Winding Loss Analysis (AC Copper Loss)
Theory: current crowds within the conductor
Overview
Our switching-supply transformer ran hot, so we changed to thicker foil — and it got hotter. We added copper, so why?
At high frequency, current no longer uses the whole conductor cross-section. The skin effect pushes it to the surface through the conductor's own field, and the proximity effect shoves it to one side through fields from neighbouring layers. In multilayer windings the proximity effect is especially strong, and thickening the conductor can enlarge counter-flowing eddy currents and increase loss. Choosing windings on DC resistance alone walks right into this trap.
Skin depth
| Frequency | Copper skin depth (20 °C) |
|---|---|
| 50 Hz | 9.3 mm |
| 1 kHz | 2.1 mm |
| 10 kHz | 0.66 mm |
| 100 kHz | 0.21 mm |
| 1 MHz | 0.066 mm |
$h$ is the foil thickness (for round wire an equivalent thickness derived from the diameter) and $\Delta$ the ratio of thickness to skin depth. Resistivity rises with temperature, so skin depth increases slightly when hot.
Dowell's equation
| Δ | 1 layer | 2 layers | 4 layers | 6 layers |
|---|---|---|---|---|
| 0.5 | 1.01 | 1.03 | 1.11 | 1.25 |
| 1.0 | 1.09 | 1.41 | 2.69 | 4.82 |
| 1.5 | 1.38 | 2.78 | 8.38 | 17.7 |
| 2.0 | 1.90 | 5.15 | 18.1 | 39.8 |
| 3.0 | 3.01 | 9.54 | 35.7 | 79.2 |
$m$ is the number of layers over which the field rises from zero to its maximum. The first term is skin effect and the second proximity effect, which grows with $m^2$. One layer at Δ = 2 costs only 1.9×, but six layers cost about 40×.
Dowell's 1966 paper
The equation comes from a 1966 paper by P. L. Dowell in Britain. Treating each winding layer as a one-dimensional conducting sheet, he solved the current distribution analytically from the field strength on either side. In an era without routine computing, a single formula for the AC resistance of multilayer windings made it a standard tool for switching-supply magnetics. Many later extensions handle round wire, Litz wire and gaps between layers.
Worked example: foil winding at 100 kHz
Varying foil thickness
For a four-layer foil winding (same turns and width), the AC-resistance factor and the AC resistance relative to the DC resistance of 0.1 mm foil were compared for different thicknesses (100 kHz, skin depth 0.209 mm):
| Foil thickness | Δ | F_R | AC resistance (0.1 mm DC = 1) |
|---|---|---|---|
| 0.1 mm | 0.48 | 1.09 | 1.09 |
| 0.2 mm | 0.96 | 2.43 | 1.22 |
| 0.3 mm | 1.44 | 7.39 | 2.46 |
| 0.5 mm | 2.40 | 26.1 | 5.23 |
At 0.5 mm there's five times the copper, yet five times the loss. That's exactly what happened to our transformer…
Right. Under these conditions 0.1–0.2 mm gives the lowest loss. Keeping thickness no more than about one skin depth (Δ ≈ 1) is the basic rule for multilayer windings. The optimum shifts with layer count and frequency, though, and real currents contain harmonics, so loss must be summed over the spectrum of the waveform.
The effect of interleaving
Winding four primary layers then four secondary layers makes the field build up over all four primary layers ($m = 4$). Interleaving as primary 2 – secondary 4 – primary 2 halves the peak field, so each section behaves as $m = 2$. For the 0.2 mm foil example, $F_R$ drops from 2.43 to 1.34 — nearly halving AC winding loss. Leakage inductance also falls, but inter-winding capacitance rises.
Assessment with FEM
Key points
- Resolve the skin depth: place at least two or three element layers within one skin depth of the conductor surface — about 0.05–0.1 mm for copper at 100 kHz.
- 2D cross-sections often suffice: model the winding window as an axisymmetric or planar 2D section and compute per-conductor loss with a frequency-domain eddy-current analysis. Go 3D only when end-turn effects matter.
- Solve per harmonic: for near-square-wave currents, solve separately at the fundamental and main harmonics and sum losses weighted by each current squared.
- Litz wire: modelling every strand is impractical, so homogenised models with an equivalent complex permeability and conductivity derived from strand diameter are commonly used.
When Litz wire doesn't help
Litz wire twists fine strands together to suppress skin and proximity effects. But with strands too fine or too numerous, insulation takes up more space, the copper fill drops, and the rise in DC resistance can outweigh the AC gain. Poor twisting also leaves current unevenly shared among strands, so it behaves like solid wire. Aim for a strand diameter of about half the skin depth, and compare losses at the actual frequency and winding arrangement before choosing.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Computing copper loss from DC resistance only | Loss badly underestimated at high frequency | Use Dowell or FEM for AC resistance |
| Assuming thicker is better | Proximity effect increases loss | Aim for Δ ≈ 1 and choose together with layer count |
| Evaluating the fundamental only | Harmonic losses missed | Sum over the main harmonics |
| Mesh coarser than the skin depth | Eddy losses underestimated | Two to three elements per skin depth |
| Ignoring resistivity rise with temperature | Loss grows at operating temperature | Use resistivity at operating temperature |
I'd like to try different frequencies and conductor thicknesses.
Use the skin depth calculator and the skin effect simulator for skin depth and current distribution, and the transformer design simulator and the transformer efficiency simulator for the overall loss split in a transformer. Core loss is covered in the Steinmetz equation (core loss), high-frequency design in high-frequency transformers, and winding arrangement versus leakage in leakage inductance.
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