Eddy Currents: Why a Core Is Sliced Up

A block of non-magnetic metal (μr = 1) sits in an alternating field that runs along it, into the page. The changing flux drives loops of current inside the solid metal — eddy currents — and they turn real power into heat. Cut the same 100 cm3 of metal into N insulated laminations and the loops shrink with the slices. The gold card is the exact slab solution; the blue card is the textbook estimate π2B2f2d2 / 6ρ, which assumes the field reaches the middle of the metal. The second band shows whether it does.

Thickness in skin depths, d/δ2.168
Textbook estimate / exact1.883x
Exact loss per unit volume13.00 MW/m3
Same metal, solid block1.300 kW
Real cut in loss vs the solid block1.000x
What the d-squared rule predicts1.000x

Where this state sitsThe metal is thicker than the field can get into. The textbook formula is high by a factor of 1.883, and past d/delta = 2.2542 making the metal thicker actually LOWERS the loss per unit volume, because the extra metal carries no current.

Eddy-current loss, exact  the full slab solution × 100 cm3
1.300 kW
Textbook estimate only  P/V = π2B2f2d2 / 6ρ
2.448 kW
Lamination thickness  d = 20.0 mm / N
20.000 mm
Skin depth  δ = sqrt(ρ / π f μ0)
9.225 mm
Peak flux density B0.50 T
Frequency f50 Hz
Laminations N1 lamination
Resistivity ρ1.68 uOhm.cm
The stack is fixed at 20.0 × 50.0 × 100.0 mm = 100 cm3, so solid and laminated always compare the same metal; d = 20.0 mm / N. μ0 = 4π × 10-7 and the metal is non-magnetic, μr = 1 — a real ferromagnetic core has a far smaller skin depth, which is why real laminations are a fraction of a millimetre. Insulation is idealised to zero thickness, and hysteresis loss is a separate mechanism this tool does not model. ρ is in uOhm.cm, where 1 uOhm.cm = 1 × 10-8 Ω·m: copper is 1.68 and nichrome 110, the two figures this site already publishes.
Three things on this screen do not divide: the two power readouts reproduce the printed ratio in only 94.4 % of settings and can even be in different units; d/δ recomputed from the two millimetre readouts disagrees with the printed figure more often than it agrees (57.48 %); and d = 20.0/N is a recurring decimal for 29 of the 40 lamination counts, so N × the printed thickness misses 20.000 mm in those 29. Read the numbers, do not multiply them.