The Skin Effect

Why the skin effect matters

At DC the current in a wire is spread evenly across the cross-section, and the resistance is simply R = ρL/A. The instant you switch to AC that comfortable picture breaks. The alternating current sets up an alternating magnetic field inside the metal, and by Faraday's and Lenz's laws that changing field induces circulating eddy currents that oppose the original flow most strongly at the axis. Current is expelled from the core and packed into a thin annulus at the surface. The higher the frequency, the thinner that annulus — and the less copper actually does any work.

• Power distribution. Even at 50/60 Hz a 30 mm-diameter conductor has a core that barely conducts; the AC resistance of a large solid busbar can be 30–60% above its DC value.
• Transformers and inductors. Switch-mode magnetics run at 20 kHz–2 MHz, where solid winding wire would waste most of its copper — the reason Litz wire exists.
• RF and microwave. At 1 GHz the current lives in the top ~2 µm of the metal, so surface finish, plating (silver, gold), and roughness dominate loss — hence silver-plated waveguides and gold-flashed connectors.
• Induction heating and wireless power. The effect is exploited on purpose: shallow δ concentrates heat in a workpiece's surface for case-hardening, or is fought in resonant coils with Litz.
• Grounding and lightning. Fast transient currents (µs rise times contain MHz content) travel on conductor surfaces, so wide flat straps beat round rods for bonding.

How it works, step by step

1. Current changes. AC of angular frequency ω = 2πf flows along the conductor axis (call it the z direction).
2. A magnetic field appears inside the metal. By Ampère's law the current encircles itself with a circumferential H-field whose magnitude grows toward the surface.
3. That field is changing. Because the current is AC, the internal B-field is time-varying.
4. Eddy currents are induced. Faraday's law says the changing B drives loops of induced EMF; Lenz's law makes those loops oppose the source current where it is strongest — at the center.
5. Current redistributes. The net effect is that current density J is largest at the surface and decays exponentially inward: J(x) = J_s · e^−x/δ · e^−jx/δ, where x is depth below the surface. The magnitude falls to 1/e ≈ 36.8% of its surface value at x = δ.
6. Resistance rises. With current squeezed into a shell of thickness ~δ, the effective area shrinks and R_AC exceeds R_DC.

The whole phenomenon falls out of Maxwell's equations. Inside a good conductor, displacement current is negligible and the field obeys the diffusion equation ∇²E = jωμσE. Its one-dimensional solution decays as e^−x/δ with

δ = √( 2 / (ω μ σ) ) = 1 / √( π f μ σ )

where each symbol has a precise meaning and unit:

Symbol	Quantity	Units	Notes
δ	Skin depth	m	Depth where |J| falls to 1/e of surface value
ω	Angular frequency	rad/s	ω = 2πf
f	Frequency	Hz
μ	Magnetic permeability	H/m	μ = μr·μ0, μ0 = 4π×10−7 H/m
σ	Electrical conductivity	S/m	Cu ≈ 5.8×107, Al ≈ 3.5×107
x	Depth below surface	m	J(x) = Js·e−x/δ

Two facts are worth burning in. First, δ scales as f^−1/2: raise the frequency 100× and the skin depth drops 10×. Second, δ scales as (μσ)^−1/2, so magnetic materials (large μ_r) have tiny skin depths — the very reason steel is a poor high-frequency conductor but a superb magnetic core, and why an ACSR line's steel core carries almost no current.

Skin depth across frequency and material

The table below tabulates δ = 1/√(πfμσ) for common conductors. Copper and aluminum are essentially non-magnetic (μ_r ≈ 1); iron is shown with an approximate low-field μ_r ≈ 1000 to illustrate how permeability collapses the skin depth.

Frequency	Copper (σ 5.8×107)	Aluminum (σ 3.5×107)	Iron (σ 1×107, μr≈1000)
50 Hz	9.3 mm	12 mm	≈ 0.71 mm
60 Hz	8.5 mm	11 mm	≈ 0.65 mm
1 kHz	2.1 mm	2.7 mm	≈ 0.16 mm
10 kHz	0.66 mm	0.85 mm	≈ 50 µm
100 kHz	0.21 mm	0.27 mm	≈ 16 µm
1 MHz	66 µm	85 µm	≈ 5 µm
100 MHz	6.6 µm	8.5 µm	≈ 0.5 µm
1 GHz	2.1 µm	2.7 µm	≈ 0.16 µm

The handy engineering shortcut for copper is δ_Cu ≈ 66/√f mm (f in Hz), or equivalently 2.1/√f mm with f in kHz. It reproduces the whole column above to within a percent.

Worked example: AC resistance of a solid round wire

Take a solid copper conductor of radius r = 5 mm carrying 100 kHz current. From the table δ = 0.21 mm, so r/δ ≈ 24 — the wire is far thicker than the skin depth. When r ≫ δ, the current behaves as if confined to a surface ring of thickness δ around a circumference 2πr, giving an effective area A_eff ≈ 2πr·δ instead of the full πr². The AC resistance is then

R_AC / R_DC ≈ r / (2δ)  (valid for r ≫ δ)

Plugging in: R_AC/R_DC ≈ 5 mm / (2 × 0.21 mm) ≈ 12. The same wire that shows about 0.022 Ω per 100 m at DC presents roughly 0.26 Ω per 100 m at 100 kHz — an order of magnitude worse — even though not one atom of copper has been removed. All that changed was the frequency. (A more exact treatment uses Bessel/Kelvin functions of the argument q = √2·r/δ; the r/(2δ) form is the high-frequency asymptote and is accurate to a few percent once r/δ ≳ 5.)

The design lesson: past r ≈ 2δ, making a solid conductor fatter buys you almost nothing at AC. That is why you subdivide the copper (Litz), hollow it out (tube), or spread it thin and wide (flat busbar) so that every part of the metal sits within a skin depth of a surface.

Countermeasure	Mechanism	Best frequency band	Trade-off
Litz wire	Many transposed strands, each ≤ δ, share surface exposure	~20 kHz – 3 MHz	Lower copper fill, higher cost, strand insulation overhead
Hollow / tubular conductor	Removes idle core copper; metal only where current flows	50 Hz – HV lines, RF tubing	Larger diameter for same R; mechanical support needed
Flat wide busbar / laminations	High perimeter-to-area ratio; foil thickness ≤ 2δ	Power to ~100 kHz	Bulkier layout, insulation between laminations
ACSR (steel-cored)	Aluminum surface carries current; steel carries tension	50/60 Hz overhead lines	Steel core is magnetic; near-zero current contribution
Surface plating (Ag, Au)	Higher-σ, non-oxidizing shell where the RF current lives	≥ 100 MHz	Thin plating must exceed a few δ; cost of noble metals

The proximity effect — the skin effect's partner in crime

The skin effect is driven by a conductor's own field. The proximity effect is driven by the fields of its neighbors. Place two current-carrying conductors side by side and each one's alternating field induces eddy currents in the other, crowding current toward one edge (opposing currents) or the far edges (parallel currents). In a tightly wound multilayer coil the proximity effect can dominate the skin effect entirely — the inner layers of a transformer winding see the summed field of every layer outside them, and their AC loss can be several times the skin-effect prediction. This is quantified by Dowell's method, which gives F_R = R_AC/R_DC as a function of the number of layers and the ratio of foil/wire thickness to δ. Litz wire attacks both effects at once: fine strands beat the skin effect, and full transposition averages out the proximity field so each strand sees the same net exposure.

Common misconceptions and failure modes

• "The current stops at the skin depth." No — δ is just the 1/e depth. Current keeps flowing (and reversing phase) deeper; about 63% of the current lives within one δ of the surface, ~86% within 2δ.
• "Bigger wire always means lower AC resistance." Once r ≳ 2δ, added core copper is nearly dead weight at AC. Fatter can even be worse per kilogram.
• "Skin effect only matters at RF." It already inflates R_AC of large power conductors at 60 Hz by tens of percent; utilities design around it.
• "Litz wire helps at any frequency." Above a few MHz the strand insulation and reduced fill outweigh the benefit; solid or tubular silver-plated conductors win.
• "Skin depth ignores the material's magnetism." μ_r is inside the square root. A steel wire's skin depth is ~13× smaller than copper's at the same frequency — great for shielding, terrible for conduction.
• "Proximity effect and skin effect are the same thing." They share physics (induced eddy currents) but have different sources — self-field versus neighbor-field — and in dense windings the proximity term is usually the larger loss.