Skin effect

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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.

Contents

[edit] 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 consideration for design of discharge tube circuits.

[edit] Mathematics

Main article: skin depth

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

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

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:

d=\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 and μr is the relative permeabilty 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, thin conductors such as wires, the resistance is approximately that of a hollow tube with wall thickness d carrying direct current. For example, for a round wire, the AC resistance is approximately:

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

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:

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

[edit] 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.

Large power transformers will be wound with conductors of similar construction to Litz wire, but of larger cross-section.

In other applications, solid conductors are replaced by tubes, which have the same resistance at high frequencies but of course are lighter.

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.

[edit] Examples

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

frequency d
60 Hz 8.57 mm
10 kHz 0.66 mm
100 kHz 0.21 mm
1 MHz 66 µm
10 MHz 21 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.

[edit] See also

[edit] External links

[edit] 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
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