What is magnetostriction?

A change in shape of magnetic materials such as iron and nickel when magnetised: the crystal lattice stretches or shrinks slightly as magnetic domains align. The inverse (Villari) effect, where force changes magnetisation, is used in force sensors. Magnetostriction causes transformer and motor noise and is exploited in actuators and sonar.

How large is magnetostriction?

Comparing saturation strain: iron about 10 ppm, nickel −33 ppm (it contracts), iron–gallium alloy (Galfenol) about 250 ppm, and terbium–dysprosium–iron alloy (Terfenol-D) about 1,600 ppm. A 100 mm rod moves about 1 μm in iron and 160 μm in Terfenol-D.

Why do transformers hum?

Core steel stretches and shrinks by magnetostriction each time it magnetises. Magnetostriction has the same sign regardless of magnetisation direction (roughly proportional to flux density squared), so on 50 Hz AC it cycles twice per period, making 100 Hz (120 Hz at 60 Hz) the fundamental tone. Steel strain is around 1 ppm, but a 1 m limb vibrates about 1 μm, radiating through the tank as sound.

Magnetostriction Analysis

Category: Coupled Analysis – Electromagnetic-Structural | Revised 2026-10-01

Theory: strain caused by magnetisation

Overview

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Is a transformer's hum the wires vibrating?

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The main cause is core magnetostriction. Core steel stretches and shrinks very slightly each time it magnetises. Since magnetostriction stretches the same way whether magnetisation is positive or negative, it cycles twice per AC period, so the fundamental tone is twice the supply frequency. Strain is tiny, around 1 ppm, but in large cores the vibration amplitude reaches micrometres and travels through oil and tank as sound. Conversely, materials with large magnetostriction make precise actuators and powerful underwater sound sources. Magnetostriction analysis couples the magnetic field with structural deformation.

Stroke, force and field

$$ \Delta L = \lambda L,\qquad F_b = E^H \lambda A,\qquad H = \frac{N I}{L},\qquad \lambda(B) \propto B^2\ \Rightarrow\ f_{hum} = 2 f_{supply} $$

λ is magnetostrictive strain, L rod length, E^H Young's modulus at constant field, A cross-section, N turns and I current. Magnetostriction doesn't depend on flux direction, so AC excitation vibrates at twice supply frequency.

Coffee Break Trivia

Joule's discovery and giant magnetostriction

Joule discovered magnetostriction in 1842 by measuring the change in length of an iron rod when magnetised. Iron and nickel show little magnetostriction, yet nickel-based sonar was built around the Second World War. In the 1970s the US Naval Ordnance Laboratory found that alloys of iron with rare earths terbium and dysprosium showed magnetostriction tens of times nickel's, naming it Terfenol-D after its elements (Ter for terbium, Fe for iron) and the laboratory (NOL).

Worked examples

Example 1: saturation magnetostriction and 100 mm rod stroke

Material	Saturation strain	100 mm rod stroke
Iron	about 10 ppm	1.0 μm
Nickel	−33 ppm	3.3 μm
Galfenol (Fe-Ga)	250 ppm	25 μm
Terfenol-D	1,600 ppm	160 μm

Example 2: Terfenol-D actuator (100 mm long, 10 mm diameter, 1,000 ppm)

Item	Value
Stroke	100 μm
Blocked force (E = 30 GPa)	2,356 N
Stiffness	2.36×10⁷ N/m
Required field	80 kA/m
MMF (current with 500 turns)	8,000 AT (16 A)
Work density (½Eλ²)	15 kJ/m³

By the same measure, the earlier piezo example (PZT stack, 890 ppm at 1.5 kV/mm, E = 48 GPa) has a work density of 19 kJ/m³, comparable in size.

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It gives force similar to piezo, but needs as much as 16 A.

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Magnetostrictive actuators are driven by magnetic field, so coils need large current. Piezo is voltage-driven and needs almost no current when static, whereas magnetostrictive coils heat continuously through resistance, raising temperature and changing properties. In return they drive at low voltage, are rugged, and handle large forces and large-amplitude vibration, suiting powerful underwater sound sources and high-force vibrators. Magnetostrictive materials also stretch more under compressive prestress, which aligns domains, so they're normally held by springs. Analysis couples magnetostrictive nonlinearity, prestress, eddy currents and heating.

Analysis workflow

Compute field distribution in the material from coil current by magnetic analysis.
Enter field- and stress-dependent (nonlinear) magnetostriction data.
Pass magnetostrictive strain to structural analysis for displacement and stress, feeding back if needed.
For AC drive include eddy currents and heating, with temperature-dependent properties.
For transformer noise, analyse from core vibration through tank vibration to sound radiation.

Coffee Break Trivia

“A transformer that should have been quiet”

At a substation, a transformer designed as low-noise was louder than expected once installed, with a particularly strong 200 Hz component. Magnetostrictive core vibration contains 100 Hz and its multiples, and 200 Hz nearly matched a tank-wall natural frequency, causing resonance. Core-only analysis showed small vibration, but analysis coupled through to the tank reproduced the cause. Adding stiffeners to shift the tank frequency brought noise down to target.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Evaluating at supply frequency	Wrong frequency	Use twice the frequency
Treating magnetostriction as linear	Wrong stroke	Use nonlinear data
Ignoring prestress	Strain underestimated	Include prestress dependence
Ignoring coil heating	Properties drift with heat	Couple heat
Core-only noise evaluation	Tank resonance missed	Include the tank

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