Skin Depth Calculator
Calculate AC current skin depth δ = √(2/μσω) in conductors.
Penetration depth for copper, aluminum, silver, gold, and steel from 50 Hz to GHz.
At DC, current flows uniformly through the cross-section of a wire. At AC, the current crowds toward the outer surface. The depth at which the current density drops to 1/e (about 37%) of its surface value is called the skin depth. Above a few MHz, this is significant enough that engineers stop thinking of conductors as solid wires and start thinking of them as hollow tubes.
The formula:
δ = √(2 / (μ · σ · ω))
Where δ is skin depth (m), μ is the magnetic permeability of the conductor (H/m), σ is its electrical conductivity (S/m), and ω = 2π·f is the angular frequency (rad/s). The penetration depth shrinks as the square root of frequency, so higher frequencies penetrate less.
A useful equivalent form for non-magnetic conductors:
δ = 1 / √(π · f · μ₀ · σ)
where μ₀ = 4π × 10⁻⁷ H/m is the permeability of free space.
Where the skin effect comes from:
AC current creates a time-varying magnetic field inside the conductor. By Faraday’s law, this field induces an EMF that opposes the current it came from. The opposition is stronger toward the center of the conductor than at the surface, so current is pushed outward. The decay is exponential: current density at depth x below the surface is
J(x) = J₀ · e^(-x/δ)
So at x = δ, current is 37% of the surface value. At x = 3δ, it’s 5%. At x = 5δ, essentially zero. For RF applications, conductors thicker than about 5δ are wasting copper; the inner mass carries almost no current.
Worked example, copper at 60 Hz:
Copper: σ = 5.96 × 10⁷ S/m, μ_r = 1 (non-magnetic), so μ = μ₀.
δ = 1 / √(π · 60 · 4π × 10⁻⁷ · 5.96 × 10⁷) = 1 / √(π × 60 × 4π × 10⁻⁷ × 5.96 × 10⁷) = 1 / √(1.412 × 10⁴) = 1 / 118.8 = 8.4 mm
So at line frequency, copper conductors thicker than about 1.7 cm (2δ) start to show measurable skin effect. For typical household wiring this is irrelevant; for power-grid bus bars and transformer windings it matters and is one reason transformer copper is split into many thin parallel strands (“Litz wire”).
Worked example, copper at 1 GHz:
Same σ. f = 10⁹ Hz.
δ = 1 / √(π × 10⁹ × 4π × 10⁻⁷ × 5.96 × 10⁷) = 1 / √(2.353 × 10¹¹) ≈ 2.1 μm
At microwave frequencies, current flows in a layer barely a few micrometers deep. A solid copper wire is overkill; what matters is the surface finish. PCB traces at GHz frequencies are typically electroplated with a thin layer of higher-conductivity material; surface roughness becomes a measurable loss mechanism.
Reference values (skin depth at common frequencies):
| Material | σ (S/m) | μ_r | δ at 60 Hz | δ at 100 kHz | δ at 1 GHz |
|---|---|---|---|---|---|
| Silver | 6.30 × 10⁷ | 1 | 8.2 mm | 0.20 mm | 2.0 μm |
| Copper | 5.96 × 10⁷ | 1 | 8.4 mm | 0.21 mm | 2.1 μm |
| Gold | 4.10 × 10⁷ | 1 | 10.2 mm | 0.25 mm | 2.5 μm |
| Aluminum | 3.50 × 10⁷ | 1 | 11.0 mm | 0.27 mm | 2.7 μm |
| Iron | 1.00 × 10⁷ | 1000 | 0.7 mm | 0.017 mm | 0.17 μm |
| Steel (carbon) | 0.7 × 10⁷ | 200 | 1.7 mm | 0.04 mm | 0.4 μm |
Notice steel and iron — magnetic permeability multiplies the effective shielding. A 1 mm steel sheet at 100 kHz attenuates an EM wave by far more than a 1 mm copper sheet would, even though copper has higher σ. This is why magnetic shielding uses ferrous materials.
RF design implications:
- AC resistance: at frequencies where δ « conductor radius, AC resistance is roughly R_DC · (a / 2δ), where a is the wire radius. A 1 mm copper wire at 1 MHz has δ ≈ 65 μm, so AC resistance is about 8× the DC value. This is why RF power amplifiers run hotter than their DC current would predict.
- Litz wire: thin enameled strands woven so each spends equal time near and far from the surface. Cancels the skin effect for frequencies in the audio and low-RF range (up to ~1 MHz). Used in switching power supply transformers and inductors.
- Hollow waveguides and tubular conductors: at GHz frequencies, you literally can use a copper-plated steel tube; the steel doesn’t carry RF current, only mechanical load.
- EMC shielding: a shielding enclosure needs to be at least 5δ thick at the lowest frequency of concern. A 1 mm copper enclosure is overkill at 1 GHz (δ ~2 μm) but borderline at 60 Hz (δ ~8 mm).
Frequency dependence:
δ ∝ 1 / √f. To halve the skin depth, quadruple the frequency. This is why the transition from “current fills the wire” to “current barely penetrates” happens over a relatively narrow frequency band for a given conductor thickness — usually a factor of 100× in frequency takes you from full penetration to negligible penetration.
When the formula breaks down:
- Anomalous skin effect: at very high frequencies and low temperatures (cryogenic), the electron mean free path becomes comparable to δ, and the bulk conductivity σ no longer applies. The current actually penetrates deeper than the classical formula predicts.
- Semiconductors: σ is much lower (10² to 10⁴ S/m), so skin depth is large at all reasonable frequencies. Silicon at 1 GHz has δ on the order of 100 μm-1 mm.
- Superconductors: σ is effectively infinite at DC; the relevant length scale is the London penetration depth (a quantum mechanical quantity, not the classical skin depth).