# Proximity effect (electromagnetism)

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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.

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.

Current crowding is a nonhomogenous distribution of current density through a conductor or semiconductor, especially at the vicinity of the contacts and over the PN junctions.

## Explanation

A changing magnetic field will influence the distribution of an electric current flowing within an electrical conductor, by electromagnetic induction. When an alternating current (AC) flows through a conductor, it creates an associated alternating magnetic field around it. The alternating magnetic field induces eddy currents in adjacent conductors, altering the overall distribution of current flowing through them. The result is that the current is concentrated in the areas of the conductor farthest away from nearby conductors carrying current in the same direction.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. Magnetic fields are observed in a wide range of size scales, from subatomic particles to galaxies. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is an example of a vector field.

An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge in one or more directions. Materials made of metal are common electrical conductors. Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases.

The proximity effect can significantly increase the AC resistance of adjacent conductors when compared to its resistance to a DC current. The effect increases with frequency. At higher frequencies, the AC resistance of a conductor can easily exceed ten times its DC resistance.

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.

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.

## Example

For example, if two wires carrying the same alternating current lie parallel to one another, as would be found in a coil used in an inductor or transformer, the magnetic field of one wire will induce longitudinal eddy currents in the adjacent wire, that flow in long loops along the wire, in the same direction as the main current on the side of the wire facing away from the other wire, and back in the opposite direction on the side of the wire facing the other wire. Thus the eddy current will reinforce the main current on the side facing away from the first wire, and oppose the main current on the side facing the first wire. The net effect is to redistribute the current in the cross section of the wire into a thin strip on the side facing away from the other wire. Since the current is concentrated into a smaller area of the wire, the resistance is increased.

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.

A transformer is a static electrical device that transfers electrical energy between two or more circuits. A varying current in one coil of the transformer produces a varying magnetic flux, which, in turn, induces a varying electromotive force across a second coil wound around the same core. Electrical energy can be transferred between the two coils, without a metallic connection between the two circuits. Faraday's law of induction discovered in 1831 described the induced voltage effect in any coil due to changing magnetic flux encircled by the coil.

Similarly, in two adjacent conductors carrying alternating currents flowing in opposite directions, such as are found in power cables and pairs of bus bars, the current in each conductor is concentrated into a strip on the side facing the other conductor.

A power cable is an electrical cable, an assembly of one or more electrical conductors, usually held together with an overall sheath. The assembly is used for transmission of electrical power. Power cables may be installed as permanent wiring within buildings, buried in the ground, run overhead, or exposed.

## Effects

The additional resistance increases power losses which, in power circuits, can generate undesirable heating. Proximity and skin effect significantly complicate the design of efficient transformers and inductors operating at high frequencies, used for example in switched-mode power supplies.

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.

In radio frequency tuned circuits used in radio equipment, proximity and skin effect losses in the inductor reduce the Q factor, broadening the bandwidth. To minimize this, special construction is used in radio frequency inductors. The winding is usually limited to a single layer, and often the turns are spaced apart to separate the conductors. In multilayer coils, the successive layers are wound in a crisscross pattern to avoid having wires lying parallel to one another; these are sometimes referred to as "basket-weave" or "honeycomb" coils. Since the current flows on the surface of the conductor, high frequency coils are sometimes silver-plated, or made of litz wire.

## Dowell method for determination of losses

This one-dimensional method for transformers assumes the wires have rectangular cross-section, but can be applied approximately to circular wire by treating it as square with the same cross-sectional area.

The windings are divided into 'portions', each portion being a group of layers which contains one position of zero MMF. For a transformer with a separate primary and secondary winding, each winding is a portion. For a transformer with interleaved (or sectionalised) windings, the innermost and outermost sections are each one portion, while the other sections are each divided into two portions at the point where zero m.m.f occurs.

The total resistance of a portion is given by ${\displaystyle R_{AC}=R_{DC}{\bigg (}Re(M)+{\frac {(m^{2}-1)Re(D)}{3}}{\bigg )}}$

RDC is the DC resistance of the portion
Re(.) is the real part of the expression in brackets
m number of layers in the portion, this should be an integer
${\displaystyle M=\alpha h\coth(\alpha h)\,}$
${\displaystyle D=2\alpha h\tanh(\alpha h/2)\,}$
${\displaystyle \alpha ={\sqrt {\frac {j\omega \mu _{0}\eta }{\rho }}}}$
${\displaystyle \omega }$ Angular frequency of the current
${\displaystyle \rho }$ resistivity of the conductor material
${\displaystyle \eta =N_{l}{\frac {a}{b}}}$
Nl number of turns per layer
a width of a square conductor
b width of the winding window
h height of a square conductor

## Squared-field-derivative method

This can be used for round wire or litz wire transformers or inductors with multiple windings of arbitrary geometry with arbitrary current waveforms in each winding. The diameter of each strand should be less than 2 δ. It also assumes the magnetic field is perpendicular to the axis of the wire, which is the case in most designs.

• Find values of the B field due to each winding individually. This can be done using a simple magnetostatic FEA model where each winding is represented as a region of constant current density, ignoring individual turns and litz strands.
• Produce a matrix, D, from these fields. D is a function of the geometry and is independent of the current waveforms.

${\displaystyle \mathbf {D} =\gamma _{1}\left\langle {\begin{bmatrix}\left|{\hat {{\vec {B}}_{1}}}\right|^{2}&{\hat {{\vec {B}}_{1}}}\cdot {\hat {{\vec {B}}_{2}}}\\{\hat {{\vec {B}}_{2}}}\cdot {\hat {{\vec {B}}_{1}}}&\left|{\hat {{\vec {B}}_{2}}}\right|^{2}\end{bmatrix}}\right\rangle _{1}+\gamma _{2}\left\langle {\begin{bmatrix}\left|{\hat {{\vec {B}}_{1}}}\right|^{2}&{\hat {{\vec {B}}_{1}}}\cdot {\hat {{\vec {B}}_{2}}}\\{\hat {{\vec {B}}_{2}}}\cdot {\hat {{\vec {B}}_{1}}}&\left|{\hat {{\vec {B}}_{2}}}\right|^{2}\end{bmatrix}}\right\rangle _{2}}$

${\displaystyle {\hat {{\vec {B}}_{j}}}}$ is the field due to a unit current in winding j
<.>j is the spatial average over the region of winding j
${\displaystyle \gamma _{j}={\frac {\pi N_{j}l_{t,j}d_{c,j}^{4}}{64\rho _{c}}}}$
${\displaystyle N_{j}}$ is the number of turns in winding j, for litz wire this is the product of the number of turns and the strands per turn.
${\displaystyle l_{t,j}}$ is the average length of a turn
${\displaystyle d_{c,j}}$ is the wire or strand diameter
${\displaystyle \rho _{c}}$ is the resistivity of the wire
• AC power loss in all windings can be found using D, and expressions for the instantaneous current in each winding:

${\displaystyle P={\overline {{\begin{bmatrix}{\frac {di_{1}}{dt}}{\frac {di_{2}}{dt}}\end{bmatrix}}\mathbf {D} {\begin{bmatrix}{\frac {di_{1}}{dt}}\\{\frac {di_{2}}{dt}}\end{bmatrix}}}}}$

• Total winding power loss is then found by combining this value with the DC loss, ${\displaystyle I_{rms}^{2}\times R_{DC}}$

The method can be generalized to multiple windings.