- Effective Internal Impedance Method for Series
Impedance Calculations of Lossy Transmission Lines: Comparison to Standard
Impedance Boundary Condition
- ABSTRACT
- I. INTRODUCTION
- II. STANDARD IMPEDANCE BOUNDARY CONDITION
- III. BOUNDARY ELEMENT METHOD FOR TWO-DIMENSIONAL FINITE CONDUCTIVITY MULTI-CONDUCTOR TRANSMISSION LINES
- IV. THE EFFECTIVE INTERNAL IMPEDANCE AND THE SURFACE RIBBON METHOD
- V. EXAMPLE: CIRCULAR CYLINDRICAL CONDUCTORS (TWIN-LEAD)
- VI. CONCLUSION
- REFERENCES
- FIGURE CAPTIONS

This article is **significantly** different from our full paper submitted
(submitted: Jan., 1999) to the IEEE Transactions on Microwave Theory and
Technique; this material is of summary nature only, and does not consitute
a "publication" of the manuscript submitted to the IEEE Transactions
on Microwave Theory and Technique.

To receive a copy of the full manuscript:

Also refer to other material of related nature on our web pages.

**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: neikirk@mail.utexas.edu

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 [11] first introduced the concept of surface impedance in electromagnetics in 1934 for the analysis of coaxial cables. In the 1940's Leontovich [9] 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 [12] 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

, (2)

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 [13] 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 [16] with the assumptions of non-radiation and quasi-static fields, the following equation is derived

, (7)

where V is a volume enclosing all the conductors, is the applied
potential on the q^{th} 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 [17].

Using Green's theorem and the power dissipation term in (7), the series
resistance of the q^{th} conductor of the transmission line is given
by the surface integral

, (9)

where is the perimeter of the q^{th} 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 q^{th}
conductor are

(10)

, (11)

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

, (12)

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 [16]. 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

, (14)

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

(17)

.

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

, (18)

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

, (19)

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

, (20)

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

. (21)

Resistance can be calculated in a manner similar to (15) using

. (22)

Internal inductance of conductor q and total external inductance are calculated using

(23)

and

. (24)

Surface inductance due to the magnetic energy stored at the impedance sheet is given by

. (25)

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 k^{th} ribbon yields

, (29)

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

, (30)

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 [20]. A similar approach was used in [21] 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

, (31)

where r is the radius of the circular conductor, and J0 and J1 are Bessel functions of the first kind [22].

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.

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**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).