Electronic Technology · 120 h · Topic 3 of 9

Magnetic materials

Without an air gap, a 30-turn ferrite inductor has 2.8 mH but saturates at 0.31 A. With half a millimeter of air gap it drops to 0.2 mH and withstands 4.3 A: the energy is stored in the air.

Hysteresis Saturation Ferrites Core losses Air gap

01Magnetization and hysteresis

A ferromagnetic material responds to the field \( H \) with a flux density \( B \) much greater than that of vacuum, but not linearly: the magnetization curve rises with a high slope, the permeability, and flattens out at saturation, where the material can no longer align more domains. When a cycle is traversed, the downward curve does not coincide with the upward one: this is the hysteresis loop, with a remanence \( B_r \) and a coercive field \( H_c \). The physical origin was covered in magnetic properties of matter.

\[ B = \mu_0\,\mu_r\,H \] The relative permeability \( \mu_r \) is not constant: it depends on the field, the temperature and the frequency, and falls to almost 1 at saturation. Soft materials have narrow loops and are used in cores; hard ones, wide loops, and are used as magnets.

02Soft magnetic materials

Material	Approximate B_sat	Frequencies	Use
Laminated silicon steel	1.5 to 2 T	50 Hz to a few kHz	Mains transformers, motors
Amorphous alloys	≈ 1.5 T	up to tens of kHz	Low-loss distribution transformers
Nanocrystalline	≈ 1.2 T	up to about 100 kHz	Common-mode chokes, switching transformers
Iron powder and powder alloys	≈ 1 T or more	up to MHz	Inductors with DC current, distributed air gap
Manganese–zinc ferrites	0.4 to 0.5 T	kHz to a few MHz	Transformers and inductors of switching power supplies
Nickel–zinc ferrites	0.3 to 0.4 T	MHz to hundreds of MHz	Interference suppressors, RF transformers

Ferrites are oxide ceramics: their high resistivity almost eliminates eddy currents and so they work at high frequency, but they saturate at less than half the flux density of iron and their saturation drops with temperature. All lose their magnetism above the Curie temperature.

Permanent magnets are chosen for their remanence, their coercive field and their maximum energy product: barium or strontium ferrite, cheap; AlNiCo, stable with temperature; and neodymium–iron–boron and samarium–cobalt, the most powerful.

03Core losses

A core in AC dissipates through hysteresis, proportional to the area of the loop and to the frequency, and through eddy currents, proportional to the square of the frequency, which are reduced by laminating the iron or using high-resistivity materials. Manufacturers give the losses per unit volume, which are fitted with the Steinmetz equation:

\[ P_v = k\,f^{\alpha}\,\hat{B}^{\beta} \] For ferrites, \( \alpha \) is between 1 and 2 and \( \beta \) between 2 and 3, with \( k \), \( \alpha \) and \( \beta \) valid over a range of frequency and flux density. Since the losses grow faster than the flux density, at high frequency one works far from saturation.

04The air gap

An inductor that carries DC current, or the transformer of a flyback supply, needs to store energy. A closed core of high permeability has a lot of inductance, but saturates with very little current. An air gap of length \( \ell_g \) adds a series reluctance that dominates the magnetic circuit:

\[ L = \frac{\mu_0 N^2 A_e}{\ell_g + \ell_e / \mu_r} \] \[ I_{sat} = \frac{B_{max}\,(\ell_g + \ell_e / \mu_r)}{\mu_0 N} \] \( A_e \) is the effective cross-section and \( \ell_e \) the effective magnetic length of the core. The maximum energy, \( \tfrac12 L I_{sat}^2 \), grows with the air gap: almost all of it is stored in the air. The flux that fringes at the edges of the gap raises the actual inductance somewhat.

Lab · ferrite inductor with an air gap

A ferrite E core with A_e = 97 mm², ℓ_e = 79 mm and μ_r = 2000, designed for a maximum flux density of 0.3 T. As a function of the air gap, the inductance (blue, left scale) and the saturation current (yellow, right scale).

Turns N =

Air gap ℓ_g =

In E cores the air gap is ground into the center leg; if instead a spacer is placed between the two halves, the air gap appears twice in the flux path. The design of power inductors and transformers continues in Power Electronics.

05In the lab

Activity 1 · Hysteresis loop

With a transformer, a sense resistor and an RC integrator, observe on the oscilloscope in XY mode the hysteresis loop of the core and drive it into saturation.

Activity 2 · Air gap

Measure with an LCR meter the inductance of a winding on a closed ferrite core and with paper spacers of different thicknesses.

Activity 3 · Saturation

Apply voltage pulses to an inductor and observe the current ramp; identify the point where it steepens because the core saturates.

06Common mistakes

Designing with the B_sat at 25 °C. In a hot ferrite the saturation is considerably lower.
Seeking maximum inductance with no air gap. With DC current, the core saturates immediately.
Forgetting that a spacer counts twice. It is on the outer legs and on the center one.
Using silicon steel at high frequency. Eddy currents heat it.

07Self-assessment

What happens to the inductance and the saturation current when the air gap is doubled?

If the air gap dominates, the inductance falls to almost half and the saturation current almost doubles.

And when the turns are doubled with the same air gap?

The inductance is multiplied by four and the saturation current is halved; the maximum energy does not change.

Why are ferrites useful at high frequency?

Because of their high resistivity, which almost eliminates eddy currents.

What distinguishes a soft material from a hard one?

The soft one has a narrow hysteresis loop, with little coercive field; the hard one, a wide one, and it retains magnetization.

08Further reading

Colonel Wm. T. McLyman. Transformer and Inductor Design Handbook. 4th ed., CRC Press, 2011. The practical design of inductors and transformers, with material data.
B. D. Cullity and C. D. Graham. Introduction to Magnetic Materials. 2nd ed., Wiley-IEEE Press, 2009. The physics and technology of magnetic materials.
Robert W. Erickson and Dragan Maksimović. Fundamentals of Power Electronics. 3rd ed., Springer, 2020. The chapters on magnetic components and their design in switching power supplies.

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