Skin Depth Calculator
Enter the conductor material (or supply your own resistivity and relative permeability) and the AC signal frequency to find the electromagnetic skin depth - the depth at which current density falls to about 37 % of its surface value. Results update as you type, with a full step-by-step derivation, a frequency-sweep chart, and a comparison across common metals.
What is skin depth?
When alternating current flows through a conductor, the electromagnetic field it generates opposes current flow in the interior, pushing most of the current toward the surface. This is the skin effect. Skin depth (usually written as the Greek letter delta) is the depth below the surface at which the current density has fallen to approximately 37 % (precisely 1/e) of its value at the surface. The result is that AC resistance is higher than DC resistance, and the difference grows with frequency. Power engineers encounter it at 50/60 Hz where it governs wire gauge for large busbars; RF engineers encounter it at megahertz and gigahertz frequencies where it determines how thin a plating layer must be to carry the signal current without significant loss.
The skin depth formula
The standard formula is delta = sqrt(rho / (pi * f * mu_0 * mu_r)), where rho is the electrical resistivity of the conductor in ohm-metres, f is the AC frequency in hertz, mu_0 is the permeability of free space (4*pi x 10^-7 H/m), and mu_r is the relative magnetic permeability of the material. An equivalent form is delta = sqrt(2*rho / (omega * mu)), where omega = 2*pi*f is the angular frequency and mu = mu_0 * mu_r is the absolute permeability. For copper at 60 Hz: delta = sqrt(1.68e-8 / (pi * 60 * 4*pi*1e-7 * 1)) = approx 8.5 mm. At 1 GHz the same formula gives about 2.1 micrometres, illustrating how dramatically skin depth shrinks with frequency. Ferromagnetic metals like iron and nickel have high relative permeability (hundreds to tens of thousands), which cuts their skin depth far below that of copper even though their resistivity is higher.
Practical design rules using skin depth
A conductor thicker than about 5 skin depths carries more than 99 % of the current, so adding extra thickness beyond that wastes material without reducing AC resistance. For PCB traces at GHz frequencies the copper skin depth is under 3 micrometres, which means that copper surface roughness, typically 0.5 to 3 um on standard FR4, is comparable to or larger than the skin depth and can dominate insertion loss. At audio and RF frequencies (1 kHz to 1 MHz) litz wire bundles multiple thin insulated strands, each much thinner than the skin depth, to minimize AC resistance. Coaxial cable shields and hollow tubular conductors exploit the same principle: because the interior carries almost no current, removing it saves weight without increasing loss. Magnetic shielding often uses high-permeability materials like mu-metal or silicon steel, and their very small skin depths at power frequencies are precisely why a thin sheet provides effective attenuation.
Material choices and how they affect skin depth
Silver has the lowest resistivity of any element (1.59 x 10^-8 ohm-m) and therefore the largest skin depth at a given frequency, making it the material of choice for low-loss RF components. Copper is nearly as good and far cheaper, which is why it dominates RF plating and microwave waveguides. Gold does not corrode, so it is used in thin coatings over copper or nickel to protect contact surfaces at high frequencies. Aluminum has a larger skin depth than copper, so aluminum busbars need slightly more cross-section to match copper AC resistance, but weight savings often more than compensate. Iron and nickel have high permeabilities (mu_r of 5000 and 600 respectively for soft grades), which makes their skin depths far smaller than their moderate resistivities would suggest - a thin iron shield at 50 Hz has a skin depth of under 0.3 mm, far less than the 8.5 mm skin depth of copper at the same frequency.
Skin depth in common conductors at selected frequencies
| Material | 60 Hz | 1 kHz | 1 MHz | 1 GHz |
|---|---|---|---|---|
| Silver | 8.27 mm | 2.02 mm | 63.9 um | 2.02 um |
| Copper | 8.50 mm | 2.08 mm | 65.7 um | 2.08 um |
| Gold | 10.2 mm | 2.50 mm | 79.1 um | 2.50 um |
| Aluminum | 10.7 mm | 2.61 mm | 82.6 um | 2.61 um |
| Tungsten | 15.5 mm | 3.79 mm | 120 um | 3.79 um |
| Nickel | 0.353 mm | 86.1 um | 2.72 um | 86.1 nm |
| Iron (soft) | 0.239 mm | 58.5 um | 1.85 um | 58.5 nm |
| Stainless Steel 316 | 2.91 mm | 712 um | 22.5 um | 712 nm |
Calculated at room temperature using the standard skin depth formula delta = sqrt(rho / (pi * f * mu_0 * mu_r)).
Frequently asked questions
Why does skin depth decrease at higher frequencies?
Higher frequency means faster-changing electromagnetic fields, which induce stronger opposing eddy currents inside the conductor. These opposing currents cancel the primary current more aggressively in the interior, compressing it into an ever-thinner surface layer. Mathematically, skin depth is proportional to 1/sqrt(f), so quadrupling the frequency halves the skin depth.
Does skin effect apply to DC current?
No. DC current does not alternate in direction, so there is no time-varying magnetic field to push current toward the surface. DC distributes uniformly across the cross-section of a conductor (subject to the conductor being uniform). Skin effect only appears with alternating or pulsed current - even at the low frequency of 50/60 Hz mains power it is significant enough to influence busbar and cable sizing for large conductors.
How do I reduce skin effect losses in a cable or PCB trace?
The main approaches are: (1) use a conductor material with lower resistivity or lower permeability so the skin depth is larger; (2) keep individual conductor dimensions at or below the skin depth - litz wire does this by bundling many thin strands; (3) for tubular conductors, remove the interior where little current flows; (4) for PCB traces at microwave frequencies, use smoother copper foil (low-profile or reverse-treated) to keep surface roughness well below the skin depth.
Why does iron have a much smaller skin depth than copper even though iron is more resistive?
Skin depth depends on both resistivity and permeability: delta = sqrt(rho / (pi * f * mu_0 * mu_r)). Iron has a relative permeability of roughly 5000, which increases the denominator enormously and more than compensates for its higher resistivity. At 60 Hz, iron has a skin depth of about 0.24 mm compared to 8.5 mm for copper. This is why iron and steel cores in transformers and motors are laminated into thin sheets separated by insulation - each lamination must be thinner than the skin depth to suppress eddy current losses.
What is the "5 skin depths" rule of thumb?
Current density falls exponentially with depth: J(d) = J0 * e^(-d/delta). At one skin depth it is 37 %, at two it is 14 %, at three it is 5 %, at four it is 2 %, and at five skin depths it is less than 1 %. So a conductor or plating layer thicker than 5 skin depths carries over 99 % of the available current. Anything thicker adds negligible electrical benefit while increasing weight and cost. This rule is widely used to size copper plating on RF connectors and waveguides.
How does skin depth affect PCB copper plating at microwave frequencies?
At 10 GHz, copper has a skin depth of about 0.65 um. Standard electrodeposited copper on PCB substrates can have surface roughness of 1 to 3 um, which is larger than the skin depth. Because current concentrates in the outermost layer, it must follow the rough contours of the copper surface, traveling a longer effective path and increasing resistive loss. This is why high-speed PCB designs above 10 GHz specify "ultra-low profile" or "reversed-treated" copper foil with roughness below 0.5 um.