Iron losses — hysteresis + eddy current
Total core loss = P_h (∝f) + P_e (∝f²·t²·σ). Loss separation by plotting P/f vs f. Mitigation via thin laminations, silicon alloying, CRGO grain orientation, amorphous metal, ferrite cores.
Step 1 — Iron loss = hysteresis (∝ f) + eddy current (∝ f²)
Reference notes
Iron loss (also called core loss or no-load loss in a transformer) has two physically distinct components: hysteresis and eddy current. Use Next → to revisit the Steinmetz hysteresis model, derive the eddy-current scaling with lamination thickness, walk through the standard loss-separation procedure, and review mitigation levers from silicon alloying to amorphous metal cores.
The two components
- Hysteresis loss Ph — from irreversible domain-wall motion as the iron is cycled around the B-H loop. Each cycle dissipates energy equal to the loop area. Steinmetz model:
- Eddy-current loss Pe — induced loop currents in the conductive iron dissipate I2R heat. For thin laminations of thickness t, conductivity σ, density ρ:
Ph is LINEAR in frequency. Pe is QUADRATIC in frequency. At 50/60 Hz the two are typically comparable; at 400 Hz aerospace eddy dominates; above ~5 kHz only ferrites are practical.
Hysteresis loss in detail
Hysteresis comes from microscopic domain-wall pinning at lattice defects, impurities, grain boundaries. Each cycle, energy goes into freeing pinned walls — converted to heat. Steinmetz (1892) empirically captured this with the Kh·f·Bmaxn form, decades before modern domain theory. Typical CRGO at 1.7 T, 50 Hz: Ph ≈ 0.5–0.7 W/kg. At 2.0 T (slight saturation): jumps to 2-3 W/kg due to steep loop area growth above the saturation knee.
Eddy currents — physical mechanism
Inside the iron, alternating flux Φ(t) through any closed loop induces an EMF by Faraday's law: V = −dΦ/dt. The iron is conductive (σ ≈ 2 MS/m for silicon-steel), so the EMFs drive circulating eddy currents. By Lenz's law, the eddy currents oppose the flux change that created them — partially shielding the interior of the lamination from the applied flux ("magnetic skin effect"). The currents dissipate I2R heat = eddy loss.
Why lamination thickness matters: Pe ∝ t2
Derivation outline: in a lamination of thickness t, the eddy-loop length scales with t; induced EMF (× area) scales with t; current I = V/R scales with t / (constant); dissipated power I2R scales with t2. So halving the lamination thickness reduces eddy loss by 4×. Quartering it, by 16×. This is the entire reason iron cores are built from many thin laminations rather than a solid block.
| Application | Lamination thickness |
|---|---|
| 50/60 Hz distribution transformer (premium CRGO) | 0.23–0.27 mm |
| 50/60 Hz general (CRGO/CRNGO) | 0.30–0.35 mm |
| Small-motor laminations | 0.50 mm |
| 400 Hz aerospace transformer | 0.10–0.15 mm |
| > 5 kHz | Switch to ferrite cores |
Loss separation procedure
Standard method (IEC, IEEE, ASTM): measure total iron loss Piron at multiple frequencies with peak flux density Bmax held FIXED.
Divide both sides by f:
Plot Piron/f versus f — this should be a straight line. Y-intercept = Kh' (hysteresis), slope = Ke' (eddy). Multiplying each by f gives the separated Ph(f) and Pe(f). Transformer test reports always include this separation.
Typical loss numbers at 50 Hz, 1.7 T
| Material | Total loss | Comment |
|---|---|---|
| CRGO 0.35 mm | ~1.0–1.1 W/kg | Standard transformer core |
| CRGO 0.23 mm (premium) | ~0.7–0.8 W/kg | High-end distribution |
| CRNGO 0.50 mm | ~3–5 W/kg | Motor laminations |
| Amorphous metal | ~0.10–0.15 W/kg | Premium low-loss DT |
| Ferrite (MnZn) at 1 kHz / 0.3 T | ~5 W/kg | SMPS, very different operating point |
Mitigation levers
- Thin laminations — direct attack on eddy (∝ t2). Practical floor ~0.10 mm for cold-rolled steel.
- Silicon alloying (3–4 % Si) — raises iron resistivity from ~10 μΩ·cm to ~50 μΩ·cm (about 4×). Since Peddy ∝ σ (or equivalently 1/(μΩ·cm of resistivity)), a 4× higher resistivity cuts eddy loss 4×. Bonus: silicon stabilizes grain growth → lower hysteresis. Limit: brittleness above ~4.5 % Si.
- Grain orientation (CRGO) — align crystal grains along the rolling direction (the [100] easy axis of BCC iron). 20-30 % lower loss for flux flowing along grain. Cores designed (mitered corners, step-lap joints) to follow grain direction.
- Amorphous metal (Metglas) — rapidly-quenched Fe-B-Si non-crystalline structure. No grain boundaries to pin domain walls → very narrow hysteresis loop. High resistivity → low eddy. ~10× lower total loss than CRGO. Trade-off: brittle, lower Bsat (~1.5 T), ~30–50 % cost premium. 3–7 year payback for distribution transformers operating 24/7.
- Ferrites (Mn-Zn, Ni-Zn ceramics) — resistivity 106× higher than iron → eddy negligible to MHz. Bsat only 0.3–0.5 T, used for switch-mode supplies and RF transformers where high f gives small physical size anyway.
Keyboard shortcuts
- The lamination panel animates the flux direction and eddy-current loops; the loss panel plots Ph, Pe, and Ptotal versus frequency to show the linear-and-quadratic scaling.