# Skin Depth Calculator

Created by Luis Hoyos
Last updated: Nov 05, 2022
Table of contents:

Understanding the skin effect and calculating the skin depth is essential for designing efficient and effective electrical circuits. With this tool, you can calculate the skin depth of copper and many other conductors; you only need to know the frequency and the material used.

## The skin effect

When an alternating current flows through a conductor, the electric field associated with the current will cause the flow of electrons to concentrate near the conductor's surface. This phenomenon is known as the skin effect, and its cause is the eddy currents. Eddy currents refer to currents in the conductor that, as a consequence of Faraday's Law, are induced by a changing magnetic field.

The skin depth is the distance from the conductor's surface at which the current density reduces to 1/e (approximately 37%) of its original value.

The skin effect reduces the effective cross-section of the conductor and, as you can observe in our wire resistance calculator, increases the effective resistance.

## Skin depth formula

To calculate the skin depth, we use the following formula:

$\footnotesize \delta = \sqrt{\frac{\rho}{\pi f \mu_0 \mu_\text r}}$

, where:

• $\delta$ — Skin depth, calculated in meters ($\text m$);
• $\rho$ — Conductor resistivity, in ohms meters ($\Omega \cdot \text m$);
• $f$ — Frequency of the AC signal, in hertz ($\text{Hz}$);
• $\mu_0 = 4\pi \times 10^{-7} \text{ H/m}$ — The permeability of free space, in henries per meter; and
• $\mu_\text r$ — Relative magnetic permeability of the conductor, a unitless quantity (learn more about it in our magnetic permeability calculator).

We can use the units we want in the calculator, and it will provide the correct result. Still, to properly use the skin depth formula, we must use the units mentioned above to warranty dimensional homogeneity. We must consider that:

• Resistivity is usually given in $\text μ \Omega \cdot \text{cm}$. To use it in the skin depth equation, we must convert it to $\text μ \Omega \cdot \text{cm}$, remembering that $1\ \text μ \Omega \cdot \text{cm} = 10^{-8}\ \Omega \cdot \text m$.
• Skin depth is usually required in $\text{mm}$ (or even $\text{μm}$ for high frequencies). To convert it to $\text{mm}$ or $\text{μm}$, remember that $1\ \text m = 10^3 \text{ mm} = 10^6 \text{ μm}$, or use our length converter.
• For radio frequency applications, frequencies are high and usually expressed in gigahertz ($\text{GHz}$). To convert them to $\text{Hz}$, remember that $1 \text{ GHz} = 10^9\text{ Hz}$.

For example, let's use the skin depth equation to calculate the skin depth of copper at $\text{Hz}$.

$\footnotesize \begin{split} \delta &= \sqrt{\frac{\rho}{\pi f \mu_0 \mu_\text r}}\\ &=\!\! \sqrt{\frac{1.678 \times 10^{-8}\ \Omega \cdot \text m}{\!\!\!(\!\pi\!) (\!60\text{ Hz}\!) (\!4\pi \!\!\times\!\! 10^{-7} \text{ H/m}\!) (\!0.999991\!)}} \\ &=0.008417 \text{ m} = 8.417 \text{ mm} \end{split}$

So, the skin depth of copper at 60 Hz is 8.417 mm.

🙋 Remember that, for copper, $\rho = 1.678 \text{ μ}\Omega \cdot \text {cm} = 1.678 \times 10^{-8}\ \Omega \cdot \text m$ and $\mu_\text r = 0.999991$.

Luis Hoyos
Material
Copper
Resistivity (ρ)
μΩ·cm
Relative permeability (μᵣ)
Frequency (f)
GHz
Skin depth (δ)
μm
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