The Faint Mechanical Hum Inside a Ferrite Core That Sneaks Onto Your Signal

A power amplifier can be electrically immaculate and still carry a ghost. The bias is clean, the supply is quiet, the layout is textbook, yet a careful look at the output spectrum shows faint sidebands creeping up around the carrier, and a sensitive ear on a demodulated signal catches a low background that should not be there. Chase it through the circuit and the trail can lead somewhere unexpected: not to a transistor, not to a power rail, but to a ferrite core sitting in a transformer or an inductor, quietly changing its physical shape in time with the magnetic field passing through it. That shape-changing is magnetostriction, and when the core flexes, it can stir the very signal it carries, writing a mechanical modulation onto an electrical waveform. It is one of the subtler noise sources in power amplifier design, and one of the easiest to overlook because it lives in the mechanical world rather than the electrical one.

The effect is small in absolute terms, a dimensional change measured in parts per million, but smallness is not the same as harmlessness. A core that changes length by a millionth of its size with each swing of the field is also a core whose inductance changes slightly with that swing, and whose motion can couple back into the windings as a tiny varying voltage. In a sensitive stage those parts-per-million dimensional wiggles become parts-per-thousand spectral impurities, and the amplifier that looked perfect on paper carries a hum it cannot quite shake.

What magnetostriction is and why a core breathes with the field

Magnetostriction is a property of magnetic materials by which they change physical dimensions under the influence of a magnetic field. The mechanism lives in the magnetic domains, the small regions within the material whose magnetization points in a common direction. When an external field is applied, the domains rotate and their boundaries shift to align with the field, and because the crystal lattice is slightly coupled to the direction of magnetization, this reorientation distorts the lattice itself. The material physically lengthens or shortens along the field direction. Remove the field and it relaxes back. Apply an alternating field and the core breathes in and out in step with it.

The magnitude is genuinely tiny. Measurements on transformer core materials show dimensional changes on the order of one micrometer per meter of length, a strain of about one part per million. Written as a strain lambda, the saturation magnetostriction of typical soft magnetic materials sits in the range

lambda_s = 1 to 30 parts per million (10^-6)

depending on the material, with many soft ferrites toward the low end. This seems negligible until you consider two things: that the change happens twice per electrical cycle and at every harmonic the field contains, and that mechanical structures have resonances where tiny periodic forces build into large motions.

Why the breathing happens at twice the signal frequency

A crucial and often confusing feature of magnetostriction is its frequency. The dimensional change does not track the field linearly. Below saturation, the magnetostrictive strain depends approximately on the square of the magnetic flux density:

lambda(B) is proportional to B^2

A squared quantity does not care about the sign of B. When the field swings positive the core lengthens, and when the field swings equally negative the core lengthens again, because the square of a negative number is positive. So the mechanical deformation completes a full cycle twice for every electrical cycle, and the dominant vibration appears at twice the signal frequency. This is why magnetostrictive hum in mains transformers is heard at 100 or 120 hertz, double the 50 or 60 hertz line frequency, rather than at the line frequency itself.

The square-law relationship also means the vibration spectrum is not a clean copy of the drive spectrum. Squaring a signal that contains several frequencies generates sums and differences of those frequencies, so a core driven by a complex waveform vibrates at a richer set of frequencies than the drive contains. The measured behavior confirms this: the vibration excited by an alternating signal has a broad spectrum and is not exactly the same as the power spectrum of the driving signal, primarily because of the nonlinear mechanical response of the magnetostrictive material. For a power amplifier handling a modulated signal, this means the core can manufacture spectral components that were never in the original, a recipe for intermodulation-like sidebands of mechanical origin.

The DC bias that wakes the effect up

A quiet detail turns magnetostriction from a curiosity into a real amplifier problem: the role of a steady bias. Because the strain follows the square of the field, a core driven by a pure alternating field around zero produces strain symmetrically and the fundamental-frequency component of the motion is suppressed, leaving mainly the double-frequency term. But add a DC bias, a standing magnetization on top of the alternating drive, and the picture changes. With a bias B0 and an alternating component b, the field is B0 plus b, and squaring it gives

(B0 + b)^2 = B0^2 + 2B0b + b^2

The middle term, 2B0b, is linear in the alternating drive and oscillates at the fundamental signal frequency, scaled by the bias. So a DC bias resurrects a strong fundamental-frequency vibration that the unbiased core suppressed, and the larger the bias, the larger this fundamental motion.

This is exactly the regime a power amplifier transformer or a biased inductor lives in, and the measurements bear out its importance. Studies of ferrite cores under DC bias found a strong correlation between core losses and magnetostrictive vibration, with the vibration increasing markedly as the DC bias rose, and proposed that the rise in core loss with bias is in part caused by the increased vibration. For the amplifier designer the message is direct: a core carrying both signal and a DC or low-frequency bias current vibrates more, at more troublesome frequencies, than the same core driven symmetrically, and the bias is often unavoidable in a real circuit.

When the mechanical resonance turns a whisper into a shout

The danger sharpens dramatically when the vibration frequency meets a mechanical resonance of the core. Every physical object has natural frequencies at which it rings, and a ferrite core is no exception. Its fundamental extensional mode, the frequency at which it naturally stretches and compresses along its length, depends on its size and stiffness through

f_resonance = (1 / 2L) * sqrt(E / rho)

where L is the relevant length, E is Young's modulus of the ferrite, and rho is its density. When the double-frequency magnetostrictive drive, or the bias-induced fundamental, lands on this mechanical resonance, the small periodic force is amplified by the mechanical quality factor of the core into a large motion.

The amplification is enormous. Measurements at the magnetomechanical resonance of ferrite cores found the core loss to be about 100 times greater at resonance than the normal losses due to hysteresis and eddy currents. A factor of one hundred turns a negligible effect into a dominant one. If the signal happens to drive the core through its mechanical resonance, whether directly or through the doubled frequency, the magnetostrictive coupling stops being a parts-per-million curiosity and becomes the loudest thing in the core. For a power amplifier this means a particular signal frequency, the one whose double hits the core resonance, suddenly shows far worse spectral purity than its neighbors, an effect baffling to anyone not thinking about mechanics.

A numerical estimate of the modulation a flexing core imposes

Quantify how a flexing core writes onto the signal. The inductance of a wound core depends on the core's dimensions and permeability, and magnetostriction changes both the dimensions and, through stress, the permeability. Take a core whose magnetostrictive strain reaches lambda of 2 parts per million at its operating flux. The fractional change in a relevant dimension is then 2 times ten to the minus six. If the inductance depends on that dimension roughly linearly, the inductance wobbles by a similar fraction at the vibration frequency:

dL / L is on the order of 2e-6 at the unstressed case

That alone is tiny. But the stress-dependence of permeability, the inverse magnetostrictive or Villari effect, amplifies it. In a magnetically soft ferrite the permeability can change by a fraction far larger than the strain, sometimes by a factor of ten or more times the mechanical strain, when the core is stressed, so the effective inductance modulation can reach

dL / L on the order of 2e-5 to 1e-4 once stress-permeability coupling is included

A reactance modulated by one part in ten thousand at the vibration frequency places sidebands on the carrier at roughly

sideband level = 20 log10(0.5 1e-4) = 20 * log10(5e-5) = -86 dBc

per sideband for a small modulation index, which sounds reassuringly low until the mechanical resonance multiplies the motion by a factor of a hundred. At resonance the same calculation with dL/L raised by 100 gives

sideband level = 20 log10(0.5 1e-2) = 20 * log10(5e-3) = -46 dBc

a pair of sidebands only 46 decibels below the carrier, firmly in the range that degrades a demanding signal. The arithmetic shows why off-resonance the effect is buried and on-resonance it dominates, and why a single unlucky frequency can stand out as far dirtier than the rest.

Telling magnetostrictive sidebands apart from electrical ones

Before a designer can silence the hum, it has to be correctly identified, because magnetostrictive sidebands masquerade as ordinary electrical noise on the spectrum analyzer. The discriminator is mechanical sensitivity, and a few targeted tests separate the two. The first is the touch test in reverse: gently clamping or pressing the suspect core while watching the offending sidebands. A purely electrical noise source is indifferent to mechanical pressure, but a magnetostrictive one changes when the core's freedom to vibrate is altered, so sidebands that shift or shrink under a clamp betray their mechanical origin.

The second test exploits the resonance signature. Sweeping the signal frequency slowly while monitoring sideband level reveals the magnetomechanical resonance as a sharp peak where the sidebands jump by the hundred-fold factor described earlier, then fall away on either side. An electrical noise mechanism produces no such narrow mechanical peak. The resonant frequency can be estimated in advance from the core geometry using the extensional-mode formula, and finding measured peaks near twice that frequency, given the square-law doubling, confirms the diagnosis. The frequency relationship itself is the third test: magnetostrictive vibration concentrates at twice the drive frequency in the unbiased case and at the fundamental when biased, so checking whether the dominant sideband spacing corresponds to the doubled frequency, and whether it strengthens when a DC bias is added, distinguishes magnetostriction from supply ripple or bias noise that would track the fundamental regardless of bias. Running these three checks turns an ambiguous spectral blemish into a definite verdict, and only once the verdict is mechanical do the mechanical remedies make sense to apply.

Designing the hum out before it reaches the signal

The defenses follow from the mechanism, and they are mostly mechanical rather than electrical. The first is material choice. Ferrite formulations differ in their saturation magnetostriction, and some compositions are specifically engineered toward near-zero magnetostriction, so selecting a low-magnetostriction core attacks the problem at its root. The second is keeping the operating point away from the conditions that worsen it: minimizing unnecessary DC bias removes the fundamental-frequency term that bias resurrects, and choosing core dimensions so that no mechanical resonance falls on twice any important signal frequency keeps the hundred-fold resonant amplification from ever engaging.

The third defense is mechanical damping and fixturing. A core free to vibrate stores and releases mechanical energy efficiently, giving a high mechanical quality factor and a sharp, strong resonance. Potting the core in a damping compound, or clamping it so its motion is constrained, lowers the mechanical quality factor and flattens the resonance, trading a tall narrow peak for a broad low one that never builds to a damaging amplitude. The same studies that found the loss surging at resonance also noted the dependence on the amplitude and nature of the vibration waveform, which is precisely what damping controls.

The deeper lesson is that a power amplifier is not purely an electrical system. It is an electromechanical one, and its magnetic components are tiny transducers whether the designer intends them to be or not. A ferrite core converts a fraction of the magnetic energy passing through it into mechanical motion, and that motion, if it finds a resonance or rides on a bias, finds its way back onto the signal. The hum is faint, born of dimensional changes measured in millionths, but the path from a millionth-of-a-meter wiggle to a sideband forty-six decibels down is real, and it runs straight through the mechanical resonance of a component most designers never think to tap and listen to. The builder who treats the core as a possible loudspeaker, and silences it accordingly, closes a door that electrical care alone cannot.