Proximity effect (electromagnetism)

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

In a conductor carrying 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.

1 Explanation
2 Dowell method for determination of losses
3 Squared-field-derivative method
4 Cables
5 See also
6 External links
7 References

Explanation

A changing magnetic field will influence the distribution of an electric current flowing within an electrical conductor. When an alternating current(AC) flows through an isolated conductor, it creates an associated alternating magnetic field. The alternating magnetic field induces eddy currents within adjacent conductors, altering the overall distribution of current flowing through them.

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

The additional resistance increases electrical losses which, in turn, generate undesirable heating. Proximity and skin effects significantly complicate the design of efficient transformers operating at high frequencies within switching power supplies.

Many methods exist for determining winding losses due to the proximity effect in transformers and inductors.

Dowell method for determination of losses

This 1D 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 m.m.f. 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

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

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

$\alpha = \sqrt{\frac{j \omega \mu_0 \eta}{\rho}}$

ω

Angular frequency of the current

ρ

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

ρ 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 dynamically change (slightly) along with the audio signal. 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 is so small as to have an unmeasurable practical impact on audio quality.