TheInfoList

Skin effect is the tendency of an alternating electric current (AC) to become distributed within a
conductor Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble like, for example, an orchestra. * Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of ...
such that the
current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...

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 A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparent ...

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 Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
. Skin effect is caused by opposing eddy currents induced by the changing
magnetic Magnetism is a class of physical attributes that are mediated by s. s and the s of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomenon of . The ...

field resulting from the alternating current. At 60 in
copper Copper is a chemical element In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elem ...

, 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 A wire is a single usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads or electricity Electricity is the set of physical phenomena associa ...
. 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 telecommunication, 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 ...

-frequency and
microwave Microwave is a form of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

circuits, transmission lines (or waveguides), and antennas. It is also important at mains frequencies (50–60 Hz) in AC systems. It is one of the reasons for preferring
high-voltage direct current A high-voltage, direct current (HVDC) electric power transmission Electric power transmission is the bulk movement of electrical energy Electrical energy is energy derived from electric potential energy or kinetic energy. When used loosely, ...
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 The ''Australian Dictionary of Biography'' (ADB or AuDB) is a national co-operative enterprise founded and maintained by the Australi ...
in 1883 for the case of spherical conductors, and was generalised to conductors of any shape by
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English autodidactic Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education without the guidance of masters (such as teach ...
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 An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'natu ...
flowing through that conductor. The charge carriers constituting that current, usually
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

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),Graf, "counterelectromotive force", Dictionary of Electronics also known as back electromotive force (back EMF), is the electromotive force In electromagnetism Electromagnetism is a branch of p ...
” (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 In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...

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 Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (math ...
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 may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the t ...

. An
electromagnetic wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

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 electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...

in a conductor 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

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 Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gr ...

, or equivalently, the
phase velocity The phase velocity of a wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or ...
in the conductor is very much slower than the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
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 of light at the interface between two media of different refractive index, refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of in ...

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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
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 Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
of the conductor : $\mu_0$ = the
permeability of free space Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field pro ...
: $\varepsilon$ =
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
of the conductor, $\varepsilon_r \, \varepsilon_0$ : $\varepsilon_r$ = relative
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
of the conductor : $\varepsilon_0$ = the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
At frequencies much below $1/\left(\rho \epsilon\right)$ 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 frequencyPlasma oscillations, also known as Langmuir waves (after Irving Langmuir Irving Langmuir (; 31 January 1881 – 16 August 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work ...
(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 $1/\left(\rho \epsilon\right)$ 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 In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...

may be described in terms of
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematici ...
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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
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 In and , a phasor (a of phase vector), is a representing a whose (''A''), (''ω''), and (''θ'') are . It is related to a more general concept called ,Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p2 ...

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 science Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the ...
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 Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
of the conductor : $\quad \mu_0$ = the
permeability of free space Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field pro ...
= 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 The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

quantity corresponding to a resistance (real) in series with the (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 c ...
, 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 may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the 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 A nautical cable is a band of tightly woven and clamped ropes, of a defined cable length, used during the age of sail for deep water anchoring, heavy lifting, ship to ship transfers and towing durin ...
or a coil, the AC resistance is also affected by proximity effect, 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 millimetre) 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 the basic mechanism by which certain materials (such as iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), formi ...
its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220
micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental orga ...
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 250px, Laminate flooring Lamination is the technique/process of manufacturing a Raw material, material in multiple layers, so that the composite material achieves improved strength of materials, strength, stability, sound insulation, visual ...

s in power transformers, increasing their losses. Iron rods work well for
direct-current Direct current (DC) is the one directional flow of electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ''posit ...
(DC)
welding Welding is a process that joins materials, usually s or s, by using high to melt the parts together and allowing them to cool, causing . Welding is distinct from lower temperature metal-joining techniques such as and , which do not the base ...

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 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 A wire is a single usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads or electricity Electricity is the set of physical phenomena associa ...
(from the
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

''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 Electrical network, circuits. A varying current in any one coil of the transformer produces a varying magnetic flux ...

s to increase their efficiency by mitigating both skin effect and proximity effect. 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 image of a single-walled carbon nanotube. Carbon nanotubes (CNTs) are tubes made of carbon with diameters typically measured in nanometers The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI ...

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 distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions ...

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 , a busbar (also bus bar) is a metallic strip or bar, typically housed inside , , and for local high current power distribution. They are also used to connect high voltage equipment at electrical switchyards, and low voltage equipment i ...

) 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 In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical ele ...

-
plated Plating is a surface covering in which a metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and co ...
to take advantage of silver's higher conductivity. This technique is particularly used at
VHF Very high frequency (VHF) is the designation for the range of s (s) from 30 to 300 (MHz), with corresponding wavelengths of ten meters to one meter. Frequencies immediately below VHF are denoted (HF), and the next higher frequencies are kn ...
to
microwave Microwave is a form of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

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 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 In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elemen ...

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 In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elements ...

, 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 cooker 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 achi ...
s, where some types of
stainless steel Stainless steel is a group of ferrous alloys that contain a minimum of approximately 11% chromium Chromium is a chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistr ...
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
micrometerMicrometer can mean: * Micrometer (device) A micrometer, sometimes known as a micrometer screw gauge, is a device incorporating a calibrated screw widely used for Accuracy and precision, accurate measurement of components in mechanical engineeri ...
: Thus at
microwave Microwave is a form of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

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 , a busbar (also bus bar) is a metallic strip or bar, typically housed inside , , and for local high current power distribution. They are also used to connect high voltage equipment at electrical switchyards, and low voltage equipment i ...

for
alternating current Alternating current (AC) is an electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'natu ...
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. Note that 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: :$L/D = \frac \ln \left\left( \frac \right\right) \,$ 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: : $L\left(f\right) = \frac \,$

*
Proximity effect (electromagnetism) In a conductor carrying 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 Direct current (DC) is the one d ...
*
Penetration depth Penetration depth is a measure of how deep light or any electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the n ...
* Eddy currents *
Litz wire Litz can refer to: * Litz wire Litz wire is a particular type of multistrand wire A wire is a single usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads or electricity Electricity is the se ...
*
Transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple Electrical network, circuits. A varying current in any one coil of the transformer produces a varying magnetic flux ...

*
Induction cooking Induction cooking is performed using direct induction heating of cooking vessels, rather than relying on indirect radiation, Convection (heat transfer), convection, or thermal conduction. Induction cooking allows high power and very rapid increase ...
*
Induction heating Induction heating is the process of heating electrically conductive materials like metals by electromagnetic induction Electromagnetic or magnetic induction is the production of an electromotive force In electromagnetism Electromagne ...

*
Magnetic Reynolds numberThe magnetic Reynolds number (Rm) is the magnetic analogue of the Reynolds number, a fundamental dimensionless group that occurs in magnetohydrodynamics Magnetohydrodynamics (MHD; also magneto-fluid dynamics or hydro­magnetics) is the study ...
* , a method for estimating skin effect resistance

# 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. * * * * * * * *