Skin effect

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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 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. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor. The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller. Increased AC resistance due to the skin effect can be mitigated by using specially woven litz wire. 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.

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

In electromagnetism, current density is the electric current per unit area of cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI units, the electric current density is measured in amperes per square metre.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

Cause

Conductors, typically in the form of wires, may be used to transmit electrical energy or signals using an alternating current flowing through that conductor. The charge carriers constituting that current, usually electrons, are driven by an electric field due to the source of electrical energy. An alternating current in a conductor produces an alternating 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” (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. [1] [2]

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. As it is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

Counter-electromotive force, also known as back electromotive force, is the electromotive force or "voltage" that opposes the change in current which induced it. CEMF is the EMF caused by magnetic induction.

An alternating current may also be induced in a conductor due to an alternating magnetic field according to the law of induction. An electromagnetic wave impinging on a conductor will therefore generally produce such a current; this explains the reflection of electromagnetic waves from metals.

Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field.

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 which usually will be distributed evenly over the cross-section of the wire.

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

Direct current (DC) is the unidirectional flow of electric charge. A battery is a good example of a DC power supply. Direct current may flow in a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.

The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalised to conductors of any shape by Oliver Heaviside in 1885. The skin effect has practical consequences in the analysis and design of radio-frequency and microwave 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. 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.

Sir Horace Lamb was an English applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics (1895) and Dynamical Theory of Sound (1910). Both of these books remain in print. The word vorticity was coined by Lamb in 1916.

Oliver Heaviside FRS was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.

Radio is the technology of using radio waves to carry information, such as sound and images, by systematically modulating properties of electromagnetic energy waves transmitted through space, such as their amplitude, frequency, phase, or pulse width. When radio waves strike an electrical conductor, the oscillating fields induce an alternating current in the conductor. The information in the waves can be extracted and transformed back into its original form.

Formula

The AC current density J in a conductor which can be derived from Maxwell-Ampere Law and Ohm's Law [3] , decreases exponentially from its value at the surface JS according to the depth d from the surface, as follows: [4] :362

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:

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

where δ 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 JS. 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 the conductor requires 2π skin depths, at which point the current density is attenuated to e−2π (-54.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in vacuum, or equivalently, the phase velocity in the conductor is very much slower than the speed of light in vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum λ0 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 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 is: [5] [6]

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

where

${\displaystyle \rho }$ = resistivity of the conductor
${\displaystyle \omega }$ = angular frequency of current = 2π × frequency
${\displaystyle \mu }$ = ${\displaystyle \mu _{r}}$${\displaystyle \mu _{0}}$
${\displaystyle \mu _{r}}$ = relative magnetic permeability of the conductor
${\displaystyle \mu _{0}}$
${\displaystyle \epsilon }$ = ${\displaystyle \epsilon _{r}}$${\displaystyle \epsilon _{0}}$
${\displaystyle \epsilon _{r}}$ = relative permittivity of the material
${\displaystyle \epsilon _{0}}$

At frequencies much below ${\displaystyle 1/\rho \epsilon }$ the quantity inside the large radical is close to unity and the formula is more usually given as:

${\displaystyle \delta ={\sqrt {{2\rho } \over {\omega \mu }}}}$.

This formula is valid at frequencies away from strong atomic or molecular resonances (where ${\displaystyle \epsilon }$ would have a large imaginary part) and at frequencies that are much below both the material's plasma frequency (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 1] For example, in the case of copper, this would be true for frequencies much below 1018 Hz.

However, in very poor conductors, at sufficiently high frequencies, the factor under the large radical increases. At frequencies much higher than ${\displaystyle 1/\rho \epsilon }$ it can be shown that the skin depth, rather than continuing to decrease, approaches an asymptotic value:

${\displaystyle \delta \approx {2\rho }{\sqrt {\epsilon \over \mu }}}$

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 (λ = 3000 m). 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 functions. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by Weeks. [7] :38

${\displaystyle \mathbf {J} _{r}={\frac {k\mathbf {I} }{2\pi R}}{\frac {J_{0}(kr)}{J_{1}(kR)}}=\mathbf {J} _{R}{\frac {J_{0}(kr)}{J_{0}(kR)}}}$

where

${\displaystyle \quad \omega }$ = angular frequency of current = 2π × frequency
${\displaystyle \quad r=}$ distance from the axis of the wire
${\displaystyle \quad R=}$ radius of the wire
${\displaystyle \quad \mathbf {J} _{r}=}$ current density phasor at distance, r, from the axis of the wire
${\displaystyle \quad \mathbf {J} _{R}=}$ current density phasor at the surface of the wire
${\displaystyle \quad \mathbf {I} =}$ total current phasor
${\displaystyle \quad J_{0}=}$ Bessel function of the first kind, order 0
${\displaystyle \quad J_{1}=}$ Bessel function of the first kind, order 1
${\displaystyle \quad k={\sqrt {\frac {-j\omega \mu }{\rho }}}={\frac {1-j}{\delta }}}$ the wave number in the conductor
${\displaystyle \quad \delta ={\sqrt {\frac {2\rho }{\omega \mu }}}}$ also called skin depth.
${\displaystyle \quad \rho }$ = resistivity of the conductor
${\displaystyle \quad \mu _{r}}$ = relative magnetic permeability of the conductor
${\displaystyle \quad \mu _{0}}$ = the permeability of free space = 4π x 10−7 H/m
${\displaystyle \quad \mu }$ = ${\displaystyle \mu _{r}}$${\displaystyle \mu _{0}}$

Since ${\displaystyle k}$ 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: [7] :40

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

This impedance is a complex quantity corresponding to a resistance (real) in series with the reactance (imaginary) due to the wire's internal self-inductance, 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 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. [8] This small component of inductance approaches a value of ${\displaystyle {\frac {\mu }{8\pi }}}$ 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 δ) can be solved as if the current flowed uniformly through a layer of thickness δ based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to δ times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter D large compared to δ, has a resistance approximately that of a hollow tube with wall thickness δ carrying direct current. The AC resistance of a wire of length L and resistivity ${\displaystyle \rho }$ is:

${\displaystyle R\approx {{L\rho } \over {\pi (D-\delta )\delta }}\approx {{L\rho } \over {\pi D\delta }}}$

The final approximation above assumes ${\displaystyle D\gg \delta }$.

A convenient formula (attributed to F.E. Terman) for the diameter DW of a wire of circular cross-section whose resistance will increase by 10% at frequency f is: [9]

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

This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a cable 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. [10]

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 its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220 micrometres 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 laminations in power transformers, increasing their losses.

Iron rods work well for direct-current (DC) welding 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's about 0.25 m. [11]

Mitigation

A type of cable called litz wire (from the German Litzendraht, braided wire) is used to mitigate the skin effect for frequencies of a few kilohertz to about one megahertz. It consists of a number of insulated wire strands woven together in a carefully designed pattern, so that the overall magnetic field acts equally on all the wires and causes the total current to be distributed equally among them. 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. [12]

Litz wire is often used in the windings of high-frequency transformers 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. [13] Conductive threads composed of carbon nanotubes [14] 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 lines 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) 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 will cause an increase in temperature rise in 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 – e.g. "ribbon" conductors, where the effects from corners is 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-plated to take advantage of silver's higher conductivity. This technique is particularly used at VHF to microwave 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 [15] . 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:

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

where

${\displaystyle \delta =}$ the skin depth in meters
${\displaystyle \mu _{r}=}$ the relative permeability of the medium (for copper, ${\displaystyle \mu _{r}}$ = 1.00)
${\displaystyle \rho =}$ the resistivity of the medium in Ω·m, also equal to the reciprocal of its conductivity: ${\displaystyle \rho =1/\sigma }$ (for copper, ρ = 1.68×10−8 Ω·m)
${\displaystyle f=}$ the frequency of the current in Hz

Gold is a good conductor with a resistivity of 2.44×10−8 Ω·m and is essentially nonmagnetic: ${\displaystyle \mu _{r}=}$ 1, so its skin depth at a frequency of 50 Hz is given by

${\displaystyle \delta =503\,{\sqrt {\frac {2.44\cdot 10^{-8}}{1\cdot 50}}}=11.1\,\mathrm {mm} }$

Lead, in contrast, is a relatively poor conductor (among metals) with a resistivity of 2.2×10−7 Ω·m, about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or ${\displaystyle {\sqrt {9}}=3}$ times that of gold.

Highly magnetic materials have a reduced skin depth owing to their large permeability ${\displaystyle \mu _{r}}$ 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 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:

Skin depth at microwave frequencies
ConductorSkin depth (μm)
Aluminum0.820
Copper0.652
Gold0.753
Silver0.634

Thus at microwave 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:

Skin depth in copper
FrequencySkin depth (μm)
50 Hz9220
60 Hz8420
10 kHz652
100 kHz206
1 MHz65.2
10 MHz20.6
100 MHz6.52
1 GHz2.06

In Engineering Electromagnetics, Hayt points out[ page needed ] that in a power station a busbar for alternating current 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 [16] or Hayt [17] 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. [18]

The inductance ${\displaystyle L_{\text{cen}}\,}$ is associated with the magnetic field in the region with radius ${\displaystyle r, the region inside the center conductor.

The inductance ${\displaystyle L_{\text{ext}}\,}$ is associated with the magnetic field in the region ${\displaystyle a, the region between the two conductors (containing a dielectric, possibly air).

The inductance ${\displaystyle L_{\text{shd}}\,}$ is associated with the magnetic field in the region ${\displaystyle b, the region inside the shield conductor.

The net electrical inductance is due to all three contributions:

${\displaystyle L_{\text{total}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}$

${\displaystyle L_{\text{ext}}\,}$ 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:

${\displaystyle L/D={\frac {\mu _{0}}{2\pi }}\ln \left({\frac {b}{a}}\right)\,}$

At low frequencies, all three inductances are fully present so that ${\displaystyle L_{\text{DC}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}$.

At high frequencies, only the dielectric region has magnetic flux, so that ${\displaystyle L_{\infty }=L_{\text{ext}}\,}$.

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 ${\displaystyle b. Only ${\displaystyle L_{\text{ext}}}$ 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 21 °C (70 °F).

Frequency (Hz)R (Ω/km)L (mH/km)G (μS/km)C (nF/km)
1172.240.61290.00051.57
1k172.280.61250.07251.57
10k172.700.60990.53151.57
100k191.630.58073.32751.57
1M463.590.506229.11151.57
2M643.140.486253.20551.57
5M999.410.4675118.07451.57

More extensive tables and tables for other gauges, temperatures and types are available in Reeve. [19] Chen [20] gives the same data in a parameterized form that he states is usable up to 50 MHz.

Chen [20] gives an equation of this form for telephone twisted pair:

${\displaystyle L(f)={\frac {l_{0}+l_{\infty }\left({\frac {f}{f_{m}}}\right)^{b}}{1+\left({\frac {f}{f_{m}}}\right)^{b}}}\,}$

Notes

1. 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.

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The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

Electromagnetic shielding is the practice of reducing the electromagnetic field in a space by blocking the field with barriers made of conductive or magnetic materials. Shielding is typically applied to enclosures to isolate electrical devices from their surroundings, and to cables to isolate wires from the environment through which the cable runs. Electromagnetic shielding that blocks radio frequency electromagnetic radiation is also known as RF shielding.

The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the space-time, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor. In particular, it can be shown that one of these scalars-- in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

The word electricity refers generally to the movement of electrons through a conductor in the presence of potential and an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals or energy travel as electromagnetic waves typically on the order of 50%–99% of the speed of light, while the electrons themselves move (drift) much more slowly.

In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed the proximity effect. This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.

A microwave cavity or radio frequency (RF) cavity is a special type of resonator, consisting of a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. The structure is either hollow or filled with dielectric material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's resonant frequencies they reinforce to form standing waves in the cavity. Therefore, the cavity functions similarly to an organ pipe or sound box in a musical instrument, oscillating preferentially at a series of frequencies, its resonant frequencies. Thus it can act as a bandpass filter, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.

The article Ferromagnetic material properties is intended to contain a glossary of terms used to describe ferromagnetic materials, and magnetic cores.

The gyrator–capacitor model is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

In fluid dynamics, a flow with periodic variations is known as pulsatile flow, or as Womersley flow. The flow profiles was first derived by John R. Womersley (1907–1958) in his work with blood flow in arteries. The cardiovascular system of chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in engines and hydraulic systems, as a result of rotating mechanisms pumping the fluid.

In probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu in 2011. The G-MVLG is a flexible distribution. Skewness and kurtosis are well controlled by the parameters of the distribution. This enables one to control dispersion of the distribution. Because of this property, the distribution is effectively used as a joint prior distribution in Bayesian analysis, especially when the likelihood is not from the location-scale family of distributions such as normal distribution.

In superparamagnetism, the Néel effect appears when a superparamagnetic material in a conducting coil is subject to varying frequencies of magnetic fields. The non-linearity of the superparamagnetic material acts as a frequency mixer, with voltage measured at the coil terminals. It consists of several frequency components, at the initial frequency and at the frequencies of certain linear combinations. The frequency shift of the field to be measured allows for detection of a direct current field with a standard coil.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

References

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