Why electrical steel has core loss: hysteresis, eddy current and excess loss

SURA M270-35A and M270-50A carry the same rating: at most 2.70 W/kg at 1.5 T and 50 Hz. On the datasheet they measure 2.47 and 2.52 W/kg, near-identical. At 1 kHz and 1.0 T the thicker grade loses 58% more. Core loss is not one mechanism but three, and each grows with frequency in its own way.

Core loss (also called iron loss) is the energy a magnetic core turns into heat each time its magnetisation is cycled, quoted in W/kg at a given peak flux density and frequency. It has three components:

The first comes straight from the B-H loop, the second from Faraday’s law acting inside the sheet, and the third from the way magnetic domains actually move. This article takes each in turn, then splits a real datasheet into the three parts.

Hysteresis loss: the area inside the B-H loop

In an AC motor the stator core is magnetised one way, then the other, 50 to 1,000+ times a second. If you plot B against H over one full cycle, the curve does not retrace itself: the demagnetising branch sits above the magnetising branch, and the two enclose an area. That closed curve is the hysteresis loop.

The area of the loop is the energy per unit volume that the steel absorbs and does not give back in one cycle:

Because the same loop is traversed every cycle, hysteresis loss is a fixed energy per cycle. Double the frequency and you double the loss. Where does the energy go? Steel is made of magnetic domains separated by domain walls. As H changes, walls move, but they snag on grain boundaries, inclusions and internal stress, then jump free. Each jump dissipates energy as heat. Fewer snags mean a narrower loop: that is why electrical steel is annealed for large grains and kept clean of carbon and inclusions.

A sanity check on real data

Later in this article we separate the M270-35A loss data into its three parts. The hysteresis part at 1.5 T comes out as 35.7 mJ/kg per cycle, a loop area of 273 J/m³ (using ρ_m = 7,650 kg/m³). If the loop were a perfect rectangle its area would be 4·H_c·B_max, which gives H_c ≈ 46 A/m. The datasheet quotes a DC coercivity of 40 A/m. For M270-50A the same estimate gives 32 A/m against a quoted 30 A/m. The loss tables and the coercivity agree, to within what a rectangle approximation can promise.

The loop also grows with flux density. In 1892 Charles Steinmetz observed that hysteresis energy per cycle scales roughly as B^1.6. Fitting the separated M270-35A hysteresis energy between 0.3 and 1.5 T gives an exponent of 1.64, remarkably close to Steinmetz’s original value.

Eddy-current loss: Faraday’s law inside the sheet

Steel conducts electricity. When the flux inside a sheet changes, it induces a voltage around every closed path in the metal that encircles it, exactly as it would in a coil. That voltage drives eddy currents that circulate inside the sheet and dissipate I²R heat. The induced voltage is proportional to the rate of change of flux, so it rises with both frequency and flux density.

For a thin sheet with sinusoidal flux that penetrates the full thickness, the classical result is:

Three levers follow directly from the formula:

1. Thickness, squared. Halving d cuts classical eddy loss by four. This is why motor cores are stacks of 0.20–0.65 mm sheets with an insulating coating, never solid blocks.
2. Resistivity. Silicon and aluminium raise resistivity. Pure iron sits near 10 µΩ·cm; M270-35A is 52 µΩ·cm, five times higher. The cost is a slightly lower saturation flux density.
3. Frequency, squared. Negligible at 50 Hz, dominant at high speed.

For M270-35A (d = 0.35 mm, ρ_e = 52 µΩ·cm) at 1.0 T the formula gives 0.13 W/kg at 50 Hz, 8.1 W/kg at 400 Hz and 50.7 W/kg at 1 kHz. The measured totals are 1.01, 17.3 and 73 W/kg. Eddy current is a rounding error at 50 Hz and most of the loss at 1 kHz.

How the B-H curve ties into eddy-current loss: skin depth

The classical formula assumes the flux is spread evenly through the sheet thickness. That stops being true at high frequency. The eddy currents produce their own field, which opposes the applied one and pushes the flux towards the sheet surfaces. The distance the flux penetrates is the skin depth:

This is where the B-H curve comes back in. Permeability is not a material constant: it is the slope of the line from the origin to your operating point on the B-H curve. On the M270-35A curve, 1.0 T needs H = 112 A/m, so μ_r ≈ 7,105. The skin depth follows:

Once δ falls below half the sheet thickness, the centre of the sheet carries less flux than the formula assumes and the classical formula over-predicts eddy loss. M270-50A shows it clearly. At 0.5 T and 1 kHz the classical formula alone predicts 24.6 W/kg. Hysteresis must be added on top. At 50 Hz, where eddy loss is small, the datasheet loss per cycle is 6.6 mJ/kg; removing the classical share (1.2 mJ/kg) leaves about 5.4 mJ/kg, almost all of it hysteresis: roughly 5.4 W/kg at 1 kHz. The two together exceed the measured 26.1 W/kg by about 15%. A formula that predicts more loss than the steel actually has is outside its validity range.

The flip side: permeability collapses as the steel saturates. At 1.5 T, μ_r of M270-35A drops to about 700, the skin depth at 1 kHz grows to 0.43 mm and flux penetrates the sheet fully again. Where you operate on the B-H curve changes how the eddy currents behave.

Excess loss: what the classical theory misses

Add hysteresis loss to classical eddy loss and you still come up short of the measured total. The classical formula assumes the sheet magnetises uniformly. In reality magnetisation changes only where domain walls move, so the eddy currents concentrate around the moving walls rather than spreading evenly. Concentrated currents dissipate more energy than the same total current spread out. The difference is the excess loss, sometimes called anomalous loss.

Giorgio Bertotti’s statistical theory of domain-wall motion (IEEE Transactions on Magnetics, 1988) showed that this term scales as (B_pk·f)^1.5. His three-term split, hysteresis + classical + excess, is the basis of the loss models in most motor-design software. We look at those models, and where they break, in core loss models compared.

Worked example: splitting M270-35A’s datasheet loss

A datasheet gives only total loss. The three parts can still be separated with a simple trick: divide loss by frequency to get energy per cycle.

At a fixed flux density, hysteresis is constant per cycle, classical eddy loss per cycle rises in a straight line, and excess loss per cycle rises as √f. The procedure:

1. Take the datasheet losses at one flux density, here 1.0 T at 50, 100, 200, 400 and 1,000 Hz, and divide each by its frequency.
2. Compute the classical term from thickness, resistivity and density. No fitting is needed: k_cl = 50.7×10⁻⁶ for M270-35A.
3. Subtract it, then fit what remains to C_h + C_ex√f. The intercept at f = 0 is the hysteresis energy per cycle: 16.4 mJ/kg.

Three physically derived terms reproduce the datasheet within 4.5% across a 20:1 frequency range. The split changes completely with frequency: at 50 Hz hysteresis is 78% of the loss; at 1 kHz classical eddy loss is 66%. The slight over-prediction at 1 kHz is the skin effect from the previous section starting to appear.

Why two “2.70 W/kg” grades part ways

Run the same separation on both grades at the rating point, 1.5 T and 50 Hz:

The totals match, but for different reasons. M270-50A has the narrower loop: lower coercivity and 29% less hysteresis loss. It spends that margin on eddy loss, because its sheet is thicker. At 50 Hz, where hysteresis dominates, the trade is even. Raise the frequency and the f² term takes over:

The ratio of classical eddy coefficients between the two grades is (0.50/0.35)² × (52/55) = 1.94. The 50 Hz rating hides that factor; your motor will not.

What this means for motor design

The frequency the stator core sees is the electrical frequency, not the shaft speed:

A 4-pole industrial motor at 1,500 rpm runs at 50 Hz. A 10-pole 12s/10p servo at 6,000 rpm runs at 500 Hz, which puts it firmly in eddy-current territory. The design priority follows from which mechanism dominates:

1. Work out the electrical frequency first. It decides which loss term you are fighting.
2. Up to about 100 Hz, hysteresis dominates. Compare grades on the 50/60 Hz loss and coercivity. Thickness matters little, and thicker sheet is cheaper to punch and stack.
3. At several hundred hertz and above, eddy current dominates. Choose thin sheet (0.35 mm or less) with high resistivity, and compare grades on the loss table at your frequency, never the 50 Hz rating.
4. Check the operating flux density. Every term grows faster than linearly with B, and permeability, and with it skin depth, changes along the B-H curve.
5. Leave margin. Datasheet losses are measured under ideal conditions: sinusoidal flux, uncut strips, no mechanical stress. Punching, press-fitting and PWM harmonics all add loss in a real motor.

Summary

Core loss is the sum of three mechanisms. Hysteresis loss is the area of the B-H loop, a fixed energy per cycle that grows in proportion to frequency; the datasheet B-H curve joins only the loop tips and cannot give it to you. Classical eddy-current loss comes from currents induced in the sheet. It grows with thickness squared and frequency squared, and it is held back by resistivity. Its validity depends on the skin depth, which the B-H curve controls through permeability. Excess loss covers the extra eddy-current loss around moving domain walls and grows as f^1.5. Dividing datasheet loss by frequency separates the three, and the split explains why grades with identical 50 Hz ratings behave so differently in a high-speed motor.