The skin depth calculator above computes the depth at which the current density in a conductor has fallen to about 37 per cent of its surface value, which is the standard measure of how far an alternating field penetrates a metal. It then uses that depth to estimate the AC resistance of a round wire and compare it with the direct-current figure, because the practical consequence of the skin effect is almost always a resistance that is higher than the handbook value.
Arb Digital publishes free engineering calculators that separate the physical quantity from the engineering approximation built on it. Skin depth is exact for the model it comes from. The AC resistance figure is an approximation whose accuracy depends on how the conductor radius compares with that depth, and this page states which regime you are in rather than quietly presenting one number as though both were equally solid.
What This Skin Depth Calculator Does
The hero figure is the skin depth in the most readable unit for its size, from nanometres up to metres. It depends only on frequency, conductivity and permeability — not on the shape or the size of the conductor. A copper sheet and a copper wire at the same frequency have the same skin depth; what differs is what that depth means for their resistance.
The grid puts the consequence next to the cause. The DC resistance is the familiar length over conductivity times area. The AC resistance uses the effective conducting annulus that the skin effect leaves available. Their ratio is the penalty factor you are paying, and the final box reports the radius expressed in skin depths, which is the single number that tells you whether the approximation being used is trustworthy.
How to Use It
- Enter the frequency of the current, not of anything else. For a switching converter the relevant frequency is the switching frequency and its significant harmonics, and the harmonics matter more than the fundamental for a square wave.
- Use the conductivity at operating temperature. Metals become less conductive as they warm, so a conductor running hot has a slightly larger skin depth and a considerably higher resistance than a cold one.
- Set relative permeability honestly. Leave it at one for the common non-magnetic conductors. For steel it is large, variable and dependent on the field strength, so any answer is indicative rather than precise.
- Enter the wire radius for the resistance figures. If your conductor is not round, the skin depth is still valid but the resistance comparison is not, because the effective area depends on the perimeter.
- Read the radius-in-skin-depths figure before trusting the ratio. Below about two, the current has not really been pushed to the surface and the annulus approximation overstates the penalty.
The Formula: How Skin Depth Is Calculated
For a good conductor, where the conduction current dominates the displacement current, the skin depth is δ = 1 ÷ √(π f μ σ), with frequency f in hertz, permeability μ = μr × 4π × 10−7 H/m, and conductivity σ in siemens per metre. The same expression is often written as √(2 ÷ ωμσ), which is identical once ω = 2πf is substituted. LibreTexts' treatment of skin depth in Ellingson's Electromagnetics II derives it and gives aluminium at 10 MHz as a worked example.
The current density does not stop at that depth. It decays exponentially, so the skin depth is the distance over which it falls by a factor of e. Roughly 63 per cent of the current flows within one skin depth of the surface and about 95 per cent within three. That exponential tail is why a shield several skin depths thick is effective even though nothing is blocked outright.
For resistance, this page uses the effective annulus model: the current is treated as flowing uniformly through a shell of thickness δ at the surface, giving an area of π(a² − (a − δ)²) for a wire of radius a, and the full circular area whenever δ exceeds a. LibreTexts' section on the impedance of a wire sets out the same substitution and shows the characteristic square-root-of-frequency growth in resistance that follows from it.
Work the defaults through by hand. At 10 MHz with σ = 3.7 × 107 S/m and μr = 1, the product πfμσ is π × 107 × 1.2566 × 10−6 × 3.7 × 107 = 1.4607 × 109. Its square root is 38,219, so δ = 26.17 µm, which matches the published aluminium example. For a 1 mm radius wire one metre long, the DC resistance is 1 ÷ (3.7 × 107 × π × 10−6) = 8.603 mΩ. The annulus area is π(10−6 − (9.7384 × 10−4)²) = 1.6225 × 10−7 m², giving an AC resistance of 166.6 mΩ and a ratio of 19.4. The radius is 38.2 skin depths, so the thin-skin approximation is well inside its valid range here.
Why Thicker Wire Stops Helping
At direct current, doubling the radius quarters the resistance because the area grows with the square. Once the skin depth is much smaller than the radius, that stops being true. The conducting shell has a fixed thickness set by frequency and material, so its area grows only in proportion to the circumference, which means doubling the radius merely halves the resistance.
The engineering consequence is that beyond a certain size, adding copper to a single round conductor is close to wasted metal at high frequency. The centre carries almost nothing. This is why high-frequency conductors are made as tubes, foils, wide flat straps or bundles of individually insulated strands rather than as thick rods, and why a hollow copper pipe can be an entirely sensible RF conductor.
It also changes how you should think about current rating. A conductor sized for a DC current will run hotter than expected on the same RMS current at high frequency, because the losses scale with the higher AC resistance. If you are sizing conductors by mass or by cross-section, the wire weight calculator gives the material side of that trade, and the conductivity and resistivity calculator converts between the two material figures this page needs.
Where the Simple Model Breaks Down
Three limits are worth knowing. The first is the good-conductor assumption. The standard formula drops the displacement current term, which is entirely safe for metals at any frequency you are likely to work at, but not for seawater, soil, semiconductors or biological tissue, where the full lossy-medium expression is needed and the simple square-root scaling fails.
The second is the annulus approximation for resistance. When the radius is comparable to or smaller than the skin depth, the current has not been displaced to the surface at all and the exact solution involves Bessel functions rather than a shell. The tool detects that case and says so instead of quietly reporting a ratio it cannot support. Below about two skin depths of radius, treat the AC and DC figures as effectively equal.
The third is proximity effect, which this page does not model at all. A nearby conductor carrying current distorts the field and pushes current towards or away from it, and in a tightly wound multi-layer inductor the proximity loss frequently exceeds the skin loss by a wide margin. That is why a transformer winding built from wire chosen purely on skin depth can still run hot. For the inductive side of that problem, the solenoid inductance calculator handles the geometry and the inductance converter the units.
Skin Depth as a Shielding Rule
Because the field decays exponentially, shielding effectiveness is naturally measured in skin depths rather than in millimetres. Each skin depth of thickness attenuates the field by a factor of e, which is about 8.7 dB. Three skin depths gives around 26 dB from absorption alone, and five gives about 43 dB, before any reflection at the surface is counted.
This is why enclosures that look flimsy work well at high frequency and why the same enclosure is useless at low frequency. A thin aluminium box is many skin depths thick at radio frequencies and a small fraction of one at mains frequency, which is exactly why magnetic shielding at 50 or 60 hertz needs high-permeability alloys rather than more aluminium. High permeability shrinks the skin depth directly, which is the mechanism those alloys exploit.
The same reasoning explains why apertures dominate real shield performance. Once the wall is several skin depths thick, the metal is no longer the leak; the seams, the ventilation holes and the cable penetrations are. Increasing the thickness beyond a handful of skin depths buys almost nothing.
Choosing Strand Diameter for Litz Wire
Litz wire divides a conductor into many individually insulated strands, transposed so that each strand spends equal time at every position in the bundle. The design rule that follows from this page is straightforward: each strand should be no more than about two skin depths across at the highest frequency of interest, so that no strand is itself operating in the crowded regime.
There is a cost. Insulation and the space between round strands mean a litz bundle carries less copper than a solid conductor of the same outside diameter, so the DC resistance is higher. Below the frequency where the skin effect bites, litz wire is strictly worse than solid wire. The crossover point is exactly what this calculator locates: run your operating frequency and your candidate strand radius, and if the AC-to-DC ratio is close to one, you are paying for stranding you do not need.
For non-sinusoidal currents the relevant frequency is not the fundamental. A square-edged current waveform in a switching converter carries significant energy in harmonics well above the switching frequency, and those harmonics see a much smaller skin depth. Sizing strands against the fundamental alone is a common and expensive mistake in converter magnetics.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using a room-temperature conductivity for a hot conductor — conductivity falls appreciably as metals warm, so both the depth and the resistance move in the direction that makes the design worse.
- Applying the good-conductor formula to a poor conductor — for soil, seawater or tissue the displacement current is not negligible and the simple square-root-of-frequency scaling does not hold.
- Sizing a strand against the fundamental of a square wave — the harmonics carry real current at frequencies where the skin depth is several times smaller, and they set the loss.
- Assuming skin effect alone explains a hot winding — proximity effect in a multi-layer coil is frequently the larger term and is not modelled here.
- Treating a permeability figure for steel as a constant — it varies with field strength, frequency and heat treatment, so a ferrous result is an order of magnitude rather than a number.
Related Free Tools From Arb Digital
The PCB impedance calculator handles the controlled-impedance geometry where these losses show up on a board, and the cable impedance calculator does the same for coaxial and twisted-pair runs. The dipole antenna calculator is the usual next step when the conductor in question is radiating rather than merely carrying, and the Ohm's law calculator turns any of these resistances into a loss figure. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the distance into a conductor at which the alternating current density has fallen to about 37 per cent of its value at the surface, which is a factor of one over e. It is not a hard boundary: the current decays exponentially, so roughly 63 per cent of the total flows within one skin depth and about 95 per cent within three.
No. It depends only on frequency, conductivity and magnetic permeability. A thin foil and a thick bar of the same metal have identical skin depths at the same frequency. What the size changes is the consequence: whether that depth represents most of the conductor or a thin shell at its surface.
Because the current is confined to a shell whose thickness shrinks as the square root of frequency, so the effective cross-section available to carry it shrinks too. Once the skin depth is much smaller than the radius, the resistance grows roughly with the square root of frequency, which is why a conductor that is comfortable at mains frequency can dissipate heavily at radio frequencies.
Because beyond a certain diameter the centre of a solid conductor carries almost no current, so extra metal adds weight and cost without reducing resistance. Splitting the conductor into many insulated strands, each no more than about two skin depths across and transposed through the bundle, keeps the whole cross-section working. Below the frequency where crowding starts, litz wire is worse than solid wire because insulation and packing reduce the copper present.
It applies the same formula, but the result should be read as an order of magnitude. The relative permeability of ferrous materials is large and varies with field strength, frequency and heat treatment, so there is no single value to enter. The general conclusion holds: high permeability makes the skin depth much smaller, which is why a steel conductor confines current far closer to its surface than copper does.
Shielding is measured in skin depths rather than in millimetres, because each skin depth of thickness attenuates the field by a factor of e, or about 8.7 decibels. Three skin depths give roughly 26 decibels of absorption and five give about 43. Beyond a handful of skin depths, seams, apertures and cable penetrations dominate the leakage rather than the metal itself.
It tells you whether the resistance comparison can be trusted. When the radius is many skin depths, the current genuinely occupies a thin shell and the annulus model is sound. When it is around one or below, the current still fills the conductor, the exact solution needs Bessel functions rather than a shell, and the AC and DC resistances are effectively the same.
This tool is provided for educational and preliminary engineering use. The skin depth expression assumes a good conductor with constant permeability, and the resistance figures use an effective-annulus approximation that ignores proximity effect and conductor shape. Verify magnetics and current-carrying design against measurement and the relevant standards before building.
