Skin effect 261642 223355581 2008-07-03T18:45:07Z Wolfkeeper 41690 /* See also */ The '''skin effect''' is the tendency of an [[alternating current|alternating electric current]] (AC) to distribute itself within a [[Conductor (material)|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 [[electrical resistance|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 [[electric power transmission|electrical power transmission and distribution]] systems. Also, it is of considerable importance when designing [[discharge tube]] circuits. {{main|skin depth}} The [[current density]] ''J'' in an ''infinitely thick'' plane conductor [[exponential decay|decreases exponentially]] with depth d from the surface, as follows: :<math>J=J_\mathrm{S} \,e^{-{d/\delta }}</math> where ''δ'' is a constant called the ''[[Skin_depth|skin depth]]''. This is defined as the depth below the surface of the conductor at which the current density decays to 1/[[e (mathematical constant)|e]] (about 0.37) of the current density at the surface (''J''<sub>S</sub>). It can be calculated as follows: :<math>\delta=\sqrt{{2\rho }\over{\omega\mu}}</math> where :ρ = [[resistivity]] of conductor :ω = [[angular frequency]] of current = 2π &times; frequency :μ = absolute [[magnetic permeability]] of conductor <math>= \mu_0 \cdot \mu_r </math>, where <math> \mu_0 </math> is the [[permeability of free space]] (4&pi;&times;10<sup>&minus;7</sup>&nbsp;[[newton|N]]/[[ampere|A]]<sup>2</sup>) and <math> \mu_r </math> 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: :<math>R={{\rho \over d}\left({L\over{\pi (D-d)}}\right)}\approx{{\rho \over d}\left({L\over{\pi D}}\right)}</math> where :L = length of conductor :D = diameter of conductor The final approximation above is accurate if ''D'' >> ''d''. A convenient formula (attributed to [[Frederick Terman|F.E. Terman]]) for the diameter ''D''<sub>W</sub> of a wire of circular cross-section whose resistance will increase by 10% at frequency ''f'' is: :<math>D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}</math> 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 (electromagnetism)|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 <math>R</math> on the order of or smaller than <math>d</math>, the assumption of exponential decrease of <math>J</math> with depth <math>\delta</math> is no longer valid. In this case, <math>J</math> must be found by solving :<math>\frac{d^2J}{dr^2} + \frac{1}{r} \frac{dJ}{dr} = j \omega \mu \sigma J</math> If we transform variables from <math>r</math> to <math>j^{-1/2}r</math>, this equation has the form of a zeroth-order [[Bessel equation]]. Using the boundary condition <math>J(R) = J_S</math> and considering that <math>J</math> must be finite at <math>r = 0</math> for a solid wire, the solution to this equation is :<math>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)},</math> where <math>J_0(x)</math> is the zeroth order Bessel function of the first kind, and <math>\mathrm{Ber}(x)</math> and <math>\mathrm{Bei}(x)</math> are [[Kelvin functions]]. The total current in the wire may be found by integrating <math>J(r)</math> from 0 to <math>R</math>. 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 :<math>H(R) = \frac{I}{2 \pi R}</math> [[Maxwell's Equations]] in cylindrical coordinates gives :<math>H(r) = \frac{1}{j \omega \mu} \frac{dE}{dr}</math> where the electric field <math>E</math> points in the direction of the current. Equating these two functions at <math>r = R</math> gives :<math>I = - \frac{2 \pi R d J_S}{\sqrt{-2j}} \frac{J_0'(\sqrt{-2j}R/d)}{J_0(\sqrt{-2j}R/d)}</math> where the prime on the <math>J_0</math> in the numerator indicates a first derivative, and we have used <math>J(r) = \sigma E(r)</math>. The impedance in the wire is given by :<math>Z = \frac{E(R)}{I} = R' + j \omega L',</math> where <math>R'</math> and <math>L'</math> are the resistance and inductance per unit length of the wire. Plugging in for <math>E(R)</math> and <math>I</math> gives :<math>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})}</math> :<math>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}</math> :<math>\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}</math> where the fundamental resistance <math>R_0</math> and unitless scaled "radius" <math>\tilde{R}</math> are given by :<math>R_0 = \frac{1}{\sigma d}</math> and :<math>\tilde{R} = \frac{\sqrt{2}R}{d}.</math> ==Mitigation==<!-- This section is linked from [[Transformer]] --> A type of cable called [[litz wire]] (from the [[German language|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 [[transformer]]s, to increase their efficiency by mitigating both skin effect and, more importantly, [[proximity effect (electromagnetism)|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 powerline|overhead power transmission lines]] often use [[Aluminium Conductor Steel Reinforced|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]]-[[Electroplating|plated]] providing a better conductor (the best possible conductor excepting only [[superconductor]]s) 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== [[Image:Vdorna globina.png|thumb|right|250px|Skin depths for some metals]] In copper, the skin depth at various frequencies is shown below. {| class="wikitable" |- !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. ==See also== *[[Proximity effect (electromagnetism)]] *[[skin depth]] *[[Tesla coil#The Skin Effect Myth|"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== *[http://www.st-andrews.ac.uk/~jcgl/Scots_Guide/audio/skineffect/page1.html Skin Effect in HiFi Cables] *[http://www.audioholics.com/education/cables/skin-effect-relevance-in-speaker-cables Skin Effect Relevance in Speaker Cables] [[Category:Electronics terms]] [[de:Skineffekt]] [[es:Efecto pelicular]] [[fr:Effet de peau]] [[it:Effetto pelle]] [[lt:Paviršinis efektas]] [[nl:Skineffect]] [[ja:表皮効果]] [[pl:Naskórkowość]] [[pt:Efeito pelicular]] [[ru:Скин-эффект]] [[sk:Skinefekt]] [[sl:Kožni pojav]] [[ta:மின் புறவோட்டம்]] [[vi:Hiệu ứng bề mặt]] [[zh:集膚效應]]