# Skin effect: Wikis

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# Encyclopedia

Skin effect is the tendency of an alternating electric current (AC) to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. That is, the electric current tends to flow at the "skin" of the conductor, at an average depth called the skin depth. The skin effect causes the effective resistance of the conductor to increase with the frequency of the current because much of the conductor does little. Skin effect is due to eddy currents set up by the AC current. At 60 Hz in copper, skin depth is about a centimetre. At high frequencies skin depth is much smaller.

Methods to minimise skin effect include using specially woven wire and using hollow pipe-shaped conductors.

## Introduction

When an electromagnetic wave interacts with a conductive material, mobile charges within the material are made to oscillate back and forth with the same frequency as the impinging fields. The movement of these charges, usually electrons, constitutes an alternating electric current, the magnitude of which is greatest at the conductor's surface. The decline in current density versus depth is known as the skin effect and the skin depth is a measure of the distance over which the current falls to 1/e of its original value. A gradual change in phase accompanies the change in magnitude, so that, at a given time and at appropriate depths, the current can be flowing in the opposite direction to that at the surface.

The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalised to conductors of any shape by Oliver Heaviside in 1885. The skin effect has practical consequences in the design of radio-frequency and microwave circuits and to some extent in AC electrical power transmission and distribution systems. Also, it is of considerable importance when designing discharge tube circuits.

The current density J in an infinitely thick plane conductor decreases exponentially with depth d from the surface, as follows:

$J=J_\mathrm{S} \,e^{-{d/\delta }}$

where δ is a constant called the skin depth. This is defined as the depth below the surface of the conductor at which the current density decays to 1/e (about 0.37) of the current density at the surface (JS). It can be calculated as follows:

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

where

ρ = resistivity of conductor
ω = angular frequency of current = 2π × frequency
μ = absolute magnetic permeability of conductor $= \mu_0 \cdot \mu_r$, where μ0 is the permeability of free space (4π×10−7 N/A2) and μr is the relative permeability of the conductor.

The resistance of a flat slab (much thicker than δ) to alternating current is exactly equal to the resistance of a plate of thickness δ to direct current. For long, cylindrical conductors such as wires, with diameter D large compared to δ, the resistance is approximately that of a hollow tube with wall thickness δ carrying direct current. That is, the AC resistance is approximately:

$R={{\rho \over \delta}\left({L\over{\pi (D-\delta)}}\right)}\approx{{\rho \over \delta}\left({L\over{\pi D}}\right)}$

where

L = length of conductor
D = diameter of conductor

The final approximation above is accurate if D >> δ.

A convenient formula (attributed to F.E. Terman) for the diameter DW of a wire of circular cross-section whose resistance will increase by 10% at frequency f is:

$D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}$

The increase in AC resistance described above is accurate only for an isolated wire. For a wire close to other wires, e.g. in a cable or a coil, the ac resistance is also affected by proximity effect, which often causes a much more severe increase in ac resistance.

Skin depth is due to the circulating eddy currents cancelling the current flow in the center of a conductor and reinforcing it in the skin.

## Material effect on skin depth

Skin depth varies as the inverse square root of the conductivity. This means that better conductors have a reduced skin depth. The overall resistance of the better conductor is lower even though the skin depth is less. This tends to reduce the difference in high frequency resistance between metals of different conductivity.

Skin depth also varies as the inverse square root of the permeability of the conductor. In the case of iron, its conductivity is about 1/7 that of copper. Its permeability is about 10,000 times greater however. The skin depth for iron is about 1/38 that of copper or about 220 micrometres at 60 Hz. Iron wire is worthless as a conductor at power line frequencies. Skin effect reduces both the effective thickness of laminations in power transformers and their losses.

Iron rods work well for direct-current (DC) welding but it is impossible to use them at frequencies much higher than 60 Hz. At a few kilohertz, the welding rod will glow red hot from skin effect losses but will barely have enough power available to sustain an arc. Only non-magnetic rods can be used for high-frequency welding.

## Mitigation

A type of cable called litz wire (from the German litzendraht, braided wire) is used to mitigate the skin effect for frequencies of a few kilohertz to about one megahertz. It consists of a number of insulated wire strands woven together in a carefully designed pattern, so that the overall magnetic field acts equally on all the wires and causes the total current to be distributed equally among them. This has the effect of reducing the effective permeability and increasing the skin depth.[1]

Litz wire is often used in the windings of high-frequency transformers, to increase their efficiency by mitigating both skin effect and, more importantly, proximity effect.

Large power transformers are wound with stranded conductors of similar construction to litz wire, but of larger cross-section.[2]

High-voltage, high-current overhead power transmission lines often use aluminum cable with a steel reinforcing core, where the higher resistivity of the steel core is largely immaterial.

In other applications, solid conductors are replaced by tubes, which have the same resistance at high frequencies but lighter weight. Very recently, researchers have been able to create extremely light cell-phone antennas using carbon-nanotubes,[3] their performance attributed to skin effect.

Solid or tubular conductors may also be silver-plated providing a better conductor (the best possible conductor except for superconductors) than copper on the "skin" of the conductor. Silver-plating is most effective at VHF and microwave frequencies, because the very thin skin depth (conduction layer) at those frequencies means that the silver plating can economically be applied at thicknesses greater than the skin depth.

## Examples

Skin depths for some metals

If the electrical resistivity of a material is equal to 1/σ and its relative permeability is defined as μ / μ0, where μ0 is the magnetic permeability of free space.

$\delta = \frac{1}{\sqrt{\pi \mu_o}} \,\sqrt{\frac{\rho}{\mu_r f}} \approx 503\,\sqrt{\frac{\rho}{\mu_r f}}\qquad\qquad(9)$

where

δ = the skin depth in metres
μ0 = ×10-7 H/m
μr = the relative permeability of the medium
ρ = the resistivity of the medium in Ω·m
f = the frequency of the wave in Hz

If the resistivity of aluminum is taken as 2.8×10-8 Ω·m and its relative permeability is 1, then the skin depth at a frequency of 50 Hz is given by

$\delta = 503 \,\sqrt{\frac{2.82 \cdot 10^{-8}}{1 \cdot 50}}= 11.9$ mm

Iron has a higher resistivity, 1.0×10-7 Ω·m, and this will increase the skin depth. However, its relative permeability is typically 90, which will have the opposite effect. At 50 Hz the skin depth in iron is given by

$\delta = 503 \,\sqrt{\frac{1.0 \cdot 10^{-7}}{90 \cdot 50}}= 2.4$ mm

Hence, the higher magnetic permeability of iron more than compensates for the lower resistivity of aluminium and the skin depth in iron is therefore one-fifth that of aluminium. This will be true whatever the frequency, assuming the material properties are not themselves frequency-dependent.

Skin depth values for some common good conductors at a frequency of 10 GHz (microwave region) are indicated below.

Conductor Skin depth (μm)
Aluminum 0.80
Copper 0.65
Gold 0.79
Silver 0.64

At microwave frequencies, most of the current in a good conductor flows in an extremely thin region near the surface. The extremely short skin depth at microwave frequencies shows that only surface coating of guiding conductor is important. A piece of glass with an evaporated silver surface 3 μm thick is an excellent conductor at these frequencies.

In copper, the skin depth at various frequencies is shown below.

Frequency Skin depth (μm)
60 Hz 8470
10 kHz 660
100 kHz 210
1 MHz 66
10 MHz 21

In Engineering Electromagnetics, Hayt points out that in a power station a bus bar for alternating current at 60 Hz with a radius larger than one-third of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons. A thin film of silver deposited on glass is an excellent conductor at microwave frequencies.

## References

1. ^ [1]
2. ^ Central Electricity Generating Board (1982). Modern Power Station Practice. Pergamon Press.
3. ^ Spinning Carbon Nanotubes Spawns New Wireless Applications
• Hayt, William Hart. Engineering Electromagnetics Seventh Edition. New York: McGraw Hill, 2006. ISBN 0-07-310463-9.
• Nahin, Paul J. Oliver Heaviside: Sage in Solitude. New York: IEEE Press, 1988. ISBN 0-87942-238-6.
• Ramo, S., J. R. Whinnery, and T. Van Duzer. Fields and Waves in Communication Electronics. New York: John Wiley & Sons, Inc., 1965.
• Terman, F. E. Radio Engineers' Handbook. New York: McGraw-Hill, 1943. For the Terman formula mentioned above.
• Ramo, Whinnery, Van Duzer (1994). Fields and Waves in Communications Electronics. John Wiley and Sons.