Skin Effect and High-Frequency Conductor Loss

At DC, current in a round conductor distributes uniformly across its cross-section. As frequency rises, eddy currents induced by the conductor's own changing magnetic field push current toward the surface, shrinking the effective conducting area and raising AC resistance well above the DC value. This single effect — skin effect — sets the copper loss budget in everything from 60 Hz power buses at high current to multi-GHz PCB traces, and it interacts directly with proximity effect in windings and with dielectric loss in transmission lines. Ignoring it leads to underestimated I²R loss, wrong thermal margins, and connector/trace designs that look fine in a DC ohmmeter check but run hot or attenuate badly at operating frequency.

Physical Origin

A time-varying current in a conductor produces a time-varying magnetic field, both outside and inside the conductor. By Faraday's law, that changing internal field induces eddy currents that oppose the change in current density at the center and reinforce it near the surface. The net effect: current density decays exponentially from the surface inward.

For a conductor with current flowing in the z-direction, current density as a function of depth x below the surface is:

J(x) = J₀ · e^(−x/δ)

where J₀ is the surface current density and δ is the skin depth — the depth at which current density falls to 1/e (≈37%) of its surface value.

Skin Depth Formula

δ = √(2ρ / (ω·μ)) = √(ρ / (π·f·μ))

• ρ — resistivity of the conductor (Ω·m)
• f — frequency (Hz), ω = 2πf
• μ — absolute permeability of the conductor (μ = μ₀·μᵣ), H/m

For copper (ρ ≈ 1.68×10⁻⁸ Ω·m, μᵣ ≈ 1, μ₀ = 4π×10⁻⁷ H/m), this simplifies to the commonly quoted approximation:

δ(copper) ≈ 66.1 / √f   [mm, f in Hz]

Two things to note: δ ∝ 1/√f, so effective conducting depth shrinks slowly (square-root law, not linear) as frequency rises; and δ ∝ 1/√μᵣ, so ferromagnetic conductors (steel, nickel-plated copper) have far shallower skin depth than nonmagnetic copper or aluminum at the same frequency.

AC Resistance and the Effective Area Approximation

Once δ is much smaller than the conductor's physical dimensions, a useful approximation treats the current as flowing uniformly through an annular "skin" of thickness δ at the surface, rather than computing the true exponential profile. For a round wire of radius a with δ ≪ a:

A_eff ≈ 2π·a·δ        (annulus area, thin-wall approximation)
R_AC ≈ ρ·l / A_eff = ρ·l / (2π·a·δ)

Compare to DC resistance R_DC = ρ·l / (π·a²). The ratio is:

R_AC / R_DC ≈ a / (2δ)     (valid when δ ≪ a)

This ratio grows linearly with frequency once skin effect dominates (since δ ∝ 1/√f, and a/(2δ) ∝ √f), meaning AC resistance itself scales as √f in the strong skin-effect regime — not linearly with f.

Worked Example: AWG18 Wire at 1 MHz

AWG18 copper wire: diameter = 1.024 mm → radius a = 0.512 mm. Resistivity ρ = 1.68×10⁻⁸ Ω·m.

Step 1 — DC resistance per meter:

R_DC = ρ / (π·a²) = 1.68×10⁻⁸ / (π × (0.512×10⁻³)²)
     = 1.68×10⁻⁸ / (8.24×10⁻⁷)
     ≈ 0.0204 Ω/m

Step 2 — skin depth at 1 MHz:

δ ≈ 66.1×10⁻³ / √(1×10⁶) mm = 66.1×10⁻³ / 1000 mm = 66.1 µm

Step 3 — check δ ≪ a: δ = 66.1 µm, a = 512 µm → δ/a ≈ 0.13, comfortably in the thin-skin regime (ratio < ~0.2 is a reasonable rule of thumb for the annulus approximation to hold well).

Step 4 — effective area and R_AC:

A_eff ≈ 2π·a·δ = 2π × 0.512×10⁻³ × 66.1×10⁻⁶
      ≈ 2.126×10⁻⁷ m²
R_AC = ρ / A_eff = 1.68×10⁻⁸ / 2.126×10⁻⁷ ≈ 0.0790 Ω/m

Step 5 — ratio check:

R_AC / R_DC = 0.0790 / 0.0204 ≈ 3.87
a / (2δ) = 0.512×10⁻³ / (2 × 66.1×10⁻⁶) ≈ 3.87   ✓ matches

So at 1 MHz, this wire's AC resistance is nearly 4× its DC value — a wire that runs cool carrying 1 A DC will dissipate ~4× the I²R loss carrying the same RMS current at 1 MHz.

Related High-Frequency Loss Mechanisms

• Proximity effect: adjacent current-carrying conductors (e.g., transformer/inductor windings, twisted pairs) distort each other's current distribution, concentrating current further and adding loss beyond isolated-conductor skin effect. This is why litz wire — many thin, individually insulated, twisted strands — is used instead of one thick conductor: each strand stays thinner than δ, keeping current spread uniformly within it.
• Surface roughness: PCB copper trace roughness (especially electrodeposited foil with a rough treated side for adhesion) forces current to follow a longer, bumpier path once δ becomes comparable to the roughness profile, adding loss beyond the smooth-surface skin-effect prediction — a significant correction above ~1 GHz.
• Dielectric loss: in PCB striplines/microstrips at multi-GHz frequencies, dielectric loss (tan δ of the substrate) grows linearly with frequency and eventually dominates over conductor skin-effect loss, which grows only as √f.

Design Implications

• Stranding and litz wire: for windings operating above tens of kHz, use strand diameter smaller than δ at the operating frequency, litz-transposed to also mitigate proximity effect.
• PCB trace width vs. thickness: at high frequency, adding copper thickness gives diminishing returns once thickness exceeds a few δ; increasing trace width (and using ground-referenced striplines) is often more effective for reducing loss.
• Connector and via plating: current in vias and connector pins at RF concentrates in the outer plating — plating material and thickness matter more than bulk conductor cross-section.
• Silver or gold plating: silver (ρ ≈ 1.59×10⁻⁸ Ω·m) plated onto copper conductors reduces surface resistance slightly at frequencies where the plating thickness exceeds δ, commonly used in RF connectors and waveguides.
• Impedance/loss modeling in SPICE: simple lumped R models miss skin effect entirely; AC analysis with frequency-dependent R (or a distributed transmission-line model) is required to predict loss correctly above the frequency where δ approaches conductor thickness.

Key Takeaways

• Skin effect confines AC current to a surface layer of depth δ = √(ρ/(π·f·μ)), which shrinks as 1/√f.
• In the strong skin-effect regime (δ ≪ a), R_AC/R_DC ≈ a/(2δ), so AC resistance grows as √f, not linearly with frequency.
• Copper skin depth is ~66 µm at 1 MHz and only ~2 µm at 1 GHz — conductor thickness beyond a few skin depths contributes negligible additional current-carrying benefit.
• Proximity effect compounds skin-effect loss in windings and closely spaced conductors; litz wire and careful trace/via layout mitigate both.
• Above several GHz, surface roughness and dielectric loss (tan δ) often outweigh pure skin-effect conductor loss and must be modeled separately.