Magnetic Hysteresis

What hysteresis means

"Hysteresis" comes from the Greek hysterein, "to lag behind." In a ferromagnet the magnetization M does not depend only on the field H applied right now; it depends on every field the material has seen. Push H up and the flux density B follows one curve. Pull H back down and B follows a higher curve. The state remembers its history — the system is path-dependent. Plot B against H through a full up-then-down cycle and you get a closed loop, the famous B-H loop first measured carefully by James Alfred Ewing in 1882.

The three field quantities are linked by

B = μ₀ (H + M)

where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space. In a paramagnet or diamagnet, M is a tiny, single-valued, linear function of H — no loop. In a ferromagnet M is large, nonlinear, and multivalued: that multivaluedness is hysteresis.

Anatomy of the loop

Start from a fully demagnetized sample and ramp H up. B climbs along the initial (virgin) magnetization curve, steeply at first, then flattening as the material approaches saturation (B_s) — all domains aligned, nothing left to flip. Now reduce H back to zero. B does not return down the virgin curve; it stays high, settling at the remanence B_r. To force B back to zero you must apply a reverse field equal to the coercivity H_c. Push further negative and the material saturates the other way; reverse again and you close the loop.

Feature	Symbol	Physical meaning
Saturation flux density	B_s	Maximum B once every domain is aligned with H
Remanence (retentivity)	B_r	B left over at H = 0 — the permanent magnet's strength
Coercivity	H_c	Reverse H needed to bring B back to zero
Loop area	∮ H dB	Energy dissipated per unit volume per cycle (core loss)
Initial permeability	μ_i = (1/μ₀)(dB/dH) at origin	Slope of the virgin curve near zero field
Energy product	(BH)_max	Quality figure for permanent magnets (kJ/m³)

Loop area is energy lost

The work done by the source per unit volume to change the magnetization is

w = ∮ H dB     (J/m³ per cycle)

Because B traces a closed loop, this integral is exactly the enclosed area. If magnetizing and demagnetizing followed the same reversible path, the up and down integrals would cancel and the area would be zero. They don't: pinning of domain walls makes the process irreversible, so the loop encloses real area and that area is dissipated as heat every cycle. Charles Steinmetz fit transformer-core loss empirically in 1892 to

P_hyst = η · f · B_max^n      (n ≈ 1.6–2.0, Steinmetz exponent)

so hysteresis power loss rises linearly with frequency f (each cycle costs the loop area) and steeply with peak flux. This is distinct from eddy-current loss, which scales as f²B² and is suppressed by laminating the core. Together they make up the core loss of a transformer or motor.

Real numbers

Material	Coercivity H_c	Remanence B_r	Typical use
Permalloy (78% Ni)	~1–4 A/m	~0.7 T	Magnetic shielding, sensitive cores
Grain-oriented silicon steel	~10–40 A/m	~1.8 T	Power-transformer cores
Soft ferrite (MnZn)	~10–30 A/m	~0.3 T	High-frequency switching cores
AlNiCo	~40–130 kA/m	~1.2 T	Sensors, guitar pickups, motors
Hard ferrite (SrFe)	~200–300 kA/m	~0.4 T	Fridge magnets, cheap motors
NdFeB (neodymium)	~800–2000 kA/m	~1.2–1.4 T	EV motors, hard drives, wind turbines

That is a span of roughly six orders of magnitude in coercivity, all from the same hysteresis physics — just different domain-wall pinning. A silicon-steel transformer might dissipate ~1 W/kg in hysteresis at 1.5 T and 50 Hz; an NdFeB magnet has a loop so fat and square that a real demagnetizing field in service barely dents it, which is exactly why it stays magnetic for decades.

The microscopic picture

A ferromagnet below its Curie temperature spontaneously magnetizes, but it breaks into magnetic domains — regions each saturated but pointing in different directions — to minimize stray-field energy. Apply H and three things happen in sequence:

• Reversible wall motion. At small fields, domain walls bow elastically; remove H and they spring back. This is the low-field, reversible part of the virgin curve.
• Irreversible wall jumps. As H grows, walls tear free of pinning sites (defects, grain boundaries, inclusions) in sudden steps — the Barkhausen jumps you can literally hear as crackle on a coil and amplifier. These are irreversible: they are what open the loop.
• Rotation. Near saturation the remaining misaligned moments rotate coherently toward H. This is largely reversible again, which is why the loop tips pinch closed.

Coercivity is set by how strongly walls are pinned. Make pinning weak (clean, large-grain, stress-free permalloy) and you get a soft magnet; make it strong (fine single-domain NdFeB grains with sharp anisotropy) and you get a hard magnet. Heat the sample past the Curie temperature (770 °C for iron, 1115 °C for cobalt, 358 °C for nickel) and thermal motion destroys the alignment entirely — the loop collapses and the material turns paramagnetic.

Modeling a loop (Jiles–Atherton, simplified)

// Toy hysteresis model: anhysteretic Langevin curve + irreversible lag.
// Demonstrates path dependence and loop-area = energy.
const Ms = 1.6e6;   // saturation magnetization, A/m
const a  = 1100;    // shaping field, A/m
const k  = 2000;    // pinning constant ~ coercivity, A/m
const alpha = 1.6e-3; // inter-domain coupling

// Anhysteretic (loss-free) magnetization: Langevin function
function anhysteretic(He) {
  const x = He / a;
  if (Math.abs(x) < 1e-6) return 0;
  return Ms * (1 / Math.tanh(x) - 1 / x);
}

// March M forward as H sweeps; delta = +1 rising, -1 falling
function step(M, H, dH) {
  const delta = Math.sign(dH);
  const He = H + alpha * M;          // effective field
  const Man = anhysteretic(He);
  const dMdH = (Man - M) / (k * delta - alpha * (Man - M));
  return M + dMdH * dH;
}

// Sweep one full cycle and integrate loop area (∮ H dB)
function cycle(Hmax = 4000, steps = 2000) {
  const mu0 = 4 * Math.PI * 1e-7;
  let M = 0, area = 0, prevB = mu0 * (0 + M);
  const path = [];
  for (let i = 0; i <= steps; i++) {
    const phase = (i / steps) * 2 * Math.PI;
    const H = Hmax * Math.sin(phase);
    const dH = Hmax * Math.cos(phase) * (2 * Math.PI / steps);
    M = step(M, H, dH);
    const B = mu0 * (H + M);
    area += H * (B - prevB);         // Riemann sum of ∮ H dB
    prevB = B;
    path.push({ H, B });
  }
  return { path, lossPerCycle: Math.abs(area) }; // J/m^3
}

const { lossPerCycle } = cycle();
console.log(`Hysteresis loss: ${lossPerCycle.toFixed(0)} J/m^3 per cycle`);
// At 50 Hz this becomes lossPerCycle * 50 W/m^3 of core

The key line is k * delta: flipping the sign of the pinning term when the sweep reverses is what makes the rising and falling branches differ. Set k = 0 and the loop collapses to the single-valued anhysteretic curve with zero area — no hysteresis, no loss.

Where hysteresis matters

• Permanent magnets. Remanence is the entire point — motors, generators, speakers, MRI, hard-disk write heads, every brushless DC fan.
• Transformer and motor cores. Here hysteresis is the enemy; soft, low-coercivity silicon steel or ferrite minimizes the loop area paid 50–60 times a second (or hundreds of kHz in switch-mode supplies).
• Magnetic recording. Hard-disk and tape media need enough coercivity to retain bits against stray fields, but not so much that the write head can't flip them. The loop's squareness sets the signal-to-noise floor.
• Inductors and chokes. The initial permeability (virgin-curve slope) sets inductance; the loop sets the loss and the saturation limit.
• Sensors. Fluxgate and Barkhausen-noise sensors read the loop directly to detect fields or material stress.
• Magnetic shielding. Mu-metal's vanishingly thin loop lets it soak up fields with almost no remanent contamination.

Common misconceptions

• Confusing B-H with M-H loops. The B-H loop keeps tilting upward forever (B = μ₀H + μ₀M never saturates because of the μ₀H term); the M-H loop genuinely flattens at M_s. Coercivity quoted as H_c (on M) and "intrinsic" H_ci can differ for hard magnets.
• Thinking remanence equals saturation. B_r is always less than B_s; only a perfectly square loop would make them equal. Squareness ratio B_r/B_s is a real design figure.
• Blaming all core heating on hysteresis. At high frequency, eddy currents (∝ f²) usually dominate; lamination and ferrites fight those, not the loop area.
• Assuming the loop is fixed. Minor loops (subloops from partial cycles) are smaller and depend on history; temperature, stress, and frequency all reshape the loop.
• Forgetting demagnetization. A finite magnet shapes its own internal field; the operating point sits on the second-quadrant load line, not at B_r itself.
• Treating soft and hard as different physics. It is one phenomenon spanning six decades of coercivity, tuned entirely by domain-wall pinning.