Steinmetz Equation: Reading Core Loss Coefficients for Ferrite Designers

A core loss budget that predicts 12 W while the bench measures 19 W is rarely a bad material. It is usually the Steinmetz equation used outside the conditions it was fitted for, and the correction costs nothing more than a unit check and a waveform adjustment.

Engineers selecting a MnZn power ferrite for a 100 kHz flyback or LLC transformer start from the same three numbers. The datasheet prints k, alpha and beta in the loss table, and those values compress thousands of measurement points into one curve. Inside the calibrated window they rank candidate grades within roughly 10 to 20 percent before anything is wound. Outside it they can point the design at the wrong grade, particularly when DC bias, dead time or trapezoidal flux enter the waveform.

What the Steinmetz Equation Actually Says

The Steinmetz equation is an empirical model that predicts time-average core loss per unit volume from excitation frequency and peak flux density, written as Pv = k f^α B_pk^β.

Empirical is the important word. Nothing in that expression follows from first principles; k, alpha and beta come from a least-squares fit to measured loss data. The form used in most design tools today is really a power equation, a practical descendant of Steinmetz's 1892 hysteresis work, where the headline result was a hysteresis exponent of roughly 1.6.

k is bound to the unit set it was fitted in. A value of 40 printed for mW/cm3, kHz and mT is not the equivalent of 40 in W/m3, Hz and T, where the same material carries an entirely different constant. Moving k between unit systems is one of the fastest ways to produce an estimate that is wrong by orders of magnitude.

The exponents carry information too. With alpha near 1.6, doubling the switching frequency multiplies loss by about 3.0. With beta near 2.4, doubling the peak flux density multiplies it by about 5.3. That asymmetry is why designers trade frequency for flux swing whenever the topology allows it.

Reading the Coefficients Before You Trust Them

The three Steinmetz coefficients are regression constants, valid only inside the frequency, flux density and temperature window of the measurement set they came from.

That window is often narrower than a summary table suggests. A grade may be characterised from 25 kHz to 500 kHz and 50 mT to 200 mT at a stated reference temperature, while the application runs a 20 mT ripple in a colder spot. The equation still returns a number. It simply stops describing the core.

• Fitted, not derived. The constants come from log-log regression over a grid of frequency and flux points and inherit the scatter of that grid.
• Unit-bound. k changes with the unit set; alpha and beta usually do not.
• Temperature-blind. A fit taken at room temperature and applied at 100 degrees Celsius can overestimate loss by 30 percent or more.
• Sinusoidal by construction. The base form cannot represent a trapezoidal or triangular flux.
• Bias-blind. A DC-biased core walks a minor loop, and the base equation does not know the loop moved.

The practical fix is to ask for the measurement set behind the constants rather than the constants alone. Loss curves taken at two or three temperatures, plotted against flux density, tell a designer far more than a three-number table ever will.

Where the Equation Breaks: Nonsinusoidal Flux

The base Steinmetz equation assumes sinusoidal excitation, so applying it directly to a switching waveform can misprice core loss by a factor of two or more in either direction.

Flux in a hard-switched transformer rises in a trapezoid, holds during the on-time and falls again, with dead time and ringing at the corners. Hysteresis loss depends on how far the material is driven around the loop, while eddy-current loss depends on the instantaneous rate of flux change. Two waveforms with identical peak flux density but different rise times will not lose identical power.

The mechanisms behind that gap are well documented, and a material-level explanation of how MnZn ferrite reduces core loss shows where hysteresis and eddy-current losses originate and which grade choices shift them.

The Extended Family: GSE, iGSE and MSE

The Generalized Steinmetz Equation extends the base model to arbitrary waveforms by integrating loss over the instantaneous rate of flux change, and it reduces to the ordinary equation when the excitation is sinusoidal.

GSE keeps the same fitted parameters but evaluates the waveform point by point instead of relying on a single frequency and peak value. That is why it is treated as a strict generalisation rather than a separate model.

1. GSE. Works from the rate of flux change and the peak-to-peak excursion; handles square and triangular excitation well.
2. iGSE. A 2002 refinement that weights loss by the instantaneous flux level, improving accuracy for duty-cycle-controlled waveforms.
3. MSE. Adds a waveform coefficient while keeping the original exponents, useful when the flux shape is stable.

High-frequency designs put extra pressure on the material itself. MnZn grades with higher resistivity hold eddy-current loss in check at hundreds of kilohertz, which is exactly where the waveform correction and the loss mechanism are both large.

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From Coefficients to a Design Check

A five-step sequence turns Steinmetz coefficients into a defensible loss estimate, and every step costs minutes rather than days.

When the corrected loss still breaks the thermal budget, the next lever is the component rather than the constant: a core and winding combination chosen for the real ripple instead of a catalogue part pushed to its limit.

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Temperature, DC Bias and Material Choice

Coefficients describe a material at one temperature and one bias level, while real cores run hot and often sit on a DC pedestal, so every estimate needs adjustment before it is trusted.

Power MnZn grades are usually characterised with a loss minimum between 80 and 100 degrees Celsius and a curve that climbs steeply below 60 degrees Celsius. A room-temperature fit therefore overestimates loss in a warm, well-cooled design and underestimates it during a cold start, which is precisely when a converter has the least margin.

DC bias adds an effect no coefficient table captures. A biased core operates on a minor loop offset from the origin, so incremental permeability falls, ripple current rises and the flux swing widens. A core running at a modest peak flux can absorb some bias with little extra loss, while one already close to saturation loses efficiency quickly.

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Frequently Asked Questions

What is the Steinmetz equation used for?

It estimates core loss per unit volume in a magnetic material from frequency and peak flux density. Designers use it to compare ferrite grades, size a core against a thermal budget and predict efficiency before a prototype exists.

What are typical Steinmetz exponents?

For power MnZn ferrites, alpha usually falls between 1.2 and 1.8 and beta between 2.0 and 3.0. A higher alpha means loss climbs faster with frequency; a higher beta means the flux swing dominates the budget.

Can it be used for non-sinusoidal waveforms?

Not in its base form. Use GSE or iGSE, which integrate over the instantaneous flux slope and reduce to the classic equation for a sine wave. The correction is essential for hard-switched converters.

How accurate is the Steinmetz equation?

Inside the fitted window it typically lands within 10 to 20 percent of measurement. Outside that window, in flux, frequency or temperature, the error can exceed a factor of two, so the worst case is always worth confirming on the bench.