Visual calculator 01 · Conductors
Skin effect
AC current crowds toward the surface of a conductor. Move the frequency and watch where it flows.
Material
AC resistance over DC resistance
Exact Bessel solution against the thin-skin approximations.
Skin depth
δ = √(2ρ / ωµ). Where δ drops below the radius, the skin takes over.
Check
Method
δ = √(2ρ / (ω µ0 µr))
k = (1 − j) / δ
J(r) = J(0) · J0(k r) (exact, round wire)
Z/Rdc = (k a / 2) · J0(ka) / J1(ka)
Rac/Rdc = Re{Z/Rdc}
thin skin: Rac/Rdc ≈ a/(2δ) + 1/4
flat: J(z) = J(0) · e^(−z/δ) · e^(−j z/δ)
J0 and J1 of complex argument: power series for |ka| ≤ 25, Hankel asymptotic expansion above (scaled, so very large a/δ does not overflow).
Reference values · q = √2·a/δ
Columns: this page's Bessel solution; an independent Kelvin-function (ber/bei) series, Rac/Rdc = (q/2)(ber·bei′ − bei·ber′)/(ber′² + bei′²); classic tabulated values; and the asymptote a/(2δ) + 1/4 + 3δ/(32a) for large q.
- S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., Wiley, 1994, Sec. 5.5–5.6 (internal impedance of a round wire).
- M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, NBS, 1964, §9.2 (asymptotic expansions) and §9.9 (Kelvin functions).
- F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, 1943, Sec. 2 (skin-effect resistance ratio table).
- Resistivities at 20 °C: annealed copper standard (IEC 60028, 1.7241 µΩ·cm); Ag 1.59, Au 2.21, Al 2.65 µΩ·cm from the CRC Handbook of Chemistry and Physics (293 K). Steel is illustrative: its µr depends strongly on field level and frequency.