Why are cores built from thin laminations?

Alternating flux drives eddy currents circulating around it inside the core, turning into heat. Thin, insulated sheets narrow the current paths, and loss falls with thickness squared. Here at 50 Hz and 1.5 T, 2.42 W/kg at 1.0 mm drops to 0.30 W/kg at 0.35 mm and 0.097 W/kg at 0.20 mm.

What changes at higher frequency?

Loss grows with frequency squared, and flux penetrates only near the sheet surface (skin depth). For silicon steel (relative permeability 3000) skin depth is 0.92 mm at 50 Hz and 0.32 mm at 400 Hz. As thickness approaches skin depth, the thickness-squared formula fails and fields must be solved. That is why high-speed motors use thin steel.

Is eddy heating a problem outside cores?

Yes. Leakage flux striking transformer tank walls, clamping parts or steel near heavy-current conductors concentrates eddy currents at the surface, causing local hot spots. Here a 5 kA/m peak leakage field on a mild-steel wall generates about 1,520 W/m², raising it about 30 K even in oil. An aluminium shield cuts this to about 29 W/m².

Eddy Current Loss Heating

Category: Coupled Analysis – Electromagnetic-Thermal | Revised 2026-10-01

Theory: heat from unwanted eddy currents

Overview

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Induction heating uses eddy currents on purpose, so why do transformers avoid them?

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Same mechanism, opposite goal. Changing flux through a conductor drives eddy currents around it, which heat it through resistance. Induction heating wants that, but in transformer and motor cores the only job is carrying flux, so eddy heat is pure loss, lowering efficiency and raising temperature. Cores are therefore stacks of thin insulated steel sheets that narrow eddy paths. Meanwhile flux leaking out of the core hits thick steel such as tank walls and fittings, concentrating eddy currents at the surface and creating local hot spots. Both are coupled problems: obtain loss distribution from the field and pass it to thermal analysis.

Lamination eddy loss and skin depth

$$ P_e = \frac{\pi^2 f^2 B_p^2 d^2}{6\rho}\quad [\mathrm{W/m^3}],\qquad \delta = \sqrt{\frac{2\rho}{\omega\mu}} $$

f is frequency, B_p peak flux density (1.5 T), d sheet thickness, ρ resistivity (0.5 μΩm for silicon steel) and μ permeability (relative 3000). The formula holds when thickness is well below skin depth δ.

Surface loss in thick steel

$$ \frac{P}{A} = \frac{1}{2}\,H_p^2\,\frac{\rho}{\delta}\qquad (\text{thick plate, linear material}) $$

H_p is peak tangential field at the surface. Magnetic saturation usually makes real loss larger, so treat this as a lower-bound guide.

Coffee Break Trivia

Adding silicon to iron

Around 1900 Hadfield in Britain found that adding a few per cent of silicon to iron raised its resistivity several-fold while keeping good magnetic properties. Higher resistivity means weaker eddy currents. Cores of thin, stacked silicon-steel sheets became the standard for transformers and motors, and core-material development still pursues higher resistivity, thinner sheets and easier flux passage.

Worked examples

Example 1: eddy loss by thickness and frequency (1.5 T)

Thickness	50 Hz	Thickness/skin depth at 50 Hz	400 Hz	Thickness/skin depth at 400 Hz
0.20 mm	0.097 W/kg	0.22	6.19 W/kg	0.62
0.35 mm	0.296 W/kg	0.38	19.0 W/kg (rough)	1.08
0.50 mm	0.605 W/kg	0.54	38.7 W/kg (rough)	1.54
1.0 mm	2.42 W/kg (rough)	1.09	155 W/kg (rough)	3.08

Skin depth is 0.92 mm at 50 Hz and 0.32 mm at 400 Hz. Entries with thickness/skin depth near or above 1 are outside the simple formula's range; actual loss is lower because flux doesn't penetrate fully. Real sheet loss also includes hysteresis and other components.

Example 2: wall hit by leakage flux (5 kA/m peak, 50 Hz)

Material	Skin depth	Surface loss	Rise in oil (h = 50 W/m²K)
Mild steel (0.15 μΩm, μr 500)	1.23 mm	1,521 W/m²	30.4 K
Aluminium shield (0.028 μΩm)	11.9 mm	29 W/m²	0.6 K

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Aluminium conducts better than steel, so why is its loss about one-fiftieth?

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Because it isn't magnetic. Steel's relative permeability is in the hundreds, so flux is squeezed into the outer millimetre or so and current crowds there; a thinner current layer means more resistance and more loss. Non-magnetic aluminium spreads current over about 12 mm and has lower resistivity. Loss scales with ρ/δ: steel's 0.15 μΩm/1.23 mm against aluminium's 0.028 μΩm/11.9 mm is about 1:50. Transformers therefore line tanks with aluminium or copper to repel leakage flux, or conversely use laminated steel magnetic shunts to absorb it and guide it along low-loss paths. Which works better depends on leakage-flux direction and magnitude, so 3D field analysis gives loss distribution and thermal analysis checks hot-spot temperatures.

Modelling workflow

Give sheet losses from measured data versus frequency and flux density.
Solve the field including core, windings and surrounding structures.
Treat thick steel with mesh finer than skin depth or a surface-impedance condition.
Pass loss distribution to thermal analysis with cooling to get temperature.
If resistivity changes with temperature, re-solve the field until converged.

Coffee Break Trivia

“Tank bolts with scorched paint”

During a routine inspection of a large transformer, paint was discoloured only around the bolts fastening the tank cover. 3D field analysis showed that near a high-current bushing, the cover plate and bolts formed a closed loop in which leakage flux induced current. Replacing part of the cover with non-magnetic stainless steel broke the loop; the predicted temperature fell, and the next inspection found no discolouration.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Simple formula at high frequency	Loss overestimated	Check skin depth
Eddy loss as total core loss	Core loss underestimated	Measured core-loss data
Coarse mesh in thick steel	Surface loss wrong	Finer than skin depth
Omitting structures	Local hot spots missed	Include fittings and bolts
Ignoring saturation	Surface loss underestimated	Nonlinear magnetisation

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