NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 205

ensures almost uniform current density through the cross section of the signal
conductor.
At dc the current will spread out as much as possible and flow through the
entire cross section of the conductor, and the resistive loss can be found with

l
σA cross section

l
σwt

R dc = = ohms (5-11)

where l is the length, w the width, t the thickness of the signal conductor, and
σ the conductivity of the metal. Note that (5-11) has neglected the dc losses of
the current return path in the reference plane. This is an adequate approximation
because at dc, the current will spread out and flow through the entire plane, which
is several orders of magnitude larger than the signal conductor. Consequently,
the cross-sectional area where the current flows in the return path will be much
larger and the associated resistance will be much smaller.

5.2.2 Frequency-Dependent Resistance in Conductors
By extending the dc equation (5-11), the frequency dependence of the resistance
in a transmission line can be approximated. Frequency-dependent resistance will
be referred to as ac resistance or skin effect resistance in the remainder of the
topic. At low frequencies, the ac resistance will be identical to the dc resistance
because the skin depth will be much greater than the thickness of the conductor.
The ac resistance will remain equal to the dc resistance until the frequency
increases to a point where the skin depth is smaller than the conductor thickness.

Microstrip Conductor Losses (Smooth Conductors) Figure 5-4 depicts the cur-
rent distribution on a microstrip line at high frequencies. Notice that the current
distribution is concentrated on the bottom edge of the transmission line. This is
because the fields between the signal line and the ground plane pull the charge to
the bottom edge, and the skin depth is much smaller than the conductor thickness.
Also notice that the current density is greater near the corners of the conductor.
This is because the charge density increases significantly in the proximity of a
sharp edge, as described in Sections 3.4.4 and 3.4.5, and the current density along
the conductor will vary in the same way. Furthermore, there is still significant
field concentration along the thickness (the t dimension in Figure 5-4) of the
conductor.

w

t

d (Skin depth)

h

Reference plane

Figure 5-4 Current distribution in a microstrip with an ideal reference plane at high
frequencies where the skin depth δ is small compared to the thickness t .