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.
The line, αr,dc=4.34(Rdc/Z0) dB/m, expresses resistive attenuation, taking into consideration only the dc resistance of the trace, Rdc (ohms/meter), and assuming a constant-impedance termination set equal to Z0. The loss at dc varies in direct proportion to Rdc. The temperature-correction factor for dc resistance is TCF=1.0039T-30, where T is the temperature in degrees Celsius. You multiply TCF times Rdc at room temperature to find the corrected dc resistance.
Next, αr,skin=4.34(R0/Z0)(ω/ω0)½ dB/m approximates resistive attenuation, taking into account only the skin effect, where R0 is the ac resistance of the line (ohms/meter) at the particular frequency ω0 (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 αr,skin, while the skin depth grows in proportion to TCF½, decreasing αr,skin. The combined effect makes αr,skin change in proportion to TCF½.
Because for small a you can approximate (1+a)½ = 1+a/2, the sensitivity of αr,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 Rdc but only an 8% increase in αr,skin.
The last line, αd=4.34(θω/v0)×(ω/ω0)-θ/π dB/m, models dielectric attenuation, where v0 is the velocity of propagation (meters/second) and tan(θ) is the loss tangent of the dielectric material, both taken at frequency ω0. 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 TCF½. 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.