Skin Depth Calculator

Created by Purnima Singh, PhD
Reviewed by Steven Wooding
Last updated: Feb 02, 2023

Use the skin depth calculator to determine the skin depth in conductors at any signal frequency.

Have you ever wondered why RF antennas are made from hollow tubes, not solid rods? If you have, then our article on the skin depth effect may help you in satisfying your curiosity.

Continue reading to know what skin effect is and the formula for skin depth calculation. You will also find an example of how to calculate skin depth in a conductor.

What is skin effect?

When a DC signal flows through a conductor, the current is distributed uniformly throughout the cross-section of the conductor (see figure 1 (a)).

On the other hand, when an AC signal flows through a conductor, the distribution of current is not uniform over the entire cross-section. The current density is higher at the surface of the conductor and decreases exponentially as we move towards its center (see figure 1 (b)).

We call this phenomenon of non-uniform distribution of an alternating current in a conductor the skin effect.

Skin effect depends on the frequency of the AC signal; the higher the frequency, the more pronounced the effect. At high frequencies, the conducting area is restricted to a very thin skin on its surface (see figure 1 (c)).

What is the cause of skin effect?

When an alternating current flows through a conductor, the magnetic flux linked with the conductor changes with the change in the magnitude and direction of the current.

According to Faraday's law of electromagnetic induction, the changing magnetic flux induces a current in the conductor. This induced current is called eddy current, and it creates an electric field that opposes any change in the alternating current flowing through the conductor. This effect is strongest at the center, and hence the current is forced to flow closer to the surface.

Skin effect increases the effective resistance of a conductor by reducing the cross-sectional area available for conducting current. In fact, the effective resistance offered by the conductor becomes more for AC than for DC. This results in a larger power loss for a given average value of alternating current as compared to the power dissipated for the same value of direct current.

Skin depth formula

The region between the outer surface and the point where the current density drops to about 1/e or 37% of its value at the surface is called the skin depth (see figure 1 (b)).

Skin depth depends on the properties of the conductor, i.e., its size, resistivity, magnetic permeability as well as the frequency of the signal. To calculate skin depth, we will use the formula:

$\scriptsize \delta = \sqrt{\frac{\rho}{\pi\ f \ \mu_0\ \mu_r}}$

where:

• $\delta$ — Skin depth;

• $\rho$ — Resistivity of the conductor;

• $f$ — Frequency of the AC signal;

• $\mu_0 = 4\pi \times 10^{-7}\ \rm H/m$ — The permeability of free space; and

• $\mu_r$ — Relative magnetic permeability of the conductor. Check out the magnetic permeability calculator to learn more about permeability.

Calculating skin depth in copper

To further understand the concept of skin depth, let us study the effect of frequency on current density distribution in copper wire.

1. The resistivity of copper at room temperature is:

$\rho= 1.678\ \rm \mu\Omega\cdot cm$,
$\rho = 1.678 \times 10^{-8}\ \Omega\cdot m$
and the relative permeability is:
$\mu_r = 0.999$.

2. We will first consider a copper conductor carrying an AC signal at a frequency of 50 Hz (typical frequency of AC signals used in homes). We can calculate the skin depth at this frequency as:

$\delta = \sqrt{\frac{1.678 \times 10^{-8}\ \Omega\cdot m}{3.14 \times 50\ \text{Hz} \times 1.257 \times 10^{-6}\ \rm H/m \times 0.999}}$

$\delta = 9.22\ \rm mm$

So, if we use a wire of diameter less than 1 cm, the skin effect is negligible for AC signals at typical frequencies used in the home.

3. Now will consider the same conductor carrying an AC signal at a frequency of 2.4 GHz (in radio frequency range). We can calculate the skin depth at this frequency by using the given values in skin depth equation:

$\delta = \sqrt{\frac{1.678 \times 10^{-8}\ \Omega\cdot m}{3.14 \times 2.4 \times 10^9\ \text{Hz} \times 1.257 \times 10^{-6}\ \rm H/m \times 0.999}}$

$\delta = 1.3308\ \rm \mu m$

Here, we note that at radio frequencies, the current flows only through an extremely thin region (within a few μm) near the surface. That is why we use hollow metal tubes in antennas and RF conductors, to save both cost and effort in managing heavy solid metal rods.

How to calculate the skin depth in a conductor using the skin depth calculator?

Now let us see how we can calculate the skin depth in copper using our skin depth calculator.

1. Using the drop-down menu, select the material, copper.

2. The resistivity and relative permeability fields will auto-populate. You can also enter these values if you know them for another material. Our conductivity to resistivity calculator can come in handy if you know the conductivity and want to determine the resisitivity.

3. Enter the frequency of the AC signal, e.g., 2.4 GHz.

4. The skin effect calculator will display the skin depth in copper at 2.4 GHz, i.e., 1.3308 μm.

FAQ

Why there is no skin effect in DC?

Skin effect depends on the frequency of the signal. Since the frequency of the DC signal is zero, the skin effect is absent in DC.

How do I reduce skin effect?

To reduce skin effects, we can implement one or more of the following suggestions:

1. Use a conductor with lower magnetic permeability.
2. Reduce the size of the conductor.
3. Use hollow tubes instead of metal rods for high-frequency signals.
4. Use specially designed woven strands of several insulated thin wires (also known as Litz wires) instead of a single thick wire.

How do I calculate skin effect?

To calculate the skin effect (or skin depth), follow the given instructions:

1. Multiply the frequency of the signal with the relative permeability of the conductor, the permeability of free space, and π.
2. Divide the resistivity of the conductor by the value obtained in step 1.
3. Take the square root of the value from step 2.
4. Congrats! You have calculated the skin depth.

How is the skin effect related to the frequency of the signal?

Skin depth is inversely proportional to the square root of the frequency of the signal. Hence, the higher the frequency, the lower the skin depth.

Purnima Singh, PhD
Material
Copper
Resistivity (ρ)
μΩ·cm
Relative permeability (μᵣ)
Frequency (f)
GHz
Skin depth (δ)
μm
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