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Skin effect, δ, and the 30δ rule

Skin depth

For a good conductor at angular frequency \(\omega\):

\[ \delta = \sqrt{\frac{2\rho}{\omega \mu}} \]

where \(\rho\) is resistivity and \(\mu = \mu_0 \mu_r\). For copper at 1 GHz, \(\delta \approx 2.1 \,\mu\mathrm{m}\).

Why it matters for lineforge's Rs accuracy

The Faraday solver assigns a single current value \(i_n\) per conductor pixel. If the pixel side \(h\) is large compared to \(\delta\), the discretization can't resolve the actual current concentration at the surface, and Rs is underestimated.

atlc2's empirical rule (which lineforge inherits):

Rs is accurate to ±1% when \(\delta \ge 30 \cdot h\).

For coarser grids, accuracy degrades to ±5%, and to worse than 5% if conductors of different voltage are too close (per atlc2's diagram, < 8 pixels for flat surfaces, < 16 pixels for corners).

"Restrict to skin depth"

atlc2 ships a checkbox; lineforge exposes the equivalent restrict_to_skin_depth=True flag. When enabled, conductor pixels deeper than \(3\delta\) from any conductor surface are blackened (treated as vacuum). Result: \(N\) shrinks drastically at high frequency, keeping the dense \(\mathcal{O}(N^3)\) Faraday solve tractable.

The 3× factor is conservative: \(e^{-3} \approx 5\%\) of the surface current remains, well below the noise floor of the FD discretization itself.

When to disable the skin restriction

  • Low-frequency / DC analysis: \(\delta\) is large; nothing is masked anyway.
  • Very thin conductors (foil): the conductor is already < \(3\delta\) thick.
  • Cross-validation against analytical formulas that don't include skin depth.

In lineforge:

result = lineforge.solve_lrs(geom, frequency="1GHz", restrict_to_skin_depth=False)

Low-frequency dispersion

A subtle consequence of skin effect: at low frequency the current spreads into the conductor interior, increasing the internal inductance contribution. At high frequency the current is concentrated at the surface, reducing \(L\). This is the well-known low-frequency dispersion effect.

lineforge's frequency-resolved Faraday solve captures this naturally: running solves at multiple frequencies and plotting \(L(f)\) shows the dispersion.

References

  • atlc2 docs §"Getting an accurate Rs": http://www.hdtvprimer.com/kq6qv/atlc2.html.
  • C. Paul, Inductance: Loop and Partial, Wiley 2010, Chapter 3.
  • D. Pozar, Microwave Engineering, 4th ed., §1.7 (skin depth, surface resistance).