Core loss models compared: Steinmetz, Bertotti and beyond — when to use each

Core-loss equations are full of odd exponents: f^1.38, B^1.6, (fB)^1.5. They are not arbitrary fitting knobs. Each one is the fingerprint of a physical mechanism, and reading them that way tells you when a model will hold and when it will fail.

In why electrical steel has core loss we split loss into three mechanisms and showed they add up to the datasheet within a few percent. In how core loss is measured we saw where the datasheet numbers come from. This article connects the two: it starts from the physics of each mechanism, shows why every popular loss model has the exponents it has, and then tests the models against the same SURA M270-35A datasheet.

Where the exponents come from

Hysteresis: f¹, and B to a power between about 1.6 and 3

Hysteresis loss is the area of the B-H loop, a fixed amount of energy per cycle. The loop does not care how fast it is traversed, so the loss per second is simply energy per cycle times cycles per second: the frequency exponent is exactly 1.

The flux-density exponent is less tidy, because the loop changes shape as it grows. At very low fields (the Rayleigh region, after Lord Rayleigh’s 1887 measurements) the loop area grows as the cube of the amplitude. Through the working range of electrical steel it grows roughly as B^1.6, the value Charles Steinmetz reported in 1892. Separating the M270-35A datasheet gives 1.64. Near the knee of the B-H curve, where each extra tenth of a tesla needs much more field, it steepens again.

Classical eddy current: exactly (fB)²

By Faraday’s law, the electric field induced inside a sheet is proportional to the rate of change of flux, dB/dt. For a sine wave the peak of dB/dt is 2πfB. The resistive loss density is σE², so it goes as the square of dB/dt:

Excess loss: (fB)^1.5

Real magnetisation changes by domain walls moving, not uniformly. Bertotti’s statistical theory (1988) treats the sheet as a population of “magnetic objects” that switch on as the rate of magnetisation rises. The extra field needed to drive them grows only as the square root of dB/dt. Power is field times rate of change of flux:

Why a single power law has “odd” exponents

The Steinmetz equation fits all of this with one power law, P = k·f^α·B^β. Its exponent α is the slope of loss against frequency on a log-log plot. Differentiate a sum of three power laws and a neat result drops out:

The Steinmetz frequency exponent is the loss-weighted average of the three physical exponents. It sits between 1 and 2 because the loss is a mixture, and it drifts upwards with frequency because the eddy-current share grows. We can test this directly. The dots below are the slopes between adjacent frequencies in the M270-35A datasheet at 1.0 T. The line is the prediction from the loss shares found in the first article.

Between 100 and 400 Hz the prediction lands within 0.02 of the datasheet. At the ends it is off by up to 0.1: the three-term split is itself a few percent out at 50 Hz, and at 1 kHz the skin effect trims the eddy current below its classical value.

The flux-density exponent β behaves the same way. At 400 Hz, where all three mechanisms contribute, the datasheet slope sits near 1.8 through the working range, the weighted average of 1.6, 2 and 1.5. It climbs past 2 at the knee, where hysteresis loops balloon:

The models

1. Steinmetz equation

Steinmetz’s 1892 law covered hysteresis alone; the frequency term was added later to make it a general empirical fit. Fitted to M270-35A between 50 Hz and 1 kHz and 0.3–1.5 T, it gives α = 1.38 and β = 1.79, both what the physics above predicts for a hysteresis-dominated mixture.

2. Steinmetz for non-sinusoidal flux: MSE and iGSE

PWM inverters and flux-weakened motors do not produce sinusoidal flux. The Modified Steinmetz Equation (Reinert, 2001) and the improved Generalised Steinmetz Equation (iGSE; Venkatachalam and co-authors, 2002) rewrite the Steinmetz law in terms of dB/dt. The same α, β and k can then be applied to any waveform, including triangular and trapezoidal flux. The rate of change of flux is exactly the quantity the eddy-current and excess mechanisms respond to.

3. Two-term loss separation

Jordan’s 1924 separation adds the classical eddy term to a Steinmetz-style hysteresis term. It has the right frequency exponents for two of the three mechanisms, but no excess term. On M270-35A the fitted eddy coefficient comes out 14% above the value calculated from thickness and resistivity: it quietly absorbs the missing excess loss.

4. Bertotti three-term model

All three physical exponents are built in. This is the electrical-steel loss model in most motor-design software. Ansys Maxwell, for example, uses k_h, k_c and k_e; its versions up to 2023 R2 fixed the hysteresis exponent at 2, and later versions let it vary.

One subtlety: when all three coefficients are fitted to total loss, the data cannot fully tell the terms apart. On M270-35A the fitted classical coefficient comes out at only 55% of its physical value, and the excess term takes up the difference. The total is still accurate; the split is not. If you want physically meaningful coefficients, fix k_c from thickness, resistivity and density, and fit only k_h, n and k_e.

5. Variable-coefficient models

If the coefficients depend on flux density, let them. Ionel and co-authors (IEEE Transactions on Industry Applications, 2007) made the Bertotti coefficients functions of B and frequency. The IEM formula (Eggers, Steentjes and Hameyer, 2012) adds a higher-order term for behaviour near saturation. The loss separation in the first article, one set of coefficients per flux density, is the simplest version of this idea.

6. Hysteresis models: Jiles–Atherton and Preisach

The models above predict loss per cycle for a given peak flux density. Hysteresis models reconstruct the whole B-H loop instead, including minor loops caused by PWM ripple and loops offset by DC bias. The Preisach model (1935) builds the loop from a distribution of elementary switching units. Jiles–Atherton (1986) uses a differential equation with five material parameters.

Head-to-head on M270-35A

All five closed-form models were fitted to the SURA M270-35A datasheet between 50 Hz and 1 kHz and 0.3–1.5 T (61 points). Each fit minimised relative error, with coefficients kept positive.

Inside the fitted range, every model is good to about 5% on average. The variable-coefficient model is best, but it uses 26 parameters (two per flux-density level) against three. If you only ever interpolate, the choice of model barely matters. The differences appear when you leave the data.

The extrapolation test

Suppose the datasheet stopped at 200 Hz, as many do. Fit each model on 50–200 Hz only, then predict 400 Hz and 1 kHz:

Steinmetz under-predicts 1 kHz loss by 20% on average. That is the unsafe direction for a thermal design. Its exponent was measured where hysteresis dominated, and it has no way to know that eddy current will take over. The fitted Bertotti model stays within 5%.

The physics-anchored Bertotti over-predicts by 20%. This is the skin effect from the first article: the classical formula assumes full flux penetration, and at 1 kHz a 0.35 mm sheet no longer has it. Physics-based coefficients are only as good as the assumptions behind them.

Fitting pitfalls

1. Fit relative error, not absolute error. Loss spans nearly three orders of magnitude, from 0.13 to 73 W/kg in this dataset. An ordinary least-squares fit in W/kg is dominated by the biggest numbers. Fitted that way, Bertotti scores R² = 0.9989, yet it misses the 50 Hz points by 21% on average, and 34% at 1.2 T and 50 Hz. It even returns a slightly negative excess coefficient. R² hides this; always look at percentage error at your operating points.
2. Keep coefficients positive. Every loss term is a physical dissipation. A negative coefficient means the fit is trading terms against each other, not describing the steel.
3. Fit the range you will use. A model fitted at 50–100 Hz says little about 1 kHz, and one fitted up to 2.5 kHz may sacrifice accuracy at 50 Hz.
4. Mind the rounding. Datasheets quote two decimals: 0.03 W/kg really means 0.025–0.035, ±17%. Drop or down-weight the lowest flux densities; we fitted from 0.3 T up.
5. Check the frequency exponent is identifiable. Data at a single frequency cannot determine α. A fitting routine will still return a number, so reject it.

Which model to use

Whatever you pick, remember the previous article: all of these are fitted to Epstein data from ideal strips. Cutting damage, stress and rotating flux are still on top.

Summary

Each loss mechanism has its own physical exponents. Hysteresis goes as f¹ because it is a fixed energy per cycle; classical eddy current goes as (fB)² from Faraday’s law and Joule heating; excess loss goes as (fB)^1.5 from domain-wall dynamics. A Steinmetz fit reports their loss-weighted average, which is why its exponents land between 1 and 2 and drift with operating point, and why it under-predicts when extrapolated upwards. The Bertotti model keeps the physical exponents separate and extrapolates far better. Its coefficients vary with flux density, which is what variable-coefficient models address. Inside the fitted range all the models are accurate to about 5%. What separates them is how far you trust them outside it, so fit on relative error and check the percentage error where you will actually operate.

References

1. C. P. Steinmetz, “On the law of hysteresis,” Transactions of the American Institute of Electrical Engineers, vol. 9, 1892.
2. Lord Rayleigh, “On the behaviour of iron and steel under the operation of feeble magnetic forces,” Philosophical Magazine, vol. 23, 1887.
3. H. Jordan, “Die ferromagnetischen Konstanten für schwache Wechselfelder,” Elektrische Nachrichten-Technik, vol. 1, 1924.
4. F. Preisach, “Über die magnetische Nachwirkung,” Zeitschrift für Physik, vol. 94, pp. 277–302, 1935.
5. D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,” Journal of Magnetism and Magnetic Materials, vol. 61, pp. 48–60, 1986.
6. G. Bertotti, “General properties of power losses in soft ferromagnetic materials,” IEEE Transactions on Magnetics, vol. 24, no. 1, pp. 621–630, 1988.
7. J. Reinert, A. Brockmeyer and R. W. De Doncker, “Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation,” IEEE Transactions on Industry Applications, vol. 37, no. 4, 2001.
8. K. Venkatachalam, C. R. Sullivan, T. Abdallah and H. Tacca, “Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters,” IEEE Workshop on Computers in Power Electronics (COMPEL), 2002.
9. D. M. Ionel, M. Popescu, M. I. McGilp, T. J. E. Miller, S. J. Dellinger and R. J. Heideman, “Computation of core losses in electrical machines using improved models for laminated steel,” IEEE Transactions on Industry Applications, vol. 43, no. 6, 2007.
10. D. Eggers, S. Steentjes and K. Hameyer, “Advanced iron-loss estimation for nonlinear material behavior,” IEEE Transactions on Magnetics, vol. 48, no. 11, p. 3021, 2012.
11. Ansys, “Core loss model for a Maxwell material,” Ansys Electronics Desktop help, 2024 R2.
12. Cogent Power / Surahammars Bruk, SURA® M270-35A datasheet. Loss data used for all fits and charts in this article.