## Skin Hot

I've just returned from a four-month stint as a visiting fellow of Oxford University. Oxford is a beautiful city—well worth the visit if you have a chance. The weather there is mild, not too hot or too cold, and moderately humid. It's a good climate for computers. This article considers weather that isn't so good. Specifically, it addresses how temperature affects the skin-effect loss in a pc-board trace.

Figure 1 illustrates the attenuation in decibels of a typical differential-stripline configuration. Three straight-line asymptotes appear on the chart.

**Figure 1—**Above 300 MHz dielectric losses dominate the performance of this differential stripline configuration.

The line, *α*_{r,dc}=4.34(*R*_{dc}/*Z*_{0}) dB/m, expresses resistive attenuation, taking into consideration only the dc
resistance of the trace, *R*_{dc} (ohms/meter), and assuming a
constant-impedance termination set equal to *Z*_{0}. The loss at dc varies in direct
proportion to *R*_{dc}. The temperature-correction factor for dc
resistance is *TCF*=1.0039^{T-30}, where *T* is the
temperature in degrees Celsius. You multiply *TCF* times *R*_{dc} at room temperature to find the corrected dc resistance.

Next, *α*_{r,skin}=4.34(*R*_{0}/*Z*_{0})(*ω*/*ω*_{0})^{½} dB/m approximates resistive attenuation, taking into account only the skin effect,
where *R*_{0} is the ac resistance of the line (ohms/meter) at the
particular frequency *ω*_{0} (radians/second). This term varies
in proportion to the square root of *TCF*. That
variation results from the changes in skin depth that occur as a function of
temperature. As the temperature rises, the bulk resistivity of the copper grows
in direct proportion to *TCF*, increasing *α*_{r,skin},
while the skin depth grows in proportion to *TCF*^{½},
decreasing *α*_{r,skin}. The combined effect makes *α*_{r,skin} change in proportion to *TCF*^{½}.

Because for small *a* you can approximate (1+*a*)^{½} = 1+*a*/2,
the sensitivity of *α*_{r,skin} to temperature is only half as
great as you might think, based on looking only at the change in dc resistance.
For example, a 40°C rise in temperature induces a 17% increase in *R*_{dc} but only an 8% increase in *α*_{r,skin}.

The last line, *α*_{d}=4.34(θ*ω*/*v*_{0})×(*ω*/*ω*_{0})^{-θ/π} dB/m, models dielectric attenuation, where *v*_{0} is the velocity
of propagation (meters/second) and tan(θ) is the loss
tangent of the dielectric material, both taken at frequency *ω*_{0}. In practice, dielectric losses vary considerably more
with temperature than do resistive losses, especially as you approach the glass
transition temperature of a pc-board material. However, this article assumes for
simplicity that the dielectric properties remain fixed. Only variations in trace
resistance are illustrated.

The figure
plots the composite loss, taking into account all three loss mechanisms under
cold, room-temperature, and hot conditions of operation. Notice that the
temperature dependence of copper has a diminishing effect on overall loss as you
go to higher and higher frequencies. In the flat-loss region below the onset of
the skin effect, the *TCF* affects loss in a direct way. Above the onset of
the skin effect, the loss varies at most only in proportion to *TCF*^{½}.
Above 1 GHz, as dielectric losses begin to dominate the behavior of
the trace, the *TCF* rapidly diminishes in importance. At these speeds,
it's mostly temperature-induced variations in dielectric loss that bother you,
not variations in the resistivity of copper.