Illustrated Physics
physics, drawn
Electromagnetism

The hum at twice the mains

A transformer hums, and the note is not the frequency of the mains but twice it: a hundred hertz in Europe, a hundred and twenty in America. The reason is that iron changes shape when it is magnetised, by a few parts in a million, and the change does not care which way the magnetisation points. A core driven at fifty cycles a second therefore stretches a hundred times a second. The same evenness makes the strain trace a butterfly against the field, puts harmonics at every even multiple of the mains and none at the odd ones, and makes the hum climb steeply as the steel nears saturation.
15 min read 5 figures What stays the sameMade into an instrumentThe shape decides

Assumes: The curve that is really a staircase · The magnet that reverses where its crystal is weakest

Anyone standing near an electricity substation, or beside the transformer of an old radio, has heard the hum. It is a steady low note, close to the G two octaves below middle C in Europe and a little under the B above that in North America. The mains there alternate at fifty and sixty cycles a second. The hum is at a hundred and a hundred and twenty: exactly twice the frequency of the current that causes it. A loudspeaker driven by an alternating current moves back and forth at the current’s frequency, and so does anything pushed by a force proportional to the current. Something about a transformer core responds to the square of what drives it.

That something was found by James Prescott Joule in 1842, three years before his better-known measurement of the mechanical equivalent of heat. He wound a coil round an iron bar, passed a current, and found with a lever magnifying the motion that the bar lengthened by a few parts in a million. Reversing the current did not make it shorten; it lengthened again. The effect, magnetostriction, is the reason for the hum, the reason the hum is at twice the mains, and the working principle of the sonar projectors in submarine hulls. It is also, through the details of which way a material stretches and why, a direct view of the coupling between an electron’s spin and the shape of its orbit.

Even in the magnetisation

The curve that is really a staircase describes how a ferromagnet is magnetised: not by turning every atomic moment a little, but by moving the walls between domains in which the moments are already fully aligned, then, at high fields, rotating what is left. The magnetisation of the whole sample follows the familiar hysteresis loop, rising along one branch and falling along another, crossing zero at the coercive field.

A loop, and the butterfly it makes. Left: magnetisation against applied field for a model ferromagnet, as a fraction of saturation, with the field in units of the coercive field — the rising branch and the falling one, crossing zero magnetisation at plus and minus one. Right: the strain the same material shows along the field, as a fraction of its saturation magnetostriction, on the same two branches. Because the strain goes as the square of the magnetisation it does not care which way the magnetisation points, so it is the same at full positive and full negative saturation, and it falls to nothing at the two points where the magnetisation passes through zero — the coercive fields. A loop traced once produces a butterfly traced once: the strain goes through two full stretches for every one cycle of the field.
Fig. 1 Left, magnetisation against applied field for a model ferromagnet, rising and falling branches crossing zero at ±1 coercive field. Right, the strain along the field on the same branches, going as the square of the magnetisation: equal at both saturations, zero at the coercive fields — the butterfly.

The strain along the field follows the magnetisation, but it follows its square. Saturated one way, the bar is longer by the saturation magnetostriction λs\lambda_s; saturated the other way, it is longer by the same amount. The strain cannot tell north from south. Between the two, where the magnetisation passes through zero at the coercive field, the strain falls to its demagnetised value. Drawn against the applied field, the strain on the two branches makes a shape like a butterfly’s wings, touching down at the two coercive fields and rising to the same height at both ends.

Evenness is required by symmetry, and the argument is short. Reversing every magnetic moment in a crystal is the operation of time reversal, under which the positions of atoms are unchanged. A strain is a statement about positions. So the state with magnetisation MM and the state with −M-M must have the same strain, and the strain can depend on the magnetisation only through even powers. The leading term is the square.

That explains the domain picture too. Moving a wall between two domains magnetised in exactly opposite directions — a 180° wall, whose thickness is the first length that belongs to the substance — changes the magnetisation without changing the strain at all, since both domains are stretched identically. Only the walls between domains at right angles, and the rotation of moments away from their easy directions, change the shape of the sample. Two specimens of the same steel can therefore have the same magnetisation and different strains, depending on how their domains are arranged, which is why real butterfly loops are less tidy than the model’s.

Two stretches for every cycle

In a transformer the flux density in the core follows the applied voltage, swinging sinusoidally between a positive and a negative peak fifty times a second. The strain follows the square of the flux, and so stretches twice a cycle.

Two stretches for every cycle. Over two cycles of 50 Hz mains, the flux density in a transformer core swinging to ±1.7 T, as a fraction of its peak, and the magnetostrictive strain of the core steel in the model, as a fraction of its own peak of 0.51 parts per million. The flux goes positive and negative once a cycle; the strain, indifferent to sign, stretches at every positive and every negative peak, four times in the two cycles drawn — a vibration at 100 Hz. Its waveform is not a sine: it lingers near zero through most of each half-cycle and rises sharply at each flux peak, as the steel approaches saturation, so it carries harmonics at 200, 300 and 400 Hz as well. That is the transformer's hum, and why its pitch is close to the G two octaves below middle C in Europe and, where the mains run at 60 Hz, between a B-flat and a B.
Fig. 2 Two cycles of 50 Hz mains: flux density swinging to ±1.7 T and the model core’s strain, peaking at 0.51 parts per million. The strain stretches at every positive and negative flux peak, four times in two cycles — a vibration at 100 Hz, lingering near zero between peaks and rising sharply near saturation.

If the strain were exactly proportional to the square of a sine, it would be a pure tone at twice the frequency: sin⁡2ωt=(1−cos⁡2ωt)/2\sin^2 \omega t = (1-\cos 2\omega t)/2. The strain of real transformer steel is not, because the material’s response grows faster than the square as the flux approaches saturation. Grain-oriented silicon steel, the material of nearly every power transformer, has a strain under a part per million at its working flux and rises steeply above about 1.7 tesla, where the remaining domains not yet aligned with the rolling direction rotate into it. The model drawn here has that shape. Its strain lingers near zero through most of each half-cycle and spikes at each peak of the flux, a waveform with strong harmonics.

The evenness survives the nonlinearity. Whatever the shape of the response, the strain at time tt and at time tt plus half a mains cycle are equal, because the flux at those two times is equal and opposite. A waveform that repeats every half-cycle contains only frequencies that are multiples of twice the mains: 100, 200, 300, 400 hertz, and nothing at 50, 150 or 250. That is a prediction a microphone beside a transformer can check, and it holds: the hum’s spectrum is a comb on the even harmonics, with the odd ones far weaker and coming from other mechanisms, such as the magnetic forces between laminations or the vibration of the windings, whose forces go as the product of current and field.

The hum grows teeth

How loud and how harsh the hum is depends on how close to saturation the core is worked.

The hum grows teeth near saturation. The harmonic content of the core's strain, in decibels relative to a thousandth of a part per million, at multiples of the 50 Hz mains, for peak flux densities of 1.3, 1.7, 1.9 T in the model of grain-oriented steel. Every odd multiple is absent, as it must be for a strain that ignores the sign of the flux; the even ones — 100, 200, 300 and 400 Hz — carry it all. Raising the peak flux from 1.3 to 1.9 T raises the 100 Hz line by 22 dB and the higher ones by more, so a core worked near saturation hums louder and harsher. Designing a quiet transformer is, first, designing one that runs its steel at lower flux, which costs more steel.
Fig. 3 The model core strain’s harmonics, in decibels, at peak flux densities of 1.3, 1.7 and 1.9 T: no odd multiples of 50 Hz at all, and the even ones at 100, 200, 300 and 400 Hz. Raising the peak flux from 1.3 to 1.9 T raises the 100 Hz line by 22 dB and the higher ones by more.

At 1.3 tesla the strain is small and nearly sinusoidal at twice the mains; at 1.9 tesla it is twenty-two decibels stronger at a hundred hertz and much richer in the higher harmonics, which the ear hears as a harsher, buzzier tone. A transformer designer chooses the working flux density as a compromise. A higher flux uses less steel for the same power, since the voltage a winding develops is proportional to the rate of change of flux through it, so the cheapest transformer runs its core as close to saturation as the losses allow. A quieter transformer runs it lower and uses more steel. Distribution transformers for city substations and for the basements of buildings are routinely specified a tenth of a tesla or two below those for rural sites, for exactly this reason.

The higher harmonics matter for a second reason: the ear’s sensitivity rises steeply from a hundred hertz to a few hundred, so a hum with a strong component at three or four hundred hertz is heard as louder than its energy suggests, and it is these components that carry furthest through the walls of a building. The low components, conversely, are the ones walls fail to stop: the wall that stops sound by its weight gains only six decibels for every doubling of frequency, so a hundred-hertz hum passes through a wall that silences voices. Steady and periodic, it is also the textbook case for active cancellation, which the silence that fits in a tenth of a wavelength shows works best on precisely such a sound: one that repeats, with a long wavelength, from a source whose spectrum does not change.

How loud a core can hum

The size of the strain sets the size of the motion, and the motion sets the sound.

How loud a core can hum. An upper estimate of the sound level at the face of a core a metre long, vibrating with the model's magnetostrictive strain, against the peak flux density at which the core is worked: each harmonic's surface velocity converted to a plane wave's intensity and the harmonics summed. At 1.2 T it is 46 dB, at 1.5 T 53, at 1.7 T 63 and at 1.9 T 75 dB: the two tenths of a tesla from 1.7 to 1.9 T add 12 dB, nearly as much as the half tesla from 1.2 to 1.7 T added (17 dB). Real cores radiate less than a plane face would and add noise from the magnetic forces between laminations, so absolute levels differ; the steepness is the point. Low-noise transformers for houses and city substations are specified at lower working flux than those in open country.
Fig. 4 An upper estimate of the sound level at the face of a metre-long core with the model’s strain, each harmonic’s surface velocity treated as a plane wave: 46 dB at 1.2 T, 53 at 1.5 T, 63 at 1.7 T and 75 at 1.9 T. The two tenths of a tesla from 1.7 to 1.9 add 12 dB, nearly as much as the half tesla below.

A strain of half a part per million in a core a metre long is a change of length of half a micrometre, a few hundredths of the thickness of a hair, repeated a hundred times a second. The surface moves at a fraction of a millimetre a second. Treated as a plane face radiating into air, that gives a sound level of about sixty decibels at the face, which is a conversation; a large substation transformer, with a core several metres across and a tank whose panels resonate — at a frequency that gets an answer out of all proportion to the drive — can be heard a hundred metres away. The estimate in the figure is an upper bound in one sense, since a real core is not a rigid piston and radiates less efficiently than a plane wave, and a lower bound in another, since real transformers add the vibration of their windings and the magnetic attraction between laminations, which also scales with the square of the flux.

What the figure shows reliably is the steepness. The last two tenths of a tesla before saturation add twelve decibels, almost as much as the half tesla before them. That is the reason for the designer’s compromise above, and the reason that a transformer driven with a voltage a few per cent above its rating, or with a direct-current offset that pushes one half of each cycle closer to saturation, becomes conspicuously louder. Geomagnetic storms induce such offsets in long transmission lines, and the transformers at their ends are heard to roar.

How much a magnetised metal stretches

Magnetostriction is not a universal constant of magnetic materials. It varies by more than three decades and changes sign from one material to another.

How much a magnetised metal stretches. Saturation magnetostriction — the fractional change of length along the magnetisation when a material is magnetised to saturation from the demagnetised state — in parts per million, on a logarithmic scale, with its sign: grain-oriented transformer steel under one, polycrystalline iron −7, an iron-based amorphous ribbon +27, nickel −33, cobalt −62, the iron–gallium alloy Galfenol about +250 and Terfenol-D, an alloy of terbium, dysprosium and iron, about +1600. Nickel and iron shorten along the field, the alloys lengthen. The range is more than three decades, and its top end comes from the rare-earth atoms, whose 4f electron clouds are strongly non-spherical and are dragged round with the spin. A Terfenol rod a metre long would lengthen by more than a millimetre — the strain sonar projectors and precision actuators use; a transformer core is made of steel chosen to stretch as little as possible.
Fig. 5 Saturation magnetostriction in parts per million, logarithmic, with sign: grain-oriented transformer steel under 1, polycrystalline iron −7, iron-based amorphous ribbon +27, nickel −33, cobalt −62, Galfenol about +250, Terfenol-D about +1600. Iron and nickel shorten along the field; the alloys lengthen.

Nickel shortens by 33 parts per million along the field and cobalt by 62; polycrystalline iron by a few, although single crystals of iron lengthen along some directions and shorten along others. Alloys of iron and gallium, called Galfenol, lengthen by a few hundred. Terfenol-D, an alloy of terbium, dysprosium and iron developed at the US Naval Ordnance Laboratory in the 1970s, lengthens by about sixteen hundred parts per million, enough that a rod a metre long would grow by more than a millimetre. Terfenol drives the projectors of low-frequency sonar and precision actuators that move by micrometres under a current; nickel was the sonar material before it.

The origin of the effect lies in the coupling between an electron’s spin and its orbital motion, and so, like the magnetism itself, in quantum mechanics: the magnetism classical physics forbids has no room for it. What holds a magnet together is the exchange interaction, which aligns spins without caring which direction they point relative to the crystal. Spin–orbit coupling breaks that indifference: the electron’s orbital cloud, if it is not spherical, prefers a particular orientation relative to the spin, so turning the spins turns the clouds. If the clouds are not spherical, turning them changes how the atoms pack. The same coupling produces the crystal’s preferred directions of magnetisation, whose role in reversal is the subject of the magnet that reverses where its crystal is weakest; magnetostriction is the way that preference changes when the crystal is strained, read the other way round. The rare-earth atoms in Terfenol have 4f electron clouds that are strongly flattened or elongated and are locked to the spin, which is why they stretch so much. The iron in transformer steel has a nearly spherical cloud and a weak coupling, and the steel is processed to orient its grains so that magnetisation along the rolling direction needs almost no rotation of the moments at all.

Making the square into a line

A material that responds to the square of its drive is a poor loudspeaker. Fed a tone at a thousand hertz, a bare Terfenol rod would sing at two thousand, with the waveform’s every feature doubled and distorted. The evenness that makes the transformer hum at twice the mains is, for a transducer, a defect to be engineered away, and the way it is done is the same trick used for every square-law device from the carbon microphone to the optical modulator: bias it.

A steady magnetic field, supplied by a permanent magnet or a direct current in the coil, moves the operating point out along one wing of the butterfly, to where the curve is steep and nearly straight. A small alternating field added on top then swings the strain up and down along that straight stretch, at the drive frequency and in proportion to the drive. The square has been made locally linear by expanding it about a point away from zero: (H0+h)2=H02+2H0h+h2(H_0 + h)^2 = H_0^2 + 2H_0 h + h^2, and for small hh the middle term dominates. The sensitivity grows with the bias, until the bias pushes the material into saturation and the wing flattens again. Choosing the bias is choosing the steepest stretch of the butterfly.

Terfenol has a second requirement, which also follows from the domain picture. Its strain comes from rotating domains from directions across the rod into directions along it, and a rod with its domains already lying along its length has nothing to rotate. So the rod is squeezed lengthwise by a spring before it is used, which, by the coupling described in the next section, turns the domains to lie across it; the field then rotates them back. A Terfenol projector in a sonar array is a rod held in compression by a stack of springs, surrounded by a coil carrying a direct current and an alternating one, with a permanent magnet to save the direct current’s power — every component there because the strain is even in the magnetisation and depends on which way the domains lie.

The same expansion explains the residual odd harmonics in a transformer’s hum. A core with a small direct-current offset, from an unbalanced rectifier load or a geomagnetic storm, is biased slightly along one wing, and the strain then contains a component at the mains frequency itself, fifty hertz, in proportion to the offset. A hum with a fifty-hertz component is a diagnosis: something is pushing a direct current through the transformer.

Squeeze a magnet, and it answers

Because the coupling is a thermodynamic one, it works both ways. If magnetising a material changes its length, then stretching it changes how easily it magnetises. That converse, found by Emilio Villari in 1865, is the Villari effect — a case of the coupling that is the same both ways, since one free energy produces both directions of the effect — and it is how a magnetostrictive torque sensor works: a shaft carrying torque has its surface stressed along helices, the stress changes the surface’s permeability along those directions, and a pickup coil reads the change without touching the shaft.

It also explains why transformer cores have to be assembled with care. Clamping the laminations too hard, or bending them as they are cut, introduces stresses that change the steel’s magnetostriction and its permeability, raising both the losses and the noise. Cores are annealed after cutting to relax the stresses the cutting introduced, and grain-oriented steel is coated with an insulating layer applied under tension, which puts the steel into a state of stress that reduces its magnetostriction further. A transformer core is a mechanical object designed around a magnetic one.

The same coupling, in the volume rather than the length, makes possible the alloy that the pendulum that must not grow when it warms needs: in invar the magnetic ordering expands the lattice by almost exactly as much as cooling would contract it, so the thermal expansion nearly vanishes below the Curie point. Magnetostriction there is not a nuisance but the reason the material exists.

A loop drawn as a tanh, a core as a curve

The magnetisation loop is a hyperbolic tangent on each branch, with the field measured in units of the coercive field; real loops have the more elaborate shape of a material’s domain structure, and their branches are traced by the jumps of individual walls. The strain is taken as λs(M/Ms)2\lambda_s (M/M_s)^2 in the butterfly, which is the leading even term and correct for an isotropic material whose domains rotate freely; real materials with 180° walls show strains that depend on the domain arrangement, not only on the net magnetisation. The core’s strain against flux density is an empirical shape, small and quadratic at low flux and rising as the fourteenth power near 2 tesla, chosen to resemble grain-oriented steel; it is a model of the trend, not the curve of any particular grade, and it ignores the hysteresis in the strain itself, which adds weak odd harmonics. The sound levels assume a rigid plane face of the core radiating into air and omit the tank, the windings and the oil.

Still open: a transformer that does not hum

The quietest transformers now use amorphous metal cores, ribbons of iron-based glass cast so fast that they have no crystal grains; their losses are a third of grain-oriented steel’s, but their magnetostriction is larger, around twenty-seven parts per million, and they are noisier for the same flux. Cobalt-based amorphous alloys have magnetostriction close to zero and almost no hum, but cobalt is expensive and their saturation flux is low. A material with the saturation flux of iron, the low losses of a glass and zero magnetostriction does not exist, and whether one can be designed — by balancing contributions of opposite sign from different atoms, as the cobalt alloys do at a cost in flux — is a question of active materials research, with the noise of the electricity grid in the balance.

The habit worth carrying away is to ask whether a response is even or odd in its cause. Magnetostriction stretches a ferromagnet by the same amount whichever way it is magnetised, so a core driven at fifty hertz stretches at a hundred, traces a butterfly against the field, and hums at every even harmonic of the mains and none of the odd ones — louder by twelve decibels for the last fifth of a tesla before saturation. A transformer is telling anyone who listens that its strain is a function of the square of its flux.

Part 8 of 8

This essay is one argument about Magnetisation. The others:

The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly. The magnet that reverses where its crystal is weakest Part 7 — A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect.

Where this goes next

A magnetisation curve, and the same curve magnified. On the left, the position of a domain wall against the applied field, over the whole of a crystal. It looks like a curve. On the right, a window 2.0% of its width, taken 42% of the way across, at the magnification a sensitive measurement reaches: the wall does not move at all while the field rises, then jumps, then stops again. The largest jump in the window covers 0.67 units of wall position in no field change at all. There is nothing smooth underneath this — the smooth curve on the left is a staircase with 1400 steps in it, drawn small. The curve that is really a staircase A magnetisation curve is drawn as a smooth line and is nothing of the sort. Measured finely enough it is a sequence of jumps of every size, audible as a crackle in a coil, with no typical jump and no smooth motion underneath. The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly. The magnet that reverses where its crystal is weakest A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect. Where cancelling a 500 Hz tone works. Sound at 500 Hz (wavelength 69 cm) arriving from every direction at random, as it does in a room or a cabin, and a small loudspeaker 50 cm to the left of a microphone (black dot), driven to cancel the sound exactly at the microphone. Contours mark where the sum is 20 dB quieter than the original (red), 10 dB quieter (blue), unchanged (grey) and 6 dB louder (black), in a plane through both. The zone at least 10 dB quieter is 12.0 cm across, at right angles to the line from speaker to microphone — 0.18 of a wavelength — and beyond the unchanged contour the cancelling source only adds sound. Silence has been made at a point, not in a room. The silence that fits in a tenth of a wavelength Two sounds of equal strength and opposite phase add to nothing, so any noise can be cancelled by playing its inverse. At one point. A microphone and a loudspeaker can silence the air exactly where the microphone is, and nowhere else for certain: in the diffuse sound of a room the quiet zone is a few tenths of a wavelength across, under ten centimetres at a kilohertz, and beyond it the cancelling speaker only adds noise. That is why noise cancellation works in a headphone and not in a room, and why it works on the drone of an engine and not on a voice.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

HarmonicHysteresisMagnetic domainMagnetocrystalline anisotropyMagnetostrictionSaturation magnetisationSpin orbit couplingTransformer