At any frequency, this skin depth calculator gives the depth in copper, silver, gold, aluminum or nickel. It also works back from a depth to the frequency that gives it. Enter a PCB trace to see how much the skin effect raises its resistance above the DC value.
Skin depth δ
66.08 µm
at 1 MHz
Assumes a conductor much thicker than the skin depth, with a smooth surface and resistivity at 20 °C. Thinner conductors do not follow this depth once δ passes half the thickness. The percentages integrate the magnitude of the current density and ignore its phase shift with depth.
Skin depth of copper (annealed) from 50 Hz to 10 GHz
| Frequency | Skin depth | In mil |
|---|---|---|
| 50 Hz | 9.35 mm | 368 |
| 60 Hz | 8.53 mm | 336 |
| 1 kHz | 2.09 mm | 82.3 |
| 10 kHz | 661 µm | 26 |
| 100 kHz | 209 µm | 8.23 |
| 1 MHz | 66.1 µm | 2.6 |
| 10 MHz | 20.9 µm | 0.823 |
| 100 MHz | 6.61 µm | 0.26 |
| 1 GHz | 2.09 µm | 0.0823 |
| 10 GHz | 661 nm | 0.026 |
Skin depth is δ = √(ρ ÷ (π f µ)). For annealed copper that is 8.53 mm at 60 Hz, 66.1 µm at 1 MHz, 6.61 µm at 100 MHz and 2.09 µm at 1 GHz. It falls with the square root of frequency, and the current density at three skin depths is 5.0 % of its surface value.
Alternating current induces eddy currents that oppose it inside the conductor, so the current density falls off exponentially with depth. Skin depth is the distance over which it falls to 1/e.
The surface resistance is the resistance per square of the skin layer. For copper Johnson quotes it as 2.61 × 10⁻⁷ × √f Ω per square:
For a rectangular trace of width w and thickness t, the skin-limited resistance per unit length spreads Rs around the perimeter. It cannot be below the DC value, so the skin effect calculator uses the larger of the two:
Johnson applies a factor of 1.69 (1 + 0.33 + 0.36) to the skin-limited figure for a typical 75 Ω microstrip: 0.33 for current pulled to the side facing the plane, 0.36 for loss in the plane itself.
For 0.006 in by 0.0015 in copper at 1 GHz, δ = 2.09 µm and Rs = 8.25 mΩ per square. Rac = Rs ÷ (2 (w + t)) = 550 mΩ/in, or 7.29 times the DC resistance of 75.4 mΩ/in. With the 1.69 microstrip factor it is 930 mΩ/in, the 0.93 Ω/in Johnson reports.
δ = 93.5 µm, against a copper thickness of 70 µm. The skin-limited formula meets the DC resistance at 4.63 MHz, so at 500 kHz Rac ÷ Rdc is 1: the current still fills the whole cross-section. Switching harmonics well above the crossover see a higher resistance.
f = ρ ÷ (π µ0 δ²). With δ = 35 µm that is 3.57 MHz. Below this frequency the foil is thinner than one skin depth, so current fills it and thicker copper lowers the resistance.
Bulk metal at 20 °C. Resistivity is listed in the first row. The skin depth chart in the calculator above plots depth against frequency for the metal you choose.
| Frequency | Copper (annealed) | Silver | Gold | Aluminum |
|---|---|---|---|---|
| ρ, µΩ·cm | 1.724 | 1.587 | 2.214 | 2.653 |
| 50 Hz | 9.35 mm | 8.97 mm | 10.6 mm | 11.6 mm |
| 60 Hz | 8.53 mm | 8.19 mm | 9.67 mm | 10.6 mm |
| 1 kHz | 2.09 mm | 2 mm | 2.37 mm | 2.59 mm |
| 10 kHz | 661 µm | 634 µm | 749 µm | 820 µm |
| 100 kHz | 209 µm | 200 µm | 237 µm | 259 µm |
| 1 MHz | 66.1 µm | 63.4 µm | 74.9 µm | 82 µm |
| 10 MHz | 20.9 µm | 20 µm | 23.7 µm | 25.9 µm |
| 100 MHz | 6.61 µm | 6.34 µm | 7.49 µm | 8.2 µm |
| 1 GHz | 2.09 µm | 2 µm | 2.37 µm | 2.59 µm |
| 10 GHz | 661 nm | 634 nm | 749 nm | 820 nm |
Skin depth of nickel (ρ = 6.93 µΩ·cm). µr = 1 is the non-magnetic limit, and 3 to 9 is the expected range at microwave frequencies quoted by Pranonsatit et al. (2011). These are not measurements of any one plating: they show how strongly the answer depends on µr. The 600 that HFSS and many textbooks list is a low-frequency figure.
| Frequency | µr = 1 | µr = 3 | µr = 9 | Gold (µr = 1) |
|---|---|---|---|---|
| 1 GHz | 4.19 µm | 2.42 µm | 1.4 µm | 2.37 µm |
| 2.4 GHz | 2.7 µm | 1.56 µm | 901 nm | 1.53 µm |
| 5.8 GHz | 1.74 µm | 1 µm | 580 nm | 983 nm |
| 10 GHz | 1.32 µm | 765 nm | 442 nm | 749 nm |
Selected initial permeability values from the survey by Lucyszyn (2008), with the skin depth each value implies. Hodsman’s 1948 figures are for 99.5 % nickel wire, with a probable error of the order of 15 %. None of these is a measurement of an ENIG layer, and Lucyszyn concludes that there is not enough data to design with confidence at microwave frequencies.
| Frequency | Measured µr | Measured by | Skin depth | At µr = 1 |
|---|---|---|---|---|
| 0.41 GHz | 20 | Arkadiew, 1919 | 1.46 µm | 6.54 µm |
| 3.356 GHz | 8.3 | Hodsman et al., 1948 | 794 nm | 2.29 µm |
| 5.062 GHz | 4.1 | Hodsman et al., 1948 | 920 nm | 1.86 µm |
| 6.522 GHz | 3 | Hodsman et al., 1948 | 947 nm | 1.64 µm |
| 8.772 GHz | 1.5 | Hodsman et al., 1948 | 1.16 µm | 1.41 µm |
| 10.084 GHz | 1 | Hodsman et al., 1948 | 1.32 µm | 1.32 µm |
| 12.3 GHz | 1.2 | Arkadiew, 1919 | 1.09 µm | 1.19 µm |
The depth formula assumes a thick, smooth, uniform conductor. Once the skin depth passes half the thickness the conductor no longer follows it: current fills the whole section. The perimeter model for a trace is a straight line on either side of the crossover, while the real curve bends smoothly, so treat values near the crossover as estimates. It also assumes even current around the rectangle, which a nearby return plane spoils, and it leaves out foil roughness, which adds loss once the skin depth is similar to the height of the copper teeth. Resistivity here is at 20 °C. Copper rises by 0.393 % per kelvin, so the skin depth grows by about 0.2 % per kelvin. Plated and electrodeposited metal often has a higher resistivity than the bulk values in the table. Copper above is the annealed standard value of 1.7241 µΩ·cm; 99.999 % pure copper is 1.678 µΩ·cm in Matula’s data.
How far alternating current penetrates a conductor: the depth at which the current density has fallen to 1/e (36.8 %) of its surface value. Current crowds toward the surface as frequency rises, so skin depth falls with the square root of frequency.
The skin depth equation is δ = √(ρ ÷ (π f µ0 µr)), with ρ in Ω·m, f in Hz and µ0 = 4π × 10⁻⁷ H/m.
8.53 mm at 60 Hz, 66.1 µm at 1 MHz and 2.09 µm at 1 GHz, for annealed copper at 20 °C. It falls with the square root of frequency, so 100 times the frequency is one tenth the depth.
Integrating the magnitude of the current density in a thick conductor puts 63.2 % within one skin depth, 95.0 % within three and 99.3 % within five, ignoring its phase shift with depth. Metal much deeper than about five skin depths carries almost nothing, which is why RF conductors can be plated or hollow.
Skin depth equals the 35 µm foil thickness at 3.57 MHz and half of it at 14.3 MHz. Current entering both faces still fills the foil until the depth nears half the thickness, so a 0.2 mm trace starts to climb around 19.7 MHz, the crossover the calculator reports, and rises with √f above that. A nearby return plane pulls current to one face and moves the rise lower in frequency.
Yes at high frequency. A 6 mil, 1.5 mil thick trace is 7.29 times its DC resistance at 1 GHz on the perimeter model, and Johnson multiplies that by 1.69 for the proximity effect and return-plane loss of a typical 75 Ω microstrip.
Rs = ρ ÷ δ gives the surface resistance per square. The perimeter model on this page spreads it around a rectangular trace: Rac = max(Rdc, Rs ÷ (2 (w + t))) per unit length, so the result cannot be below the DC value, ρ ÷ (w t).
Nickel has about 4 times the resistivity of copper and is ferromagnetic. Its permeability shrinks the skin depth, both properties raise the surface resistance, and at high frequency the current runs in the outer layers of the conductor, which on an exposed ENIG pad is the finish. Published microwave permeabilities run from about 1 to 20, and one group fitted µr = 8 to its own electroplated nickel. None of those is an ENIG layer, so the calculator asks for it.
Among non-magnetic metals, the lowest resistivity gives the smallest depth: silver is 63.4 µm at 1 MHz, against 66.1 µm for copper and 82 µm for aluminum. A ferromagnetic conductor can be much smaller again.
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