Winding loss: skin and proximity effects

Intent

• Explain the skin effect in one conductor and the proximity effect between the layers of a winding.
• Estimate a layered winding’s ac resistance with Dowell’s factor, and for thin layers with its series.
• Choose the layer thickness and the number of layers, knowing that a thicker layer can lose more.

Theory

Skin effect

An alternating current crowds towards the surface of a conductor: its density falls by a factor $e$ for every skin depth (skin depth) into the conductor [Erickson & Maksimović 2020, Ch. 10]. A conductor much thicker than a skin depth carries the current in a surface layer, and its resistance is higher than at dc.

Proximity effect

In a winding of several layers, each layer sits in the field of the layers beside it, which induces eddy currents in it. Their loss adds to that of the layer’s own current, and it grows quickly with the number of layers [Erickson & Maksimović 2020, Ch. 10].

Dowell’s factor

Dowell treated each layer as a foil spanning the window’s breadth, with a one-dimensional field across the layers [Dowell 1966]. The ratio of the ac to the dc resistance (Dowell's factor) then depends on the number of layers $M_\ell$ from zero MMF to its peak and on the layer thickness measured in skin depths:

$$\varphi = \frac{h}{\delta}$$

$\varphi$

layer thickness over skin depth

$\delta$

skin depth

$h$

conductor layer thickness

Foil thickness in skin depths✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• definition

Notes

for round wire, `wind.phi_round` gives the equivalent foil's $\varphi$

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: eddy currents in winding conductors)

Equation id

wind.phi_foil · equations.yaml entry

A layer of round wire is replaced by an equivalent foil (layer porosity, round wire as a foil) [Erickson & Maksimović 2020, Ch. 10].

Thin layers

For layers up to about a skin depth thick, the first terms of the series of Dowell’s factor [Dowell 1966] are enough. Derived here, they lie at most about 4 % above the factor:

$$F_R \approx 1 + \frac{\left(5 M_\ell^{2} - 1\right) \varphi^{4}}{45}$$

$F_R$

ac resistance factor

$M_\ell$

layers from zero to peak MMF

$\varphi$

layer thickness over skin depth

AC resistance factor of thin layers (Dowell's series)✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• foil layers spanning the winding breadth, a sinusoidal current, and a one-dimensional field in the window
• layers thin compared with the skin depth (φ up to about 1): the leading terms of Dowell's factor

Notes

the first terms of Dowell's factor in powers of $\varphi$: the excess loss grows as $\varphi^4$ and, through the proximity effect, as $M_\ell^2$. For $\varphi \le 1$ it lies above the exact factor by at most about 4 %

Source

Dowell 1966, the one-dimensional model of foil layers whose factor this series expands; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: eddy currents in winding conductors); derived here as the series of Dowell's factor

Derivation

step-by-step sympy derivation

Equation id

wind.dowell_low · equations.yaml entry

The excess resistance grows as the fourth power of the thickness and as the square of the number of layers.

The thickness of least loss

At a given frequency the skin depth is fixed. A thicker layer then has a lower dc resistance, but a larger $F_R$. Measured against a layer one skin depth thick carrying the same current as dc, the loss is:

$$P/P_\delta = \frac{F_R}{\varphi}$$

$P/P_\delta$

loss relative to a one-skin-depth layer at dc

$F_R$

ac resistance factor

$\varphi$

layer thickness over skin depth

Loss of a winding layer at a given frequency, against its thickness✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• foil layers spanning the winding breadth, a sinusoidal current, and a one-dimensional field in the window
• the layers' breadth and turns fixed, so a layer's dc resistance is inversely proportional to its thickness

Notes

the loss relative to that of a layer one skin depth thick carrying the same current as dc: a thicker layer lowers the dc resistance ($1/\varphi$) and raises $F_R$

Source

Dowell 1966; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: eddy currents in winding conductors); derived here

Derivation

step-by-step sympy derivation

Equation id

wind.loss_rel · equations.yaml entry

With the series for thin layers, the loss is least at:

$$\varphi_\mathrm{opt} \approx \left(\frac{15}{5 M_\ell^{2} - 1}\right)^{\frac{1}{4}}$$

$\varphi_\mathrm{opt}$

layer thickness of least loss, in skin depths

$M_\ell$

layers from zero to peak MMF

Layer thickness of least loss✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• foil layers spanning the winding breadth, a sinusoidal current, and a one-dimensional field in the window
• layers thin compared with the skin depth (φ up to about 1): the leading terms of Dowell's factor
• the layers' breadth and turns fixed, so a layer's dc resistance is inversely proportional to its thickness

Notes

at a given frequency a thicker layer has less dc resistance but a larger $F_R$; their product, the loss, is least at this thickness, where $F_R = 4/3$. The more layers, the thinner the best layer. From the series of Dowell's factor: within 2 % of the exact optimum for two or more layers; for one layer the exact optimum is $\varphi = \pi/2$

Source

Dowell 1966; Hurley et al. 2000, the optimum layer thickness, for any periodic current waveform; derived here for a sinusoidal current

Derivation

step-by-step sympy derivation

Equation id

wind.phi_opt · equations.yaml entry

There $F_R = 4/3$, whatever the number of layers. The more layers, the thinner the best layer. Hurley, Gath and Breslin give the optimum for any periodic current, from the rms values of the current and of its derivative [Hurley et al. 2000].

Converter currents

A converter’s winding current is not sinusoidal. Its dc part sees the dc resistance, and each harmonic sees Dowell’s factor at its own frequency; the loss is the sum over the harmonics [Erickson & Maksimović 2020, Ch. 10]. An inductor with a small ripple carries mostly dc, so $F_R$ matters little. A flyback winding carries pulses, whose harmonics are large.

Interleaving

Splitting a transformer’s primary and placing the secondary between the halves lowers the peak MMF in the window and the number of layers from zero to peak MMF, and with them the proximity loss [Erickson & Maksimović 2020, Ch. 10]. It also lowers the leakage (see leakage inductance).

Worked example

An example foil winding, three layers from zero to peak MMF:

Foil winding example

Symbol	Value	How it is obtained
$\rho_{20}$ resistivity at 20 °C	17.2 nΩ·m	given
$\alpha_{20}$ temperature coefficient at 20 °C	0.00393 1/K	given
$T_w$ winding temperature	100 °C	given
$f$ frequency	100 kHz	given
$h$ conductor layer thickness	0.3 mm	given
$M_\ell$ layers from zero to peak MMF	3	given
$\rho_w$ conductor resistivity	22.7 nΩ·m	Resistivity of a conductor at its temperature
$\delta$ skin depth	0.24 mm	Skin depth of a conductor
$\varphi$ layer thickness over skin depth	1.25	Foil thickness in skin depths
$F_R$ ac resistance factor	3.19	AC resistance factor of a foil winding (Dowell)
${F_R}_{\mathrm{low}}$ ac resistance factor	3.4	AC resistance factor of thin layers (Dowell's series)
$P/P_\delta$ loss relative to a one-skin-depth layer at dc	2.55	Loss of a winding layer at a given frequency, against its thickness
$\varphi_\mathrm{opt}$ layer thickness of least loss, in skin depths	0.764	Layer thickness of least loss
${F_R}_{\mathrm{opt}}$ ac resistance factor	1.33	AC resistance factor of a foil winding (Dowell)
${P/P_\delta}_{\mathrm{opt}}$ loss relative to a one-skin-depth layer at dc	1.74	Loss of a winding layer at a given frequency, against its thickness

At 100 kHz and 100 °C, copper’s skin depth is 0.24 mm, so the 0.3 mm foil is 1.25 skin depths thick. Three layers give $F_R$ of 3.19. The series gives 3.4: too high, because the foil is thicker than a skin depth. For three layers the loss is least at 0.764 skin depths, where the factor is 1.33. The relative loss falls from 2.55 to 1.74: the thinner foil loses less, although its dc resistance is higher.

Try it

▶ Open in the equation explorer: AC resistance factor of a foil winding (Dowell)
Preset from the example dowell-foil

Sweep $\varphi$: with the example’s three layers, the factor stays close to 1 up to about half a skin depth, has nearly doubled at one skin depth, and rises steeply beyond. Change $M_\ell$ from 3 to 1 and to 6 and compare.

Bench exercise

Measure the series resistance of a transformer’s primary with its secondary shorted, from a low frequency to a few times the switching frequency, and well below the self-resonance (see Measuring magnetics). The secondary’s current then opposes the primary’s, as in operation, and the reading holds both windings: the primary’s resistance plus the secondary’s divided by the square of the turns ratio. At the low end, where the short already takes the current (see Measuring magnetics) but neither resistance has started to rise, it is the two dc resistances so combined. Divide by that value, not by the primary’s dc resistance alone, and compare the rise with the two windings’ Dowell factors, each for its own number of layers, weighted by their shares of that value.

Gotchas

• More copper can lose more. Past the thickness of least loss, a thicker foil or a thicker wire raises the loss at the switching frequency.
• Count the layers from zero MMF. $M_\ell$ is the number of layers from a point of zero MMF to the peak; interleaving halves it [Erickson & Maksimović 2020, Ch. 10].
• Next to a gap the field is not one-dimensional. Dowell’s factor assumes a one-dimensional field in the window (Dowell's factor), but the flux fringes around a gap [Dixon 2001]. Its loss estimate does not hold for turns close to a gap.

Go deeper

• Book chapter Basic Magnetics Theory (Ch. 10) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  Magnetic circuits, the transformer model with magnetizing and leakage inductance, and core and winding loss mechanisms.
• Course Magnetics for Power Electronic Converters — University of Colorado Boulder (Coursera) · retrieved 2026-09-23
  Magnetic circuits, core and winding losses, and inductor and transformer design.
• Book Magnetics Design Handbook (SLUP132) — L. H. Dixon, Texas Instruments (Unitrode) · retrieved 2026-09-23
  Lloyd Dixon's magnetics design handbook, reproduced by TI from the 2001 Unitrode edition. The ti.com copy requires a free myTI login.

Quiz

1. What is the proximity effect?

   1. The current crowding to the surface of a single conductor
   2. The capacitance between adjacent turns
   3. Eddy currents induced in a layer by the field of the other layers
   4. Core loss next to the winding
   Answer and explanation

   Correct answer: C Each layer of a winding sits in the field of the layers beside it; the eddy currents that field induces add loss, growing quickly with the number of layers.

2. A foil is 0.3 mm thick, and the skin depth at the frequency of interest is 0.2 mm. What is the foil's thickness in skin depths?

   1. 1.5
   2. 0.67
   3. 0.06
   4. 3
   Answer and explanation

   Correct answer: A The thickness divided by the skin depth: 0.3/0.2.

3. For layers thinner than a skin depth, how does the excess ac resistance grow with the layer thickness?

   1. In proportion to the thickness
   2. As the fourth power of the thickness
   3. As the square of the thickness
   4. It does not grow
   Answer and explanation

   Correct answer: B The first term of Dowell's series beyond 1 is proportional to $\varphi^4$.

4. For three layers the loss is least at about 0.76 skin depths. What happens to the loss if the layers are made thicker than that?

   1. It rises: the ac resistance factor grows faster than the dc resistance falls
   2. It keeps falling, because the dc resistance falls
   3. It stays the same
   4. It falls to the dc loss
   Answer and explanation

   Correct answer: A At a fixed frequency the loss is proportional to $F_R/\varphi$: past the optimum, $F_R$ grows faster than $\varphi$.

5. From the thin-layer series, what is the ac resistance factor at the thickness of least loss?

   1. 1
   2. 2
   3. It grows with the number of layers
   4. 4/3, whatever the number of layers
   Answer and explanation

   Correct answer: D At the optimum the fourth-order term equals one third, for any number of layers.

6. How does interleaving a transformer's windings lower the proximity loss?

   1. It shortens the mean length per turn
   2. It raises the skin depth
   3. It lowers the peak MMF and the number of layers from zero to peak MMF
   4. It lowers the dc resistance
   Answer and explanation

   Correct answer: C Placing the secondary between the halves of the primary halves the peak MMF in the window, and with it the layer count that sets the proximity effect.