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Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. Skin depth depends on the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of the alternating current; as frequency increases, current flow moves to the surface, resulting in less skin depth. Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance. Skin effect is caused by opposing
eddy current Eddy currents (also called Foucault's currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction or by the relative motion of a conductor in a mag ...
s induced by the changing
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
field resulting from the alternating current. At 60 Hz in
copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...
, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller. Increased AC resistance caused by the skin effect can be mitigated by using specially woven
litz wire Litz wire is a particular type of multistrand wire or cable used in electronics to carry alternating current (AC) at radio frequencies. The wire is designed to reduce the skin effect and proximity effect losses in conductors used at frequencies ...
. Because the interior of a large conductor carries so little of the current, tubular conductors such as pipe can be used to save weight and cost. The skin effect has practical consequences in the analysis and design of
radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
-frequency and
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
circuits, transmission lines (or waveguides), and antennas. It is also important at mains frequencies (50–60 Hz) in AC electrical power transmission and distribution systems. It is one of the reasons for preferring
high-voltage direct current A high-voltage direct current (HVDC) electric power transmission system (also called a power superhighway or an electrical superhighway) uses direct current (DC) for electric power transmission, in contrast with the more common alternating curre ...
for long distance power transmission. The effect was first described in a paper by
Horace Lamb Sir Horace Lamb (27 November 1849 – 4 December 1934)R. B. Potts,, ''Australian Dictionary of Biography'', Volume 5, MUP, 1974, pp 54–55. Retrieved 5 Sep 2009 was a British applied mathematician and author of several influential texts on ...
in 1883 for the case of spherical conductors, and was generalized to conductors of any shape by
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
in 1885.


Cause

Conductors, typically in the form of wires, may be used to transmit electrical energy or signals using an
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
flowing through that conductor. The charge carriers constituting that current, usually
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s, are driven by an electric field due to the source of electrical energy. A current in a conductor produces a magnetic field in and around the conductor. When the intensity of current in a conductor changes, the magnetic field also changes. The change in the magnetic field, in turn, creates an electric field which opposes the change in current intensity. This opposing electric field is called “
counter-electromotive force Counter-electromotive force (counter EMF, CEMF, back EMF),Graf, "counterelectromotive force", Dictionary of Electronics is the electromotive force (EMF) manifesting as a voltage that opposes the change in current which induced it. CEMF is the EMF c ...
” (back EMF). The back EMF is strongest at the center of the conductor, and forces the conducting electrons to the outside of the conductor, as shown in the diagram on the right."These emf's are greater at the center than at the circumference, so the potential difference tends to establish currents that oppose the current at the center and assist it at the circumference" "To understand skin effect, you must first understand how eddy currents operate..." Regardless of the driving force, the current density is found to be greatest at the conductor's surface, with a reduced magnitude deeper in the conductor. That decline in current density is known as the ''skin effect'' and the ''skin depth'' is a measure of the depth at which the current density falls to 1/e of its value near the surface. Over 98% of the current will flow within a layer 4 times the skin depth from the surface. This behavior is distinct from that of
direct current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
which usually will be distributed evenly over the cross-section of the wire. An alternating current may also be ''induced'' in a conductor due to an alternating magnetic field according to the law of
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
. An
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
impinging on a conductor will therefore generally produce such a current; this explains the reflection of electromagnetic waves from metals. Although the term "skin effect" is most often associated with applications involving transmission of electric currents, the skin depth also describes the exponential decay of the electric and magnetic fields, as well as the density of induced currents, inside a bulk material when a plane wave impinges on it at normal incidence.


Formula

The AC current density in a conductor decreases exponentially from its value at the surface according to the depth from the surface, as follows: : J=J_\mathrm \,e^ where \delta is called the ''skin depth''. The skin depth is thus defined as the depth below the surface of the conductor at which the current density has fallen to 1/ e (about 0.37) of S. The imaginary part of the exponent indicates that the phase of the current density is delayed 1 radian for each skin depth of penetration. One full
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
in the conductor requires 2 skin depths, at which point the current density is attenuated to e−2 (1.87×, or −54.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, or equivalently, the phase velocity in the conductor is very much slower than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in a vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum of about 300 m, whereas in copper, the wavelength is reduced to only about 0.5 mm with a phase velocity of only about 500 m/s. As a consequence of
Snell's law Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
and this very tiny phase velocity in the conductor, any wave entering the conductor, even at grazing incidence, refracts essentially in the direction perpendicular to the conductor's surface. The general formula for the skin depth when there is no dielectric or magnetic loss is:The formula as shown is algebraically equivalent to the formula found on page 130 : \delta= \sqrt \; \sqrt ~ where : \rho =
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of the conductor : \omega =
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of current = 2\pi f ~, where f is the frequency. : \mu = permeability of the conductor, \mu_r \, \mu_0 : \mu_r = relative
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
of the conductor : \mu_0 = the
permeability of free space The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
: \varepsilon =
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the conductor, \varepsilon_r \, \varepsilon_0 : \varepsilon_r = relative
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the conductor : \varepsilon_0 = the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
At frequencies much below 1/(\rho \epsilon) the quantity inside the large radical is close to unity and the formula is more usually given as: : \delta=\sqrt ~. This formula is valid at frequencies away from strong atomic or molecular resonances (where \epsilon would have a large imaginary part) and at frequencies that are much below both the material's
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
(dependent on the density of free electrons in the material) and the reciprocal of the mean time between collisions involving the conduction electrons. In good conductors such as metals all of those conditions are ensured at least up to microwave frequencies, justifying this formula's validity.Note that the above equation for the current density inside the conductor as a function of depth applies to cases where the usual approximation for the skin depth holds. In the extreme cases where it doesn't, the exponential decrease with respect to the skin depth still applies to the ''magnitude'' of the induced currents, however the imaginary part of the exponent in that equation, and thus the phase velocity inside the material, are altered with respect to that equation. For example, in the case of copper, this would be true for frequencies much below  Hz. However, in very poor conductors, at sufficiently high frequencies, the factor under the large radical increases. At frequencies much higher than 1/(\rho \epsilon) it can be shown that the skin depth, rather than continuing to decrease, approaches an asymptotic value: : \delta \approx \sqrt ~. This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much larger than the skin depth itself. For instance, bulk silicon (undoped) is a poor conductor and has a skin depth of about 40 meters at 100 kHz ( = 3 km). However, as the frequency is increased well into the megahertz range, its skin depth never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors, such as undoped silicon, the skin effect doesn't need to be taken into account in most practical situations: Any current is equally distributed throughout the material's cross-section, regardless of its frequency.


Current density in a round conductor

When the skin depth is not small with respect to the radius of the wire, current density may be described in terms of
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by: : \mathbf_r = \frac \frac = \mathbf_R \frac where : \quad \omega =
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of current = 2π × frequency : \quad r = distance from the axis of the wire : \quad R = radius of the wire : \quad \mathbf_r = current density phasor at distance, r, from the axis of the wire : \quad \mathbf_R = current density phasor at the surface of the wire : \quad \mathbf = total current phasor : \quad J_0 = Bessel function of the first kind, order 0 : \quad J_1 = Bessel function of the first kind, order 1 : \quad k = \sqrt = \frac the
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
in the conductor : \quad \delta = \sqrt also called skin depth. : \quad \rho =
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of the conductor : \quad \mu_r = relative
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
of the conductor : \quad \mu_0 = the
permeability of free space The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
= 4π x 10−7 H/m : \quad \mu = \mu_r \mu_0 Since k is complex, the Bessel functions are also complex. The amplitude and phase of the current density varies with depth.


Impedance of round wire

The ''internal'' impedance per unit length of a segment of round wire is given by: : \mathbf_ = \frac \frac . This impedance is a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
quantity corresponding to a resistance (real) in series with the reactance (imaginary) due to the wire's internal self-
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
, per unit length.


Inductance

A portion of a wire's inductance can be attributed to the magnetic field ''inside'' the wire itself which is termed the ''internal inductance''; this accounts for the inductive reactance (imaginary part of the impedance) given by the above formula. In most cases this is a small portion of a wire's inductance which includes the effect of
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
from magnetic fields ''outside'' of the wire produced by the current in the wire. Unlike that ''external'' inductance, the internal inductance is reduced by the skin effect, that is, at frequencies where the skin depth is no longer large compared to the conductor's size. This small component of inductance approaches a value of \frac (50 nH/m for non-magnetic wire) at low frequencies, regardless of the wire's radius. Its reduction with increasing frequency, as the ratio of the skin depth to the wire's radius falls below about 1, is plotted in the accompanying graph, and accounts for the reduction in the telephone cable inductance with increasing frequency in the table below.


Resistance

The most important effect of the skin effect on the impedance of a single wire, however, is the increase of the wire's resistance, and consequent losses. The effective resistance due to a current confined near the surface of a large conductor (much thicker than ) can be solved as if the current flowed uniformly through a layer of thickness based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter large compared to , has a resistance ''approximately'' that of a hollow tube with wall thickness carrying direct current. The AC resistance of a wire of length and resistivity \rho is: : R\approx \approx The final approximation above assumes D \gg \delta. A convenient formula (attributed to F.E. Terman) for the diameter of a wire of circular cross-section whose resistance will increase by 10% at frequency is: : D_\mathrm = This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a
cable Cable may refer to: Mechanical * Nautical cable, an assembly of three or more ropes woven against the weave of the ropes, rendering it virtually waterproof * Wire rope, a type of rope that consists of several strands of metal wire laid into a hel ...
or a coil, the AC resistance is also affected by
proximity effect Proximity effect may refer to: * Proximity effect (atomic physics) * Proximity effect (audio), an increase in bass or low frequency response when a sound source is close to a microphone * ''Proximity Effect'' (comics), a comic book series written by ...
, which can cause an additional increase in the AC resistance.


Material effect on skin depth

In a good conductor, skin depth is proportional to square root of the resistivity. This means that better conductors have a reduced skin depth. The overall resistance of the better conductor remains lower even with the reduced skin depth. However the better conductor will show a higher ratio between its AC and DC resistance, when compared with a conductor of higher resistivity. For example, at 60 Hz, a 2000 MCM (1000 square millimeter) copper conductor has 23% more resistance than it does at DC. The same size conductor in aluminum has only 10% more resistance with 60 Hz AC than it does with DC. Skin depth also varies as the inverse square root of the permeability of the conductor. In the case of iron, its conductivity is about 1/7 that of copper. However being
ferromagnetic Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220
micrometer Micrometer can mean: * Micrometer (device), used for accurate measurements by means of a calibrated screw * American spelling of micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures; ...
s at 60 Hz. Iron wire is thus useless for AC power lines (except to add mechanical strength by serving as a core to a non ferromagnetic conductor like aluminum). The skin effect also reduces the effective thickness of
lamination Lamination is the technique/process of manufacturing a material in multiple layers, so that the composite material achieves improved strength, stability, sound insulation, appearance, or other properties from the use of the differing materials ...
s in power transformers, increasing their losses. Iron rods work well for
direct-current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
(DC)
welding Welding is a fabrication (metal), fabrication process that joins materials, usually metals or thermoplastics, by using high heat to melt the parts together and allowing them to cool, causing Fusion welding, fusion. Welding is distinct from lower ...
but it is impossible to use them at frequencies much higher than 60 Hz. At a few kilohertz, the welding rod will glow red hot as current flows through the greatly increased AC resistance resulting from the skin effect, with relatively little power remaining for the arc itself. Only
non-magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
rods can be used for high-frequency welding. At 1 megahertz the skin effect depth in wet soil is about 5.0 m; in seawater it is about 0.25 m.


Mitigation

A type of cable called
litz wire Litz wire is a particular type of multistrand wire or cable used in electronics to carry alternating current (AC) at radio frequencies. The wire is designed to reduce the skin effect and proximity effect losses in conductors used at frequencies ...
(from the
German German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
''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. With the skin effect having little effect on each of the thin strands, the bundle does not suffer the same increase in AC resistance that a solid conductor of the same cross-sectional area would due to the skin effect. Litz wire is often used in the windings of high-frequency
transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
s to increase their efficiency by mitigating both skin effect and
proximity effect Proximity effect may refer to: * Proximity effect (atomic physics) * Proximity effect (audio), an increase in bass or low frequency response when a sound source is close to a microphone * ''Proximity Effect'' (comics), a comic book series written by ...
. Large power transformers are wound with stranded conductors of similar construction to litz wire, but employing a larger cross-section corresponding to the larger skin depth at mains frequencies. Conductive threads composed of
carbon nanotube A scanning tunneling microscopy image of a single-walled carbon nanotube Rotating single-walled zigzag carbon nanotube A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers. ''Single-wall carbon na ...
s have been demonstrated as conductors for antennas from medium wave to microwave frequencies. Unlike standard antenna conductors, the nanotubes are much smaller than the skin depth, allowing full utilization of the thread's cross-section resulting in an extremely light antenna. High-voltage, high-current
overhead power line An overhead power line is a structure used in electric power transmission and distribution to transmit electrical energy across large distances. It consists of one or more uninsulated electrical cables (commonly multiples of three for three-p ...
s often use aluminum cable with a steel reinforcing core; the higher resistance of the steel core is of no consequence since it is located far below the skin depth where essentially no AC current flows. In applications where high currents (up to thousands of amperes) flow, solid conductors are usually replaced by tubes, completely eliminating the inner portion of the conductor where little current flows. This hardly affects the AC resistance, but considerably reduces the weight of the conductor. The high strength but low weight of tubes substantially increases span capability. Tubular conductors are typical in electric power switchyards where the distance between supporting insulators may be several meters. Long spans generally exhibit physical sag but this does not affect electrical performance. To avoid losses, the conductivity of the tube material must be high. In high current situations where conductors (round or flat
busbar In electric power distribution, a busbar (also bus bar) is a metallic strip or bar, typically housed inside switchgear, panel boards, and busway enclosures for local high current power distribution. They are also used to connect high volt ...
) may be between 5 and 50 mm thick the skin effect also occurs at sharp bends where the metal is compressed inside the bend and stretched outside the bend. The shorter path at the inner surface results in a lower resistance, which causes most of the current to be concentrated close to the inner bend surface. This causes an increase in temperature at that region compared with the straight (unbent) area of the same conductor. A similar skin effect occurs at the corners of rectangular conductors (viewed in cross-section), where the magnetic field is more concentrated at the corners than in the sides. This results in superior performance (i.e. higher current with lower temperature rise) from wide thin conductors (for example, "ribbon" conductors) in which the effects from corners are effectively eliminated. It follows that a transformer with a round core will be more efficient than an equivalent-rated transformer having a square or rectangular core of the same material. Solid or tubular conductors may be
silver Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...
-
plated Plating is a surface covering in which a metal is deposited on a conductive surface. Plating has been done for hundreds of years; it is also critical for modern technology. Plating is used to decorate objects, for corrosion inhibition, to improv ...
to take advantage of silver's higher conductivity. This technique is particularly used at VHF to
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
frequencies where the small skin depth requires only a very thin layer of silver, making the improvement in conductivity very cost effective. Silver plating is similarly used on the surface of waveguides used for transmission of microwaves. This reduces attenuation of the propagating wave due to resistive losses affecting the accompanying eddy currents; the skin effect confines such eddy currents to a very thin surface layer of the waveguide structure. The skin effect itself isn't actually combatted in these cases, but the distribution of currents near the conductor's surface makes the use of precious metals (having a lower resistivity) practical. Although it has a lower conductivity than copper and silver, gold plating is also used, because unlike copper and silver, it does not corrode. A thin oxidized layer of copper or silver would have a low conductivity, and so would cause large power losses as the majority of the current would still flow through this layer. Recently, a method of layering non-magnetic and ferromagnetic materials with nanometer scale thicknesses has been shown to mitigate the increased resistance from the skin effect for very high frequency applications. A working theory is that the behavior of ferromagnetic materials in high frequencies results in fields and/or currents that oppose those generated by relatively nonmagnetic materials, but more work is needed to verify the exact mechanisms. As experiments have shown, this has potential to greatly improve the efficiency of conductors operating in tens of GHz or higher. This has strong ramifications for 5G communications.


Examples

We can derive a practical formula for skin depth as follows: : \delta=\frac = \sqrt = :: = \frac \approx 503\,\sqrt \approx 503\,\frac, where : \delta = the skin depth in meters :\alpha = the attenuation in \frac :\mu_0 = the permeability of free space : \mu_r = the
relative permeability In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be ...
of the medium (for copper, \mu_r = ) :\mu = the permeability of the medium : \rho = the resistivity of the medium in Ω·m, also equal to the reciprocal of its conductivity: \rho = \frac (for copper, ρ = ) :\sigma = the conductivity of the medium (for copper, \sigma \approx ) : f = the frequency of the current in Hz
Gold Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile met ...
is a good conductor with a resistivity of and is essentially nonmagnetic: \mu_r = 1, so its skin depth at a frequency of 50 Hz is given by : \delta = 503 \,\sqrt= 11.1\,\mathrm
Lead Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cu ...
, in contrast, is a relatively poor conductor (among metals) with a resistivity of , about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or \sqrt = 3 times that of gold. Highly magnetic materials have a reduced skin depth owing to their large permeability \mu_r as was pointed out above for the case of iron, despite its poorer conductivity. A practical consequence is seen by users of induction cookers, where some types of
stainless steel Stainless steel is an alloy of iron that is resistant to rusting and corrosion. It contains at least 11% chromium and may contain elements such as carbon, other nonmetals and metals to obtain other desired properties. Stainless steel's corros ...
cookware are unusable because they are not ferromagnetic. At very high frequencies the skin depth for good conductors becomes tiny. For instance, the skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a
micrometer Micrometer can mean: * Micrometer (device), used for accurate measurements by means of a calibrated screw * American spelling of micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures; ...
: Thus at
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver 3  μm thick evaporated on a piece of glass is thus an excellent conductor at such frequencies. In copper, the skin depth can be seen to fall according to the square root of frequency: In ''Engineering Electromagnetics'', Hayt points out that in a power station a
busbar In electric power distribution, a busbar (also bus bar) is a metallic strip or bar, typically housed inside switchgear, panel boards, and busway enclosures for local high current power distribution. They are also used to connect high volt ...
for
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
at 60 Hz with a radius larger than one-third of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons.


Skin effect reduction of the internal inductance of a conductor

Refer to the diagram below showing the inner and outer conductors of a coaxial cable. Since the skin effect causes a current at high frequencies to flow mainly at the surface of a conductor, it can be seen that this will reduce the magnetic field ''inside'' the wire, that is, beneath the depth at which the bulk of the current flows. It can be shown that this will have a minor effect on the self-inductance of the wire itself; see Skilling or Hayt for a mathematical treatment of this phenomenon. The inductance considered in this context refers to a bare conductor, not the inductance of a coil used as a circuit element. The inductance of a coil is dominated by the mutual inductance between the turns of the coil which increases its inductance according to the square of the number of turns. However, when only a single wire is involved, then in addition to the "external inductance" involving magnetic fields outside of the wire (due to the total current in the wire) as seen in the white region of the figure below, there is also a much smaller component of "internal inductance" due to the portion of the magnetic field inside the wire itself, the green region in figure B. That small component of the inductance is reduced when the current is concentrated toward the skin of the conductor, that is, when the skin depth is not much larger than the wire's radius, as will become the case at higher frequencies. For a single wire, this reduction becomes of diminishing significance as the wire becomes longer in comparison to its diameter, and is usually neglected. However the presence of a second conductor in the case of a transmission line reduces the extent of the external magnetic field (and of the total self-inductance) regardless of the wire's length, so that the inductance decrease due to the skin effect can still be important. For instance, in the case of a telephone twisted pair, below, the inductance of the conductors substantially decreases at higher frequencies where the skin effect becomes important. On the other hand, when the external component of the inductance is magnified due to the geometry of a coil (due to the mutual inductance between the turns), the significance of the internal inductance component is even further dwarfed and is ignored.


Inductance per length in a coaxial cable

Let the dimensions ''a'', ''b'', and ''c'' be the inner conductor radius, the shield (outer conductor) inside radius and the shield outer radius respectively, as seen in the crossection of figure A below. For a given current, the total energy stored in the magnetic fields must be the same as the calculated electrical energy attributed to that current flowing through the inductance of the coax; that energy is proportional to the cable's measured inductance. The magnetic field inside a coaxial cable can be divided into three regions, each of which will therefore contribute to the electrical inductance seen by a length of cable. The inductance L_\text \, is associated with the magnetic field in the region with radius r < a \, , the region inside the center conductor. The inductance L_\text \, is associated with the magnetic field in the region a < r < b \, , the region between the two conductors (containing a dielectric, possibly air). The inductance L_\text \, is associated with the magnetic field in the region b < r < c \, , the region inside the shield conductor. The net electrical inductance is due to all three contributions: : L_\text = L_\text + L_\text + L_\text\, L_\text \, is not changed by the skin effect and is given by the frequently cited formula for inductance ''L'' per length ''D'' of a coaxial cable: : L/D = \frac \ln \left( \frac \right) \, At low frequencies, all three inductances are fully present so that L_\text = L_\text + L_\text + L_\text\, . At high frequencies, only the dielectric region has magnetic flux, so that L_\infty = L_\text\, . Most discussions of coaxial transmission lines assume they will be used for radio frequencies, so equations are supplied corresponding only to the latter case. As the skin effect increases, the currents are concentrated near the outside of the inner conductor (''r''=''a'') and the inside of the shield (''r''=''b''). Since there is essentially no current deeper in the inner conductor, there is no magnetic field beneath the surface of the inner conductor. Since the current in the inner conductor is balanced by the opposite current flowing on the inside of the outer conductor, there is no remaining magnetic field in the outer conductor itself where b < r < c \, . Only L_\text contributes to the electrical inductance at these higher frequencies. Although the geometry is different, a twisted pair used in telephone lines is similarly affected: at higher frequencies the inductance decreases by more than 20% as can be seen in the following table.


Characteristics of telephone cable as a function of frequency

Representative parameter data for 24 gauge PIC telephone cable at . More extensive tables and tables for other gauges, temperatures and types are available in Reeve. Chen gives the same data in a parameterized form that he states is usable up to 50 MHz. Chen gives an equation of this form for telephone twisted pair: : L(f) = \frac \,


Anomalous skin effect

For high frequencies and low temperatures the usual formulas for skin depth break down. This effect was first noticed by
Heinz London Heinz London (Bonn, Germany 7 November 1907 – 3 August 1970) was a German-British physicist. Together with his brother Fritz London he was a pioneer in the field of superconductivity. Biography London was born in Bonn in a liberal Jewish-Ge ...
in 1940, who correctly suggested that it is due to the mean free path length of the electrons reaching the range of the classical skin depth.R. G. Chambers, ''The Anomalous Skin Effect'', Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 215, No. 1123 (Dec. 22, 1952), pp. 481-497 (17 pages) https://www.jstor.org/stable/99095
Mattis–Bardeen theory The Mattis–Bardeen theory is a theory that describes the electrodynamic properties of superconductivity. It is commonly applied in the research field of optical spectroscopy on superconductors. It was derived to explain the anomalous skin effect ...
was developed for this specific case for metals and
superconductors Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
.


See also

*
Proximity effect (electromagnetism) In electromagnetics, proximity effect is a redistribution of electric current occurring in nearby parallel electrical conductors carrying alternating current flowing in the same direction which causes the current distribution in the conductor to ...
*
Penetration depth Penetration depth is a measure of how deep light or any electromagnetic radiation can penetrate into a material. It is defined as the depth at which the intensity of the radiation inside the material falls to 1/e (about 37%) of its original valu ...
*
Eddy current Eddy currents (also called Foucault's currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction or by the relative motion of a conductor in a mag ...
s *
Litz wire Litz wire is a particular type of multistrand wire or cable used in electronics to carry alternating current (AC) at radio frequencies. The wire is designed to reduce the skin effect and proximity effect losses in conductors used at frequencies ...
*
Transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
*
Induction cooking Induction cooking is performed using direct induction heating of cooking vessels, rather than relying on indirect radiation, convection, or thermal conduction. Induction cooking allows high power and very rapid increases in temperature to be ach ...
*
Induction heating Induction heating is the process of heating electrically conductive materials, namely metals or semi-conductors, by electromagnetic induction, through heat transfer passing through an induction coil that creates an electromagnetic field within th ...
*
Magnetic Reynolds number In magnetohydrodynamics, the magnetic Reynolds number (Rm) is a dimensionless quantity that estimates the relative effects of advection or induction of a magnetic field by the motion of a conducting medium to the magnetic diffusion. It is the mag ...
* Wheeler Incremental Inductance Rule, a method for estimating skin effect resistance


Notes


References

* * * * Nahin, Paul J. ''Oliver Heaviside: Sage in Solitude''. New York: IEEE Press, 1988. . * Ramo, S., J. R. Whinnery, and T. Van Duzer. ''Fields and Waves in Communication Electronics''. New York: John Wiley & Sons, Inc., 1965. * * * * * * * *


External links


Conductor Bulk Resistivity & Skin Depths
{{Authority control Electromagnetism Electrical parameters