Skin Depth Calculator

What the skin depth calculator does

At high frequencies, alternating current flows only near the surface of a conductor, and the skin depth measures how far in it penetrates. This calculator finds the skin depth from the material's resistivity and permeability and the frequency, with presets for common metals.

Below is what skin depth is, the equation behind it, why it shrinks as frequency rises, and a worked example.

How to use it

1. Pick a material preset, or enter the resistivity and relative permeability yourself.
2. Enter the frequency of the alternating current.
3. Press Calculate for the skin depth, or Reset to clear it.

What skin depth and the skin effect are

When direct current flows through a wire, it spreads evenly across the whole cross-section. But when alternating current flows, especially at high frequency, something different happens: the current crowds toward the outer surface of the conductor and avoids the centre. This is called the skin effect, because the current flows mainly in a thin skin near the surface. The skin depth is the measure of how thick that skin is, the depth below the surface at which the current has fallen to a certain fraction of its surface value.

The skin effect happens because the changing current creates changing magnetic fields inside the conductor, and these induce swirling currents that oppose the flow in the interior, pushing the current outward. The higher the frequency, the more pronounced this becomes, and the thinner the conducting skin. At low frequencies the skin depth is large and the effect is negligible, but at radio and microwave frequencies it can confine the current to an extremely thin surface layer. This calculator computes the skin depth for a given material and frequency, which is essential for understanding high-frequency conductors.

The equation it uses

The skin depth is given by:

δ = √(2ρ ÷ (ωμ))

which can be written as δ = √(ρ ÷ (π f μ)). Here δ is the skin depth, ρ is the resistivity of the material, f is the frequency, ω is the angular frequency (two pi times the frequency), and μ is the magnetic permeability of the material, the product of the permeability of free space and the material's relative permeability. The skin depth grows with the square root of the resistivity and shrinks with the square root of the frequency and the permeability. The calculator evaluates this, taking the material properties from a preset or from your own entries.

Why it shrinks as frequency rises

The defining behaviour of skin depth is that it decreases as frequency increases, falling with the square root of the frequency. At low frequencies the current penetrates deeply, filling most of a typical wire, but as the frequency climbs the current is squeezed into an ever-thinner surface layer. Raise the frequency by a factor of a hundred and the skin depth shrinks tenfold. By the time you reach the gigahertz frequencies of modern wireless, the skin depth in a good conductor is only a couple of micrometres, far thinner than a human hair.

The material properties also play a role. A higher resistivity gives a larger skin depth, so current penetrates further into a poorer conductor, while a higher magnetic permeability shrinks it, which is why magnetic metals like nickel and iron have very shallow skin depths. The calculator's presets capture these differences between metals such as copper, aluminium, gold, and silver. Understanding that skin depth falls with frequency, and depends on the material, is the key to predicting how current will distribute itself in a conductor at any given frequency, which the calculator makes concrete.

What the skin effect means in practice

The skin effect has real consequences for anything carrying high-frequency current. Because the current uses only a thin surface layer, the effective cross-section of the conductor is reduced, which raises its resistance at high frequency well above its direct-current value. A wire that carries current easily at low frequency can become surprisingly resistive at radio frequencies, simply because most of its metal is going unused. This extra resistance causes losses that matter in radio, microwave, and high-speed electronics.

Engineers respond to the skin effect in several ways. Since the interior of a conductor carries little high-frequency current, solid wires are sometimes replaced with hollow tubes or with bundles of many fine insulated strands, called litz wire, to increase the useful surface area. Conductors are sometimes plated with a thin layer of a very good conductor like silver, since only the surface matters. The skin effect also governs electromagnetic shielding, where a thin conducting layer can block high-frequency fields. Knowing the skin depth tells designers how thick a conductor or shield needs to be and how much resistance to expect, which is why this calculation is a staple of high-frequency design.

Units and precision

The calculator takes the resistivity in ohm-metres or related units, the relative permeability as a number, and the frequency in units from hertz up to gigahertz, returning the skin depth in metres and its smaller multiples. It offers presets for common metals with their resistivity and permeability filled in, or you can enter custom values. It uses the standard value for the permeability of free space. The relationship is exact for a good conductor, the regime in which the skin-depth formula applies.

A worked example

Suppose you want the skin depth in copper at 1 gigahertz, a typical wireless frequency.

Using copper's resistivity and a relative permeability of about one, the skin depth comes out to δ ≈ 2.1 micrometres. So at 1 gigahertz, the current in a copper conductor flows almost entirely within about two micrometres of the surface. The same copper at a much lower 1 megahertz would have a skin depth of around 66 micrometres, deeper but still thin, showing how the penetration shrinks as frequency climbs.

Questions people ask

How do you calculate skin depth?

Use δ = √(2ρ ÷ (ωμ)), from the resistivity, the angular frequency, and the permeability. It can also be written δ = √(ρ ÷ (πfμ)).

What is the skin effect?

The tendency of alternating current to crowd toward the surface of a conductor at high frequency, avoiding the centre, so the current flows in a thin surface skin.

Why does skin depth fall with frequency?

Because higher-frequency currents induce stronger opposing currents in the conductor's interior, pushing the flow outward. The depth falls with the square root of frequency.

Why does the skin effect matter?

It raises a conductor's resistance at high frequency, since only a thin layer carries current. This drives the use of hollow conductors, litz wire, and surface plating.

References

A quick note on where the physics comes from. The skin effect and the skin-depth formula are standard electromagnetism, set out in Georgia State University's HyperPhysics and in standard references. The permeability of free space follows NIST. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. HyperPhysics, Skin Effect and Skin Depth. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/skindepth.html
2. National Institute of Standards and Technology (NIST), Fundamental Physical Constants, vacuum magnetic permeability. https://physics.nist.gov/cgi-bin/cuu/Value?mu0
3. Wikipedia, Skin effect. https://en.wikipedia.org/wiki/Skin_effect