What is skin effect and why does it matter?

Skin effect is the tendency of alternating current to flow near the surface of a conductor. The skin depth delta = 1/sqrt(pi*f*mu_0*sigma) is the depth at which the current density has decayed to 1/e (37 percent) of its surface value. For copper at 1 GHz: delta = 2.1 micrometers. At 10 GHz: delta = 0.66 micrometers. At 100 GHz: delta = 0.21 micrometers. This means that at microwave frequencies, only a thin surface layer carries current, and the conductor's DC resistance becomes irrelevant. The effective RF resistance is R_s = 1/(sigma*delta) = sqrt(pi*f*mu_0/sigma) ohms per square. This surface resistance determines the loss in transmission lines, waveguides, and resonant cavities. A practical implication is that plating a base metal with a few skin depths of copper or silver provides the same RF performance as a solid copper or silver conductor, since the current only flows in the surface layer.

Which conductor material is best for RF?

Silver has the highest conductivity (6.3 times 10^7 S/m) and therefore the lowest surface resistance, making it the theoretical best conductor for RF applications. However, silver tarnishes (silver sulfide formation) which increases surface resistance, so it requires protective overcoating. Copper (5.8 times 10^7 S/m) is the practical choice for most RF applications: excellent conductivity, readily available, and relatively stable. Gold (4.1 times 10^7 S/m) has lower conductivity but is completely corrosion resistant, making it the standard for connector contacts, bond wires, and MMIC metallization where long-term reliability is critical. Gold plating thickness of 1 to 5 micrometers is standard. Aluminum (3.5 times 10^7 S/m) is used for waveguides and satellite antenna reflectors where weight is critical (density 2.7 versus 8.9 g/cm3 for copper). Its native oxide layer provides natural corrosion protection but slightly increases surface resistance.

How does surface roughness affect RF performance?

Surface roughness becomes significant when the roughness height approaches the skin depth. At frequencies where the RMS roughness is comparable to or greater than the skin depth, the current path is elongated by following the surface contours, increasing the effective resistance. The Hammerstad-Jensen roughness correction factor is: K_r = 1 + (2/pi)*arctan(1.4*(R_q/delta)^2), where R_q is the RMS surface roughness and delta is the skin depth. For R_q = 1 micrometer at 10 GHz (delta = 0.66 micrometers for copper): K_r = 1 + 0.64*arctan(1.4*(1/0.66)^2) = approximately 1.8, meaning the conductor loss is 80 percent higher than for a perfectly smooth surface. This is why PCB copper foil surface finish matters greatly at mmWave frequencies: standard ED copper has R_q of 2 to 5 micrometers, while HVLP (hyper very low profile) foil has R_q of 0.3 to 0.5 micrometers. For 5G mmWave PCBs at 28 GHz, HVLP copper can reduce trace loss by 30 to 50 percent compared to standard copper.

RF Materials

Conductor

/kon-duk-ter/

Material with high conductivity (σ > 10⁶ S/m) that guides current and EM waves. At RF: skin effect confines current to surface layer δ = 1/√(πfμσ). Copper at 1 GHz: δ = 2.1 μm. Surface resistance R_s = 1/(σδ) determines TL/waveguide loss. Materials: Ag (σ=6.3×10⁷, lowest R_s), Cu (5.8×10⁷, practical standard), Au (4.1×10⁷, corrosion-free), Al (3.5×10⁷, lightweight). Surface roughness degrades loss above 10 GHz.

Cu σ: 5.8×10⁷ S/m

δ@1G: 2.1 μm

R_s: 1/σδ

Understanding RF Conductors

At DC and low frequencies, a conductor is characterized by its bulk resistivity: thicker wires have lower resistance. At RF frequencies, the skin effect changes everything. Current concentrates in a thin surface layer whose thickness decreases with frequency, making the conductor's surface properties (smoothness, plating material) far more important than its cross-sectional area or bulk properties.

This has profound implications for RF hardware design. A copper microstrip trace at 28 GHz only uses the outermost 0.4 micrometers of copper for current flow. If the copper surface is rough (typical PCB manufacturing creates 2-5 micrometer roughness), the current path is elongated, increasing loss by 50-100% compared to smooth copper. Selecting the right copper foil profile (HVLP or rolled) and surface finish is critical for mmWave PCB performance.

Conductor Equations

Skin depth:
δ = 1/√(πfμ₀σ) (meters)
Cu at 1 GHz: δ = 2.1 μm
Cu at 10 GHz: δ = 0.66 μm
Cu at 100 GHz: δ = 0.21 μm

Surface resistance:
R_s = 1/(σδ) = √(πfμ₀/σ) Ω/sq
Cu at 1 GHz: R_s = 0.0083 Ω/sq
Cu at 10 GHz: R_s = 0.026 Ω/sq

Roughness correction:
K_r = 1+(2/π)arctan(1.4(R_q/δ)²)
R_q=1μm, 10 GHz: K_r ≈ 1.8
80% more loss than smooth surface

RF Conductor Material Comparison

Material	σ (S/m)	δ @ 10 GHz	R_s @ 10 GHz	Application
Silver	6.3×10⁷	0.63 μm	0.025 Ω/sq	High-Q cavities
Copper	5.8×10⁷	0.66 μm	0.026 Ω/sq	PCB, connectors
Gold	4.1×10⁷	0.79 μm	0.031 Ω/sq	Connectors, MMIC
Aluminum	3.5×10⁷	0.85 μm	0.034 Ω/sq	Waveguide, antenna
Brass	1.5×10⁷	1.30 μm	0.051 Ω/sq	Connectors (plated)

Common Questions

Frequently Asked Questions

What is skin effect?

AC current concentrates at conductor surface. δ = 1/√(πfμσ). Cu at 1 GHz: 2.1 μm. At 10 GHz: 0.66 μm. At 100 GHz: 0.21 μm. Only surface layer carries current. RF resistance = R_s = 1/(σδ) Ω/sq. Practical: plating 3-5 skin depths of Cu or Ag gives same RF performance as solid conductor.

Best material for RF?

Ag: lowest R_s but tarnishes (needs overcoat). Cu: practical standard, excellent σ, stable. Au: corrosion-free, standard for connectors/MMIC bond wires, 1-5 μm plating. Al: lightweight (density 2.7 vs Cu 8.9 g/cm³), used for waveguide/satellite antennas. Choice = trade-off between loss, corrosion, weight, and cost.

Surface roughness impact?

Significant when roughness ≈ skin depth. K_r = 1+(2/π)arctan(1.4(R_q/δ)²). R_q=1μm at 10 GHz (δ=0.66μm): 80% more loss. Standard PCB copper: R_q=2-5 μm. HVLP copper: R_q=0.3-0.5 μm. At 28 GHz: HVLP reduces trace loss 30-50% vs standard. Critical for 5G mmWave PCB design.

Related Terms

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