Hong Life CalculatorsUnderstanding the skin depth calculator — concept and principle
At high frequencies, alternating current does not flow evenly through a conductor; it crowds into a thin layer near the surface. This calculator finds the skin depth, the depth at which the current density falls to about 37% (1/e) of its surface value. Engineers use it to choose wire gauge, Litz wire, PCB copper and shield thickness.
How it is calculated
For a good conductor, δ = √(ρ / (π f μ)), where μ = μ₀ μᵣ and μ₀ = 4π × 10⁻⁷ H/m.
- f (Hz): frequency. Higher frequency gives a thinner skin.
- ρ (Ω·m): resistivity. Copper at room temperature is about 1.68 × 10⁻⁸ Ω·m; aluminium about 2.65 × 10⁻⁸ Ω·m.
- μᵣ: relative permeability, 1 for copper and aluminium, much larger for iron and steel, which makes their skin depth far smaller.
The result is shown in micrometres.
The principle
A changing current creates a changing magnetic field inside the conductor, which induces eddy currents (Faraday's law). These eddy currents oppose the current in the centre and reinforce it near the surface. Solving Maxwell's equations for a good conductor gives fields that decay exponentially with depth, as e^(−x/δ). Because δ scales with 1/√f, a tenfold rise in frequency shrinks the skin depth by about 3.16 times. For AC resistance, a thick conductor behaves roughly as if all current flowed uniformly in a layer one skin depth thick.
Worked example
- μ = 1 × 4π × 10⁻⁷ ≈ 1.2566 × 10⁻⁶ H/m.
- π f μ = π × 10⁶ × 1.2566 × 10⁻⁶ ≈ 3.948.
- ρ / (π f μ) = 1.68 × 10⁻⁸ / 3.948 ≈ 4.255 × 10⁻⁹ m².
- δ = √(4.255 × 10⁻⁹) ≈ 6.523 × 10⁻⁵ m.
- Result: 65.23 µm for copper at 1 MHz.
| Frequency (Hz) | Skin depth (µm) |
|---|---|
| 250,000 | 130.4682293 |
| 500,000 | 92.25496965 |
| 750,000 | 75.3258673 |
| 1e+06 | 65.23411464 |
| 1.25e+06 | 58.34716592 |
| 1.5e+06 | 53.26343156 |
| 2e+06 | 46.12748483 |
How to read the result
If the conductor is much thicker than about 2δ to 3δ, the extra copper carries little current and AC resistance rises above DC resistance. Typical copper values: about 8.4 mm at 60 Hz, about 2.1 µm at 1 GHz. For shielding, each skin depth of thickness attenuates the field by about 8.7 dB, so several skin depths give strong absorption. For conductors much thinner than δ, current is nearly uniform and skin effect can be ignored.
Common mistakes
- Entering frequency in MHz instead of Hz (an error of √10⁶ = 1000×).
- Using μᵣ = 1 for steel or nickel plating, which greatly overestimates skin depth.
- Forgetting that resistivity rises with temperature, so hot copper has a slightly larger δ.
- Assuming skin depth is the only loss: in coils, the proximity effect from neighbouring turns often dominates.
Frequently asked questions
What is the skin depth of copper at 50 Hz?
About 9.2 mm, which is why large busbars and cables for power distribution are sometimes hollow or stranded.
Why use Litz wire?
Litz wire uses many thin, insulated, woven strands, each thinner than δ, so current is shared evenly and AC resistance stays low up to a few hundred kHz.
Sources: Pozar, Microwave Engineering, section on conductors and skin depth · Ramo, Whinnery and Van Duzer, Fields and Waves in Communication Electronics