Dynkin's formula

Magnitude of current density in the windings of a 20kHz transformer.

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 as the proximity effect. This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.

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 an isolated 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 furthest away from nearby conductors carrying current in the same direction.

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.

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.

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.

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.

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 $R_{AC}=R_{DC}{\bigg (}Re(M)+{\frac {(m^{2}-1)Re(D)}{3}}{\bigg )}$

The ratio of AC to DC resistance for a portion of a strip winding at different frequencies (δ is Skin depth). It can be seen that increasing the number of layers dramatically increases the resistance at high frequencies.

R_DC 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

$M=\alpha h\coth(\alpha h)\,$

$D=2\alpha h\tanh(\alpha h/2)\,$

$\alpha ={\sqrt {\frac {j\omega \mu _{0}\eta }{\rho }}}$

$\omega$ Angular frequency of the current

$\rho$ resistivity of the conductor material

$\eta =N_{l}{\frac {a}{b}}$

N_l 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.

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

${\hat {{\vec {B}}_{j}}}$ is the field due to a unit current in winding j

<.>_j is the spatial average over the region of winding j

$\gamma _{j}={\frac {\pi N_{j}l_{t,j}d_{c,j}^{4}}{64\rho _{c}}}$

$N_{j}$ 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.

$l_{t,j}$ is the average length of a turn

$d_{c,j}$ is the wire or strand diameter

$\rho _{c}$ 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:

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

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

The method can be generalized to multiple windings.

Cables

Proximity effect can also occur within electrical cables. For example, if the conductors are a pair of audio speaker wires, their currents have opposite direction, and currents will preferentially flow along the sides of the wires that are facing each other. The AC resistance of the wires will change (slightly) along with the frequency of the audio signal, though for any frequency, the amplitude of the current will still be linearly proportional to the voltage. Some believe that this will potentially introduce distortion and degrade stereo imaging. However, it can be shown that, for reasonable conductor sizes, spacing, and length, this effect has a small impact on audio quality.

See also

• Skin effect

External links

• Skin Effect, Proximity Effect, and Litz Wire Electromagnetic effects
• Skin and Proximity Effects and HiFi Cables

References

• Terman, F.E. Radio Engineers' Handbook, McGraw-Hill 1943—details electromagnetic proximity and skin effects
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