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proximity_effect

Proximity effect

Stan Zurek, Proximity effect, Encyclopedia Magnetica, E-Magnetica.pl

Proximity effect - a phenomenon causing non-uniform distribution of AC current in multi-turn windings or nearby conductors, which can lead to significant increase of power loss, as compared to DC current.

Proximity effect in round wires, with the same amplitude of sinusoidal current: same phase (top), opposite phase (middle), and skin effect in a single wire (bottom)

Proximity effect can be several orders of magnitude greater than skin effect, and it grows exponentially with the number of layers of wire in the winding.1)2)3)4)5)

Non-uniform current distribution can cause high concentration of current near a surface which leads to elevated loss, typically termed proximity loss.6)

Because of the proximity implications, the currents flowing in the same direction appear to “repel” each other (direct proximity effect), and currents in the opposing directions are “drawn” together (reverse proximity effect).7)

The effect is also important at power frequency (50 or 60 Hz) in thick conductors such as three-phase busbars. Skin effect alone limits thickness of busbars to below 10 mm, and parallel connection of closely positioned conductors leads to significant increase in power loss. In the worst case the power loss can increase faster than the effective cross-sectional area, which is a particular problem for busbars operating at very high currents (>1kA).8)

Interleaving of primary and secondary windings, or increasing separation between the conductors reduces proximity loss.9)10)11)

Proximity effect can be important in all applications where there is some variation of magnetic field, including permanent-magnet DC motors, because of ripples in the field penetrating the DC windings.12)

Explanation of proximity effect

The underlying cause of the proximity effect is similar to skin effect, because in both cases there are eddy currents induced in the body of the current-carrying conductor. These currents increase power lost in the conductor, and are equivalent to increase in resistance (hence AC resistance is greater than DC resistance).13)

Mechanism of proximity effect, for clearer illustration shown here for a pair of flat conductors; if the currents have the same amplitudes both wires act on each other in the same way14)

However, in the case of the proximity effect, the eddy currents are induced because the magnetic field from the nearby conductor penetrates the body of the wire perpendicularly to the axis of the wire. This induces eddy currents, which cancel near the symmetry plane, but at the surfaces they flow in the same axis as the main current, but with opposing directions at the two edges of the wire.15)

At the edge at which the direction of eddy currents is the same as the main current they add, causing elevated local current density. At the other edge, the direction of eddy currents is in the opposing direction to the main current, so they subtract causing lower local current density.

Local power dissipation is proportional to the square of current density, hence the higher local current causes disproportionately higher losses, even though the net current trough the wire remains the same (e.g. if driven by a current source).16)

Effect of layers

Proximity effect is exacerbated by multiple layers, because of the distribution of magnetic field caused by each layer.17)18) The presence of magnetic core means that the outer field is effectively “short-circuited”, so the magnetic field above the outermost layer is very low.19) (The implications are slightly different for the cores with distributed gap, because the zero-field occurs in the middle layer, not the outermost, but similar logic applies.)

Explanation of proximity effect for multiple layers in a winding20)

The image shows an example for three layers of winding. For clarity of illustration, the layers are made as solid conductors (e.g. tape). The operating conditions (frequency) is such that the penetration depth δ is 1/3 of the thickness of the conductor d (hence Q = d / δ = 3).

Net current in each conductor is the same (I = 1 A, in this example).

In the outermost layer (no. 3) only the net current flows (1 A), pushed to one side. The magnetomotive force (MMF) exists between layers, and must be balanced by a surface current (1 A) in the deeper layer (no. 2), but the direction of this surface current is opposite to the main current. This is because the MMF cannot penetrate deeper than just 1/3 of the thickness of the conductor.

Therefore, in order for the net current in layer 2 to remain the same as in layer 3, the surface current on the left side must be double (2 A).

The situation repeats for the innermost layer (no. 1). The outer surface sustains the balance of the MMF from the next layer (2 A), and because the net current remain the same, the innermost surface, of the innermost layer has 3 A of current.

The power loss P is proportional to the square of current, and adding up all the contributions, from all the surface currents gives a value of “19”.

However, in a comparable DC case with also 1 A in each layer, the current would flow through the whole thickness of the conductor, not just 1/3 of it, so also power loss per layer would be only 1/3, and thus total DC loss for all 3 layers would have a value of “1”.

Therefore, in this example, proximity loss is greater by a factor of 19, as compared to DC current (for which proximity loss would not occur).

Using the calculator of Dowell coefficient (included below), and setting the values so that frequency factor Q = 3 and porosity η = 1 (because of solid conductor) gives a value of “20.42”. The agreement with the simplistic model is very good, bearing in mind that Dowell's equation takes into account continuous distribution of of magnetic field, rather than uniform distribution within some fraction of thickness.

Interleaved windings in transformers

Proximity effect in a transformer with normal and interleaved primary and secondary windings, the graphs show relative amplitude of current density distribution J and magnetomotive force MMF

The increased local current density is proportional to the square of magnetic field amplitude penetrating the winding, which is directly proportional to the absolute value of magnetomotive force (MMF) generated by the turns of the winding.

In a transformer, if all layers of the given winding are wound on top of each other, then the MMF builds up through the whole winding thickness, resulting with the biggest amplitude on one side of the winding, and minimum at the other side. Therefore, the full number of layers contribute to the proximity effect.

The magnetic field amplitude between the windings can be reduced by interleaving the primary and secondary windings. By symmetrically “sandwiching” one winding inside the other the MMF can change polarity, and thus lower its absolute amplitude.

Splitting the winding in half (as shown in the image) the number of effective layers is also halved and this reduces the proximity effect roughly by a factor of 4.21) Then each half of the winding can be treated as a separate sub-winding for the calculation of the proximity factors.

Interleaving can be applied in multiple stages, but for optimum gain the layers should be split in a way which minimises the amplitude of MMF in all places.22)23) For “double-sandwiching” arrangement such as S1-P1-S2-S3-P2-S4 would be optimal if the MMF values were such that S1+S2+P1 and S3+S4=P2, and so on.

Difference between transformers and inductors

Proximity effect has the biggest impact in the volume of space where the magnetomotive force has the largest amplitude. In an “idealised” transformer (lossless core, ampere-turns perfectly balanced between primary and secondary winding), the maximum occurs between the two windings. So MMF builds up through all the layers, and full number of layers should be used in Dowell-like calculations. Each winding (primary, secondary) should be analysed separately.

In an inductor with a distinct air gap (gapped inductor) MMF builds up towards the air gap, and also full number of layers of a a given winding should be used for proximity calculations.

However, in an inductor with a distributed gap (e.g. as in powder core) the MMF drop gets distributed uniformly throughout the core. The lowest MMF is therefore in the middle of the winding. As a consequence, only half of the number of layers should be used for proximity calculation. As an added benefit, the effect of the narrow gap is removed, which greatly reduces loss in the layer, which would have been otherwise penetrated by the fringing flux around the air gap. If not prevented, the loss from the layer which is nearest the gap can overshadow other losses.

Proximity effect in a transformer (due to leakage inductance) and inductor with narrow or distributed gap (cross-section view of a core window, with the colours indicating amplitude magnetic field strength)

Skin effect

Mechanism of skin effect

Skin effect also involves inducing eddy currents, but they are caused by the current within the same conductor, with the magnetic field acting within the volume of the conductor.

Skin depth for various materials

Because the magnetic field acts around the axis of the conductor the eddy currents are induced in such a way that near the wire axis they flow in the direction opposite to the main current, thus lowering the current density.

Near the surface, the direction of eddy currents coincides with the main current, increasing the local current density.

With the same operating conditions (same wire diameter, frequency and current amplitude) the proximity effect produces much larger non-uniformity than the skin effect. For a stand-alone wire, if its radius is smaller than the the skin depth then skin effect is negligible (Q = d/δ < 2).

But the proximity effect is exacerbated by the number of layers (or number of nearby conductors), and the fact that the pertinent magnetic field penetrates full width of the conductor (twice the area as compared to skin effect, for which also the distribution of eddy currents is concentric). The combined consequences can increase the equivalent AC resistance to drastically high levels (refer to Dowell's curves below).

With multiple conductors or layers, the proximity loss is so much higher that the skin effect calculation can be neglected.24)

Dowell's curves

Proximity effect was quantified analytically by Dowell in his original paper published in 1966.25) The equation (included below) was derived for transformer windings located in a core window. The turns in a winding were approximated by rectangular “wires”, and the distance between the wires in a given layer was taken into account by introducing a unitless porosity factor $η = N · a/b$ (where: N - number of turns per layer, a - conductor width, b - winding width). Dowell's equation returns a value for the whole winding, without the need of summing over all the layers, as it is the case of other approaches presented in the literature. However, the equations presented for example by Nan and Sullivan26) can be exactly equivalent in terms of calculated values (if the same input conditions are applied).

Dowell's equations are derived on the basis of complex numbers, and allow calculation of relative proximity resistance (real part) and proximity inductance (imaginary part), referred to the DC condition (where the AC effects are negligible).

The values calculated from Dowell's equation are typically plotted as a family of curves, with both axes normalised, so that they can be applied to any winding, any wire thickness, at any frequency. A given family of values is only valid for a given metal with fixed resistivity (e.g. copper), and the specific assumed porosity factor.

The horizontal axis is the frequency factor Q = d / δ is the ratio of wire diameter d to the skin depth at the given analysed frequency f. Skin depth is proportional to reciprocal of √f, so if wire diameter decreases by a factor of 0.5, but the frequency increases by factor of 4, then Q remains the same, and also a given Dowell curve remains the same. For this reason, the range of Q from 0.1 to 10 covers a very wide practical range, because thinner wires are typically used at higher frequencies, and this provides a kind of implicit “double normalisation”.27) (In his original paper, Dowell used normalisation with respect to Q2, which is less useful for direct application. Hence, nowadays the Q factor is used almost exclusively.)28)29)30)31)32)

The vertical axis represents the proximity factor K = RAC / RDC, a ratio of the expected AC resistance to the DC resistance, which can be calculated directly from the length and area (and resistivity) of wire used in a given winding. The AC resistance cannot be smaller than the DC resistance, so the K factor values are never lower than unity.

Three input values are needed to order to read the K value from the graph, with some pre-processing: frequency and metal resistivity are needed to calculate the skin depth δ, then wire diameter d is required for calculating Q = d/δ - this is the value to be selected from the horizontal axis.

Following a vertical line upward to the given curve for the specific number of layers allows reading out the corresponding K value from the vertical axis. (This is also shown in the smaller example image in the calculator below.)

Family of Dowell's curves for copper at 20°C, with porosity factor $\eta$ = 0.785

Accuracy of calculations with analytical equations such as Dowell's is limited, as it is the case for most analytical calculations of complex structures:

  • The exact physical configuration cannot be represented in all details. For example, conductors are often wound in a helical way, rather than as strictly parallel rings.
  • Calculations are typically based on mean length of turn (MLT), which does not take into account the fact that turns in upper layers are longer. This has a different impact on transformer or on inductor windings. If the core window is not utilised uniformly, then lower DC resistance is obtained than it would be suggested by the MLT value.
  • Performance of high-frequency magnetic components is often limited by temperature rise due to losses. The cooling conditions are different inside the coil and outside of it, which means that there can be a considerable difference of temperature between the hot spot and outer part of the winding. The variation in temperature changes both DC resistance and penetration depth (and hence the way in which the proximity effect acts locally).
  • Magnetic field generated by interconnections between the windings and terminations to the outside.

Various corrections were proposed in the literature, both by using finite-element modelling or experiments on real magnetic components.33)34) They can provide superior accuracy35) as compared to the original Dowell method, or those based on the Bessel equations. However, they involve semi-empirical fitting factors and their applications to a generic problem is not proved.

However, applicability of such corrections is limited to the conditions which were used to derive the corrections. From a practical view point, prototyping is always required to ensure that other unforeseen effects do not present problems. For example, in high-voltage applications, increased temperature leads to increased current leaking through the electric insulation, which in the worst-case condition can lead to a thermal runaway.

Loss calculation

For sinusoidal waveform with a DC offset, the proximity loss can be calculated by extracting the AC component, from the total RMS current $I_{RMS}$ and the DC component (mean value of current) $I_{DC}$:

$$I_{AC,RMS} = \sqrt{{I_{RMS}}^2 - {I_{DC}}^2 }$$ (A)

and calculating the loss components as:

$$P_{proximity} = K_{Dowell} · R_{DC} · {I_{AC,RMS}}^2$$ (W)
$$P_{DC} = R_{DC} · {I_{DC}}^2$$ (W)
$$P_{total} = P_{DC} + P_{proximity}$$ (W)

If multiple harmonics are present then the AC component must be extracted for each significant harmonic, losses calculated separately, and the added up to a total figure.36)

Calculator

The following calculator can be used for estimating the proximity effect at a given frequency.

This calculator uses exactly the same equation as it was used for the large image with the family of curves above - the default input values are the same.

However, automatic recalculation is provided for different metal (Cu, Al), temperature of windings, and porosity factor.

Calculator of proximity effect from Dowell curves

Proximity effect in windings of transformers and inductors can be estimated by the normalised unitless factor $K = \frac{R_{ac}}{R_{dc}}$, as proposed in the original paper by Dowell in 1966.

$$ K = \frac{R_{ac}}{R_{dc}} = Re \left\{ \alpha·h· \coth(\alpha·h) \right\} + \frac{m^2-1}{3} · Re \left\{ 2·\alpha·h· \tanh \left( \frac{\alpha·h}{2} \right) \right\} $$ (unitless)
where: $Re \{ \ldots \}$ - function returning real component of a complex number (unitless), $\alpha = \sqrt{i· \omega · \mu_0 · \sigma · \eta}$ - inverse skin depth factor (1/m), $i$ - imaginary number $\sqrt{-1}$ (unitless), $\omega = 2·\pi·f$ - angular frequency (Hz), $f$ - frequency (Hz), $\mu_0$ - magnetic permeability of vacuum (H/m), $\sigma$ - conductivity of wire (S/m), $\eta = N · a / b$ - porosity factor (unitless), $N$ - number of turns per layer (unitless), $a$ - wire width or diameter (m), $b$ - winding or layer width (m), $h$ - wire height, thickness or diameter (m), $m$ - total number of layers in the analysed winding (unitless).

wire diameter d =       frequency f (kHz) =

type of wire:       temperature T = {-50...+200°C}

layers N (unitless) = {0.5, 1, 2, 3...300}       porosity η (unitless) = {0.01 ... 1}

      Proximity factor K = Rac/Rdc = (unitless)

Auxiliary values:    freq. factor Q = (unitless)       skin depth δ = (mm)

resistivity ρ = (Ω·m)

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Normalised Dowell's curves plotted vs. frequency factor Q=d/δ

Notes on using this calculator

  • wire diameter - Dowell's equation is derived for rectangular wires, so there are two values $a$ (wire width) and $h$ (wire thickness), which for round wires can be both assumed equal to the diameter d of the wire
  • frequency - frequency of a sinusoidal current (for distorted currents each harmonic has to be assessed separately)
  • type of wire - automatically changes resistivity and thus the skin depth
  • temperature - automatically recalculates resistivity, by using fixed temperature coefficient of 0.393%/°C (same for Cu and Al); limited to range -50…+200°C (-58…+392°F)
  • porosity - for rectangular wires the porosity factor is calculated as $\eta = N·a/b$ but for round wires a good approximation is by scaling with the effective copper area in a given layer; the default value $\eta = \pi / 4$ = 0.785 means typical round wires with some insulation; $\eta$=1 would mean edge-to-edge packing of rectangular wires without any insulation; $\eta$=0.83 would mean closest possible packing of round wires (as per Dixon paper)
  • layers - number of layers in the analysed winding, e.g. if a transformer has 3 layers in the primary winding, and 10 layers in secondary winding, use “3” when analysing primary, and “10” for secondary; the value “0.5” corresponds to a coherent winding with primary and secondary turns interleaved within the same layer; other values are rounded to whole numbers
  • Auxiliary values - these are provided as useful outputs of the calculation:
    • freq. factor Q=d/δ - ratio of wire diameter d to skin depth $\delta$, used as the normalised variable for the horizontal axis of the graph, so the K=f(Q) value can be read out directly from the graph
    • skin depth $\delta$ - ordinary skin depth as calculated for a single wire due to skin effect
    • resistivity - value of resistivity as used in the equations, scaled by temperature coefficient; base values are used for 20°C: Cu = 1.71×10-8 Ω·m, Al = 2.79×10-8 Ω·m (if different values are needed for a particular type of Cu or Al this can be adjusted just by changing the temperature value

In any real applications the Dowell values should be treated as an approximation, with some discrepancy to be expected, because real conditions such as the exact temperature distribution between the layers changes resistivity, the innermost turns are shorter than the outermost, placing of wires is typically helical (rather than strictly parallel), twist in a Litz wire is not taken into account, and so on.

Litz wire

Litz wire twisted from 11 bunches, each containing 15 strands of 0.25 mm enamelled wire two-level_litz_wire_11branchx15strandx0.25mm_magnetica.jpg

Litz wire is used for preventing excessive skin-effect losses, because the insulated strands have much smaller thickness which helps in reducing the Q factor.

However, from proximity effect viewpoint, the number of strands translates into significantly higher number of effective layers, so the losses typically get worse first, before they can be lowered - and this can only be obtained by decreasing the strand diameter to much smaller value than it would be dictated by the skin effect alone. For example, strand diameter 0.32 mm is recommended37) for an upper limit of 1 kHz, even though the skin depth for copper at 1 kHz is over 2 mm.

This apparent discrepancy arises because of the number of effective layers. For example, consider a case in which the conductor is made of solid copper, with square cross-section, 5 x 5 mm. At 10 kHz, for a single layer the proximity factor would be 7.6, which is quite elevated and reduction of this value by use of Litz wire could be attempted.

But if it was replaced by a hypothetical Litz wire, made with square strands, each 1 x 1 mm, then there would be a matrix of 25 strands (5×5) packed into the same volume, so the effective number of layers becomes 5, and the resulting proximity factor is 13.1, which is worse than for a solid wire, which is 5x thicker.

Only further reduction of strand thickness can reduce the loss. For example, reducing the strands to 0.5 mm would produce a matrix of 10×10, with 10 effective layers, giving the proximity factor of 4.6 (and further reduction of strand thickness would improve it more).

However, for the Litz wire to give the expected benefits, it must be twisted in such a way, that each strand takes all the possible positions within the total wire cross-sectional area, so that the impedance for each strand is the same. Only then the strand currents are forced to be equal.

Just a parallel connection of several strands, which are not twisted or interleaved correctly can force all the current to flow in the strand/wire/conductor with the lowest impedance, which can produce excessive losses, far worse than it would be the case if a solid conductor was used.38)

Ordinary stranded wire (with no isolation between strands) typically is not twisted in a sufficient way to be useful in high frequency applications (e.g. the inner strands remain always at the centre of the bundle).

Continuously transposed conductors

In applications such as windings of high-power AC motors and generators (and others) the windings are subjected to varying magnetic field, which also would produce elevated losses if solid conductors were used.

In order to reduce such losses, similar concept as for the Litz wire is followed. The whole “wire” is made from several “strands”, whose positions are rotated through all possible positions within the cross-section of such combined conductor, and the whole construction is called continuously transposed conductor.

Busbars

Electricity is typically distributed through a medium voltage (MV) cable or overhead line and is transformed to low voltage (LV). The LV side of the transformer can deliver very high rated current, so immediately at the output of the transformer (and throughout an LV distribution board) thick conductors such as busbars are required.

Typical busbars connections in a low-voltage substation busbars_in_a_substation_magnetica.jpg

Skin depth at 50 Hz is 9 mm, so busbars are typically made as wide flat conductors, with thickness not exceeding this value.

Wide busbars positioned in the same plane are penetrated by the perpendicular component of the magnetic field from the nearby phases, and the distribution of current density can change significantly. For the illustrated case with 6 x 100 mm busbars, separated by 100 mm the increase in loss is by 1.17 (as compared to a DC case). Reducing the spacing to 50 mm would increase the factor to 1.23.

This effect scales linearly with the amplitude of current, so the relative changes remain the same. But the overall problem is quite complex a general analytical solution does not exist.39)

FEM simulation of three-phase busbars made as rectangular copper conductors 6 x 100 mm (spaced by 100 mm), carrying 50 Hz current - proximity effect significantly alters the current distribution (for this particular case the simulated proximity factor value is K = 1.17)
Busbars connected in parallel (same single phase), 6 x 100 mm flat bars, spaced by 6 mm - proximity factor calculated with FEM is K=1.38 (vs. 1.35 given by Ducluzaux40))

To boost the ampacity several such conductors can be connected in parallel, typically with some gap between them to aid cooling. However, multiple conductors in very close proximity increase the losses.

For example, if three 6 x 100 mm conductors are spaced by 6 mm, and connected in parallel (so the current from the same phase flows in all of them) then the current density is concentrated more at the farthest edges, increasing the loss disproportionately, so that the proximity factor is K=1.4. Separation by a larger distance lowers the loss, but paralleling more bars increases it, with some significant effects.41)

Dowell's equation cannot be used for estimating the effects in busbars, because it was derived for transformers (for which the magnetic core on the outside of windings magnetically “short-circuits” the MMF). However, proximity effect can be calculated even for very intricate conductor configurations by finite-element modelling software42), such as non-commercial Finite Element Magnetics Method.43)

Proximity loss in time domain

Dowell's curves represent data which is directly useful in frequency domain.

R-L network can be used for simulation of proximity effect in time domain

However, for a given design, the curves can be approximated effectively by a network of resistors and inductors (a concept similar to Cauer network), with the model accuracy sufficient for practical purposes. Iterative calculations can be used to derive the equivalent R-L network.44)

Such R-L network can be then used directly in an electric circuit simulation such as SPICE, so that the proximity effects can be simulated in time domain, with direct calculation of losses, for an arbitrary shape of the current waveform, with automatic inclusion of all higher harmonics.45)

In the R-L network shown, “Rdc” represents the resistance at DC. Higher-order resistors have higher resistance values. Higher order inductors have decreasing values. So as the frequency increases the impedance of the inductors increases, and the current is forced to flow through the higher-order resistors, automatically increasing the incurred instantaneous power loss. SPICE functionality can be then used to calculate average loss, which includes all the harmonics, by definition of the time-domain simulation.

Just a few branches (e.g. 4 or 5) are sufficient to cover frequency range spanning from DC To 1 MHz.46) Some commercial software makes use of this approach, and can create a specific SPICE subcircuit based on a design of a given electronic transformer or inductor.47)

Leakage inductance

In magnetically-coupled windings, the currents flowing in opposing directions reduce the volume of space between the location of “centre of gravity” of current and therefore this leads to reduction of leakage inductance with increasing frequency.48)49)

However, this happens at the expense of pushing out all the energy from within the conductors, at the same time drastically increasing power loss in the conductors, so cannot be used as a practical way of reducing leakage inductance.50)

$$ K_L = \frac{L_{ac}}{L_{dc}} = \frac{3· Im\left\{ \alpha·h· \coth(\alpha·h)\right\} + (m^2 -1)· Im \left\{ 2·\alpha·h· \tanh \left( \frac{\alpha·h}{2} \right) \right\}}{m^2 · |{\alpha}^2·h^2|} $$ (unitless)
where: $Im \{ \ldots \}$ - function returning imaginary component of a complex number (unitless), $\alpha = \sqrt{i· \omega · \mu_0 · \sigma · \eta}$ - inverse skin depth factor (1/m), $i$ - imaginary number $\sqrt{-1}$ (unitless), $\omega = 2·\pi·f$ - angular frequency (Hz), $f$ - frequency (Hz), $\mu_0$ - magnetic permeability of vacuum (H/m), $\sigma$ - conductivity of wire (S/m), $\eta = N · a / b$ - porosity factor (unitless), $N$ - number of turns per layer (unitless), $a$ - wire width or diameter (m), $b$ - winding or layer width (m), $h$ - wire height, thickness or diameter (m), $m$ - total number of layers in the analysed winding (unitless).
Normalised leakage inductance calculated from the Dowell's equation; the curve for 3 layers is very close to that of infinite number of layers (generated for copper at 20 °C, with porosity factor η = 0.785)

See also

References


37) Litz Wire, MWS Wire Industries, {accessed 2021-04-09}
proximity_effect.txt · Last modified: 2021/05/02 22:15 by stan_zurek

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