Core Loss Steinmetz⚠ unverified
Electrical / Power Electronics · Compute core loss using the Steinmetz equation
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| k | k | — | 1.0 | Steinmetz coefficient (material-specific constant) |
| alpha | α | — | 1.0 | Frequency exponent (dimensionless) |
| beta | β | — | 1.0 | Flux-density exponent (dimensionless) |
| B | B | T | 1.0 | Peak flux density |
| f | f | Hz | 1.0 | Frequency |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | P | W | Core power loss density, in watts per unit volume (W) |
The science & history
Understanding the Parameters
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$k, \alpha, \beta$ — manufacturer or curve-fit constants for a given core material (and often a restricted $B$–$f$ range). They are not universal constants of nature. The live calculator labels $k$ as dimensionless; in practice $k$ carries units so that $k B^\alpha f^\beta$ yields W, W/kg, or W/m$^3$ depending on how the datasheet defines it — match $k$ to the same basis you want for $P$.
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$B$ — peak flux density (tesla). Higher $B$ raises loss steeply ($\alpha$ often near 2).
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$f$ — switching or excitation frequency. Higher $f$ raises eddy and hysteresis loss ($\beta$ typically slightly above 1 for many ferrites).
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$P$ — predicted core dissipation. Add copper (winding) loss separately for total magnetics loss.
Derivation (Approaching a Proof) — and why it is not a proof
What is fundamental. Core loss arises from (1) hysteresis (area of the $B$–$H$ loop per cycle times frequency) and (2) eddy currents (induced $E$ fields driving $I^2R$ in conductive core material, scaling roughly as $B^2 f^2$ for classical thin laminations). Maxwell + material constitutive laws give those mechanisms in principle.
What Steinmetz does. Charles Steinmetz (and later refinements) observed that measured core loss over practical ranges collapses to a compact power law $P \propto B^\alpha f^\beta$. The exponents absorb the mix of hysteresis and eddy mechanisms and the non-ideal shape of real loops. There is no unique derivation of $\alpha,\beta$ from first principles for arbitrary materials; they are fit parameters. The calculator simply evaluates that fit.
Limitations (be honest in design):
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Best for sinusoidal $B(t)$. Square-wave / PWM excitation needs modified Steinmetz (MSE), improved generalized Steinmetz (iGSE), or measured loss maps.
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Valid only inside the $B$–$f$ window of the fit; extrapolating past saturation or extreme $f$ fails.
- Does not separate copper loss, gap fringing, or DC-bias effects.
History
Charles Proteus Steinmetz published empirical core-loss laws in the late 19th / early 20th century for AC machinery and transformers. The form remains the industry default for quick ferrite/iron-powder estimates; modern design tools still quote Steinmetz coefficients alongside more refined models for high-frequency SMPS magnetics.
Related Concepts: Magnetizing Inductance, Inductor Stored Energy, Transformer Turns Ratio, Conduction Loss, Switching Loss
Notes: Registry calculator core-loss-steinmetz (unverified). Empirical — not a first-principles
identity. Confirm whether $P$ is total watts or density (W/m$^3$ or W/kg) against the $k$ you enter.
$k$ labelled dimensionless in the registry (Track B unit/metadata cleanup).