B-H curve & hysteresis

Reference notes

Magnetic materials respond nonlinearly to applied field, with both saturation and history-dependent hysteresis. Use Next → to walk through the virgin magnetization curve, the hysteresis loop, the parameters that define it (B_r, H_c, B_sat), the soft-vs-hard material distinction, and the Steinmetz model for hysteresis loss in transformer and motor cores.

Magnetic basics

• B (flux density) — measured in tesla (T) or Wb/m². The "B-field" that produces force and induces voltage via Faraday's law.
• H (field strength) — measured in A/m. The "applied field" set by NI through a magnetic circuit.
• Permeability μ = μ_0 · μ_r. μ₀ = 4π × 10⁻⁷ H/m (defined). μ_r is the dimensionless relative permeability.
  • μ_r = 1 for vacuum / air.
  • μ_r ≈ 1000 – 5000 for low-grade iron.
  • μ_r ≈ 10 000 – 50 000 for high-grade transformer silicon-steel.
  • μ_r > 100 000 for some permalloys.

Virgin magnetization curve & saturation

Starting from a fully demagnetized state, B rises with H along the virgin curve:

1. At low H, B rises nearly linearly. Magnetic domains in the iron progressively align with the applied field via domain-wall motion.
2. The slope dB/dH flattens as more domains align.
3. Eventually almost all domains are aligned with H — B saturates at B_sat. Adding more H produces only tiny additional B.

Hysteresis loop

Reduce H back toward zero. Surprise: B does NOT trace back along the virgin curve. Domain-wall motion is partly irreversible — domains get pinned at lattice defects, impurities, grain boundaries — so the material "remembers" having been magnetized. The result is a closed loop on the B-H plane traversed each cycle of an AC excitation.

Three key loop parameters

• B_sat — peak flux density achievable.
• B_r (remanence) — B retained at H = 0 after reaching saturation. Tells you how strong a residual magnetization the material holds.
• H_c (coercivity) — magnitude of negative H required to force B back to zero. Tells you how difficult the material is to demagnetize.

The area enclosed by the loop has units of joules per cubic meter — energy dissipated as heat in the iron per cycle of magnetic excitation.

Soft vs hard magnetic materials

• Soft magnetic materials — small loop, low H_c, low loss per cycle. Designed for AC use where the loop is traversed many times per second. Examples: silicon-steel (CRGO, CRNGO), permalloy, soft ferrite, amorphous metal. Used in transformers, motor stators, inductors.
• Hard magnetic materials — large loop, high H_c, high B_r. Designed to RETAIN magnetization → permanent magnets. Examples: AlNiCo (high B_r, modest H_c), ceramic hard ferrite (cheap, ubiquitous), samarium-cobalt (high temperature), neodymium-iron-boron (strongest, B_r ≈ 1.3–1.5 T, H_c ≈ 1000 kA/m).

For a hard magnet, the specification of interest is the demagnetization curve — the second quadrant of the hysteresis loop where the magnet operates against an air-gap reluctance load.

Steinmetz hysteresis loss model

Energy lost per cycle = area enclosed by the loop, times core volume. For a sinusoidal excitation at peak B_max and frequency f:

Three engineering implications:

1. Linear in frequency. Doubling f doubles P_h at fixed B_max. 400 Hz aerospace transformers run hot.
2. Highly nonlinear in B_max. Operating at 1.8 T instead of 1.6 T (12.5 % more flux) gives ~25 % more hysteresis loss for n ≈ 1.8. The penalty for pushing B near saturation is steep.
3. Linearity allows loss separation. Total iron loss = hysteresis (∝ f) + eddy-current (∝ f²). Measuring P_iron at multiple f at fixed B_max and fitting P = a·f + b·f² separates the two components.

Eddy-current loss — the other half of core loss

Time-varying B induces voltage loops in the conductive iron, dissipating I²R as heat. Together with hysteresis, this constitutes the total core loss of a transformer or motor:

where t = lamination thickness, σ = electrical conductivity. Mitigated by (a) thin insulated laminations 0.23–0.50 mm typical, (b) high-resistivity silicon-iron alloys (3-4 % Si raises ρ by 4× over pure Fe), (c) amorphous metal cores for further improvement.

Saturation in transformer operation

Transformer designers operate B_max below B_sat by a comfortable margin (typically 1.6–1.8 T peak vs 2.0 T saturation for CRGO). Why? Once into saturation:

• μ_r collapses → magnetizing current spikes 5–10× normal (inrush phenomenon).
• Current waveform becomes peaky / non-sinusoidal → harmonic injection.
• Hysteresis loss climbs steeply (Steinmetz n ≈ 1.7).
• Magnetostriction noise rises audibly.
• Local hot-spots from flux concentration.

This is why a transformer fed by a sudden 110 %-of-rated voltage event can draw enormous inrush current.

Keyboard shortcuts

• The B-H panel animates H sinusoidally; the loop area is shaded in step 6 to show the per-cycle energy loss.