Skin Effect Formulas: Skin Depth

Skin effect formulas: skin depth from resistivity and frequency for copper and aluminium, with a worked example at 50 Hz and a table by frequency.

The formulas

δ is the depth in metres at which the current density has fallen to about 37% of its value at the surface. ρ is the resistivity in Ω·m, f the frequency in hertz and μ0 = 4π × 10⁻⁷ H/m. The relative permeability of copper and aluminium is 1. Steel has a high permeability and a much smaller skin depth. The resistivities are 1.72 × 10⁻⁸ Ω·m for copper and 2.82 × 10⁻⁸ Ω·m for aluminium at 20 °C.

Worked example: copper at 50 Hz

1. ω = 2π × 50 = 314.2 rad/s
2. δ = √(2 × 1.72 × 10⁻⁸ / (314.2 × 4π × 10⁻⁷)) = 9.335 mm
3. For aluminium at 50 Hz, δ = 11.95 mm, and for copper at 60 Hz, δ = 8.521 mm

Skin depth by frequency

Frequently asked questions

What does the skin effect do to a conductor?

It pushes the AC current toward the surface. The inner part carries less current, so the AC resistance is higher than the DC resistance for a thick conductor.

When does it matter at power frequency?

For conductors with a radius comparable to the skin depth, which means a cross-section of several hundred mm² for copper at 50 Hz. Large cables use stranded or segmented conductors to reduce the effect.

Why is the skin depth smaller at high frequency?

The changing magnetic field drives eddy currents that oppose the current in the core of the conductor. The effect increases with frequency.

Does the skin effect change the voltage drop?

Slightly, for large conductors. Cable data sheets give the AC resistance, which includes the skin and proximity effects, so use that value for voltage drop.

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