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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. The skin effect causes the effective resistance of the conductor to increase with the frequency of the current. Skin effect is due to eddy currents set up by the AC current.

Introduction


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.

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

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

Effect on impedance of round wires


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wires with radius R on the order of or smaller than d, the assumption of exponential decrease of J with depth \delta is no longer valid.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

     \frac{d^2J}{dr^2} + \frac{1}{r} \frac{dJ}{dr} = j \omega \mu \sigma J

If we transform variables from r to j^{-1/2}r, this equation has the form of a zeroth-order Bessel equation. Using the boundary condition J(R) = J_S and considering that J must be finite at r = 0 for a solid wire, the solution to this equation is

     J(r) = J_S \frac{J_0(\sqrt{-2j}r/d)}{J_0(\sqrt{-2j}R/d)} = J_S \frac{\mathrm{Ber}(\sqrt{2}r/d) + j \mathrm{Bei}(\sqrt{2}r/d)}{\mathrm{Ber}(\sqrt{2}R/d) + j \mathrm{Bei}(\sqrt{2}R/d)},

where J_0(x) is the zeroth order Bessel function of the first kind, and \mathrm{Ber}(x) and \mathrm{Bei}(x) are Kelvin functions.

The total current in the wire may be found by integrating J(r) from 0 to R. 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

     H(R) = \frac{I}{2 \pi R}

Maxwell's Equations in cylindrical coordinates gives

     H(r) = \frac{1}{j \omega \mu} \frac{dE}{dr}

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

     I = - \frac{2 \pi R d J_S}{\sqrt{-2j}} \frac{J_0'(\sqrt{-2j}R/d)}{J_0(\sqrt{-2j}R/d)}

where the prime on the J_0 in the numerator indicates a first derivative, and we have used J(r) = \sigma E(r). The impedance in the wire is given by

     Z = \frac{E(R)}{I} = R' + j \omega L',

where R' and L' are the resistance and inductance per unit length of the wire. Plugging in for E(R) and I gives

     Z = \frac{j R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R}) + j \mathrm{Bei}(\tilde{R})}{\mathrm{Ber}'(\tilde{R}) + j \mathrm{Bei}'(\tilde{R})}

     R' = \frac{R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R})\mathrm{Bei}'(\tilde{R}) - \mathrm{Bei}(\tilde{R})\mathrm{Ber}'(\tilde{R})}{\mathrm{Ber}'(\tilde{R})^2 + \mathrm{Bei}'(\tilde{R})^2}

     \omega L' = \frac{R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R})\mathrm{Ber}'(\tilde{R}) + \mathrm{Bei}(\tilde{R})\mathrm{Bei}'(\tilde{R})}{\mathrm{Ber}'(\tilde{R})^2 + \mathrm{Bei}'(\tilde{R})^2}

where the fundamental resistance R_0 and unitless scaled "radius" \tilde{R} are given by

     R_0 = \frac{1}{\sigma d}

and

     \tilde{R} = \frac{\sqrt{2}R}{d}.

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 stranded 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 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


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

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 thin film of silver deposited on glass is an excellent conductor at microwave frequencies.

See also


Proximity effect (electromagnetism)
skin depth
"The Skin Effect Myth" for Tesla coils
Surface wave
Litz wire

References


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.

External links


Skin Effect in HiFi Cables
Skin Effect Relevance in Speaker Cables

    
   
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