The skin effect
Direct current spreads evenly across a wire. Alternating current does not: the changing magnetic field inside the conductor pushes the current toward the surface. As frequency rises, current flows in an ever thinner outer layer. The skin depth is the depth at which the current density has fallen to 1/e, about 37%, of its value at the surface.
The formula
δ = √(ρ ÷ (π × f × μ0 × μr))
where ρ is the resistivity, f the frequency, μ0 = 4π × 10⁻⁷ H/m and μr the relative permeability (1 for copper, aluminum, silver, gold and brass).
Typical values for copper
- 50 Hz: about 9.2 mm
- 60 Hz: about 8.4 mm
- 10 kHz: about 0.65 mm
- 1 MHz: about 65 µm
- 100 MHz: about 6.5 µm
- 10 GHz: about 0.65 µm
Skin depth falls with the square root of frequency: 100 times the frequency gives one tenth the depth.
AC resistance of a wire
When the wire diameter d is much larger than δ, the current effectively flows in a ring of thickness δ, and the AC resistance rises roughly as:
R ac ÷ R dc ≈ d ÷ (4δ) + 1/4
For a 1 mm copper wire at 1 MHz: 1 ÷ (4 × 0.065) + 0.25 ≈ 4.1. The wire has four times its DC resistance. When d is less than about two skin depths, the effect is small.
Practical consequences
- Power lines: at 50 or 60 Hz, conductors thicker than about 2 cm gain little; utilities use stranded aluminum around a steel core.
- Switch-mode power supplies: transformer windings at 100 kHz use Litz wire, many thin insulated strands, or copper foil.
- RF: silver plating helps because only the surface carries current. Hollow tubes work as well as solid rods.
- Shielding: a shield several skin depths thick blocks high-frequency fields.
Magnetic materials such as steel have μr of hundreds or thousands, so their skin depth is far smaller. The calculator accepts μr for that case, but permeability varies with field strength.