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Skin Depth Calculator

Calculate skin depth and surface resistance for copper, aluminum, and other conductors at any frequency. Essential for RF shielding and PCB design. Free, instant results.

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Formula

δ=2ωμσ=1πfμ0μrσ\delta = \sqrt{\frac{2}{\omega \mu \sigma}} = \sqrt{\frac{1}{\pi f \mu_0 \mu_r \sigma}}

Reference: Griffiths, "Introduction to Electrodynamics" 4th ed., Chapter 9

δ— Skin depth (m)
ω— Angular frequency (2πf) (rad/s)
μ— Magnetic permeability (μ₀·μᵣ) (H/m)
σ— Electrical conductivity (S/m)

How It Works

Skin depth calculator computes AC current penetration depth for any conductor material and frequency — RF circuit designers, EMC engineers, and PCB layout specialists use this to optimize trace thickness, shielding effectiveness, and high-frequency conductor performance. The skin depth delta = sqrt(2*rho/(omega*mu)) = sqrt(rho/(pi*f*mu)) represents the depth at which current density falls to 1/e (37%) of its surface value, per Jackson's 'Classical Electrodynamics' (3rd ed.) and IEEE Standard 1597.1.

For copper at room temperature (rho = 1.68e-8 ohm-m), skin depth follows delta_Cu = 66/sqrt(f_MHz) micrometers. At 1 MHz, delta = 66 um; at 100 MHz, delta = 6.6 um; at 1 GHz, delta = 2.1 um; at 10 GHz, delta = 0.66 um. This explains why PCB traces behave differently at RF: a 35 um (1 oz) copper trace carries current through its full thickness at 1 MHz but only the outer 2 um at 1 GHz — effectively reducing conductor cross-section by 15x.

Surface roughness becomes critical when comparable to skin depth: Ra = 1 um roughness causes 10-15% resistance increase at 1 GHz (delta = 2.1 um) per Hammerstad's model. Premium RF laminates specify Ra < 0.5 um (rolled annealed copper) versus standard ED copper at Ra = 2-3 um. Silver plating (rho = 1.59e-8) provides 3% improvement; gold plating (rho = 2.44e-8) is 20% worse than copper but prevents oxidation critical for connector contacts.

Worked Example

Problem: Design PCB trace for 5.8 GHz WiFi with minimum RF loss, comparing standard 1 oz copper versus ENIG finish.

Skin depth analysis:

  1. Calculate skin depth at 5.8 GHz:
delta_Cu = 66/sqrt(5800) = 0.87 um = 870 nm

  1. Standard 1 oz copper (35 um thick):
- Thickness/delta = 35/0.87 = 40 skin depths — RF current uses only outer ~3*delta = 2.6 um - Effective resistance increase versus DC: R_AC/R_DC = t/(2*delta) = 35/(2*0.87) = 20x - Surface roughness (ED copper, Ra = 2 um): roughness/delta = 2.3 — significant! - Roughness penalty per Hammerstad: 1 + (2/pi)arctan(1.4(Ra/delta)^2) = 1.67 (67% increase)
  1. ENIG finish (0.1 um Au over 5 um Ni):
- Gold skin depth at 5.8 GHz: delta_Au = 66*sqrt(2.44/1.68)/sqrt(5800) = 1.05 um - 0.1 um gold layer < delta_Au — current penetrates to nickel underlayer - Nickel resistivity: 6.99e-8 ohm-m (4.2x copper) - delta_Ni = 66*sqrt(4.2)/sqrt(5800) = 1.78 um - Current flows primarily in nickel: additional loss approximately 4x versus pure copper
  1. Recommendation:
- Use immersion silver (rho = 1.59e-8) or OSP with low-roughness copper (Ra < 0.5 um) - Immersion silver: delta_Ag = 0.84 um, 3% better than copper - Total loss reduction versus ENIG: approximately 4 dB/m at 5.8 GHz
  1. Trace width for 50 ohms on 0.2 mm FR4 (er = 4.3): W = 0.38 mm
- Loss with low-Ra copper: 0.15 dB/cm at 5.8 GHz - Loss with ENIG: 0.35 dB/cm — unacceptable for > 5 cm traces

Practical Tips

  • ✓For RF PCBs above 1 GHz, specify rolled annealed (RA) copper with Ra < 1 um surface roughness — standard electrodeposited (ED) copper roughness dominates loss above 3 GHz
  • ✓Conductor thickness beyond 3 skin depths provides negligible improvement — 35 um copper is adequate at 1 GHz (delta = 2.1 um), but 70 um (2 oz) may be needed at 100 MHz (delta = 6.6 um) for low loss
  • ✓For magnetic shielding, skin depth in steel or mu-metal is much smaller due to high permeability — at 60 Hz, delta_steel approximately equals 0.5 mm versus 8.5 mm for copper; thin steel provides effective low-frequency shielding

Common Mistakes

  • ✗Ignoring skin effect in high-frequency power calculations — DC resistance is meaningless above 1 MHz; a 10 AWG wire with 3.3 mohm/m DC resistance shows 33 mohm/m at 100 MHz due to skin effect
  • ✗Assuming linear current distribution instead of exponential decay — current density at depth d is J(d) = J_surface * exp(-d/delta); 63% of current flows in the first skin depth, 86% in two skin depths, 95% in three
  • ✗Overlooking surface roughness at microwave frequencies — standard PCB copper (Ra = 2 um) causes 50-100% resistance increase above 5 GHz; specify low-profile copper (Ra < 0.5 um) for RF traces
  • ✗Using gold plating on RF conductors — gold's higher resistivity (1.45x copper) increases loss; gold is for corrosion protection on contacts, not for RF current conduction

Frequently Asked Questions

Skin depth is inversely proportional to sqrt(frequency): delta = sqrt(rho/(pi*f*mu)). Doubling frequency reduces skin depth by factor of sqrt(2) = 1.41. For copper: 1 MHz = 66 um, 10 MHz = 21 um, 100 MHz = 6.6 um, 1 GHz = 2.1 um, 10 GHz = 0.66 um. This sqrt(f) dependence means skin effect transitions gradually — there's no sharp cutoff where it 'turns on'. Skin effect becomes significant when delta approaches conductor thickness: for 35 um PCB copper, this occurs around 4 MHz.
No — skin depth depends on both resistivity and magnetic permeability: delta = sqrt(rho/(pi*f*mu)). Non-magnetic conductors (copper, aluminum, silver, gold) have mu = mu_0 and differ only by resistivity: delta_Ag/delta_Cu = sqrt(1.59/1.68) = 0.97 (silver 3% better). Magnetic conductors (iron, nickel, steel) have mu >> mu_0, dramatically reducing skin depth: at 1 MHz, delta_Cu = 66 um but delta_steel approximately 0.5 um for mu_r = 100. This makes steel effective for magnetic shielding but poor for RF conductors.
Key applications: (1) PCB loss: trace resistance at RF is t/(2*delta) times DC resistance; dominates insertion loss above 1 GHz. (2) Shielding: enclosure skin depth determines SE at low frequencies; 1 mm copper provides 30 dB at 100 Hz but only where skin depth (660 um) fits twice. (3) Cable design: Litz wire (many thin insulated strands) reduces skin effect below 500 kHz. (4) Connector plating: thin silver/gold over copper provides corrosion protection without excessive loss since RF current stays in outer 3*delta. (5) Waveguide: interior surface plating (50 um silver) captures essentially all current at microwave frequencies.
Partially: (1) Litz wire uses many thin strands (each < 2*delta at operating frequency) individually insulated, then twisted — reduces AC/DC resistance ratio from 10x to < 2x at 100 kHz-1 MHz; ineffective above 2 MHz. (2) Higher-conductivity plating: silver provides 3% improvement over copper. (3) Surface smoothing: reducing Ra roughness from 2 um to 0.3 um saves 50% resistance at 10 GHz. (4) Wider conductors: doubling width halves resistance without fighting skin effect. (5) Hollow conductors: tube with wall thickness 3-5*delta has same RF resistance as solid conductor with far less weight — used in high-power antenna elements.
Skin effect becomes significant when skin depth is comparable to conductor dimension. For 10 AWG wire (2.6 mm diameter): at 60 Hz, delta_Cu = 8.5 mm >> 2.6 mm — negligible skin effect. At 10 kHz, delta = 0.66 mm — moderate effect, R_AC/R_DC approximately 1.5. At 100 kHz, delta = 0.21 mm — substantial effect, R_AC/R_DC approximately 5. Power transformers (50-60 Hz) use solid conductors; switching power supplies (100 kHz+) require Litz wire or foil conductors. Audio frequencies (20 Hz-20 kHz) have negligible skin effect in typical wire gauges but measurable effect in large power cables.

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