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Encyclopedia > 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. City lights viewed in a motion blurred exposure. ... In science and engineering, conductors are materials that contain movable charges of electricity. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... For other uses, see Frequency (disambiguation). ... An eddy current is a phenomenon caused by a moving magnetic field intersecting a conductor or vice-versa. ...

1 Introduction
2 Effect on impedance of round wires
3 Mitigation
4 Examples
5 See also
6 External links
7 References

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. Sir Horace Lamb FRS (November 29, 1849 - December 4, 1934) was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics (1895) and Dynamical Theory of Sound (1910). ... Year 1883 (MDCCCLXXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... Oliver Heaviside (May 18, 1850 â€“ February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... This article is about the type of Electromagnetic radiation. ... Power line redirects here. ... Gas filled tubes (also known as Discharge Tubes) are arrangements of electrodes in a gas within an insulating, temperature-resistant envelope. ...

Main article: skin depth

The current density J in an infinitely thick plane conductor decreases exponentially with depth d from the surface, as follows: 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. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

$J=J_mathrm{S} ,e^{-{d/delta }}$

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 (J_S). It can be calculated as follows: e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

$delta=sqrt{{2rho }over{omegamu}}$

where

ρ = resistivity of conductor

ω = angular frequency of current = 2π × frequency

μ = absolute magnetic permeability of conductor $= mu_0 cdot mu_r$ , where

μ 0

is the permeability of free space (4π×10⁻⁷ N/A²) and

μ r

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: Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of electric current. ... It has been suggested that this article or section be merged into Angular velocity. ... This article is in need of attention. ... This article is in need of attention. ... For other uses, see Newton (disambiguation). ... For other uses, see Ampere (disambiguation). ...

$R={{rho over d}left({Lover{pi (D-d)}}right)}approx{{rho over d}left({Lover{pi D}}right)}$

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 D_W of a wire of circular cross-section whose resistance will increase by 10% at frequency f is: Frederick Emmons Terman (born June 7, 1900 in English, Indiana; died December 19, 1982) is widely credited (together with William Shockley) with being the father of Silicon Valley. ...

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

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. For other uses, see Cable (disambiguation). ... A changing magnetic field will influence the distribution of an electric current flowing within an electrical conductor. ...

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 $δ$ 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 In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ...

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

where $J 0 (x)$ is the zeroth order Bessel function of the first kind, and $Ber(x)$ and $Bei(x)$ are Kelvin functions. The Kelvin functions BerÎ½(x) and BeiÎ½(x) are the real and imaginary parts, respectively, of where x is real, and is the Î½th order Bessel function of the first kind. ...

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 In physics, Ampères law is the magnetic equivalent of Gausss law, discovered by André-Marie Ampère. ...

$H(R) = frac{I}{2 pi R}$

Maxwell's Equations in cylindrical coordinates gives For thermodynamic relations, see Maxwell relations. ...

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

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 $J (r) = σ E (r)$ . 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. Litz wire is a special type of wire used in electronics. ... For other uses, see Transformer (disambiguation). ... A changing magnetic field will influence the distribution of an electric current flowing within an electrical conductor. ...

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. This article or section is in need of attention from an expert on the subject. ... Aluminium Conductor Steel Reinforced (or ACSR) cable is a specific type of high-capacity, high-strength stranded cable used in overhead power lines. ...

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. This article is about the chemical element. ... Electroplating is the process of using Davd lloyd current to coat an electrically conductive object with a relatively thin layer of metal. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... Very high frequency (VHF) is the radio frequency range from 30 MHz to 300 MHz. ... This article is about the type of Electromagnetic radiation. ...

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. 1500 amp busbars within a power distribution rack for a large building A busbar (often pronounced buzz bar) refers in electrical power distribution to thick strips of Copper or other material that conduct electricity around a switchboard, distribution board or other electrical apparatus. ... City lights viewed in a motion blurred exposure. ...

External links

Skin Effect in HiFi Cables
Skin Effect Relevance in Speaker Cables

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).

Categories: Electronics terms

Results from FactBites:

Compact Skin Effect Circuit (0 words)
	The skin effect has been extensively studied, although many methods treat the skin effect at high frequencies only, failing to predict the properties of transmission lines at low frequencies.
	Skin effect lumped circuit models in which the elements are frequency independent have been used [1-3], but tend to produce very large ladder circuits.
	The equivalent circuit accurately models the skin effect in circular cross section conductors up to a frequency corresponding to a 100 skin depth radius conductor.

Skin Effect (580 words)
	Skin effect is the tendency for high-frequency currents to flow on the surface of a conductor.
	Proximity effect is the tendency for current to flow in other undesirable patterns---loops or concentrated distributions---due to the presence of magnetic fields generated by nearby conductors.
	Where skin effect would also be a problem, more complex litz constructions can be used to ensure equal strand currents.

More results at FactBites »

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