7), the series resistance of the qth conductor of the transmission line is given by the surface integral
where is the perimeter of the qth conductor, Ht is the tangential magnetic field, Ez is the longitudinal electric field, is the total series resistance of conductor q, and lq is the length of conductor q. Using Green's theorem and the magnetic energy stored interior and exterior to the conductors, the external series inductance for the transmission line and the internal series inductance of the qth conductor are
where Lext is the total external inductance of the transmission line and is the internal inductance of conductor q.
If the internal characteristics of a medium are represented by only the exterior fields at the surface of a medium via the SIBC, then a two or more media, multi-region problem can be replaced by a single medium, exterior problem, and the problem may be greatly simplified. Hence, the complete problem is replaced by a one medium, exterior problem. Using the SIBC as defined by (1), the surface impedance on conductor q is given by
where is a position on the conductor perimeter .
For the surface boundary element method (SBEM), if the exact position and frequency dependent surface impedance is known, then the SBEM gives the same exterior electric and magnetic fields as the original problem, faithfully satisfying the surface equivalence theorem . Therefore, the original problem can be regarded as the equivalent problem where fields in the exterior medium are the same as the original problem, while sheet currents exists on the conductor surface(s), the electric field intensity is continuous across the conductor surface, and interior magnetic fields are zero. Since the interior magnetic fields are exactly zero, the medium filling the conductor interior becomes irrelevant; essentially, the conductor interior regions are excluded from the SBEM problem.
By using the surface impedance given by (12) to define the relationship between the exterior electric and magnetic fields at the conductor surface, (7) becomes
where is the sheet current density at the conductor boundaries and S is the surface of all the conductors, used now instead of V since sheet current exist only on the surfaces of the conductors. The second term of the right-hand side in (14) represents the magnetic energy stored only in exterior regions (since the interior magnetic fields are zero), and external inductance can be calculated using equation (10). The first term of the right-hand side in (14) consists of real and imaginary terms. The real term corresponds to power dissipated in the system and can be equated to resistance, while the imaginary term corresponds to the magnetic energy stored at the surface due to discontinuities in magnetic fields in the surface equivalent problem.
To summarize, the original problem can be replaced by an equivalent, one medium problem by applying the SIBC (12) to the BEM. As long as the exact surface impedance as a function of frequency is known, the following relationships will be exactly satisfied
Unfortunately, accurately predicting the surface impedance of (12) for coupled, lossy multi-conductor lines can become as complicated as solving the original, full problem, particularly at low and middle frequencies. Thus, the usefulness of the SIBC is limited mainly to high frequencies, where the surface impedance is given approximately by (2).
It would be useful to find some technique in which surface characteristics can be more easily approximated than the standard surface impedance boundary condition. Instead of developing complicated SIBC models to capture the coupling and non-localized field effects between multiple conductors at low and middle frequency it is possible to develop a formulation in which isolated conductor surface impedance is a useful approximation. In this approach, the conductor interior is not excluded from the domain of solution, but rather the conductor interior is replaced by the exterior medium and the Green's function of the exterior medium is used also for the interior region in the equivalent problem. The conductor is now modeled as an impedance sheet at the conductor surface, the electric field is assumed continuous across the surface of conductor, and a sheet current is assumed at the surface of conductor. For the exterior region, the boundary integral equation becomes
where is the exterior magnetic field and is the electric field at the conductor perimeter . For the interior region in the equivalent problem, the boundary integral equation (5) becomes
where is the interior magnetic field in the equivalent problem, and the Green's function of the exterior medium is used instead of the Green's function of the conductor.
For this new equivalent problem the relationship between the tangential electric field and the difference of the tangential magnetic fields across the impedance sheet is called the effective internal impedance (the EII, to distinguish it from the SIBC), and is then
where is the EII. If and in (20) are set to the exact (i.e., identical to those in the original problem) external magnetic and electric fields and in the EII problem is calculated from (19) (again using the exact electric field from the original problem), then an "exact" EII can be a posterior determined. This EII is consistent with (i.e., given the correct EII, it can be used to calculate) the exterior magnetic and electric fields of the original problem. The EII does, however, produce a finite interior magnetic field in the equivalent problem, unlike the "null" interior magnetic field obtained using the SIBC and the BEM.
To complete the statement of the EII formulation, power applied, power dissipated, and magnetic energy stored in the equivalent problem are found using
Resistance can be calculated in a manner similar to (15) using
Internal inductance of conductor q and total external inductance are calculated using
Surface inductance due to the magnetic energy stored at the impedance sheet is given by
Finally, the sum of internal, external, and surface inductances from (23), (24), and (25) gives the total series inductance for the transmission line in the EII formulation. Using the exact exterior magnetic and electric fields from the original problem, it can be shown that (22) is the same as (9) (i.e., the resistances are identical), (24) is the same as (10) (i.e., the external inductances are identical), and the sum of (23) and (25) is the same as (11) (i.e., the internal inductances are identical).
Each conductor perimeter can be further broken into segments , where these pieces represent current-carrying "ribbons" of width . If the ribbons are narrow enough that the sheet current density is constant across each ribbon, integrating over the kth ribbon yields
where is voltage drop along ribbon k, is the EII averaged over the width of ribbon k, is the length of ribbon k, and . Equation (29) can be expressed as an N x N matrix equation
where is an N x N diagonal matrix made up of the `s. This approach is called the surface ribbon method (SRM) [18-20]. One advantage of the SRM is that the number of ribbons that is required on each conductor can be quite small while still yielding accurate results . A similar approach was used in  for transient analysis of multi-conductor transmission lines, although only a simple resistance was used for the internal impedance of the conductors.
A simple test case for the EII in the SRM is closely coupled, parallel circular wires, or "twin lead" transmission line. For the two-dimensional geometry shown in Fig. 2, the "full" (i.e., full solution both exterior and interior to the conductors) BEM (FBEM), the BEM using the surface impedance of an isolated conductor as an approximation for the SIBC (I-SBEM), and the SRM using the surface impedance of an isolated conductor as an approximation for the EII (I-SRM), have been applied to calculate the series resistance and inductance of twin lead. The surface impedance of an isolated circular wire used in the I-SBEM and I-SRM is found by solving the Helmholtz equation for the cylindrically-symmetric conductor, giving
where r is the radius of the circular conductor, and J0 and J1 are Bessel functions of the first kind .
The most challenging test case is for closely spaced twin lead. For a gap between the outer surfaces of the two conductors equal to 20% of the wire radius, Figure 5 shows the series impedance calculated using the FBEM, I-SBEM, and I-SRM. If the exact surface impedance (rather than the isolated) is used as the SIBC in the SBEM or the exact EII is used in the SRM, the surface techniques give identical external fields and series impedance to full BEM over the entire frequency range. Using the isolated conductor surface impedance as an approximation for the SIBC and EII, the calculated low frequency resistances per unit length are almost all identical to the DC resistance of the twin lead, regardless of technique. At both high and low frequency the inductance calculated using the I-SRM is very close to the inductance found using the FBEM. The surface techniques are significantly more computationally efficient, and by using the isolated conductor surface impedance as an approximation for the EII in the SRM, little accuracy is lost over the whole low-to-high frequency range.
In this paper we have shown a new surface impedance technique that can be accurately approximated using the isolated conductor surface impedance, even at low frequency. This effective internal impedance (EII) can be combined with the surface ribbon method (SRM) allowing numerically efficient and accurate calculation of the series impedance of lossy multi-conductor transmission lines. This technique can also be easily applied to any lossy multi-conductor line structure using polygonal cross-section conductors by using an appropriate effective internal impedance model.
 J. Jin, The Finite Element Method in Electromagnetics: John Wiley & Sons, Inc., 1993.
 A. Darcherif, A. Raizer, J. Sakellaris, and G. Meunier, "On the use of the surface impedance boundary concept in shielded and multiconductor cable characterization by the finite element method," IEEE Trans. Magnetics, vol. 28, pp. 1446-1449, 1992.
 T. H. Fawzi, M. T. Ahmed, and P. E. Burke, "On the use of impedance boundary conditions in eddy current problems," IEEE Trans. Magnetics, vol. MAG-21, pp. 1835-1840, 1985.
 E. M. Deely, "Avoiding surface impedance modification in BE methods by singularity-free representations," IEEE Trans. Magnetics, vol. 28, pp. 2814-1816, 1992.
 J. R. Mosig, "Arbitrary shaped microstrip structures and their analysis with a mixed potential integral equation," IEEE Trans. Microwave Theory Tech., vol. 36, pp. 314-323, 1988.
 B. J. Rubin, "An electromagnetic approach for modeling high-performance computer packages," IBM J. Res. Develop., vol. 34, pp. 585-600, 1990.
 J. G. Maloney and G. S. Smith, "The use of surface impedance concepts in the finite-difference time-domain method," IEEE Trans. Antennas Prop., vol. 40, pp. 39-48, 1992.
 J. H. Beggs, R. J. Luebbers, K. S. Yee, and K. S. Kunz, "Finite-difference time-domain implementation of surface boundary conditions," IEEE Trans. Antennas Prop., vol. 40, pp. 49-56, 1992.
 M. A. Leontovich, "On the approximate boundary conditions for electromagnetic fields on the surface of well conducting bodies," Investigations of Propagation of Radio Waves, vol. B. A. Vvdensky, Ed. Moscow: Academy of Sciences USSR, pp. 5-20, 1948.
 D. J. Hoppe and Y. Rahmat-Samii, Impedance Boundary Conditions in Electromagnetics: A SUMMA Book, 1995.
 S. A. Schelkunoff, "The electromagnetic theory of coaxial transmission lines and cylindrical shields," Bell System Technical Journal, vol. 13, pp. 532-579, 1934.
 T. B. Senior, "Impedance boundary conditions for imperfectly conducting surfaces," Applied Science Research, vol. 8, pp. 418-436, 1960.
 K. M. Mitzner, "An integral equation approach to scattering from a body of finite conductivity," Radio Science, vol. 2, pp. 1459-1470, 1967.
 R.-B. Wu and J.-C. Yang, "Boundary integral equation formulation of skin effect problems in multiconductor transmission lines," IEEE Trans. Magnetics, vol. 25, pp. 3013-3015, 1989.
 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 Trans. Microwave Theory Tech., vol. 39, pp. 1338-1347, 1991.
 C. A. Balanis, Advanced Engineering Electromagnetics: John Wiley & Sons, Inc., 1989.
 M. H. Nayfeh and M. K. Brussel, Electricity and Magnetism: John Wiley & Sons, 1985.
 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.
 B.-T. Lee, E. Tuncer, and D. P. Neikirk, "Efficient 3-D Series Impedance Extraction using Effective Internal Impedance," IEEE 4th Topical Meeting on Electrical Performance of Electronic Packaging, Portland, OR, Oct. 2-4, 1995, pp. 220-222.
 B.-T. Lee and D. P. Neikirk, "Minimum Segmentation in the Surface Ribbon Method for Series Impedance Calculations of Microstrip Lines," IEEE 5th Topical Meeting on Electrical Performance of Electronic Packaging, Napa, CA, October 28-30, 1996, pp. 233-235.
 A. K. Agrawal, H. L. Price, and S. H. Gurbaxani, "Transient Response of Multiconductor Transmission Lines Excited by a Nonuniform Electromagnetic Field," IEEE Transactions on Electromagnetic Compatibility, vol. EMC-22, pp. 119-129, 1980.
 S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics, 2nd ed. New York: Wiley, 1984.
Figure 2: Geometry of twin lead used to test the EII combined with the surface ribbon method (SRM). 0 degrees corresponds to the "outside" point on the surface, and 180 degrees corresponds to the inside face.
Figure 5: Comparison of resistance and inductance calculated using various techniques. Twin lead is closely coupled: 1 mm radius, 0.2 mm spacing, and copper conductivity. Solid line: full boundary element method (FBEM) solution; dot-dashed line: surface boundary element method using the surface impedance of an isolated conductor as an approximation for the SIBC (I-SBEM); Dashed line: surface ribbon method using the surface impedance of an isolated conductor as an approximation for the EII (I-SRM).