Conductor Thickness
Why Conductor Thickness Is Counted in Skin Depths
At DC, the resistance of a flat conductor falls in direct proportion to its cross-sectional area, so doubling the thickness halves the resistance. At RF the picture inverts. The skin effect forces current into a surface layer whose effective depth, the skin depth δs, shrinks as the inverse square root of frequency. Metal buried more than a skin depth below the surface carries an exponentially decaying current and contributes almost nothing to conduction. As a result, conductor thickness at microwave frequencies is meaningful only when expressed as a multiple of δs at the lowest operating frequency.
The practical consequence is a clear floor and a flat plateau. As thickness climbs from zero, the AC resistance drops quickly until the conductor reaches about three skin depths, where it is within roughly two percent of its asymptotic value; by five skin depths the residual error is a fraction of a percent. Adding metal beyond that does not lower RF loss, though it does keep raising the DC current capacity and the mechanical robustness of the trace. This is why microstrip and stripline circuits on copper-clad laminate are almost always electrically thick: even half-ounce copper at 17 μm exceeds eight skin depths at 1 GHz.
The case that bites engineers is the opposite extreme. Thin-film metallizations, sputtered seed layers, and lightly plated traces can fall below three skin depths at the bottom of their band, where the truncated current profile drives AC resistance upward and pushes conductor loss into the path that erodes resonator unloaded Q. The standard remedy is to electroplate copper or gold up to several skin depths over the thin seed metal, recovering the bulk surface resistance without the cost or stress of a fully thick deposition.
Governing Equations
δs = 1 / √(π × f × μ × σ) (for copper: ≈ 66 μm / √fMHz)
Surface resistance (thick conductor):
Rs = 1 / (σ × δs) = √(π × f × μ / σ) Ω/sq
Surface impedance of a finite-thickness conductor:
Zs(t) = Rs,∞ × coth((1 + j) t / δs) → Re(Zs) within ~2% once t ≥ 3δs, <1% by 5δs
Where f = frequency, μ = permeability (≈ 4π × 10−7 H/m for copper), σ = conductivity (5.8 × 107 S/m for copper), t = physical conductor thickness. Example: at 10 GHz, δs ≈ 0.66 μm and Rs ≈ 0.026 Ω/sq, so t = 3.3 μm (5δs) captures over 99% of the achievable conductance.
Copper Thickness vs. Skin Depths by Frequency
| Conductor / Weight | Thickness | δs @ 1 GHz (Cu 2.1 μm) | δs @ 10 GHz (0.66 μm) | δs @ 50 GHz (0.30 μm) | RF Verdict |
|---|---|---|---|---|---|
| Sputtered seed | 0.3 μm | 0.14δs | 0.45δs | 1.0δs | Too thin; plate up |
| Thin-film Au | 2 μm | 1.0δs | 3.0δs | 6.7δs | Marginal at 1 GHz |
| Half-ounce Cu | 17 μm | 8.1δs | 26δs | 57δs | Electrically thick |
| One-ounce Cu | 35 μm | 17δs | 53δs | 117δs | Thick; high current |
| Two-ounce Cu | 70 μm | 33δs | 106δs | 233δs | Thick; power lines |
Frequently Asked Questions
How thick does a conductor need to be at RF frequencies?
About three to five skin depths captures essentially all of the achievable RF performance. For copper, one skin depth is roughly 2.1 μm at 1 GHz, 0.66 μm at 10 GHz, and 0.30 μm at 50 GHz, so a half-ounce 17 μm layer is over eight skin depths at 1 GHz and more than fifty at 10 GHz. For most microwave printed circuits the plated copper is therefore electrically as good as infinitely thick.
Why does conductor thickness barely affect RF resistance above a few skin depths?
The skin effect confines current to a thin surface layer whose depth shrinks with √f, and the current density decays exponentially below it. Once the metal is several skin depths thick, the surface impedance equals the bulk Rs and is set by conductivity and frequency, not physical depth. That is why a 35 μm one-ounce trace and a 70 μm two-ounce trace show nearly identical loss at 10 GHz; the extra metal only helps DC current handling.
When does a conductor become too thin for RF use?
Trouble starts below roughly three skin depths. Below two skin depths the AC resistance climbs sharply, and below one it approaches the much larger DC value. Sub-micrometer thin-film metallizations on alumina or GaAs at low bands can show several times the loss of a thick conductor, directly lowering resonator unloaded Q. The fix is to electroplate copper or gold up to at least three to five skin depths at the lowest operating frequency.