Cut open a thick copper bus bar carrying a radio-frequency signal and you would find, if you could measure it, that the core of the metal is doing almost nothing. The current has migrated outward, packing itself into a thin shell near the surface. The centre is still copper, still a perfectly good conductor — yet at high frequency it carries a vanishing share of the load.
This is the skin effect, and it is one of those phenomena that feels counterintuitive until you see the mechanism. It is also expensive: it makes a wire's resistance climb with frequency, drives engineers to use hollow tubes and braided Litz wire, and sets hard limits on the efficiency of RF coils, power-line conductors, and motor windings. This article explains where the effect comes from, how to estimate the skin depth, and how to read its consequences.
Why this calculation matters
At DC, current spreads itself evenly across a conductor's cross-section, and resistance is simply length over the product of conductivity and area. That comfortable picture breaks the moment the current alternates. As frequency rises, the usable cross-section shrinks, the effective resistance rises, and a conductor that looked generously sized on paper starts to run hot.
The consequences reach across electrical engineering. Power utilities accept the skin effect when they choose stranded aluminium conductors with a steel core — the steel carries mechanical load, the aluminium shell carries current. RF designers obsess over it because a coil's quality factor depends directly on conductor losses. Switching power supplies, induction heaters, and high-speed digital traces all live with it. If you size a conductor or predict a loss budget without accounting for skin depth, you will be optimistic by a margin that grows with every decade of frequency.
The core formula
The skin effect arises from electromagnetic induction acting on the conductor itself. An alternating current sets up an alternating magnetic field. That changing field induces eddy currents inside the metal, and those eddy currents oppose the original current most strongly at the centre and least strongly near the surface. The net result is that current is pushed outward.
The characteristic length scale of this crowding is the skin depth, written delta. It is the depth at which the current density has fallen to about 37 percent (1/e) of its surface value:
delta = 1 / sqrt(pi * f * mu * sigma)
Here f is the frequency in hertz, mu is the magnetic permeability of the conductor in henries per metre, and sigma is the electrical conductivity in siemens per metre. For a non-magnetic metal such as copper, mu equals the permeability of free space, mu_0 = 4 * pi * 1e-7 H/m.
Two features deserve attention. First, delta scales as one over the square root of frequency: raise the frequency by a factor of 100 and the skin depth drops by a factor of 10. Second, delta shrinks for high-conductivity and high-permeability materials. That last point surprises people — a better conductor confines current to a thinner shell, so iron, despite being conductive, makes a poor RF conductor because its high permeability collapses the skin depth dramatically.
Once you know delta, you can estimate the AC resistance of a round wire of radius R. When delta is much smaller than R, the current behaves as if it flowed uniformly through a shell of thickness delta, so the AC-to-DC resistance ratio is roughly R divided by twice delta:
R_ac / R_dc ~= R / (2 * delta) (valid when delta << R)
A worked example
Take a copper conductor carrying a 1 MHz signal. Copper has a conductivity of sigma = 5.8e7 S/m, and being non-magnetic its permeability is mu = 4 * pi * 1e-7 H/m = 1.2566e-6 H/m.
Step 1 — assemble the term under the square root.
pi * f * mu * sigma = pi * 1e6 * 1.2566e-6 * 5.8e7
= 2.29e8
Step 2 — take the square root and invert.
sqrt(2.29e8) = 15130
delta = 1 / 15130 = 6.6e-5 m
So the skin depth is about 66 micrometres — roughly the diameter of a human hair.
Step 3 — read the physical meaning. At 1 MHz, the current crowds into the outer 66 micrometres of the conductor. If the wire is, say, 1 mm in radius, the copper at the centre carries almost no current at all. The conductor is electrically equivalent to a thin copper tube, and the metal you paid for in the core is dead weight.
This is exactly why high-frequency coils are often wound from hollow copper tubing or from Litz wire — many fine, individually insulated strands. If each strand is thinner than a skin depth, the whole bundle carries current efficiently across its full cross-section instead of wasting the interior.
Common mistakes
Forgetting the square-root, not linear, dependence. Skin depth does not halve when you double the frequency — it falls by a factor of about 1.41. Many quick estimates go wrong because the engineer assumed a linear trend.
Using free-space permeability for magnetic conductors. The formula needs the conductor's own permeability. For copper and aluminium, mu_0 is correct. For steel, the relative permeability can be in the hundreds or thousands, which shrinks delta by one to two orders of magnitude. A steel wire is a far worse RF conductor than its conductivity alone suggests.
Applying the thin-shell resistance formula when delta is comparable to R. The estimate R_ac/R_dc ~= R/(2*delta) holds only when the skin depth is much smaller than the wire radius. Near the transition, where delta and R are similar, the ratio approaches one and the simple formula overshoots. Use the full Bessel-function solution, or a calculator, in that regime.
Assuming a thicker wire always helps. Once the conductor radius exceeds a few skin depths, adding more copper to the core does almost nothing for AC resistance. The fix at high frequency is more surface area — flat ribbon, tubing, or stranding — not more bulk metal.
Ignoring temperature. Conductivity falls as a conductor heats, which slightly increases the skin depth. For precise loss budgets in components that run warm, use the conductivity at operating temperature, not the room-temperature value.
Try the interactive NovaSolver calculator
Working one skin depth by hand is instructive; sweeping frequency, conductivity, and wire radius to see how the AC resistance penalty develops is where intuition really forms. The Skin Effect Simulator — AC Current Surface Concentration on NovaSolver takes frequency, electrical conductivity, relative permeability, and wire radius, and returns the skin depth delta, the delta-to-radius ratio, the AC/DC resistance ratio, and the frequency at which delta equals R — alongside a cross-section current-density map and a log-log plot of skin depth versus frequency.
Related calculators
- Skin Effect & Skin Depth Calculator — a focused tool for the skin depth itself when you want a quick number across different metals.
- Transmission Line · Reflection Coefficient · Impedance Matching — because conductor loss feeds directly into the attenuation of a real RF line.
- Series RLC Circuit Simulator — AC Impedance and Resonance — to see how a frequency-dependent resistance reshapes the impedance of a resonant circuit.
You can browse the full set in the electromagnetics tools hub.
Closing note
The skin effect is a clean example of electromagnetism quietly reshaping an everyday assumption. Current does not fill a wire uniformly once it alternates; it retreats to the surface, and the retreat deepens with every rise in frequency. The skin depth formula gives you the length scale in a single line, and once you have it the design responses follow naturally — tubes instead of rods, ribbons instead of round wire, Litz bundles instead of solid strands. Compute delta first, compare it against your conductor radius, and let that ratio tell you how much of your copper is actually working.
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