How Do You Model Core Loss for Switching (Nonsinusoidal) Waveforms?

Published by West Coast Magnetics, July 2026, based on “Improved Calculation of Core Loss with Nonsinusoidal Waveforms” by Jieli Li, Tarek Abdallah, and Charles R. Sullivan of Dartmouth College (IEEE IAS 2001), which we host (available for download below); content reviewed and confirmed current as of publication.

Core loss in a ferrite is usually estimated with the Steinmetz equation, \(P_v = k \, f^{\alpha} \, \hat{B}^{\beta}\), but that equation is fit to sinusoidal excitation, so it mispredicts loss for the nonsinusoidal flux a switching converter actually produces. The Generalized Steinmetz Equation (GSE), introduced by Charles Sullivan’s group at Dartmouth, extends it to any waveform by working from the instantaneous rate of flux change; it reduces to the ordinary Steinmetz equation for a sine wave and captures DC-bias sensitivity without extra parameters. Its 2002 refinement, the improved GSE (iGSE), is now the standard way to predict core loss from real converter waveforms.

The parameters k, α, and β come from sinusoidal core-loss data, so the Steinmetz equation is only as good as those parameters and only strictly valid for the sine wave they were measured with. The GSE keeps the parameters but frees the waveform.

Core loss is one of the two loss mechanisms that set the size, efficiency, and temperature of an SMPS magnetic (the other is winding loss). Predicting it accurately is what lets a designer pick a core and a flux-density operating point with confidence. The trouble is that the standard tool, the Steinmetz equation, was never built for switching waveforms. This article explains the model, why it fails for real converter waveforms, and the Dartmouth extension that fixes it.

The Steinmetz Equation, and Where It Breaks

The Steinmetz equation predicts the time-average core-loss power per unit volume as:

\[ P_v = k \, f^{\alpha} \, \hat{B}^{\beta} \]

where B̂ is the peak flux amplitude, f is the excitation frequency, and k, α, and β are parameters fit to a material’s measured loss. It is the workhorse of magnetics design, and it is what core manufacturers publish loss curves against.

The catch is in how those parameters are obtained: from sinusoidal excitation only. A switching power converter does not drive its magnetics with a sine wave. A square-wave voltage produces a triangular or trapezoidal flux waveform, often with a DC bias, and those waveforms lose differently than a sine of the same peak and frequency. Applying sinusoidal Steinmetz parameters to them introduces real error, and DC bias shifts the loss further.

It helps to remember that core loss is not one mechanism. It is conventionally split into static hysteresis loss, classical eddy-current loss, and excess (anomalous) loss. The Steinmetz power law lumps all three into one fitted curve, which is convenient but is exactly why it does not travel well from sinusoidal data to an arbitrary waveform.

The Generalized Steinmetz Equation (GSE)

The Dartmouth paper starts from a physically motivated hypothesis: instantaneous core loss depends on both the rate of change of flux, dB/dt, and the instantaneous flux, B(t). Written to stay consistent with the Steinmetz exponents, that is:

\[ P_v(t) = k_1 \left| \dfrac{dB}{dt} \right|^{\alpha} |B(t)|^{\beta-\alpha} \]

Averaging it over one period of the flux waveform gives a formula usable for any waveform:

\[ P_v = \dfrac{1}{T} \int_{0}^{T} k_1 \left| \dfrac{dB}{dt} \right|^{\alpha} |B(t)|^{\beta-\alpha} \, dt \]

This is the Generalized Steinmetz Equation. Three properties make it useful:

• It reduces to ordinary Steinmetz for a sine wave. Choosing the instantaneous exponents as α and β−α makes the GSE reproduce Pv = k·f^α·B̂^β for sinusoidal excitation, so you reuse the same material parameters you already have.
• It captures DC-bias sensitivity for free. Because loss is integrated from the instantaneous waveform, the GSE produces DC-bias effects without any extra measurements or curve-fitting.
• It avoids the anomalies of the earlier Modified Steinmetz Equation (MSE). The MSE defined an “equivalent frequency” by averaging (dB/dt)² over a flux excursion, but that produced non-physical results for some waveforms; the GSE does not.

The GSE can also be recast into an equivalent frequency and equivalent amplitude that you then plug into the ordinary Steinmetz equation, which is convenient for reusing existing tooling. Validated against measurements on MnZn power ferrite, the GSE tracks nonsinusoidal, varying-duty-cycle loss better than either plain Steinmetz or the MSE.

From GSE to iGSE

The GSE has a known limitation: it does not fully account for the history of the flux waveform, which matters when a waveform contains minor loops. The same Dartmouth group addressed this in 2002 with the improved Generalized Steinmetz Equation (iGSE), which works from the peak-to-peak flux excursion and the waveform history. The iGSE is now the form most widely used in converter design tools and core-loss calculators. The 2001 GSE paper linked below is where the approach originates.

What This Means for Your Design

The practical takeaway is simple: do not size a switching-converter core from raw sinusoidal Steinmetz numbers. Use a waveform-aware model (the GSE or, in practice, the iGSE) that reflects your actual duty cycle, ripple, and DC bias. It changes the predicted loss, and therefore the flux-density operating point and the core you should choose.

This is the kind of modeling West Coast Magnetics designs to, and validates by measurement rather than trusting a datasheet curve; our method for separating a component’s core loss from its copper loss is the bench side of the same problem. Getting the core-loss model right feeds directly into choosing a core for a high-power SMPS transformer, while the winding side of loss is covered in skin effect and proximity effect in windings. WCM’s connection to this research is direct: we collaborate with Professor Sullivan’s group at Dartmouth on high-frequency magnetics and partner in the Dartmouth and UC San Diego Power Management Integration Center (PMIC), which keeps foundational power-magnetics research close to the custom SMPS transformers and inductors we build.

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