Conductor Loss
How Skin Effect Drives Conductor Loss
At DC, current spreads uniformly across the full cross-section of a conductor, so the resistance is simply the bulk resistivity divided by the area. At radio frequencies this picture breaks down. The time-varying magnetic field inside the metal induces eddy currents that oppose current flow in the core and reinforce it near the surface, confining the conduction to a layer roughly one skin depth (δ) thick. For copper, δ falls from about 2.1 μm at 1 GHz to 0.66 μm at 10 GHz and 0.38 μm at 30 GHz. Because the effective conducting cross-section shrinks as frequency rises, the surface resistance Rs = 1/(σδ) climbs in proportion to √f.
Conductor loss is the part of total line attenuation that comes from this surface resistance. For a given line geometry the conductor attenuation constant αc is proportional to Rs divided by the characteristic impedance, so it also scales as √f. This is distinct from dielectric loss, which scales linearly with frequency and depends on the substrate loss tangent. The two combine additively, and their differing slopes (√f versus f) are exactly what engineers exploit to separate the two contributions from a swept measurement.
The practical consequence is that low-loss line designs minimize Rs by using high-conductivity metals (silver, then copper, then gold or aluminum), by widening conductors to lower the current density, and by controlling the metal surface finish. Silver plating a copper line lowers loss only marginally because the conductivities differ by a few percent, whereas a poorly chosen rough nickel barrier under gold can add several tenths of a dB at millimeter-wave frequencies.
Conductor Loss and Surface Roughness
Real conductors are never perfectly smooth. Electrodeposited copper foil on a PCB has peaks and valleys with an RMS height of 0.3 to 3 μm depending on the foil grade. When the RMS roughness becomes comparable to the skin depth, the current is forced along a longer, contoured path and the effective surface resistance rises. The widely used Hammerstad roughness correction multiplies Rs by a factor that saturates at 2 once RMS roughness grows well beyond the skin depth; it is already about 1.8 when the roughness roughly equals the skin depth. At 30 GHz, where δ is only 0.38 μm, even a 1 μm roughness can nearly double the conductor loss, which is why very-low-profile (VLP) and hyper-VLP copper foils are specified for 28 GHz and 77 GHz designs.
Governing Equations
Rs = 1 / (σδ) = √(πfμ / σ) [Ω/square]
Skin depth:
δ = 1 / √(πfμσ)
Conductor attenuation (general):
αc ≈ Rs / (2 × Z0 × w) [Np/m]
Roughness-corrected resistance:
Rs,rough = Rs × KSR, KSR = 1 + (2/π) × arctan[1.4 (Δ/δ)2]
Where σ = conductivity, μ = permeability, f = frequency, Z0 = characteristic impedance, w = conductor width, Δ = RMS surface roughness. For copper σ ≈ 5.8 × 107 S/m, giving Rs ≈ 26 mΩ/square at 10 GHz. Total loss: α = αc + αd.
Conductor Loss Across Common Lines and Metals
| Case | Conductivity σ (S/m) | Skin depth at 10 GHz | Rs at 10 GHz | Relative loss | Note |
|---|---|---|---|---|---|
| Silver | 6.3 × 107 | 0.64 μm | ~25 mΩ/sq | 0.96x | Lowest, costly plating |
| Copper | 5.8 × 107 | 0.66 μm | ~26 mΩ/sq | 1x (baseline) | Standard PCB conductor |
| Gold | 4.1 × 107 | 0.79 μm | ~31 mΩ/sq | 1.19x | Wire-bond, corrosion-free |
| Aluminum | 3.5 × 107 | 0.85 μm | ~34 mΩ/sq | 1.30x | Light, low-cost waveguide |
| Rough Cu (Δ=1μm) | 5.8 × 107 | 0.66 μm | ~47 mΩ/sq | 1.8x | Roughness KSR ≈ 1.8 |
Frequently Asked Questions
Why does conductor loss increase with the square root of frequency?
RF current crowds into a thin surface layer whose thickness is the skin depth, which shrinks as 1/√f. The shrinking effective cross-section raises surface resistance Rs as √f. For copper, δ is about 2.1 μm at 1 GHz and 0.66 μm at 10 GHz, and Rs rises from roughly 8.2 to 26 mΩ/square. Since attenuation tracks Rs, the conductor term scales as √f, so a line losing 0.1 dB/m at 1 GHz can exceed 0.3 dB/m at 10 GHz.
How much does surface roughness add to conductor loss at millimeter-wave frequencies?
Roughness forces current along a longer path, raising effective Rs. The Hammerstad correction saturates at a factor of 2 once RMS roughness exceeds the skin depth, and is already about 1.8 when the two are comparable. With standard electrodeposited copper at 1 to 2 μm RMS roughness and a 30 GHz skin depth of only 0.38 μm, roughness can nearly double conductor loss. That is why low-profile and VLP foils below 0.5 μm RMS are specified for 28 GHz and 77 GHz designs.
How do you separate conductor loss from dielectric loss in a measured line?
Total attenuation is the sum of the two terms, and they have different signatures: conductor loss scales as √f, dielectric loss linearly with f. Fitting α(f) = A√f + Bf isolates A (conductor) from B (dielectric). A multiline TRL calibration on two or more line lengths cancels connector and launch effects. In practice conductor loss dominates below roughly 10 to 20 GHz on low-loss laminates, with dielectric loss taking over higher up.