Skin effect

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The skin effect is the propensity of alternating electric current (AC) to be distributed inside the conductor so that the greatest current density is near the surface of the conductor, and decreases with greater depth in the conductor. The electric current flows primarily to the "skin" conductor, between the outer surface and the level called skin depth . The skin effect causes the effective resistance of the conductor to increase at a higher frequency where the skin depth is smaller, thereby reducing the effective conductor cross section. The skin effect is caused by the opposite eddy current induced by the changing magnetic field generated from alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies, the depth of the skin becomes much smaller. Increased AC resistance due to skin effect can be reduced by using special woven wire. Because large interior conductors carry so little current, tubular conductors such as pipes can be used to save weight and cost.

Video Skin effect

Penyebab

The conductor, usually in cable form, can be used to transmit electrical energy or signals using alternating current flowing through the conductor. The charge carrier states that a current, usually an electron, is driven by an electric field due to a source of electrical energy. The alternating current in the conductor produces a magnetic field alternating in and around the conductor. When the intensity of the current in the conductor changes, the magnetic field also changes. Changes in the magnetic field, in turn, create an electric field that opposes changes in the current intensity. This opposite electric field is called "counter-electromotive force" (rear EMF). The rear EMF is the strongest at the center of the conductor, and forces the conductor electrons out the conductor, as shown in the diagram on the right.

Alternating current can also be induced in the conductor because the magnetic field alternates according to the law of induction. Electromagnetic waves that hit the conductor will produce such currents; this explains the reflection of electromagnetic waves of metal.

Regardless of the driving force, the current density is found to be greatest on the surface of the conductor, with the magnitude being reduced more deeply to the conductor. The current density decrease is known as skin effect and skin depth is a measure of the depth at which the current density falls to 1/e of its value near the surface. More than 98% of the current will flow in the layer 4 times the depth of the skin from the surface. This behavior is different from the direct current behavior that would normally be distributed evenly over the wire cross section.

The effect was first described in a paper by Horace Lamb in 1883 for the case of a round conductor, and generalized to conductors of any kind by Oliver Heaviside in 1885. The skin effect has practical consequences in the analysis and design of radio frequencies and microwave circuits, or waveguides), and antennas. It is also important at the main frequency (50-60 Hz) in the transmission and distribution system of AC power. Although the term "skin effect" is most often associated with applications involving the transmission of electrical current, the depth of the skin also describes the exponential decay of electric and magnetic fields, as well as the induced current density, in bulk materials when the plane waves override at normal events.

Maps Skin effect

Formula

Densitas arus AC J dalam konduktor menurun secara eksponensial dari nilainya di permukaan J _S menurut kedalaman d dari permukaan, sebagai berikut:

${\ displaystyle J = J _ {\ mathrm {S}} \, e ^ {- {(1 j) d/\ delta}}}  ÂÂ$

where ? is called skin depth . The skin depth is thus defined as the depth below the surface of the conductor where the current density falls to 1/e (about 0.37) of J _S . The imaginary part of the exponent shows that the phase of the current density is delayed 1 radian for each depth of skin penetration. One full wavelength in a conductor takes 2? the depth of the skin, at which point the current density is attenuated to e ^-2? (-54.6Ã, dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in a vacuum, or the equivalent, the phase velocity in the conductor is much slower than the speed of light in a vacuum. For example, 1 MHz radio waves have wavelengths in a vacuum? ₀ is 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 and this very small phase velocity in the conductor, any wave entering the conductor, even in the pasture incidence, the refraction is essentially perpendicular to the surface of the conductor.

Rumus umum untuk kedalaman kulit adalah:

${\ displaystyle \ delta = {\ sqrt {{\ rho} \ over {\ omega \ mu}}} \; \; {\ sqrt {{\ sqrt {1 \ kiri ({\ rho \ omega \ epsilon} \ right) ^ {2}}} \ rho \ omega \ epsilon}}}  ÂÂ$

Pada frekuensi jauh di bawah ${\ displaystyle 1/\ rho \ epsilon}  ÂÂ$ kuantitas di dalam radikal besar dekat dengan kesatuan dan rumus biasanya lebih diberikan sebagai:

${\ displaystyle \ delta = {\ sqrt {{\ rho} \ over {\ omega \ mu}}}}  ÂÂ$ .

This formula applies at frequencies far from strong atomic or molecular resonances (where ${\ displaystyle \ epsilon}$ will have a large imaginary part) and at frequencies far below both the plasma frequency of the material (depending on the free electron density in the material) and the reciprocity of the time the bridge between collisions involving conduction electrons. In good conductors such as metals all such conditions are ascertained at least to microwave frequencies, justifying the validity of this formula. For example, in case of copper, this would be true for frequencies well below 10 ¹⁸ Ã, Hz.

However, on a very bad conductor, at a fairly high frequency, the factor is below the large radical increase. At frequencies are much higher than ${\ displaystyle 1/\ rho \ epsilon}$ may indicate that the depth of the skin, rather than continuing to decline, approximates the asymptotic value:

${\ displaystyle \ delta \ roughly {2 \ rho} {\ sqrt {\ epsilon \ over} Â Â$

This departure from the usual formula only applies to materials whose conductivity is rather low and at frequencies where the wavelength of the vacuum is not much greater than the depth of the skin itself. For example, undoped silicon is a poor conductor and has a skin depth of about 40 meters at 100 kHz (? = 3000 m). However, since the frequency is well-increased to the megahertz range, the depth of the skin never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors such as unsoped silicon, skin effects need not be taken into account in most practical situations: each current is equally distributed across the material section regardless of its frequency.

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Current density in spherical conductor

When the skin depth is not small in relation to the radius of the wire, the current density can be explained in terms of the Bessel function. The current density of the circular wire is far from the influence of another field, as the distance function of the axis is given by Sunday.

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathbf{J}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂrÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂkÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathbf{I}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂRÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂJ_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â0Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂkÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âr}{}}$                                                       J                                   1                                             ()               k               R                                       =                              J                                R                                                                               J                                   0                                             ()               k               r                                                       J                                   0                                             ()               k               R                                         {\ displaystyle \ mathbf {J} _ {r} = {\ frac {k \ mathbf {I}} {2 \ pi R}} {\ frac {J_ {0} (kr)} {J_ {1} (kR)}} = \ mathbf {J} _ {R} {\ frac {J_ {0} (kr)} {J_ {0} (kR)} }}  Â

Karena ${\ displaystyle k}  ÂÂ$ sangat kompleks, fungsi Bessel juga kompleks. Amplitudo dan fase kerapatan arus bervariasi dengan kedalaman.

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Impedansi kabel bulat

The internal impedansi per satuan panjang segmen kawat bulat diberikan oleh:

${\ displaystyle \ mathbf {Z} _ {int} = {\ frac {k \ rho} {2 \ pi R}} {\ frac {J_ {0} (kR )} {J_ {1} (kR)}}}  ÂÂ$ .

This impedance is a complex quantity corresponding to the (apparent) resistance in series with reactance (imaginary) because of the internal inductance of the wire itself, per unit length.

Inductance

A portion of the wire inductance can be attributed to the magnetic field inside the wire itself called the internal inductance ; this account is for inductive reactance (the imaginary part of the impedance) given by the above formula. In most cases this is a small part of the wire inductance which includes the induced effect of the magnetic field 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 the frequency at which the skin depth is no longer large compared to the size of the conductor. This small component of the inductance is close to the value of ${\ displaystyle {\ frac {\ mu} {8 \ pi}}}$ at low frequency, regardless of the radius of the wire. Reduced with increasing frequency, since the ratio of skin depth to the radius of the wire drops below about 1, plotted in the accompanying graph, and contributes to the reduction of 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 in wire resistance, and its consequent losses. Resistance is effective because the limited current near the surface of the large conductor (much thicker than ? ) can be solved as the current flows evenly through the thickness layer . based on the DC resistivity of the material. The effective cross-sectional area is approximately equal to ? times the circumference of the conductor. So a long cylindrical conductor like a wire, has a large diameter D compared to ? , has resistance approximately that a vacuum tube with wall thickness ? bring direct current. Air resistance from long cable L and resistivity ${\ displaystyle \ rho}$ are:

Pendekatan terakhir di atas mengasumsikan ${\ displaystyle D \ gg \ delta}  ÂÂ$ .

Rumus yang nyaman (dikaitkan dengan FE Terman) untuk diameter D _W dari kawat penampang melingkar yang resistensinya akan meningkat 10% pada frekuensi f adalah:

${\ displaystyle D _ {\ mathrm {W}} = {\ frac {200 ~ \ mathrm {mm}} {\ sqrt {f/\ mathrm {Hz}}}} }  ÂÂ$

This formula for increasing AC resistance is only accurate for isolated wires. For nearby cables, e.g. in a cable or coil, the AC resistance is also affected by the proximity effect, which can lead to an additional increase in the AC resistance.

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Effect of material on skin depth

In good conductors, the depth of the skin is proportional to the square root of resistivity. This means that the better conductor has a reduced skin depth. Overall better conductor resistance remains lower even with reduced skin depth. But a better conductor will show a higher ratio between AC and DC resistance, when compared to higher resistivity conductors. For example, at 60 Hz, a 2000 MCM copper conductor (1000 millimeters square) has 23% more resistance than in DC. The size of the same conductor in aluminum has only 10% higher resistance with AC 60 Hz than with DC.

The depth of the skin also varies as the inverse square root of the conductor permeability. In terms of iron, its conductivity is about 1/7 of the copper. However, the ferromagnetic permeability is about 10,000 times larger. This reduces the depth of the skin to iron to about 1/38 of copper, about 220 micrometers at 60 Hz. Iron wire is thus useless for AC power lines (except for adding mechanical strength by serving as the core for non ferromagnetic conductors such as aluminum). The skin effect also reduces the effective thickness of the laminate in the power transformer, increasing their losses.

Iron rods work well for direct current welding (DC) but it is not possible to use them at frequencies much higher than 60 Hz. At some kilohertz, the welding rod will glow red hot as the current flows through the 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 depth of skin effect in wet soil is about 5.0 m, in sea water about 0.25 m.

src: www.zettlermagnetics.com

Mitigation

A type of cable called litz wire (from German Litzendraht, wire mesh) is used to reduce the skin effect to a frequency of several kilohertz to about one megahertz. It consists of a number of isolated wire strands woven together in a carefully designed pattern, so that the overall magnetic field acts the same on all cables and causes the total current to be distributed evenly between them. With skin effect having little effect on every thin strand, the bundle does not experience the same increase in AC resistance that the solid conductor of the same cross-sectional area will be caused by skin effect.

Wire Litz is often used in high frequency transformer scrolls to improve their efficiency by reducing skin effect and proximity effect. The large power transformer is wrapped in a wound conductor with a construction similar to the litz wire, but uses a larger cross section according to the greater skin depth at the main frequency. Conductive yarns comprising carbon nanotubes have been demonstrated as conductors for antennas from mid-to-microwave waves. Unlike standard antenna conductors, the nanotubes are much smaller than the depth of the skin, allowing full utilization of the cross-section of the yarn to produce a very lightweight antenna.

High voltage power cable, high currents above often use aluminum wires with steel reinforcing cores; the higher resistance of the steel core is of no consequence as it lies deep below the skin depth where there is essentially no AC current flow.

In applications where high currents (up to thousands of amperes) flow, solid conductors are usually replaced by a tube, completely removing the interior of the conductor where a small current flows. This hardly affects the AC resistance, but greatly reduces the weight of the conductor. High strength but low tube weight substantially increases range capability. The tubular conductor is typical in the electrical power switch where the distance between the supporting insulators may be several meters. Long spans generally show physical sag but this does not affect electrical performance. To avoid losses, the conductivity of the tube material must be high.

In today's high situations where conductors (busbar round or flat) may be between 5 and 50 mm thick skin effect also occurs in sharp turns where the metal is compressed inside the bend and extends outside the bend. Shorter paths on the inner surface result in a lower resistance, which causes most of the current to be concentrated close to the inner corner surface. This will cause an increase in temperature rise in the region compared to the straight (unbound) region of the same conductor. The same skin effect occurs in the corner of the rectangular conductor (seen in the cross section), where the magnetic field is more concentrated in the corner than on the side. This results in superior performance (ie higher current with lower temperature rise) than wide thin conductors - ie. The "ribbon" conductor, in which the effect of the angle is effectively removed.

Therefore, a transformer with a spherical core would be more efficient than a transformer with an equivalent value that has a square or rectangular core of the same material.

Solid or tubular condensors can be gilded silver to take advantage of higher silver conductivity. This technique is mainly used in VHF for microwave frequencies where small skin depths require only a very thin silver coating, making conductivity enhancement extremely cost effective. Silver coating is also used on the surface of the waveguides used for microwave transmission. This reduces propagation wave attenuation due to resistive losses affecting the eddy currents that accompany it; the skin effect limits the eddy current to a very thin surface layer of the waveguide structure. The skin effect itself is not actually resisted in these cases, but the current distribution near the surface of the conductor makes the use of precious metals (having low resistivity) practical. Despite having lower conductivity than copper and silver, gold plating is also used, because unlike copper and silver, it does not corrode. A thin oxidized copper or silver oxidation layer will have a low conductivity, thus causing a large power loss because most of the current will flow through this layer.

Recently, non-magnetic and ferromagnetic material coating methods with nanometer-scale thickness have been shown to reduce the increased resistance of skin effect to very high frequency applications. The working theory is that the behavior of high frequency ferromagnetic materials produces opposing fields and/or currents generated by relatively nonmagnetic materials, but more work is needed to verify the exact mechanism. As this experiment shows, it has the potential to increase the efficiency of conductors operating in tens of GHz or higher. It has powerful consequences for 5G communications.

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Example

Kita dapat memperoleh formula praktis untuk kedalaman kulit sebagai berikut:

${\ displaystyle \ delta = {\ sqrt {{2 \ rho} \ over {(2 \ pi f) (\ mu_ {0} \ mu _ {r}) }}} \ kira-kira 503 \, {\ sqrt {\ frac {\ rho} {\ mu_ {r} f}}}}  ÂÂ$

dimana

${\ displaystyle \ delta =}  ÂÂ$ kedalaman kulit dalam satuan meter

${\ displaystyle \ mu _ {r} =}  ÂÂ$ permeabilitas relatif medium (untuk tembaga, ${\ displaystyle \ mu _ {r}}  ÂÂ$ = 1,00 )

${\ displaystyle \ rho =}  ÂÂ$ resistivitas medium dalam? Â · m, juga sama dengan kebalikan konduktivitasnya: ${\ displaystyle \ rho = 1/\ sigma}  ÂÂ$ (untuk tembaga,? = 1,68 ÃÆ' - 10 ^-8 Â? Â · m )

${\ displaystyle f =}  ÂÂ$ frekuensi arus dalam Hz

Emas adalah konduktor yang baik dengan resistivitas 2,44 ÃÆ' - 10 ^-8 Â? M dan pada dasarnya bukan magnetik: < span> ${\ displaystyle \ mu _ {r} =}  ÂÂ$ 1, sehingga kedalaman kulitnya pada frekuensi 50 Hz diberikan oleh

$            {\displaystyle         ?        =        503                          \sqrt{            \frac{                \mathrm{2,44}                ?                {10}^{                    -                    8                  }              }{                1                ?                50              }          }                =        \mathrm{11,1}                          \mathrm{m}          \mathrm{m}              }        \{\backslash \; displaystyle\; \backslash \; delta\; =\; 503\; \backslash ,\; \{\backslash \; sqrt\; \{\backslash \; frac\; \{2.44\; \backslash \; cdot\; 10\; ^\; \{-\; 8\}\}\; \{1\; \backslash \; cdot\; 50\}\}\}\; =\; 11.1\; \backslash ,\; \backslash \; mathrm\; \{mm\}\}\;  ÂÂ$

Timbal, sebaliknya, adalah konduktor yang relatif buruk (antara logam) dengan resistivitas 2,2 ÃÆ' - 10 ^-7 Â? Â · m , sekitar 9 kali dari emas. Kedalaman kulitnya di 50 Hz juga ditemukan sekitar 33 mm, atau ${\ displaystyle {\ sqrt {9}} = 3}  ÂÂ$ kali dari emas.

Very high magnetic material has reduced skin depth due to its large permeability ${\ displaystyle \ mu _ {r}}$ as shown above for iron cases, even though the conductivity is worse. Practical consequences are seen by induction cooker users, where some types of stainless steel cookware can not be used because they are not ferromagnetic.

At very high frequencies, the skin depth for a good conductor becomes small. For example, the skin depth of some common metals at a frequency of 10 GHz (microwave region) is less than micrometers:

Thus at microwave frequencies, most of the current flows in a very thin region near the surface. Losing Ohmic from waveguides on microwave frequencies therefore depends only on the surface layer of the material. A thick silver coating yawning on a piece of glass is thus an excellent conductor of such frequency.

In copper, the depth of the skin can be seen down by the square root of the frequency:

In Electromagnetic Engineering , Hayt points out that in generating a busbar for alternating current at 60 Hz with a radius greater than one-third of an inch (8 mm) is a copper waste, and in practice the bus bar for AC current weight rarely more than half an inch (12 mm) thick except for mechanical reasons.

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Reduction of skin effect from a conductor's internal inductance

See the diagram below showing the inner and outer conductors of the coaxial cable. Since the skin effect causes currents at high frequencies to flow mainly on the surface of the conductor, it can be seen that this will reduce the magnetic field inside the wire , that is, below the depth at which most of the current flows. It can be shown that this will have a small effect on the self-inductance of the wire itself; see Skilling or Hayt for the mathematical treatment of this phenomenon.

Note that the inductance considered in this context refers to a bare conductor, not a coil inductance used as a circuit element. The inductance of the coil is dominated by the mutual inductance between coil windings which increases the inductance according to the square of the number of turns. However, when only one wire is involved, then in addition to "external inductance" involving a magnetic field outside the wire (due to the total current in the wire) as seen in the white area of ​​the image below, there is also a component much smaller than "inductance internal "due to the magnetic field part inside the wire itself, the green region in figure B. The small component of the inductance decreases as the current is concentrated towards the skin of the conductor, which is, when the skin depth is not greater than the radius of the wire, as would be the case at higher frequency.

For one wire, this reduction becomes lessened as the length of the wire increases compared to its diameter, and is usually ignored. But the presence of a second conductor in the case of a transmission line reduces the level of the external magnetic field (and total self induced) regardless of the length of the wire, resulting in inductance decreasing because the skin effect can still be important, For example, in the case of twisted pair phones, below, the conductor inductance decreases substantially at higher frequencies where skin effect becomes important. On the other hand, when the external component of the inductance is magnified due to the geometry of the coil (due to mutual inductance between windings), the significance of internal inductance components is further dwarfed and ignored.

Inductance per length in coaxial cable

Let the dimensions a , b , and c be the inner conductor radius, the protector (outer conductor) inside the radius and the outer radius of each shield respectively, as seen in the crossing of figure A below.

For the given current, the total energy stored in the magnetic field must be equal to the calculated electrical energy associated with the current flowing through the inductance of the coax; that energy is proportional to the indication of the cable being measured.

The magnetic field inside the coaxial cable can be divided into three regions, each of which will contribute to the electrical inductance seen by the cable length.

Induktansi ${\ displaystyle L _ {\ text {cen}} \,}  ÂÂ$ dikaitkan dengan medan magnet di wilayah tersebut dengan radius ${\ displaystyle r & lt; a \,}  ÂÂ$ , wilayah di dalam pusat konduktor.

Induktansi $            {\displaystyle    Â}$

Source of the article : Wikipedia