7.5 Eddy currents

Fundamentals of eddy currents

Eddy currents are loops of electric current that form inside conductors whenever the magnetic flux through them changes. They show up constantly in physics and engineering, sometimes as a useful tool (induction cooking, electromagnetic braking) and sometimes as an unwanted source of energy loss (transformer cores, motors). This section covers how they form, why they flow the direction they do, and what secondary effects they produce.

Definition and formation

When a conducting material experiences a changing magnetic field, Faraday's law tells us an EMF is induced. Because the conductor provides a continuous path for charge to flow, this EMF drives circular loops of current within the material itself. These are eddy currents.

• They form in any conductor: solid metal plates, wires, and even conducting liquids like mercury.
• The currents flow in closed loops within the body of the conductor, oriented perpendicular to the changing magnetic field.
• Their strength depends on two main factors: how fast the magnetic field is changing and how easily current flows through the material (its conductivity).

Lenz's law application

Lenz's law determines the direction of eddy currents: they always flow in the direction that opposes the change in magnetic flux that created them.

This opposition has real, observable consequences. If you drop a strong magnet through a copper tube, eddy currents form in the tube walls and create a magnetic field that pushes back against the falling magnet, slowing it dramatically. The magnet drifts down far slower than free fall. This is a direct demonstration of energy conservation: the kinetic energy the magnet loses gets converted into thermal energy in the copper via the eddy currents.

Induced magnetic fields

Because eddy currents are moving charges, they generate their own magnetic fields. These secondary fields oppose the original changing field, which produces several effects:

• Screening effect: The induced fields partially cancel the external field inside the conductor, reducing how deeply the field penetrates. This is why electromagnetic shielding works.
• Skin effect contribution: At high frequencies, eddy currents push AC current toward the surface of a conductor, increasing effective resistance. (More on this in the Advanced Concepts section.)

Properties of eddy currents

Circular flow patterns

Eddy currents form closed loops within the conductor, flowing in planes perpendicular to the direction of the changing magnetic field. In a simple flat plate moving through a uniform field, you can picture the currents as swirling circles. In three-dimensional objects, the current distribution becomes more complex, but the closed-loop pattern remains.

Dependence on conductor size

Larger conductors allow eddy currents to flow through bigger loops, which means lower resistance and stronger currents. This is why a solid metal block develops much stronger eddy currents than a thin wire in the same changing field. The magnitude scales with the cross-sectional area available for current flow. This fact is the key reason lamination works as a reduction strategy.

Frequency dependence

Higher-frequency field changes produce stronger eddy currents and greater power dissipation. The relationship isn't perfectly linear, though, because of the skin effect: at high frequencies, the currents concentrate near the conductor's surface rather than penetrating deeply. This means the effective conducting volume shrinks as frequency rises, which complicates the overall behavior.

Factors affecting eddy currents

Three main physical quantities control how strong eddy currents will be in a given situation.

Material conductivity

Eddy current strength is directly proportional to the material's electrical conductivity. Copper and aluminum, being excellent conductors, develop strong eddy currents. Poor conductors and semiconductors produce much weaker ones. Temperature matters here too: as a metal heats up, its resistivity increases and its conductivity drops, which weakens the eddy currents.

Magnetic field strength

Stronger external magnetic fields induce larger EMFs and therefore stronger eddy currents. In most situations the induced current scales with field strength, but in ferromagnetic materials, magnetic saturation can occur at high field strengths, limiting further increases.

Rate of field change

This is the factor that Faraday's law highlights directly: $\varepsilon = -\frac{d\Phi_B}{dt}$. Faster changes in flux produce larger induced EMFs. The rate of change can come from rapid relative motion between the conductor and a magnet, or from a high-frequency alternating current producing the field. Faster changes also mean the eddy currents penetrate less deeply into the conductor (smaller skin depth).

Applications of eddy currents

Electromagnetic braking

Eddy current brakes work without physical contact, so there's no friction-based wear. A conductor (usually a metal disc or rail) moves through a magnetic field, eddy currents form, and Lenz's law guarantees those currents produce a force opposing the motion.

• High-speed trains and roller coasters use them for smooth deceleration.
• Industrial machinery uses them to control rotation speed.
• Exercise equipment like stationary bikes uses adjustable magnetic braking for resistance control.

The braking force is proportional to speed: it's strong when you're moving fast and drops to zero as you stop, which makes for very smooth deceleration.

Induction heating

An alternating magnetic field induces eddy currents in a conductive workpiece, and the resistance of the material converts that current into heat (Joule heating). No direct contact is needed between the heat source and the object.

• Induction cooktops heat the pan itself rather than a burner surface, making them faster and more energy-efficient.
• Metal processing uses induction heating for melting, forging, and heat treatment.
• Medical applications include hyperthermia therapy, where targeted heating treats certain conditions.

Metal detection

Metal detectors work by generating an alternating magnetic field from a coil. When this field encounters a metallic object, eddy currents form in the metal and produce their own secondary field, which the detector's receiver coil picks up.

• Security screening at airports and public venues
• Food industry quality control (detecting metal contaminants)
• Archaeological surveys for locating buried metallic objects
• Non-destructive testing (NDT) for finding cracks and defects in metal parts

Eddy current losses

Energy dissipation mechanisms

Eddy currents convert electromagnetic energy into thermal energy through Joule heating ($P = I^2 R$). In devices like transformers, motors, and generators, this is wasted energy that reduces efficiency. The power lost to eddy currents increases with the square of the frequency, which is why high-frequency applications require especially careful design.

Heat generation

The heat comes from the resistance the eddy currents encounter as they circulate through the conductor. In high-power equipment, this can cause significant localized temperature increases, potentially leading to thermal stress or material damage. Cooling systems in large transformers and motors exist partly to manage eddy current heating.

Efficiency reduction

Eddy current losses directly reduce the efficiency of electromagnetic devices. In a transformer, for example, some of the input power heats the core instead of being transferred to the secondary winding. The same issue affects electric motors, generators, and wireless power transfer systems. Minimizing these losses is a central concern in electrical engineering design.

Reducing eddy currents

Lamination techniques

This is the most common and important reduction method. Instead of using a solid metal core, you stack thin sheets (laminations) of the core material, each one electrically insulated from its neighbors by a thin coating.

1. The insulating layers break up the large cross-sectional area that eddy currents would otherwise flow through.
2. Each thin lamination can only support small current loops with high resistance.
3. The result is dramatically reduced eddy current magnitude and power loss.

Laminated cores are standard in transformers and electric motor stators. The technique works well for frequencies up to several kilohertz.

Material selection

Choosing the right core material makes a big difference:

• Silicon steel (iron alloyed with a few percent silicon) has higher resistivity than pure iron, which reduces eddy currents while maintaining good magnetic permeability. This is the standard transformer core material.
• Ferrite cores and powdered iron cores have very high resistivity and are used for high-frequency applications (above several kHz) where laminations alone aren't sufficient.
• There's always a trade-off: you want high magnetic permeability (to carry flux efficiently) but also high resistivity (to suppress eddy currents).

Geometric considerations

Beyond material choices, the physical shape of conductors matters:

• Minimizing the cross-sectional area perpendicular to the changing field reduces the available path for eddy currents.
• Using thin or hollow conductors limits current loop size.
• Slotted or segmented designs in rotating machinery interrupt eddy current paths.

Measurement and detection

Eddy current testing

Eddy current testing (ECT) is a non-destructive evaluation technique. A probe coil carrying AC current is placed near the test material. The coil's alternating field induces eddy currents in the material, and any defect (crack, void, corrosion) disrupts the normal eddy current flow pattern. This disruption changes the impedance of the probe coil, which the instrument detects.

ECT can find surface and near-surface defects and is widely used in aerospace, automotive, and manufacturing quality control.

Non-destructive evaluation

The major advantage of eddy current methods is that they inspect materials without damaging them or requiring disassembly. Components can be tested while still in service. The process is fast enough for automated production-line inspection and can be combined with other NDT methods (ultrasonic, radiographic) for more thorough analysis.

Thickness measurements

Eddy current instruments can measure the thickness of non-conductive coatings (paint, anodizing) on conductive substrates, and also the thickness of thin metal sheets and foils. The technique works because coating thickness affects how the eddy currents couple between the probe and the substrate, changing the measured impedance. This is common in quality control for plating and coating processes.

Eddy currents in everyday life

Household appliances

• Induction cooktops are the clearest everyday example: the cooktop coil generates a high-frequency alternating field, eddy currents form in the ferromagnetic pot, and Joule heating cooks the food. The cooktop surface itself stays relatively cool.
• Electric motors in appliances like vacuum cleaners and washing machines experience eddy current losses in their cores, which is why those cores are laminated.

Transportation systems

• Maglev trains use eddy currents as part of their levitation and guidance systems.
• Electric and hybrid vehicles use eddy current effects in regenerative braking to recover kinetic energy.
• Traditional vehicle speedometers often use an eddy current mechanism: a spinning magnet induces currents in an aluminum disc, producing a torque proportional to speed.

Industrial processes

• Induction furnaces melt metals using large-scale eddy current heating.
• Electromagnetic stirring in metallurgy uses eddy current forces to mix molten metals.
• Eddy current separators in recycling plants sort non-ferrous metals (aluminum cans, copper wire) from waste streams by repelling them with induced eddy currents.

Advanced concepts

Skin effect

At DC or low frequencies, current distributes fairly uniformly across a conductor's cross-section. As frequency increases, eddy currents within the conductor itself push the current toward the surface. The skin depth $\delta$ is the depth at which the current density falls to about 37% of its surface value:

$\delta = \sqrt{\frac{2\rho}{\omega \mu}}$

where $\rho$ is resistivity, $\omega$ is angular frequency, and $\mu$ is magnetic permeability. At 60 Hz in copper, the skin depth is about 8.5 mm. At 1 MHz, it drops to about 0.066 mm. This effect increases the effective resistance of conductors at high frequencies and influences the design of high-frequency transformers and transmission lines.

Proximity effect

When two or more conductors carrying AC current are close together, their magnetic fields interact and cause non-uniform current distribution in each conductor. Current crowds toward or away from the adjacent conductor depending on the direction of current flow. This increases effective resistance and power losses beyond what the skin effect alone would cause. It's an important consideration in the design of power cables, bus bars, and transformer windings.

Eddy currents in semiconductors

Semiconductors have much lower conductivity than metals, but eddy currents still form in them. These currents are influenced by the density of free charge carriers (electrons and holes) and play a role in devices like Hall effect sensors and magnetoresistive sensors. High-frequency semiconductor circuit design must account for eddy current effects in the substrate material.

Mathematical analysis

Faraday's law application

Faraday's law is the starting point for all eddy current calculations:

$\varepsilon = -\frac{d\Phi_B}{dt}$

This says the induced EMF equals the negative rate of change of magnetic flux. The negative sign reflects Lenz's law (the induced EMF opposes the flux change). This applies whether the conductor is stationary in a time-varying field or moving through a static field.

Induced EMF calculations

For a straight conductor of length $l$ moving at velocity $v$ perpendicular to a uniform magnetic field $B$:

$\varepsilon = Blv$

For a rectangular loop partially inside a uniform field region, this same expression gives the EMF as one side of the loop moves through the field boundary. The key is identifying which part of the conductor is "cutting" field lines and at what rate.

Power loss equations

The power dissipated by eddy currents in a thin conducting plate exposed to an alternating magnetic field is:

$P = \frac{\pi^2 B^2 f^2 d^2 \sigma V}{6}$

where $B$ is peak magnetic field strength, $f$ is frequency, $d$ is plate thickness, $\sigma$ is electrical conductivity, and $V$ is the volume of the plate.

Notice what this equation tells you about reducing losses:

• Power scales with $d^2$: halving the plate thickness cuts eddy current losses by a factor of 4. This is exactly why lamination is so effective.
• Power scales with $f^2$: doubling the frequency quadruples the losses, which is why high-frequency applications need ferrite cores instead of laminated steel.
• Power scales with $\sigma$: using higher-resistivity materials (lower $\sigma$) directly reduces losses.