NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 206

The skin effect will cause the cross-sectional area where the current is flowing
to decrease as the frequency increases. Consequently, the frequency-dependent
losses in the conductor can be approximated using the dc resistance formula by
setting t = δ :

πµf
σ

l
σwδ =

l
σw √ 2 /ωµσ =

l
w

R ac = ohms (5-12)

Note that the approximation is valid only when the skin depth is smaller than the
conductor thickness. Notice that the ac resistance is directly proportional to the
square root of the frequency f and inversely proportional to the conductivity σ .
Equation (5-12) assumes that all the current is flowing in the first skin depth,
which is not correct. Section 5.1.2 defines the skin depth such that only about
63% of the current density is contained in this depth. To test the validity of
equation (5-12), the effective area of an exponential decay can be calculated by
integrating e − αz

0 δ to z =∞ δ and comparing it to the case where all
the current is confined to one skin depth. To visualize the differences, refer to
Figure 5-5, which plots penetration depth into a conductive medium in terms of
skin depths versus the total current density. If 100% of the current is assumed
to flow within one skin depth, the area under the curve is Jδ =

from z =

1, where J is
the current density and δ is the skin depth. Integrating the wave decay term e − αz
from z = 0 δ to z =∞ δ , the area under the curve also yields Jδ :

J z =∞ δ
z =

J
α = Jδ

e − αz dz =

0 δ

Since the effective areas under each curve are identical, it is a valid approximation
to assume that all the current is flowing in an area confined by the conductor width

1

0.9

J d = 1

0.8

0.7

0.6

z = ∞d

J
a = J d

e −a z dz = 0.5 J

z = 0 d

0.4

0.3

0.2

0.1

0

0 1 2 3 4 5 6 7 8 9 10

Skin depth, d

Figure 5-5 If all the current is approximated to be in one skin depth, the total area
under the curve is identical to the realistic behavior, where the current density decays
exponentially with increasing skin depths.