# skin effect

The 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.

## Introduction

The skin effect causes the effective resistance of the conductor to increase with the frequency of the current. The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalized 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.

Main article: skin depth

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

where d 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:

where
ρ = resistivity of conductor
ω = angular frequency of current = 2π × frequency
μ = absolute magnetic permeability of conductor , where is the permeability of free space (4π×10−7 N/A2) and is the relative permeability of the conductor.
The resistance of a flat slab (much thicker than d) to alternating current is exactly equal to the resistance of a plate of thickness d to direct current. For long, cylindrical conductors such as wires, with diameter D large compared to d, the resistance is approximately that of a hollow tube with wall thickness d carrying direct current. That is, the AC resistance is approximately:

where
L = length of conductor
D = diameter of conductor

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

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:

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.

## Effect on impedance of round wires

For isolated round wires with radius on the order of or smaller than , the assumption of exponential decrease of with depth is no longer valid. In this case, must be found by solving

If we transform variables from to , this equation has the form of a zeroth-order Bessel equation. Using the boundary condition and considering that must be finite at for a solid wire, the solution to this equation is

where is the zeroth order Bessel function of the first kind, and and are Kelvin functions.

The total current in the wire may be found by integrating from 0 to . It may more easily be found by relating it to the derivative of the electric field at the surface of the wire via its magnetic field. Ampere's Law at the wire surface gives an azimuthal magnetic field

Maxwell's Equations in cylindrical coordinates gives

where the electric field points in the direction of the current. Equating these two functions at gives

where the prime on the in the numerator indicates a first derivative, and we have used . The impedance in the wire is given by

where and are the resistance and inductance per unit length of the wire. Plugging in for and gives

where the fundamental resistance and unitless scaled "radius" are given by

and

## 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. 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 conductors of similar construction to litz wire, but of larger cross-section.

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.

Solid or tubular conductors may also be silver-plated providing a better conductor (the best possible conductor excepting only 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

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

frequency d
60 Hz8.57 mm
10 kHz0.66 mm
100 kHz0.21 mm
1 MHz66 µm
10 MHz21 µm

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 1/3rd of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than 1/2 inch (12 mm) thick except for mechanical reasons. A possible solution to this problem consists of using cables with multiple insulated conductors. A thin film of silver deposited on glass is an excellent conductor at microwave frequencies.

## References

• William Hart Hayt, Engineering Electromagnetics Seventh Edition,(2006), McGraw Hill, New York ISBN 0-07-310463-9
• Paul J. Nahin, Oliver Heaviside: Sage in Solitude, (1988), IEEE Press, New York, ISBN 0-87942-238-6
• Terman, F.E. Radio Engineers' Handbook, McGraw-Hill 1943 -- for the Terman formula mentioned above
• S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley & Sons, Inc., New York (1965).
alternating current (AC) is an electrical current whose magnitude and direction vary cyclically, as opposed to direct current, whose direction remains constant. The usual waveform of an AC power circuit is a sine wave, as this results in the most efficient transmission of
In science and engineering, conductors, such as copper or aluminum, are materials with atoms have loosely held valence electrons. See electrical conduction.

## Conductors in context

Electrical resistance is a measure of the degree to which an object opposes an electric current through it. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.
FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. A sequel, titled Amplitude was released in 2003.
Sir Horace Lamb FRS (29 November 1849 – 4 December 1934) was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics (1879) and Dynamical Theory of Sound (1910).
18th century - 19th century - 20th century
1850s  1860s  1870s  - 1880s -  1890s  1900s  1910s
1880 1881 1882 - 1883 - 1884 1885 1886

:
Subjects:     Archaeology - Architecture -
Oliver Heaviside

Portrait of Oliver Heaviside (1850-1925) by Frances Hodge
Born May 18 1850
Camden Town, London, England
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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
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## Definition

A newton
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proximity effect. The combination of skin and proximity effects significantly increases the AC resistance of the conductor when compared to its resistance to a DC current. At higher frequencies, the AC resistance of a conductor can easily exceed ten times its DC resistance.
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:

for an arbitrary real or complex number α.
The Kelvin functions Berν(x) and Beiν(x) are the real and imaginary parts, respectively, of

where x is real, and is the νth order Bessel function of the first kind.