## What does skin depth mean?

Skin depth is the distance created from the surface of a conductor due to the accumulation of current such that the cross-sectional area of the current flowing path is reduced such that ${R}_{ac}>{R}_{dc}$ where ${R}_{ac}$ is the AC resistance and ${R}_{dc}$ is the DC resistance. The effect is known as the skin effect. The flux distribution decreases as we move from the inner to the outer layer but current increases from the inner to the outer layer. The skin depth depends on the frequency of the current passing through the conductor. With the increase in frequency, the electric current flows only on the surface, which results in less skin depth. The effective cross-sectional area of the conductor reduces and the effective resistance/impedance increases due to the skin effect. As the internal resistance/impedance increases, the conductivity of the conductor decreases because conductivity is always inversely proportional to that of resistance and vice versa. The changing magnetic field due to alternating current produces eddy currents which are opposed by the skin effect. The skin depth is 8.5 mm at 60 Hz frequency. The skin depth goes on decreasing as the frequency advances. It is the smallest at higher frequencies. Skin effect also reduces the effective internal reactance.

A cylindrical conductor is shown where we can see the distribution of current flow. The skin depth δ is defined as the depth where the current density is only 1/e of the value at the surface. The skin depth depends on the magnetic properties and electrical properties of the conductor. The Litz wire is used to compensate for the effect of increased AC resistance. The skin effect is the main reason for preferring high voltage direct current distribution in long-distance power flow transmission.

## Causes of skin effect

The wire-type conductors are used in the transmission of electrical energy or signals using an alternating current. The electrons are driven apart due to the electric field created nearby the copper conductor surface. The magnetic field changes with the change in the current density. This change in the magnetic field produces an electric field that opposes the current density change. This electric field is known as back electromotive force (EMF). This force is stronger inside the copper conductor and is weaker at the surface. An alternating current can be induced due to the magnetic field according to the law of induction. Such a current is produced due to the electromagnetic wave impinging on the copper conductor.

## Material effect on skin depth

The skin depth is proportional to the square root of resistivity in good conductors. This only means better conductors have skin depth in a reduced amount. The total resistance of the better conductor remains less even with the reduced skin depth. Skin depth is also inversely proportional to the root of the magnetic permeability of the conductor. Iron wire is useless for alternating power lines because the iron wire is a ferromagnetic material and its permeability is 10,000 times greater and the skin depth is reduced. The effective thickness of lamination in power transformers also effectively reduces due to skin effect increasing losses.

## Mitigation

A type of wire called Litz wire is used to mitigate higher frequencies of a few kilos Hz to about 1 mega Hz. It is carefully woven with several wires or cable strands so that the overall magnetic field acts equally on all the wires and the current is distributed equally all over the conductor surface. Litz wire is used mostly in a transformer at higher frequencies to compensate for skin effect and proximity effect.

Solid conductors which may be tubular are always silver plated to get the advantage of silver's high conductivity. This is generally used in microwave frequencies where only a thin layer of silver is used for small skin depth, making its conductivity cost-effective. Silver plating is also used on the surface of waveguides which is used in the transmission of microwaves.

## Formula

According to the depth 'd' from the surface, the alternating current density "${J}_{S}$" decreases from outside to inside.

$J={J}_{S}{e}^{-(1+j)d/\delta}$

where $\delta $ = skin depth

The general formula of skin depth is given by -

$\delta =\sqrt{\frac{2\rho}{\omega \mu}}\sqrt{\sqrt{1+{\left(\rho \omega \epsilon \right)}^{2}}+\rho \omega \epsilon}$

where,

- $\rho $ = resistivity of the conductor
- $\omega $ = angular frequency = 2$\pi $ f, where f is the frequency.
- $\mu $ = permeability of the conductor = ${\mu}_{r}{\mu}_{o}$ where, ${\mu}_{r}$ = relative permeability of the conductor and ${\mu}_{o}$ = the permeability of free space
- $\epsilon $ = permittivity of conductor = ${\epsilon}_{r}{\epsilon}_{o}$ where, ${\epsilon}_{r}$ = relative permittivity of the conductor and ${\epsilon}_{o}$ = the permittivity of free space

The formula reduces at a frequency much below 1/$\mathrm{\rho \epsilon}$ to,

$\delta =\sqrt{\frac{2\rho}{\omega \mu}}$

At frequency much larger than 1/$\mathrm{\rho \epsilon}$ the formula is written as,

$\delta \approx 2\rho \sqrt{\frac{\epsilon}{\mu}}$

A tool is designed to calculate the skin depth based on the frequency of the signal and the conductive material used and this tool is called as "Skin Depth Calculator".

## Context and Applications

The topic is significant in the professional exams for Undergraduate, Graduate, and Postgraduate courses:

- Bachelors in Technology (Electrical Engineering)
- Masters in Technology (Electrical Engineering)

## Practice Problems

Question 1: Which of the following has the highest skin depth?

- Al
- Ag
- Au
- Cu

Answer: Option a

Explanation: Al has the highest skin depth because it has the lowest conductivity among the other options given.

Question 2: At what frequencies, skin effect is noticeable?

- Audio
- Low
- High
- All of the above

Answer: Option c

Explanation: The skin effect is only noticeable at high frequency. This is because, with the increase in frequency, the current flows only near the surface.

Question 3: In which case is the AC resistance of a conductor greater than the DC resistance?

- Skin Effect
- Proximity Effect
- Both a and b
- Neither a nor b

Answer: Option a

Explanation: In the case of the skin effect, the AC resistance is greater than the DC resistance because, in the case of DC, there is no rate of change of current.

Question 4: Where does the skin effect exist among all the options stated below?

- Low voltage dc overhead transmission
- High voltage dc overhead transmission
- Cable carrying DC
- AC transmission

Answer: Option d

Explanation: The skin effect exists only in AC transmission systems due to the rate of change of current with time, whereas in the case of DC, the current does not change with time.

Question 5: Which value decreases in a conductor due to the skin effect?

- Inductance
- Resistance
- Capacitance
- None of the above

Answer: Option b

Explanation: The area of the current flowing path is reduced due to the accumulation of current on the surface. Hence, the effective resistance of the conductor decreases due to the skin effect.

## Related Concepts

- Proximity Effect
- Penetration depth
- Eddy currents
- Transformer
- Litz wire

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