Skin Depth
Calculate the skin depth of an alternating current in a conductor from frequency, resistivity and relative permeability. Enter the material properties to see how deep AC current actually penetrates.
Find How Deep AC Current Penetrates a Conductor
Skin Depth Formula
Where:
- \(\delta\) — skin depth (m)
- \(\rho\) — electrical resistivity of the conductor (Ω·m)
- \(f\) — frequency of the alternating current (Hz)
- \(\omega = 2\pi f\) — angular frequency (rad/s)
- \(\mu = \mu_0 \mu_r\) — absolute permeability (H/m), with \(\mu_0 = 4\pi \times 10^{-7}\) H/m
The formula gives the distance from the surface at which the current density falls to \(1/e \approx 37\%\) of its surface value. Because \(\delta\) is proportional to \(1/\sqrt{f}\), higher frequencies confine the current to an ever thinner layer near the surface.
Skin Depth – Calculation Example
Example: a copper conductor carries an alternating current at 1 MHz. Copper has a resistivity of \(1.68 \times 10^{-8}\) Ω·m at 20 °C and is non-magnetic, so \(\mu_r = 1\).
\[ \mu = \mu_0 \mu_r = 4\pi \times 10^{-7} \times 1 = 1.257 \times 10^{-6}\ \text{H/m} \]
\[ \delta = \sqrt{\frac{1.68 \times 10^{-8}}{\pi \times 1 \times 10^{6} \times 1.257 \times 10^{-6}}} = \sqrt{4.26 \times 10^{-9}} \]
Result: \(\delta \approx 6.52 \times 10^{-5}\) m ≈ 65.2 µm. At 1 MHz, effectively all the current in a copper wire flows within about 0.2 mm of the surface (roughly \(3\delta\)).
How to Use the Skin Depth Calculator
- Enter the Frequency (f) of the alternating current and choose its unit — Hz, kHz, MHz or GHz.
- Enter the Resistivity (ρ) of the conductor material. The default is copper (\(1.68 \times 10^{-8}\) Ω·m); select Ω·mm²/m, µΩ·m or µΩ·cm if your data comes in those units.
- Enter the Relative Permeability (μr). Use 1 for copper, aluminium, silver, gold and other non-magnetic metals; use 100–5000 for ferromagnetic materials such as iron or steel.
- Read the Skin Depth in metres, the readable value in mm or µm, and the full calculation details showing the values substituted into the formula.
What Is Skin Depth and Why It Matters
When direct current flows through a conductor, it distributes itself uniformly over the whole cross-section. Alternating current behaves differently: the time-varying magnetic field generated by the current induces eddy currents that oppose the current in the centre of the conductor and reinforce it near the surface. This phenomenon, called the skin effect, causes AC current density to decrease exponentially with depth below the surface.
The skin depth \(\delta\) quantifies this decay. At a depth of one \(\delta\) the current density is down to about 37% of its surface value, and about 95% of the total current flows within a layer of thickness \(3\delta\). At 50/60 Hz the skin depth in copper is roughly 8–9 mm, so the effect is negligible in household wiring. At 1 MHz it shrinks to about 65 µm, and at 1 GHz it is only about 2 µm — meaning a thick conductor carries high-frequency current almost entirely in a thin surface film.
The practical consequences appear everywhere in electrical engineering. At high frequencies, the effective resistance of a conductor increases because only a fraction of its cross-section is used, which is why RF conductors are often hollow, silver-plated, or made as tubes and Litz wire. Skin depth also matters for PCB trace design, coaxial cables, shielding effectiveness, induction heating and non-destructive testing of metals. Note that ferromagnetic conductors suffer far more: because their permeability is hundreds or thousands of times higher, their skin depth at the same frequency can be 10–100× smaller than that of copper, which is why steel is a poor choice for RF conductors.
Assumptions, Limitations and Practical Tips
- Plane surface assumed: the formula is exact for a flat, semi-infinite conductor. It is a good approximation for wires and traces whose radius or thickness is at least 3–4 times \(\delta\); for conductors comparable in size to \(\delta\), the current fills the cross-section and the actual penetration is larger.
- Good conductor: the formula assumes the conductor is much less lossy than the surrounding medium, which is true for ordinary metals at power, audio and radio frequencies.
- DC resistivity used: \(\rho\) is taken as constant. This is accurate when \(\delta\) is large compared with the electron mean free path — valid for bulk metals at normal frequencies, but not in the extreme far-infrared or in very cold, very pure samples.
- Temperature dependence: resistivity rises with temperature (about 0.4% per °C for copper), so a hot winding has a slightly larger skin depth than a cold one.
- Proximity effect ignored: nearby conductors carrying current distort the current distribution further and increase losses; this calculator covers only the isolated-conductor skin effect.
- Iron and steel: the permeability of ferromagnetic materials depends on frequency and field strength, so treat results for \(\mu_r \gg 1\) as estimates.
If you are sizing conductors for AC circuits, combine this tool with the Voltage Drop calculator to check that the effective resistance stays acceptable. For frequency-dependent behaviour in circuits, see the AC Reactance and RL Time Constant calculators, and for high-frequency board layout use the PCB Trace Width calculator.
Typical Skin Depths in Copper (μr = 1)
| Frequency | Skin depth δ |
|---|---|
| 50 Hz | 9.3 mm |
| 1 kHz | 2.1 mm |
| 100 kHz | 0.21 mm |
| 1 MHz | 65 µm |
| 100 MHz | 6.5 µm |
| 1 GHz | 2.1 µm |
Frequently Asked Questions
How do I calculate skin depth using conductivity instead of resistivity?
If your data sheet gives conductivity σ in S/m instead of resistivity, use \( \delta = 1/\sqrt{\pi f \mu \sigma} \). Copper, for example, has \( \sigma \approx 5.96 \times 10^7 \) S/m, which is simply the reciprocal of its resistivity (\( \rho = 1/\sigma \)). You can either convert σ to ρ yourself or enter \( 1/\sigma \) into the resistivity field in Ω·m.
How thick does a wire need to be before the skin effect matters?
A practical rule of thumb: the skin effect becomes significant when the conductor's radius exceeds about 2–3 skin depths. Below that, current still fills most of the cross-section and the AC resistance is close to the DC value. For copper at 50 Hz this means wires thinner than roughly 15–20 mm are barely affected, which is why household wiring ignores the effect entirely, while RF coils at MHz frequencies need Litz wire or plating.
Why is the skin depth in steel so much smaller than in copper?
Skin depth scales as \( 1/\sqrt{\mu_r} \), so a relative permeability of 500 shrinks the depth by a factor of about 22 compared with the same resistivity. Steel combines high permeability with high resistivity, yet the permeability usually dominates, confining current to a very thin layer. Keep in mind that μr in ferromagnetic metals varies with frequency and field strength, so treat calculated values for steel as order-of-magnitude estimates only.
How does temperature affect the skin depth result?
Resistivity of metals increases roughly linearly with temperature — about 0.4% per °C for copper near room temperature. Since \( \delta \propto \sqrt{\rho} \), a hot conductor has a slightly larger skin depth than a cold one, though the change is small: heating copper from 20 °C to 100 °C increases δ by only about 7%. For most designs the 20 °C value is adequate; only worry about temperature in tightly wound coils that run hot.
What are common mistakes when using the skin depth formula?
The most frequent errors are: entering resistivity in the wrong units (Ω·mm²/m differs from Ω·m by a factor of 10⁶); using μr values from DC magnetization curves for high-frequency steel; forgetting that the formula assumes a flat, semi-infinite surface and applying it to wires thinner than about 3δ; and confusing the depth at which current density drops to 37% with the layer carrying all the current — in reality about 95% flows within 3δ.
Does skin depth depend on the current amplitude or conductor shape?
The classical skin depth formula depends only on frequency, resistivity and permeability — not on how much current flows. Amplitude only matters indirectly in ferromagnetic materials, where strong fields change the effective permeability. Shape does matter: the formula is exact for a flat surface, so for round wires or PCB traces it is an approximation that becomes less accurate as the conductor dimensions approach δ, because the current then fills the cross-section more uniformly.
How do I convert between the resistivity units used in the calculator?
The four supported units relate as follows: \( 1\ \Omega \cdot \text{mm}^2/\text{m} = 1\ \mu\Omega \cdot \text{m} = 10^{-6}\ \Omega \cdot \text{m} \), and \( 1\ \mu\Omega \cdot \text{cm} = 10^{-8}\ \Omega \cdot \text{m} \). Copper's \( 1.68 \times 10^{-8}\ \Omega \cdot \text{m} \) equals 0.0168 Ω·mm²/m, 0.0168 µΩ·m... in fact 1.68 µΩ·cm. If unsure, pick the unit matching your source data and the calculator converts it automatically.