Skin Depth & Skin Effect: Why AC Current Crowds the Surface

What skin effect is

Skin effect is the tendency of alternating current to concentrate near a conductor's surface as frequency rises. The mechanism is self-inflicted: the AC creates a changing magnetic field inside the conductor, that changing field induces eddy currents, and those eddy currents oppose the flow in the centre while reinforcing it near the surface. The higher the frequency, the more the current is pushed outward — so less of the copper actually carries current, and the effective AC resistance rises above the DC value.

Skin depth: the one number that quantifies it

Skin depth δ is the depth at which the current density has fallen to about 37% (1/e) of its value at the surface — a convenient single measure of how far the current penetrates. It's given by:

δ = √( ρ / (π · f · µ₀ · µr) )

where ρ is the material's resistivity, f the frequency, µ₀ = 4π×10⁻⁷ H/m the permeability of free space, and µr the material's relative permeability. Higher frequency or higher permeability → less penetration; higher resistivity → more.

The key insight: it depends on material and frequency, not size

Notice what's not in that formula: the conductor's diameter or shape. Skin depth is a property of the material and frequency alone. A 1 mm wire and a 100 mm busbar of the same copper at the same frequency have identical skin depth. What changes with size is how much that skin depth matters: when δ is comparable to or larger than the conductor, skin effect is negligible; when δ is much smaller than the conductor, most of the cross-section is wasted and the AC/DC resistance ratio climbs.

Magnetic materials are a trap

Non-magnetic conductors — copper, aluminium, silver, gold, brass, austenitic stainless — have µr = 1, and skin depth is well-behaved. Ferromagnetic materials (steel, nickel) have µr ≫ 1, which sharply reduces skin depth. But their µr is not a fixed constant: it depends on field strength, saturates well below the DC values often quoted, and is itself frequency-dependent. So a skin-depth number for a magnetic material should be treated as illustrative only, not a precise design value.