What is skin depth and how does it change with frequency?

Skin depth (delta) is the distance from the conductor surface at which the current density drops to 1/e (37%) of its surface value. It is given by delta = sqrt(2 rho / (omega mu)), where rho is resistivity, omega = 2 pi f, and mu is permeability. For copper at room temperature: delta = 66.1 / sqrt(f_Hz) micrometers. At 1 MHz: 66 micrometers. At 1 GHz: 2.1 micrometers. At 10 GHz: 0.66 micrometers. At 77 GHz (automotive radar): 0.24 micrometers. The current decays exponentially: at 3 skin depths, only 5% of the surface current remains. Effectively, all current flows in a shell about 3 to 5 skin depths thick.

Why does surface roughness matter at microwave frequencies?

Standard PCB copper foil has an RMS surface roughness of 1 to 6 micrometers (depending on the foil type: ED, RTF, VLP, or HVLP). When the skin depth approaches the roughness height, the current must follow the tortuous surface contour instead of a straight path, increasing the effective path length and resistance. At 10 GHz, the skin depth in copper is 0.66 micrometers. Standard ED foil with 3 micrometer RMS roughness forces the current to traverse a surface 40 to 80 percent longer than a smooth surface, increasing conductor loss by the same factor. This is why mmWave PCB designs specify HVLP (very low profile) foil with roughness below 0.5 micrometers.

Does skin effect apply to gold-plated and silver-plated conductors?

Yes, but the plating thickness matters relative to the skin depth. Gold plating on connectors is typically 0.5 to 1.5 micrometers thick. At 1 GHz (skin depth 2.1 um for copper, 2.5 um for gold), the RF current penetrates through the gold into the underlying nickel barrier layer, which has much higher resistivity (7 times copper). This is why gold-plated connectors have higher loss than expected: the nickel barrier dominates the RF resistance. Silver plating is better because silver has the lowest resistivity of any metal (1.6 micro-ohm-cm vs. copper's 1.7) and is usually plated thick enough (5+ micrometers) to contain the current entirely.

Electromagnetics

Skin Effect

At DC, current flows uniformly through the entire cross-section of a copper wire. At 10 GHz, the same wire might as well be a hollow tube: 99% of the current flows in the outer 2 micrometers, a shell thinner than a single red blood cell. The interior copper contributes nothing. This is the skin effect, caused by the magnetic field of the AC current inducing eddy currents that oppose the flow in the conductor's interior. The practical consequence is that conductor resistance, and therefore loss, increases as the square root of frequency. It is the fundamental reason why RF cables, PCB traces, and waveguides have frequency-dependent attenuation.

Category: Electromagnetics

Copper at 1 GHz: δ = 2.1 μm

Loss Dependence: ∝ √f

When the Interior of the Wire Becomes Useless

Skin depth:
δ = √(2ρ / (ωμ)) = √(ρ / (πfμ))

For copper (ρ = 1.68×10⁻⁸ Ω·m, μ = μ₀):
δ = 66.1 / √(f_Hz) μm

Surface resistance:
R_s = ρ/δ = √(πfμρ) Ω/square

R_s for copper: 2.6×10⁻⁷ × √f Ω/square
At 10 GHz: R_s = 0.026 Ω/square

Skin Depth by Material and Frequency

Material	ρ (μΩ·cm)	δ at 100 MHz	δ at 1 GHz	δ at 10 GHz	δ at 77 GHz
Silver	1.59	6.3 μm	2.0 μm	0.63 μm	0.23 μm
Copper	1.68	6.6 μm	2.1 μm	0.66 μm	0.24 μm
Gold	2.44	7.9 μm	2.5 μm	0.79 μm	0.28 μm
Aluminum	2.65	8.2 μm	2.6 μm	0.82 μm	0.30 μm
Nickel (connector barrier)	6.84	13.2 μm	4.2 μm	1.32 μm	0.48 μm

Proximity Effect: Skin Effect's Companion

When two conductors carry current in opposite directions (like the signal and ground traces of a microstrip line), the magnetic field from each conductor forces the current in the other conductor toward the facing surfaces. This proximity effect concentrates the current distribution beyond what skin effect alone predicts. In a microstrip line at 10 GHz, the current crowds into the bottom surface of the signal trace (facing the ground plane) and the top surface of the ground plane (facing the signal trace). The effective current-carrying area is smaller than the skin depth alone would suggest, increasing conductor loss by 10 to 30% compared to isolated-conductor skin-effect calculations. Electromagnetic simulators that include both skin and proximity effects are essential for accurate loss prediction above 5 GHz.

Common Questions

Frequently Asked Questions

How does skin depth change with frequency?

δ = 66.1/√f μm for copper. At 1 MHz: 66 μm. At 1 GHz: 2.1 μm. At 77 GHz: 0.24 μm. Current decays exponentially; at 3δ, only 5% remains. Effectively all current is in a shell 3 to 5 skin depths thick.

Why does surface roughness matter?

Standard PCB copper has 1 to 6 μm RMS roughness. When skin depth approaches roughness height (above ~5 GHz), current follows the tortuous surface, increasing path length 40 to 80%. mmWave designs require HVLP foil (<0.5 μm roughness).

Gold or silver plating?

Gold plating (0.5 to 1.5 μm) over nickel is thinner than skin depth above 1 GHz, so RF current penetrates into the lossy nickel barrier. Silver has the lowest resistivity and is plated thicker (5+ μm), keeping current in the low-loss layer. Silver tarnishes but the oxide is conductive.

Related Terms

← Signal Integrity Slotline →

Trace Design

Skin Depth & Trace Loss Calculator

Enter frequency, conductor material, and trace geometry to compute skin depth, surface resistance, and transmission line conductor loss per unit length.

Calculate Skin Depth