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Skin Effect Crisis

Posted July 16th, 2008 · 4 Comments · Calculator

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Updated October 29, 2012: ADS now includes a measurement-hardened surface roughness model called multi-level hemisphere. Please see my later posting Conductor Surface Roughness. Thanks!

In serial links, you hit the “skin effect crisis” when high frequency components of the EM waves in the dielectric induce eddy currents in the metal traces and can’t penetrate the whole thickness of conductor. The effective cross section is thus reduced, and the resistance at those high frequencies goes up compared to lower frequency components. Here is an order-of-magnitude calculator that illustrates the effect in terms of relative attenuation at DC and the high frequency end of the spectrum.

Thickness: in μm or in mil or in oz/ft2
Solution-side surface roughness (RMS): in μm or in mil
Width: in μm or in mils
Length: in cm or in inches
Time and frequency: Bit rate in gigabit/s and Rise time as % of bit period or
fknee in GHz
Outputs: Resistance at DC (Ω) Resistance at fknee (Ω)
Attenuation at DC (dB) Attenuation at fknee (dB)
Skin depth at fknee (μm)
Hammerstad depth at fknee (μm) (see note below)

The above JavaScript assumes:

  • Copper traces (16.78×10-9Ω.m resisitivity) in a differential pair, with equal length of positive and negative polarity, driving a 100 Ω load.
  • The resistance given assumes the round trip (i.e. double the resistance Rtrace of one of the pair of traces).
  • Attenuation is that of a voltage divider: 20*log10(Rload/(2*Rtrace+Rload)).
  • fknee is 0.5/rise time.
  • The width is greater than the skin depth (i.e. fringing effects are ignored).
  • If the skin depth is about equal to the thickness, this simple calculation won’t be very accurate.
  • At fknee and for smooth surfaces, the current flows on the upper and lower skin of the trace i.e. the effective cross section is twice the skin depth times width.
  • The empirical roughness model is from Microstrip Handbook, E. O. Hammerstad, Edited by F. Bekkadal, ELAB Report No. STF44 A74169, University of Trondheim, Norway, 1975. Hammerstad depth is the actual skin depth divided by a unitless loss factor = (1 + (2/π) arctan(0.7 * variance / (skin depth)2). (Variance is the square of the RMS roughness.) The loss factor varies between 1 (perfectly smooth) and approaches 2 in the limit of roughness >> skin depth. For one rough surface (the solution side) and at fknee, the current flows in a effective cross section of (Hammerstad depth plus skin depth) times width. Assumes RMS roughness << thickness. Eric Bogatin has an article on smooth copper foils for extra low loss. Eric pointed out to me an error in an earlier version: the roughness is one-sided, not two-sided. The drum side is assumed to be smooth. Only the solution side that is rough.

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