Skin Depth Calculator
Enter frequency, relative permeability and conductivity to compute skin depth δ=√(2/(ω·μ·σ)). f=60 Hz, Cu (μr=1, σ=5.96e7) → δ≈8.5 mm; f=1 MHz → δ≈66 μm.
Input Data
Results
At a glance:Skin depth (skin depth) δ is the depth at which the amplitude of an electromagnetic wave (or alternating current) inside a good conductor decays to 1/e (≈37%) of its surface value: δ=√(2/(ω·μ·σ)), where δ is skin depth (m), ω angular frequency (rad/s, ω=2πf), μ the medium permeability (H/m, μ=μ₀·μr, μ₀=4π×10⁻⁷ H/m, μr relative permeability) and σ the conductivity (S/m). Widely written δ=√(2/(2πf·μ₀·μr·σ)). Physical meaning: an AC current in a conductor is not uniform across the section — the changing magnetic field induces eddy currents that cancel the interior field, confining the current to a thin outer layer, the 'skin effect'. The skin depth is that layer's thickness; below it the current is negligible. Skin depth drops as √(1/f), so higher frequency crowds the current into a thinner layer, shrinking the effective conducting area, raising AC resistance and increasing loss/heating. Example: copper μr=1, σ=5.96e7 S/m at f=60 Hz → ω=377, δ=√(2/(377×4πe-7×5.96e7))=√(2/2.825e4)=√(7.08e-5)=8.41e-3 m=8.41 mm; at f=1 MHz δ=√(60/1e6×...)≈66 μm. History: Kelvin predicted it in 1887; Heaviside formalised the theory. Classic uses: (1) transformer and inductor laminations (stack thin iron sheets to shorten the flux path and cut eddy loss); (2) busbar skin-effect compensation; (3) RF shielding and enclosures; (4) eddy-current non-destructive testing; (5) induction heating; (6) waveguide inner-wall design. Notes: (1) for ferromagnetic materials μr is large and frequency-dependent — take the operative value; (2) the formula assumes a good conductor (σ≫ωε); (3) high frequency makes δ very small, so thick conductors waste material (use hollow/tubular); (4) keep SI units.
Formula
Skin depth: δ = √(2 / (ω·μ·σ)) = √(2 / (2πf·μ₀·μr·σ))
ω = 2πf (rad/s), μ = μ₀·μr, μ₀=4π×10⁻⁷ H/m
Skin effect: AC current crowds into a layer of thickness δ
δ ∝ √(1/f): higher frequency → thinner skin
$$\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \sqrt{\frac{2}{2\pi f\,\mu_0\mu_r\,\sigma}}, \quad \omega = 2\pi f, \quad \mu = \mu_0\mu_r$$How to Use
- Enter frequency f (Hz).
- Enter relative permeability μr (Cu/Al 1) and conductivity σ (Cu 5.96e7).
- The tool gives δ=√(2/(2πf·μ₀·μr·σ)) in m, mm and μm, plus ω and μ₀.
Case Studies
Power-frequency copper busbar
HK mains 50 Hz, copper μr=1, σ=5.96e7 → δ≈9.2 mm.
A busbar thicker than ~2δ carries almost all current in the outer ~18 mm — the inner core is wasted.
At 1 MHz δ≈66 μm, so a solid 1 cm bar conducts only in a thin skin — use hollow or stranded (Litz) wire.
Transformer lamination and induction heating
Transformer cores use 0.1–0.35 mm silicon steel; at 50 Hz iron δ≈10 mm, so laminations far thinner than δ suppress eddy currents.
Induction heating uses 10–100 kHz, where δ drops to <1 mm, concentrating heat in the surface layer.
RF shielding: a few-skin-depth copper enclosure blocks most field (1 MHz δ≈66 μm, 1 mm wall ≈15δ).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.