Compact Equivalent Circuit Model for the Skin Effect*
S. Kim and D. P. Neikirk
Department of Electrical and Computer Engineering
Electrical Engineering Research Laboratory
The University of Texas at Austin
Austin, Texas 78712
*This material is abstracted from the manuscript submitted to the 1996 IEEE-MTT-S International Microwave Symposium, held in San Francisco, California, June 16-21, 1996. For the final version of this paper, please see the conference procedings:
S. Kim and D. P. Neikirk, "Compact Equivalent Circuit Model for the Skin Effect," IEEE 1996 IEEE-MTT-S International Microwave Symposium, editor: R. G. Ranson, Vol. 3, San Francisco, California, June 17-21, 1996, pp. 1815-1818.
Our poster presented at the 1996 International Microwave Symposium (held in San Francisco, June 17-21, 1996) can be down-loaded here (pdf file is approximately 100K in size).
Abstract
Rules for determining a compact circuit model consisting of four resistors and three inductors to accurately predict the skin effect have been developed. The circuit is easily constructed from the geometry, producing a response that matches exact results over a frequency range from dc to very high frequencies.
The skin effect has been extensively studied, although many methods treat the skin effect at high frequencies only, failing to predict the properties of transmission lines at low frequencies. Time domain analysis for digital signal propagation requires models covering the entire frequency range from dc to 1/[[tau]]rise, where [[tau]]rise is the rise time of the signal. Skin effect lumped circuit models in which the elements are frequency independent have been used [1-3], but tend to produce very large ladder circuits. Yen et al. [3] introduced a more compact circuit model, but this method failed to accurately capture the skin effect at high frequencies and did not establish clear rules governing the choice of component values. Here we present a modification of Yen's method using simple rules for selecting the values of resistors and inductors for a four deep ladder circuit model. The equivalent circuit accurately models the skin effect in circular cross section conductors up to a frequency corresponding to a 100 skin depth radius conductor. We also show equivalent circuits for the series impedance per unit length for coax and twin lead, including both proximity and skin effects.
For normal lossy transmission lines, since conductor loss increases as
frequency increases, for wide bandwidth digital signals the transmission
line acts somewhat like a low pass filter. As the signal propagates along
the line and high frequency components are attenuated, the effective bandwidth
decreases, and hence an "electrically short" length becomes longer.
This has been used to reduce the size of lumped ladder models for long transmission
lines through the use of non-uniform lumping [4,5]. This approach can also
be used to generate compact circuit models for the skin effect. Yen et
al. [3] introduced a constant resistance ratio (RR) to try and capture
this low pass characteristic. Figure 1 shows a schematic illustration of
our compact ladder model. Here a circular cross section conductor is divided
into four concentric rings, each ring represented by one ladder section.
For the resistance values each ring is chosen so that 
, 
, where RR is
a constant to be determined. We first require that the dc resistance of
the ladder be equal to the actual dc resistance of the conductor Rdc, and
also take the first resistance to be 
. For a four section
ladder these two constraints lead to the requirement
.
Thus, for a given selection of 
, the resistance ratio
RR is fixed by solution of this cubic equation. The inductance values are
determined in a similar fashion, with 
, 
, again requiring
that the low frequency inductance of the ladder be equal to the actual low
frequency internal inductance (Llf) of the wire. Using 
this leads to
the constraint that
,
This equation can be solved for LL, once RR has been obtained from the
solution of the cubic polynomial above and 
has been selected.
To match the high frequency resistance of the conductor, the resistance
of the outermost ring (R1) is most critical; we have found that if the maximum
frequency of interest is 
, R1 should be chosen so that
, where 
.
To ensure the frequency response from dc to 
is well modeled,
we have found that the inductance values must be chosen using 
. Once the dimensions of the conductor and 
are specified,
all the values of the components in the ladder are fixed, and the response
of this circuit from dc to 
will match the skin effect. Figure
2 illustrates the advantage of using this method over a uniform lumping
method. For instance, over 100 uniform R-L sections are necessary to represent
the response of a 50 skin-depth radius circular conductor to similar levels
of accuracy as a four section ladder using the procedure discussed above.
We have found that for 
the local error is not worse than
15%, and with 
up to 100, the error is still not worse than
25%. This corresponds to wires with radii between about 10 and 100 skin
depths at 
for less than 15% error, and up to 200 skin
depths at 25% error.
This approach can be used to model a coaxial line including skin effect in both the center and shield conductors. The exact solution for the series impedance per unit length, including skin effect is [6],
where a is the inner conductor radius, b the inner and c the outer radius
of the shield, and 
. The series impedance equivalent circuit
is shown in Fig. 3a, using the rules presented above applied to both inside
and outside conductor, one ladder for the inner conductor using Llf = uo/8[[pi]],
and another for the outer shield using [7]
.
Figure 3b shows the result of circuit modeling for a maximum frequency up to that corresponding to a 100 skin-depth inner conductor radius. Both resistance and inductance are in excellent agreement with the exact results.
For twin lead, when two conductors are very closely separated both the
skin effect, and proximity effect cause series resistance to increase. To
approximate the proximity effect with a simple equivalent circuit, we find
the fraction of the circular conductor 
that contributes
half the flux at high frequency (Fig. 4a):
.
Two ladders with values determined using the rules presented above are
constructed, with one ladder weighted by 
(representing
the inner face of the conductors), the other weighted by 
(representing
the outer faces of the conductors); they are then connected in parallel,
as shown in Fig. 4b. Figure 4c shows the equivalent circuit compared to
a conformal mapping method [8]. The conformal mapping method over-estimates
the low frequency inductance; at low frequencies our circuit model is actually
in better agreement with exact results.
Rules for determining a compact circuit model consisting of four resistors and three inductors that accurately predicts the skin effect have been developed. Each element value can be easily calculated from the geometry and conductivity to cover a frequency range from dc to very high frequencies. Since this model only contains frequency independent elements, it is easy to implement in conventional circuit simulators. This model can also be used to capture proximity effects, and can be generalized for application to rectangular conductor geometries.
This work was sponsored in part by the Advanced Research Projects Agency Application Specific Electronic Module and Mixed Signal Packaging programs, and the Texas Advanced Technology Program.
[1] H. A. Wheeler, "Formulas for the skin-effect," Proceedings of the Institute of Radio Engineers, vol. 30, pp. 412-424, 1942.
[2] T. V. Dinh, B. Cabon, and J. Chilo, "New skin-effect equivalent circuit," Electronic Letters, vol. 26, pp. 1582-1584, 1990.
[3] C.-S. Yen, Z. Fazarinc, and R. L. Wheeler, "Time-Domain Skin-Effect Model for Transient Analysis of Lossy Transmission Lines," Proceedings of the IEEE, vol. 70, pp. 750-757, 1982.
[4] T. Dhaene and D. D. Zutter, "Selection of lumped element models for coupled lossy transmission lines," IEEE Transactions on Computer Aided Design, vol. 11, pp. 805-815, 1992.
[5] N. Gopal, E. Tuncer, Dean P. Neikirk, and L. T. Pillage, "Non-uniform lumping models for transmission line analysis," IEEE Topical Meeting on Electrical Performance of Electronic Packaging, Tucson, AZ, 1992, pp. 119-121.
[6] S. Ramo, J. R. Whinnery, and T. V. Duzer, "Fields and Waves in Communication Electronics," 2nd ed. New York: Wiley, 1984, p. 181.
[7] S. Ramo, J. R. Whinnery, and T. V. Duzer, "Fields and Waves in Communication Electronics," 2nd ed. New York: Wiley, 1984, p. 106.
[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, vol. 42, pp. 1807-1815, 1994.
(a) (b)
Figure 1: Schematic illustration of four ladder compact circuit model;
drawing is to scale for 
= 10, giving RR = 1.66.
Figure 2: Comparison between uniform lumping and our compact model. Black
line: exact result from solving Helmholtz equation; Red line: four ladder
circuit modeling; 
: uniform ladder, 100 sections; 
: uniform ladder, 50 sections; 
: uniform ladder, 30
sections;
(a)
(b)
Figure 3: Example of coax line impedance calculation; (a) series impedance skin effect equivalent circuit. (b) Calculated total series impedance for a coaxial line with a = 0.1 mm, b = 0.23 mm, c = 0.25 mm. Blue line: exact result; red line: compact circuit model, fmax = 5 GHz.
(a)
(b)
(c)
Figure 4: Example of twin lead equivalent circuit including external
inductance and the proximity effect. (a) geometry of twin lead; 
represents half the flux coupling the two conductors; (b) equivalent
circuit; (c) sample calculation for r = 1 mm, d = 1.2 mm: blue line: conformal
mapping [8]; red line: circuit model.
