Steinmetz equation
Theory: core loss in one formula
Overview
A transformer design report says the core loss was “estimated with Steinmetz.” What did they actually do?
They took the manufacturer's measured loss data, taken at various frequencies and flux densities, summarised it with just three coefficients k, α and β, and read off the loss density at the design point. It is not a field equation; it is an empirical fit to measurements. So the key question is always: over what range was the data that produced these coefficients measured?
The original equation and its coefficients
For sinusoidal excitation, the core loss per unit volume $P_v$ is written in terms of frequency $f$ and the peak flux density $\hat{B}$:
| Symbol | Meaning | Typical range |
|---|---|---|
| $k$ | Material coefficient; its value depends on the unit system | Only meaningful with fixed units (see example below) |
| $\alpha$ | Frequency exponent | About 1.1–1.9 for ferrites |
| $\beta$ | Flux-density exponent | About 2–3 |
| $\hat{B}$ | Peak flux density (not peak-to-peak) | T |
Why is α larger than 1? If the material traces the same loop every cycle, loss should just be proportional to frequency.
Good instinct. Pure hysteresis loss is loop area × frequency, so α would be 1. As frequency rises, eddy-current loss (roughly ∝ $f^2$) and excess loss from domain-wall motion join in. Squeezing all of that into one exponent gives an in-between α of 1–2. Treat α as the average slope over the fitted frequency band, not a physical constant.
Getting a feel with numbers
Take a ferrite with $\alpha=1.4$ and $\beta=2.6$, and choose $k$ so that the loss is 80 kW/m³ (= 80 mW/cm³) at 100 kHz and 0.1 T. That gives $k \approx 3.2$ with $P_v$ in W/m³, $f$ in Hz and $\hat{B}$ in T. These are illustrative values; in a real design you fit them to the manufacturer's data.
| Condition | Loss density | What it tells you |
|---|---|---|
| 100 kHz, 0.1 T | 80 kW/m³ | About 1.6 W for a 20 cm³ core |
| Frequency doubled | ×2.64 → about 211 kW/m³ | $2^{1.4}$ |
| Flux density doubled | ×6.06 → about 485 kW/m³ | $2^{2.6}$ |
| 200 kHz, 0.05 T | About 34.8 kW/m³ | Raising frequency while lowering B (more turns) cuts core loss |
Because β is larger than α, loss is far more sensitive to flux density. That is why raising the switching frequency shrinks a transformer: at a fixed voltage $\hat{B}$ falls in inverse proportion to frequency, and that wins over the rise in $f^{\alpha}$. Skin and proximity effects in the winding grow the other way, so the optimum is set together with Winding loss.
Relation to loss separation (Bertotti)
For electrical steel the loss is usually split into hysteresis, classical eddy-current and excess (anomalous) terms:
The second term is the classical eddy-current loss of a lamination of thickness $d$ and conductivity $\sigma$ under sinusoidal excitation, the only term whose coefficient follows from material constants. For 0.35 mm steel with $\sigma = 2\times10^6$ S/m and density 7650 kg/m³ at 50 Hz and 1.5 T it is about 0.30 W/kg. At 400 Hz the $f^2$ dependence lifts it to about 19 W/kg, and thinning the sheet to 0.20 mm brings it back to about 6.2 W/kg, which is exactly why high-speed motors use thin laminations.
That term assumes the sheet is much thinner than the skin depth. With a high relative permeability, above a few hundred hertz the skin depth approaches the sheet thickness and the formula overestimates. Seeing the single Steinmetz equation as a local approximation of these three terms makes it much easier to judge where it may be used.
Steinmetz and the habit of “a usable formula first”
Charles P. Steinmetz, who emigrated from Germany to the United States, published an empirical law in 1892 stating that hysteresis loss grows roughly with the 1.6th power of flux density. The theoretical explanation came much later with domain theory, but having a usable formula from measurements first made quantitative design of AC transformers and generators possible. Today's ferrite equation is an extension of it, and its character — a design-ready summary rather than a complete physical explanation — has not changed.
Beyond sine waves: the iGSE
Where the original equation fails
In inverters and DC-DC converters the flux is nearly triangular. Can't I just plug in $\hat{B}$ and the switching frequency?
That throws away the waveform. Even with the same peak, intervals where the flux changes quickly add loss. Methods that evaluate loss from the rate of change $dB/dt$ were developed for this; the standard one is the iGSE (improved Generalized Steinmetz Equation, Venkatachalam et al., 2002). Its practical strength is that it reuses the same k, α and β fitted from sine-wave data.
The iGSE formula
$\Delta B$ is the peak-to-peak flux density over the cycle and $T$ the period. $k_i$ is chosen so that the iGSE reproduces the original equation for a sine wave; for $\alpha=1.4$ and $\beta=2.6$, $k_i \approx k/17.2$.
Worked example: triangular flux
For a triangular flux that rises over $DT$ and falls over $(1-D)T$, the integral can be done in closed form:
With the same ferrite (100 kHz, 0.1 T peak), compared with a sine wave of the same peak:
| Waveform | Loss density | Ratio to sine |
|---|---|---|
| Sine | 80 kW/m³ | 1.00 |
| Triangle D=0.5 | About 74.6 kW/m³ | 0.93 |
| Triangle D=0.2 | About 84.7 kW/m³ | 1.06 |
| Triangle D=0.1 | About 100 kW/m³ | 1.26 |
The more lopsided the duty, the faster the flux swings in one interval and the higher the loss. Reading a sine-wave loss curve directly for a low-D circuit such as a flyback underestimates the loss by 20–30%.
Minor loops, zero-voltage intervals and DC bias
- Minor loops: when a cycle contains smaller loops, the iGSE splits the waveform into the major loop and minor loops and evaluates each with its own $\Delta B$. Using only the overall $\Delta B$ underestimates the loss.
- Zero-voltage intervals: while the flux is held constant, as in the freewheeling interval of a full bridge, relaxation in the material still dissipates energy. The iGSE counts this as zero, which is why corrections such as the i²GSE (Mühlethaler et al., 2012) were proposed.
- DC bias: with a superimposed DC field, as in an output choke, the loss rises for the same $\Delta B$. Coefficients are normally measured without bias, so you need biased measurements or a correction factor.
Using it in FEM
Computing loss as a post-process
- Run the magnetic analysis (transient, or frequency-domain for pure sine excitation) and obtain the flux density in every element over one electrical period. In a transient run, use a period after the start-up transient has died out.
- For each element extract $\hat{B}$ or $\Delta B$ and $dB/dt$, and evaluate the loss density $p_v$ with a Steinmetz-type formula.
- Integrate over the core volume to obtain the total core loss:
So the loss isn't solved inside the field analysis, it's computed afterwards?
Right. A Steinmetz-type formula only gives the magnitude of the loss, so it does not change the field solution (one-way coupling). If you want hysteresis inside the field computation, you need a different approach that builds a hysteresis model such as the play model or Jiles-Atherton into the material law. To get the temperature rise, the usual route is to pass the loss distribution to a thermal analysis as a heat source, as covered in Core loss (iron loss) analysis.
Choosing the time step and the analysed period
- The iGSE uses $dB/dt$, so the time step must resolve PWM edges and zero-voltage intervals. Several tens of steps per switching period is a reasonable starting point.
- Even with sinusoidal excitation, saturation distorts the waveform and a frequency-domain analysis will miss the harmonics. If distortion is significant, use a transient analysis.
- In motor teeth and yokes the flux vector rotates. Evaluating only one component underestimates loss, so evaluate the two orthogonal components separately or use loss data measured under rotating fields.
Settings in common tools
| Tool | Main settings | Notes |
|---|---|---|
| Ansys Maxwell | Core-loss model: Power Ferrite for ferrites (Cm, x, y, equivalent to k, α, β) or loss-separation coefficients (Kh, Kc, Ke) for electrical steel | Check the unit system of the coefficients (W/m³ or kW/m³) in the input dialog |
| COMSOL Multiphysics | In the AC/DC Module magnetic-fields interface, add a loss calculation and choose a Steinmetz or loss-separation model | In the frequency domain the sine-wave assumption applies |
| JMAG | Iron-loss calculation evaluates loss from the material's measured loss data over frequency and flux density; a hysteresis-model option is also available | Check the measurement temperature and how processing degradation is handled |
Dialog names and features change between versions, so confirm them in the manual for your version. When moving coefficients between tools, first check that the form of the equation ($\hat{B}$ or $\Delta B$) and the units match.
Forgetting the temperature dependence of the coefficients
Ferrite loss changes strongly with temperature, and many grades are tuned to reach their minimum loss around 80–100 °C. Using coefficients fitted at room temperature for a transformer running at 100 °C tends to overestimate the loss and over-design the cooling. Manufacturers often express the temperature dependence as a quadratic correction such as $P_v = k f^{\alpha} \hat{B}^{\beta}(c_0 - c_1 T + c_2 T^2)$. Always check the temperature at which a datasheet value was measured.
Common mistakes and verification
Common mistakes
| Mistake | Effect | Fix |
|---|---|---|
| Putting peak-to-peak $\Delta B$ into the $\hat{B}$ formula | About 6× overestimate for β=2.6 | Check which quantity the formula expects |
| Mixing up mW/cm³ and W/kg | Off by orders of magnitude or by the density | 1 mW/cm³ = 1 kW/m³; divide by density for W/kg |
| Extrapolating beyond the fitted range | Large errors where α and β change | Use separate coefficients per frequency band |
| Summing losses computed per harmonic | Superposition fails because β≠2 | Evaluate the time waveform with the iGSE |
| Ignoring processing degradation | Punching, stacking and clamping stress raise the real loss | Calibrate a correction factor against measurements |
How to verify the result
- With sine excitation and a simple geometry (a toroidal core), check that the FEM result reproduces the datasheet loss curve. If it does not, the coefficients or units are wrong.
- For electrical steel, compare with values measured by standard methods such as the Epstein frame test (IEC 60404-2).
- On the real device, back-calculate total loss from efficiency or temperature-rise measurements and compare it with the FEM core loss plus winding loss.
I'd like to try out how the coefficients behave myself.
The hysteresis loop simulator shows how loop area relates to loss, and the magnetic materials simulator compares materials. For the balance between core and copper loss in a whole transformer, try the transformer design simulator and the transformer efficiency simulator; design considerations at high frequency are covered in High-frequency transformers.
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