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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: $Re \{ ... \}$ - 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.

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calculator/proximity_effect_from_dowell_curves.txt · Last modified: 2023/06/17 22:17 by

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