NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 214

The surface impedance can be defined as the ratio of the electric field at the
surface and the total current density:

E 0
J T

Z s =

(5-27)

Substituting (5-26) into (5-27) and expressing the current density in terms of the
conductivity and the electric field ( J 0 = σ E 0 ) yields

E 0

1
σδ

Z s =

1 ) = ( 1

+ j) ohms / square (5-28)

2 σ E 0 δ(j −

1

−

The surface impedance is expressed in terms of an area of unit width and unit
length, so the term ohms/square is used. Note the similarity of the real part of
(5-28) to equation (5-12), which was the series resistance caused by the skin effect
of a transmission line. For the case where l = w , the real part of equations (5-28)
and (5-12) are identical. If the real part of (5-28) is resistance, the imaginary part
must therefore be reactance (the impedance of inductance). Since the impedance
of inductance is jωL , (5-28) can be expressed in terms of a series resistance
(due to the skin effect) and a series inductance (the internal inductance):

Z s = R ac + jωL internal (5-29)

Therefore, the internal inductance can be calculated directly from the ac resis-
tance :

R ac
ω

L internal = (5-30)

Equation (5-30) highlights an important relationship between the skin effect resis-
tance (ac resistance) and the internal inductance. As the skin effect forces the
current to the periphery of the conductor, the resistance increases; however,
since current ceases to flow in the interior of the conductor, the inductance must
decrease. Figure 5-11 plots both the internal inductance and the ac resistance.
Note that when the ac resistance becomes significant, the internal inductance is
almost negligible.

Example 5-2 Calculate the total inductance and the resistance at 2 GHz of a
microstrip transmission line constructed with copper of conductivity σ = 5 . 8 ×

10 7 ( · m ) − 1 , a dielectric constant of ε r =

4 . 0 and the following cross-sectional

dimensions: t = 0 . 5 mil, h = 2 mils, w = 3 mils.

SOLUTION
Step 1: Determine if the ac or dc resistance should be used. The methodology
presented in Example 5-1 could be used, however, it is easier simply to calculate