Magnetic materials and the B-H curve

Intent

• Relate flux density to field intensity through the permeability.
• Explain saturation, and why an inductor must be checked at its peak current.
• Compute an inductor’s peak flux density from its inductance, peak current, turns and core area.

Theory

Permeability

In a linear magnetic material the flux density is proportional to the field intensity. The factor is the permeability, written as the permeability of free space times the relative permeability of the material [Erickson & Maksimović 2020, Ch. 10]:

$$B = \mu_0 \mu_r H$$

$B$

magnetic flux density

$H$

magnetic field intensity

$\mu_r$

relative permeability

$\mu_0$

permeability of free space

Flux density in a linear magnetic material✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• linear (unsaturated) magnetic material

Notes

valid below saturation; a ferrite is close to linear only over part of its B-H curve

Source

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

Equation id

mag.B_H · equations.yaml entry

Saturation and hysteresis

The linear relation holds only up to the saturation flux density. Beyond it the material’s permeability collapses toward that of free space, and an inductor’s inductance falls sharply [Erickson & Maksimović 2020, Ch. 10]. The B-H curve of a real core also shows hysteresis: going up and coming down, the flux density follows different paths, and the area between them is energy lost in every cycle (see the core-loss page) [Erickson & Maksimović 2020, Ch. 10].

Ferrites and iron alloys

Ferrites saturate at a lower flux density than iron alloys, but their high resistivity keeps eddy-current loss low, which suits them to high switching frequencies [Erickson & Maksimović 2020, Ch. 10].

Peak flux density of an inductor

An inductor’s flux linkage is its inductance times its current; divided by the turns and the core area, it gives the flux density. At the peak current [Dixon 2001], [Erickson & Maksimović 2020, Ch. 10]:

$$B_\mathrm{pk} = \frac{L I_\mathrm{pk}}{N A_e}$$

$B_\mathrm{pk}$

peak flux density

$A_e$

effective core area

$I_\mathrm{pk}$

peak current

$L$

inductance

$N$

number of turns

Peak flux density from inductance and peak current✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• linear (unsaturated) magnetic material
• flux density uniform over A_e

Source

Dixon 2001; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory)

Derivation

step-by-step sympy derivation

Equation id

mag.B_pk · equations.yaml entry

This must stay below the saturation flux density at the largest peak current the inductor will ever carry, including start-up and load transients.

Worked example

An example core material and inductor:

Core-material and inductor example

Symbol	Value	How it is obtained
$\mu_r$ relative permeability	2000	given
$H$ magnetic field intensity	100 A/m	given
$L$ inductance	100 µH	given
$I_\mathrm{pk}$ peak current	2 A	given
$N$ number of turns	20	given
$A_e$ effective core area	100 mm²	given
$B$ magnetic flux density	251 mT	Flux density in a linear magnetic material
$B_\mathrm{pk}$ peak flux density	100 mT	Peak flux density from inductance and peak current

In a material with the given relative permeability, a field of 100 A/m produces a flux density of 251 mT. The inductor reaches 100 mT at its peak current; compare this with the saturation flux density in the data sheet of the core material.

Try it

▶ Open in the equation explorer: Peak flux density from inductance and peak current
Preset from the example ferrite-bh

Sweep $N$ at fixed inductance: more turns lower the peak flux density. On a given core, though, the inductance itself grows with the square of the turns (inductance from the inductance factor); the gap page shows how a gap resolves this.

Bench exercise

Discharge a charged capacitor into the inductor through a transistor for a fixed time, and watch the inductor current with a current probe. While the core is linear the current ramps in a straight line; when the core saturates, the slope rises sharply. Read the current at the knee, compare it with the peak-flux equation and the data-sheet saturation flux density, and limit the pulse length so that the current stays within the parts’ ratings.

Gotchas

• Saturation is abrupt. Past saturation the inductance collapses and the current rises steeply [Erickson & Maksimović 2020, Ch. 10]; design for the worst-case peak current, not the average.
• A core’s permeability is not a precise number. With a gap, the inductance depends mostly on the gap length instead of the material [Erickson & Maksimović 2020, Ch. 10] (see the gap page).

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.
• 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.
• 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.

Quiz

1. In a linear material with a relative permeability of 1000, the field intensity is $200\,\mathrm{A/m}$. What is the flux density?

   1. $0.025\,\mathrm{T}$
   2. $0.25\,\mathrm{T}$
   3. $2.5\,\mathrm{T}$
   4. $0.0025\,\mathrm{T}$
   Answer and explanation

   Correct answer: B The flux density is $\mu_0 \mu_r H$: $4\pi \times 10^{-7} \times 1000 \times 200$, about $0.25\,\mathrm{T}$.

2. A $50\,\mu\mathrm{H}$ inductor with 10 turns on a core of $0.5\,\mathrm{cm^2}$ effective area carries a peak current of $4\,\mathrm{A}$. What is its peak flux density?

   1. $0.04\,\mathrm{T}$
   2. $0.2\,\mathrm{T}$
   3. $0.4\,\mathrm{T}$
   4. $4\,\mathrm{T}$
   Answer and explanation

   Correct answer: C The flux linkage is $200\,\mu\mathrm{Wb}$ turns; divided by 10 turns and $5 \times 10^{-5}\,\mathrm{m^2}$ it gives $0.4\,\mathrm{T}$.

3. What happens to an inductor's inductance when its core saturates?

   1. It rises sharply
   2. It falls sharply
   3. It is unchanged
   4. It becomes negative
   Answer and explanation

   Correct answer: B Past saturation the permeability collapses toward that of free space, and the inductance with it.

4. Why must an inductor's flux density be checked at its peak current rather than its average current?

   1. Because saturation depends on the instantaneous flux density, which is largest at the peak current
   2. Because the average current is always zero
   3. Because core loss depends only on the peak
   4. It need not be: the average is enough
   Answer and explanation

   Correct answer: A The core saturates as soon as the flux density exceeds the limit, even for a moment in every period.

5. Why are ferrite cores used at high switching frequencies despite their lower saturation flux density?

   1. Their high resistivity keeps eddy-current loss low
   2. They never saturate
   3. They have no hysteresis
   4. They are always the cheapest option
   Answer and explanation

   Correct answer: A Eddy currents are induced by the changing flux; a high resistivity limits them, which matters more as the frequency rises.

6. An inductor is rebuilt on a core with a larger winding window: same inductance, same peak current, twice the turns. What happens to the peak flux density if the core area is unchanged?

   1. It doubles
   2. It halves
   3. It is unchanged
   4. It quadruples
   Answer and explanation

   Correct answer: B At fixed inductance and current, the peak flux density is inversely proportional to the turns.