Eddy Currents

From Faraday's law to swirling loops

Faraday's law of induction says that any change in magnetic flux through a closed loop drives an EMF around that loop:

EMF = − dΦ/dt

In a wire loop the EMF drives a current; in a piece of bulk metal there are no wires, but there are countless closed paths through the conductor. Faraday's law applies to every one of them. Whichever paths have the largest dΦ/dt have the largest induced EMF, and currents flow along those paths in eddies — closed swirling loops, named after the eddies in flowing water that they resemble.

Léon Foucault demonstrated the effect in 1855 by dropping a copper disc between the poles of an electromagnet: spinning the disc became markedly harder when the magnet was on, and the disc grew warm. The currents are still sometimes called Foucault currents in French and Russian literature.

Lenz's law and the direction of opposition

The minus sign in Faraday's law is Lenz's law. The induced current flows whichever way its own magnetic field opposes the change in external flux. The opposition has two consequences in eddy current practice:

1. Energy dissipation. If you push a conductor through a magnetic field, you do work against the eddy currents, and that work appears as resistive heat in the conductor. This is the operating principle of every induction heater and electromagnetic brake.
2. Force on the conductor. The eddy currents in a moving conductor generate their own field, which interacts with the external field to produce a force opposing the motion. This is what lifts a maglev train when it moves above a track of permanent magnets, and what slows a copper coin falling through a magnetised tube.

If the law worked the other way around — if the induced current re-enforced the change — energy would be created out of nothing, violating the first law of thermodynamics. Lenz's law is essentially a statement of energy conservation in induced systems.

The thin-sheet loss formula

For a thin conducting sheet of thickness d in a sinusoidal magnetic field of peak amplitude B and frequency f, the time-averaged power dissipated per unit volume is:

P/V  =  (π² B² f² d²) / (6 ρ)

where ρ is the resistivity of the material. The result holds when d is much smaller than the skin depth δ = √(ρ/(πfμ)). Three observations:

• Quadratic in field B. Doubling the magnetic flux density quadruples the loss. This sets a hard ceiling on how much flux you can drive through a transformer core before iron loss exceeds copper loss.
• Quadratic in frequency f. Doubling the operating frequency quadruples the loss. This is why 60 Hz line transformers can use thicker laminations than 400 Hz aircraft transformers, which in turn use thicker cores than 100 kHz switchmode supplies (which need ferrite, not steel).
• Quadratic in thickness d. Cutting the lamination thickness in half cuts loss by a factor of four. This is the lever transformer designers pull to reduce core loss without reducing flux.
• Inversely proportional to resistivity ρ. A material with 10× higher resistivity has 10× less eddy loss. Silicon steel (ρ ≈ 5 × 10⁻⁷ Ω·m) is used for line-frequency transformers; ferrite (ρ ≈ 10⁵ Ω·m) is essential for high-frequency converters.

Worked example: laminated vs solid steel core

A 50 Hz transformer core operates at peak flux B = 1.5 T. Compare a solid steel block with cross-section 100 mm × 100 mm against a laminated core made of 0.3 mm sheets. Silicon steel has ρ = 4.7 × 10⁻⁷ Ω·m. The thin-sheet formula gives loss per unit volume:

P/V (solid, d = 0.1 m)
   = π² × 1.5² × 50² × 0.1² / (6 × 4.7e-7)
   = 9.87 × 2.25 × 2500 × 0.01 / 2.82e-6
   = 555 / 2.82e-6
   = 1.97 × 10⁸ W/m³  (and the formula breaks down — too thick to be 'thin')

P/V (laminated, d = 0.3 mm = 3e-4 m)
   = 9.87 × 2.25 × 2500 × 9e-8 / 2.82e-6
   = 5.0 × 10⁻³ / 2.82e-6
   = 1770 W/m³  ≈  1.77 kW/m³

The solid block formula is unreliable (the field doesn't penetrate fully — eddy currents shield the interior at this thickness), but the laminated value is honest. For a 1 m³ transformer core that is 1.77 kW of pure heat. A real M-grade silicon steel of equivalent grain orientation has slightly different prefactors, and the ratio of laminated to solid loss is closer to 10⁵ — but even the simple formula captures the essential point: lamination is the difference between a working transformer and one that melts.

The other half of transformer iron loss is hysteresis, which scales as B^n × f with n ≈ 1.6–2. For modern grain-oriented steel hysteresis and eddy losses are roughly comparable at 50 Hz, so reducing one without the other is futile.

Worked example: induction cooktop pan

A typical 2 kW domestic induction hob drives a coil at 24 kHz, producing a peak field of 0.05 T at the bottom of a steel pan. The pan base is roughly 200 mm diameter, 1.5 mm thick, with an effective resistivity 5 × 10⁻⁷ Ω·m for ferritic stainless steel. Putting numbers into the thin-sheet formula:

P/V = π² × 0.05² × 24000² × 0.0015² / (6 × 5e-7)
    = 9.87 × 2.5e-3 × 5.76e8 × 2.25e-6 / 3e-6
    = 0.0320 / 3e-6
    = 1.07 × 10⁴  W/m³  per the formula — but the skin depth at 24 kHz is

δ = √(ρ/(πfμ_r μ₀)) = √(5e-7/(π × 24000 × 100 × 4π e-7))
  = √(5e-7 / 9.5e-3) ≈ 230 μm

The skin depth is about 230 μm, much smaller than the 1.5 mm pan thickness. Most of the eddy current concentrates in a thin surface layer. Total dissipation is the surface power loss times the area: roughly 50 W/cm² × 200 cm² = 10 kW peak loading capability, with 2 kW typically delivered. The pan heats from the bottom face up, with no contact heating from the cooktop surface. Aluminium pans don't work because their high conductivity makes the skin depth even smaller (50 μm) but the resulting eddy currents are too weak to dissipate enough power — that is also why induction hobs include "magnet test" detection that refuses to power up if the pan isn't ferromagnetic.

Where eddy currents show up

• Induction cooktops. Coils underneath a glass-ceramic surface drive 20–50 kHz alternating fields. Steel pans absorb ~2–3 kW directly into their bottoms with 90% electrical efficiency. The hob itself stays cool because the glass is non-magnetic and high-resistivity. Modern units sense pan presence and size, modulating power within tens of milliseconds.
• Eddy current braking. The Shanghai Maglev uses linear induction brakes capable of stopping a 350 km/h train without contact. Roller coasters such as Top Thrill Dragster use copper fins moving past neodymium magnets for fail-safe braking — drag force ∝ v, so the brakes work without electrical power. ICE high-speed trains in Germany use eddy-current brakes for emergency stopping.
• Transformer and motor iron losses. Every utility transformer dissipates 0.1–0.5% of throughput power as iron loss, split between hysteresis and eddy currents. The US grid loses about 30 TWh annually to transformer iron alone — billions of dollars of waste heat. Amorphous-metal cores cut eddy losses by ~70% over silicon steel and are slowly displacing it in distribution transformers.
• Non-destructive testing of metal. Eddy current probes detect cracks, corrosion thinning, and metallurgical changes in aircraft skins and pipelines. A coil's impedance changes as eddy currents in the test piece interact with hidden flaws. Required for periodic airworthiness inspections of every commercial aircraft fuselage.
• Coin selectors and metal detectors. Vending machines and slot machines distinguish coins by the eddy-current signature of their alloy and dimensions. A pulsed coil drives transient currents in the coin; the decay time identifies copper vs nickel vs zinc. Walk-through metal detectors at airports use the same principle, often combined with multiple frequencies to discriminate ferromagnetic from non-ferromagnetic objects.

Variants and extensions

• Foucault dissipation. Same physics as ordinary eddy currents but emphasising the heat-loss aspect rather than the force. Used in older European literature interchangeably with "eddy currents".
• Hysteresis loss. The other half of iron loss in a transformer core. Magnetic domains rearrange irreversibly as the field cycles, dissipating energy proportional to the area of the B-H loop. Combined eddy + hysteresis loss is what's tabulated as "core loss" or "iron loss" on transformer datasheets.
• Skin effect. At high frequency, AC current in a solid conductor is confined to a surface layer of depth δ = √(ρ/(πfμ)). The same physics that drives eddy currents in a transformer core drives currents to the surface of a single thick wire. This is why high-frequency power conductors use Litz wire — many fine, individually insulated strands.
• Superconducting persistent currents. In a superconducting ring with no resistance, the induced eddy current does not decay. The flux through the ring is locked in. Persistent currents in MRI magnets maintain 1.5–3 T for years without external power.
• Lamination as engineering shortcut. By orienting laminations parallel to the field direction and insulating them, eddy currents are forced into thin loops of minimum extent, cutting loss by (d_total/d_lam)². Modern grain-oriented silicon steel has the lattice aligned to maximise permeability along the rolling direction.

Common pitfalls

• Ignoring the skin effect. The thin-sheet formula assumes uniform field penetration. Above the skin depth, currents shield the interior and the simple d² scaling no longer applies. For solid bars at high frequency, loss saturates at a value set by surface area and skin depth, not by total volume.
• Confusing eddy current loss with hysteresis loss. Both contribute to transformer iron loss, but they scale differently with frequency: eddy as f², hysteresis as f. Doubling frequency makes the two losses approach each other if eddy initially dominated. Engineering choices (lamination, grain orientation, alloying with silicon) target each separately.
• Forgetting that lamination must be insulated. Stacked laminations with no insulation between them behave electrically like a solid block — the eddy currents simply jump from sheet to sheet. The thin oxide or varnish layer between sheets is what prevents this and is typically only a few microns thick.
• Treating eddy current braking as constant force. The drag force scales with velocity: F = kv. Fast objects feel huge drag, slow ones almost nothing. This is great for service braking (smooth deceleration) but useless for parking brakes (no holding force at zero speed). Eddy brakes always need a friction or mechanical backup for final stopping.
• Underestimating the heating. Even brief eddy current exposure can heat metal alarmingly. Pacemakers and metallic surgical implants are MRI-restricted partly because of induced eddy currents heating the implant tissue interface. Always check induction-vs-implant compatibility.