Core loss

Intent

• Name the two mechanisms of core loss: hysteresis and eddy currents.
• Estimate the core loss per unit volume with the Steinmetz equation from the flux amplitude and the frequency.
• Explain why, at high switching frequencies, core loss rather than saturation limits the flux swing.

Theory

Hysteresis and eddy currents

In every period the flux density traces the core’s B-H loop, and the energy enclosed by the loop is lost as heat: the hysteresis loss [Erickson & Maksimović 2020, Ch. 10]. The changing flux also induces currents in the core material itself, which dissipate power in its resistance: the eddy-current loss [Erickson & Maksimović 2020, Ch. 10]. Steinmetz found that the hysteresis loss per cycle grows with a power of the peak flux density [Steinmetz 1984].

The Steinmetz equation

For a given material, the measured loss per unit volume is fitted over a range of frequencies and flux densities by [Erickson & Maksimović 2020, Ch. 10]:

$$P_v = k f^{\alpha} B_\mathrm{ac}^{\beta}$$

$P_v$

core loss per volume

$B_\mathrm{ac}$

ac flux-density amplitude

$\alpha$

Steinmetz frequency exponent

$\beta$

Steinmetz flux-density exponent

$f$

frequency

$k$

Steinmetz coefficient

Core loss per unit volume (Steinmetz form)✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• sinusoidal excitation (empirical fit)

Convention

$B_\mathrm{ac}$ is the peak AC amplitude, half the peak-to-peak swing $\Delta B$; $k$, $\alpha$, $\beta$ are fitted per material and valid only over the fitted range

Notes

Steinmetz's original law gives loss per cycle proportional to $B^{1.6}$; the frequency term is the empirical extension used with ferrite data

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: core loss); Steinmetz 1984

Equation id

loss.steinmetz · equations.yaml entry

$B_\mathrm{ac}$ is the peak ac flux density, half of the peak-to-peak swing given by Faraday’s law (flux swing). The coefficients hold only for the material, the temperature and the range over which they were fitted, and for sinusoidal excitation; for the rectangular voltages of a converter the equation gives an estimate.

The loss of the whole core is the loss per unit volume times the core’s effective volume, with the flux density taken as uniform over it [Erickson & Maksimović 2020, Ch. 10]:

$$P = P_v V_e$$

$P$

power

$P_v$

core loss per volume

$V_e$

effective core volume

Core loss of a whole core✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• flux density uniform over the effective volume V_e

Notes

the flux density is taken as uniform over the effective volume $V_e = A_e l_e$

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: core loss)

Equation id

loss.core · equations.yaml entry

Why the swing is limited at high frequency

At a fixed voltage and duty ratio, the flux swing is inversely proportional to the switching frequency. At high switching frequencies the swing that keeps the core loss acceptable is usually well below the saturation flux density, so core loss, not saturation, sets the flux swing in a transformer design [Erickson & Maksimović 2020, Ch. 12].

Worked example

Example Steinmetz coefficients at one operating point:

Example Steinmetz parameters

Symbol	Value	How it is obtained
$k$ Steinmetz coefficient	1 W/m³	given
$f$ frequency	100 kHz	given
$\alpha$ Steinmetz frequency exponent	1.5	given
$B_\mathrm{ac}$ ac flux-density amplitude	100 mT	given
$\beta$ Steinmetz flux-density exponent	2.5	given
$V_e$ effective core volume	3000 mm³	given
$P_v$ core loss per volume	100 kW/m³	Core loss per unit volume (Steinmetz form)
$P$ power	300 mW	Core loss of a whole core

With these coefficients, a peak ac flux density of 100 mT at 100 kHz dissipates 100 kW/m³ per unit volume. A core with an effective volume of 3000 mm³ then loses 300 mW in total.

Try it

▶ Open in the equation explorer: Core loss per unit volume (Steinmetz form)
Preset from the example core-loss

Sweep $B_\mathrm{ac}$ on its logarithmic axis: on a log-log plot the loss is a straight line whose slope is the exponent $\beta$. Halving the flux amplitude divides the loss by two to the power $\beta$.

Bench exercise

Drive a transformer’s primary with a square wave, secondary open, and record the primary voltage and current with an oscilloscope. The average of their product over a period is the core loss plus a small winding loss. Deskew the voltage and current probes first: the magnetizing current is mostly reactive, so a small timing error changes the result a lot. Repeat at half the voltage and compare the ratio of the losses with two to the power $\beta$.

Gotchas

• The coefficients are empirical. They hold only for the material, the temperature, and the range of frequency and flux density over which they were fitted [Erickson & Maksimović 2020, Ch. 10].
• Use half the peak-to-peak swing. $B_\mathrm{ac}$ is the amplitude, half of $\Delta B$; using the whole swing overstates the loss by two to the power $\beta$.
• A filter inductor with small ripple has a small swing. Its core loss is then modest, and the dc flux, which causes no core loss, sets its saturation limit [Erickson & Maksimović 2020, Ch. 10].

Go deeper

• Book chapter Basic Magnetics Theory (Ch. 10) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  Magnetic circuits, the transformer model with magnetizing and leakage inductance, and core and winding loss mechanisms.
• Course Magnetics for Power Electronic Converters — University of Colorado Boulder (Coursera) · retrieved 2026-09-23
  Magnetic circuits, core and winding losses, and inductor and transformer design.
• Book Magnetics Design Handbook (SLUP132) — L. H. Dixon, Texas Instruments (Unitrode) · retrieved 2026-09-23
  Lloyd Dixon's magnetics design handbook, reproduced by TI from the 2001 Unitrode edition. The ti.com copy requires a free myTI login.

Quiz

1. Which two mechanisms make up core loss?

   1. Hysteresis and eddy currents
   2. Winding resistance and skin effect
   3. Leakage inductance and fringing
   4. Saturation and fringing
   Answer and explanation

   Correct answer: A Hysteresis loses the energy enclosed by the B-H loop in every cycle; eddy currents dissipate power in the resistance of the core material.

2. At a fixed frequency, the flux amplitude in a core is halved. With a Steinmetz exponent $\beta = 2.5$, by roughly what factor does the core loss fall?

   1. 2
   2. 2.5
   3. 5.7
   4. 32
   Answer and explanation

   Correct answer: C The loss scales with the amplitude to the power $\beta$, so it falls by $2^{2.5}$, about 5.7.

3. A transformer's flux density swings $0.2\,\mathrm{T}$ peak-to-peak. Which value goes into the Steinmetz equation as $B_\mathrm{ac}$?

   1. $0.05\,\mathrm{T}$
   2. $0.1\,\mathrm{T}$
   3. $0.2\,\mathrm{T}$
   4. $0.4\,\mathrm{T}$
   Answer and explanation

   Correct answer: B $B_\mathrm{ac}$ is the peak ac amplitude, half of the peak-to-peak swing.

4. Why is the Steinmetz equation only an estimate for a converter's rectangular waveforms?

   1. Its coefficients are fitted for sinusoidal excitation over a limited range
   2. It ignores the frequency
   3. It applies only to air-core inductors
   4. It gives the winding loss, not the core loss
   Answer and explanation

   Correct answer: A The coefficients come from measurements with sinusoidal flux; other waveforms and conditions outside the fitted range change the loss.

5. At high switching frequencies, what usually limits the flux swing in a transformer?

   1. Saturation
   2. Core loss
   3. The winding capacitance
   4. Nothing
   Answer and explanation

   Correct answer: B The swing that keeps the core loss acceptable is usually well below the saturation flux density.

6. A core material loses $10^5\,\mathrm{W/m^3}$ at the operating point. What is the loss in a core of $2\,\mathrm{cm^3}$?

   1. $0.02\,\mathrm{W}$
   2. $0.2\,\mathrm{W}$
   3. $2\,\mathrm{W}$
   4. $20\,\mathrm{W}$
   Answer and explanation

   Correct answer: B The volume is $2 \times 10^{-6}\,\mathrm{m^3}$, and the loss per unit volume times the volume is $0.2\,\mathrm{W}$.