Eddy current

This article is about the electrical phenomenon. For the Ted McKeever comic, see Eddy Current (comics).

Electromagnetism
Electricity Magnetism
Electrostatics Electric charge Coulomb's law Electric field Electric flux Gauss's law Electric potential energy Electric potential Electrostatic induction Electric dipole moment Polarization density
Magnetostatics Ampère's law Electric current Magnetic field Magnetization Magnetic flux Biot–Savart law Magnetic dipole moment Gauss's law for magnetism
Electrodynamics Lorentz force law emf Electromagnetic induction Faraday’s law Lenz's law Displacement current Maxwell's equations EM field Electromagnetic radiation Liénard–Wiechert potential Maxwell tensor Eddy current
Electrical Network Electrical conduction Electrical resistance Capacitance Inductance Impedance Resonant cavities Waveguides
Covariant formulation Electromagnetic tensor EM Stress-energy tensor Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted

Eddy currents (also called Foucault currents) are electric currents induced in conductors when a conductor is exposed to a changing magnetic field; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electrons, or current, within the body of the conductor. These circulating eddies of current have inductance and thus induce magnetic fields. These fields can cause repulsive, attractive, propulsion and drag effects. The stronger the applied magnetic field, or the greater the electrical conductivity of the conductor, or the faster the field changes, then the greater the currents that are developed and the greater the fields produced.

The term eddy current comes from analogous currents seen in water when dragging an oar breadthwise: localised areas of turbulence known as eddies give rise to persistent vortices. Somewhat analogously, eddy currents can take time to build up and can persist for very short times in conductors due to their inductance.

Eddy currents, like all electric currents, generate heat as well as electromagnetic forces. The heat can be harnessed for induction heating. The electromagnetic forces can be used for levitation, creating movement, or to give a strong braking effect. Eddy currents can also have undesirable effects, for instance, power loss in transformers. In this application, they are minimised with thin plates, by lamination of conductors or other details of conductor shape.

Self-induced eddy currents are responsible for the skin effect in conductors. The latter can be used for non-destructive testing of materials for geometry features, like micro-cracks. A similar effect is the proximity effect, which is caused by externally-induced eddy currents.

History

The first person to observe current eddies was François Arago (1786–1853), the 25th Prime Minister of France, who was also a mathematician, physicist and astronomer. In 1824 he observed what has been called rotatory magnetism, and the fact that most conductive bodies could be magnetized; these discoveries were completed and explained by Michael Faraday (1791–1867).

In 1834, Heinrich Lenz stated Lenz's law, which says that the direction of induced current flow in an object will be such that its magnetic field will oppose the magnetic field that caused the current flow. Eddy currents develop secondary flux that cancels a part of the external flux.

French physicist Léon Foucault (1819–1868) is credited with having discovered Eddy currents. In September, 1855, he discovered that the force required for the rotation of a copper disc becomes greater when it is made to rotate with its rim between the poles of a magnet, the disc at the same time becoming heated by the eddy current induced in the metal.

Explanation

As the circular plate moves down through a small region of constant magnetic field directed into the page, eddy currents are induced in the plate. The direction of those currents is given by Lenz's law, i.e. so that the plate's movement is hindered. When a conductor moves relative to the field generated by a source, electromotive force can be generated around loops within the conductor. These EMFs acting on the resistivity of the material generate a current around the loop, in accordance with Faraday's law of induction. These currents dissipate energy, and create a magnetic field that tends to oppose changes in the current- they have inductance.

Eddy currents are created when a conductor experiences changes in the magnetic field. If either the conductor is moving through a steady magnetic field, or the magnetic field is changing around a stationary conductor, eddy currents will occur in the conductor. Both effects are present when a conductor moves through a varying magnetic field, as is the case at the top and bottom edges of the magnetized region shown in the diagram. Eddy currents will be generated wherever a conducting object experiences a change in the intensity or direction of the magnetic field at any point within it, and not just at the boundaries.

The swirling current set up in the conductor is due to electrons experiencing a Lorentz force that is perpendicular to their motion. Hence, they veer to their right, or left, depending on the direction of the applied field and whether the strength of the field is increasing or declining. The resistivity of the conductor acts to damp the amplitude of the eddy currents, as well as straighten their paths. Lenz's law encapsulates the fact that the current swirls in such a way as to create an induced magnetic field that opposes the phenomenon that created it. In the case of a varying applied field, the induced field will always be in the opposite direction to that applied. The same will be true when a varying external field is increasing in strength. However, when a varying field is falling in strength, the induced field will be in the same direction as that originally applied, in order to oppose the decline.

An object or part of an object experiences steady field intensity and direction where there is still relative motion of the field and the object (for example in the center of the field in the diagram), or unsteady fields where the currents cannot circulate due to the geometry of the conductor. In these situations charges collect on or within the object and these charges then produce static electric potentials that oppose any further current. Currents may be initially associated with the creation of static potentials, but these may be transitory and small.

Eddy currents generate resistive losses that transform some forms of energy, such as kinetic energy, into heat. This reduces efficiency of iron-core and other devices that use changing magnetic fields. Eddy currents are minimized in these devices by selecting materials that have low electrical conductivity (e.g., Ferrite )or by using thin sheets of magnetic material, known as laminations. Electrons cannot cross the insulating gap between the laminations and so are unable to circulate on wide arcs. Charges gather at the lamination boundaries, in a process analogous to the Hall effect, producing electric fields that oppose any further accumulation of charge and hence suppressing the eddy currents. The shorter the distance between adjacent laminations (i.e., the greater the number of laminations per unit area, perpendicular to the applied field), the greater the suppression of eddy currents.

The conversion of input energy to heat is not always undesirable, however, as there are some practical applications. One is in the brakes of some trains known as eddy current brakes. During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The eddy currents meet resistance as charges flow through the metal, thus dissipating energy as heat, and this acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion. Induction heating makes use of eddy currents to provide heating of metal objects.

Strength of eddy currents

Under certain assumptions (uniform material, uniform magnetic field, no skin effect, etc.) the power lost due to eddy currents can be calculated from the following equation:

$P = \frac{\pi^2 B_p^2 d^2 f^2 }{6k \rho D}$

where:

• k,
  • 1 for thin sheets
  • 2 for thin wires
• P, power dissipation (W/kg)
• B_p, peak flux density (T)
• d, thickness of the sheet or diameter of the wire (m)
• f, frequency (Hz)
• ρ, resistivity (Ωm)
• D,specific density (kg/m³)

It should be borne in mind that these equations are valid only under the so-called quasi-static conditions, where the frequency of magnetisation does not result in the skin effect; i.e., the electromagnetic wave fully penetrates the material.

Therefore, the following things usually increase the size and effects of eddy currents:

• stronger magnetic fields, increases flux density B
• faster changing fields (due to faster relative speeds or otherwise), increases the frequency f
• thicker materials, increases the thickness d
• lower resistivity materials (aluminium, copper, silver etc.)

Some things reduce the effects:

• weaker magnets, lower B
• slower changing fields (slower relative speeds), lower f
• thinner materials, lower d
• slotted materials so that currents cannot circulate, reduced d or coefficient in the denominator (6, 12, etc.)
• laminated materials so that currents cannot circulate, reduced d
• higher resistance materials (silicon rich iron, etc.), reduced p

Skin effect

In very fast changing fields due to skin effect the equations shown above are not valid because the magnetic field does not penetrate the material uniformly. However, in any case increased frequency of the same value of field will always increase eddy currents, even with non-uniform field penetration.

The penetration depth can be calculated from the following equation::$\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$ where:* δ, penetration depth (m),

• f, frequency (Hz)
• μ, magnetic permeability (H/m)
• σ, electrical conductivity (S/m)

Applications

Repulsive effects and levitation

In a fast varying magnetic field the induced currents, in good conductors, particularly copper and aluminium, frequently exhibit diamagnetic-like repulsion effects on the magnetic field, and hence on the magnet and can create repulsive effects and even stable levitation, albeit with reasonably high power dissipation due to the high currents this entails.

They can thus be used to induce a magnetic field in aluminum cans, which allows them to be separated easily from other recyclables.With a very strong handheld magnet, such as those made from neodymium, one can easily observe a very similar effect by rapidly sweeping the magnet over a coin with only a small separation. Depending on the strength of the magnet, identity of the coin, and separation between the magnet and coin, one may induce the coin to be pushed slightly ahead of the magnet - even if the coin contains no magnetic elements. Perfect conductors allow lossless conduction that allows eddy currents to form on the surface of the conductor that exactly cancel any changes in the magnetic field applied to the object after the material's resistance went to zero, thus allowing magnetic levitation. Superconductors are a subclass of perfect conductors in that they also exhibit the Meissner Effect, an inherently quantum mechanical phenomenon that is responsible for expelling any magnetic field lines present during the superconducting transition, thus making the magnetic field zero in the bulk of the superconductor.

Attractive effects

In some geometries the overall force of eddy currents can be attractive, for example, where the flux lines are past 90 degrees to a surface, the induced currents in a nearby conductor cause a force that pushes a conductor towards an electromagnet.

Identification of metals

In coin operated vending machines, eddy currents are used to detect counterfeit coins, or slug coin. The coin rolls past a stationary magnet, and eddy currents slow its speed. The strength of the eddy currents, and thus the retardation, depends on the conductivity of the coin's metal. Slugs are slowed to a different degree than genuine coins, and this is used to send them into the rejection slot.

Vibration | Displacement | Position Sensing

Eddy currents are used by some noncontact position/displacement sensors. Vibration is simply a repeating change in position. High-Performance eddy-current displacement sensors are capable of resolving a target's position within a few nanometers. There are also inexpensive "proximity" sensors which only indicate the presence/absence of a conductive object in front of the sensor. These are also called inductive sensors and may use changing inductance rather than eddy current principles. A primary advantage to this type of displacement sensor is the immunity to dirt and liquids in the measurement area. This technology was originally pioneered in the 1930s by researchers at General Electric using vacuum tube circuitry.

Electromagnetic braking

Eddy currents are used for braking at the end of some roller coasters. This mechanism has no mechanical wear and produces a very precise braking force. Typically, heavy copper plates extending from the car are moved between pairs of very strong permanent magnets. Electrical resistance within the plates causes a dragging effect analogous to friction, which dissipates the kinetic energy of the car. The same technique is used in electromagnetic brakes in railroad cars and to quickly stop the blades in power tools such as circular saws.

Structural testing

Eddy current techniques are commonly used for the nondestructive testing (NDE) and condition monitoring of a large variety of metallic structures, including heat exchanger tubes, aircraft fuselage, and aircraft structural components..

Side effects

Eddy currents are the root cause of the skin effect in conductors carrying AC current.

Similarly, in magnetic materials of finite conductivity eddy currents cause the confinement of the majority of the magnetic fields to only a couple skin depths of the surface of the material. This effect limits the flux linkage in inductors and transformers having magnetic cores.

Other applications

• Metal detectors
• Adjustable-speed driveEddy current drives*Eddy-current testing
• Electricity meter*Eddy current brakes
• Induction heating
• Proximity sensor (Displacement sensors)
• Traffic Detection Systems
• Vending machines (detection of coins)
• Coating Thickness Measurements
• Eddy current separator for metal separation
• Mechanical speedometers
• Safety Hazard and defect detection applications

Diffusion equation

The derivation of a useful equation for modeling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's circuital law, providing an expression for the magnetic field $\scriptstyle H$ surrounding a current density $\scriptstyle J$,

${\color{white}-}\nabla \times \mathbf{H} = \mathbf{J}$

The curl is taken on both sides of the equation,

${\color{white}-}\nabla \times \left(\nabla \times \mathbf{H} \right) = \nabla \times \mathbf{J}$

and using a common vector calculus identity for the Vector calculus identities Curl of the curlresults in

${\color{white}-}\nabla \left( \nabla \cdot \mathbf{H} \right) - \nabla^2\mathbf{H} = \nabla \times \mathbf{J}$

From Gauss's law for magnetism, $\scriptstyle \nabla \cdot \mathbf{H} \;=\; 0$, which drops a term from the expression and gives

$-\nabla^2\mathbf{H}=\nabla\times\mathbf{J}$

Using Ohm's law, $\scriptstyle \mathbf{J} \;=\; \sigma \boldsymbol{\Epsilon}$, which relates current density $\scriptstyle J$ to electric field $\scriptstyle \Epsilon$ in terms of a material's conductivity $\scriptstyle \sigma$, and assuming isotropic homogeneous conductivity, the equation can be written as

$-\nabla^2\mathbf{H}=\sigma\nabla\times\boldsymbol{\Epsilon}$

The differential form of Electromagnetic induction,$\scriptstyle \nabla \,\times\, \boldsymbol{\Epsilon} \;=\; -\frac{\partial \mathbf{B}}{\partial t}$, provides an equivalence for the change in magnetic flux $\scriptstyle B$ in place of the curl of the electric field, so that the equation can be simplified to

${\color{white}-}\nabla^2\mathbf{H} = \sigma \frac{\partial \mathbf{B}}{\partial t}$

By definition, $\scriptstyle \mathbf{B} \;=\; \mu_0\left(\mathbf{H} \,+\, \mathbf{M}\right)$, where $\scriptstyle M$ is the magnetization of a material, $\scriptstyle \mu_0$ is the vacuum permeability, and the diffusion equation finally appears as

${\color{white}-}\nabla^2\mathbf{H} = \mu_0 \sigma \left( \frac{\partial \mathbf{M} }{\partial t}+\frac{\partial \mathbf{H}}{\partial t} \right)$

References

• Fitzgerald, A. E.; Kingsley, Charles Jr. and Umans, Stephen D. (1983). Electric Machinery (4th ed. ed.). Mc-Graw-Hill, Inc.. pp. 20. ISBN 0-07-021145-0. 
• Sears, Francis Weston; Zemansky, Mark W. (1955). University Physics (2nd ed. ed.). Addison-Wesley. pp. 616–618. 

Further reading

• Stoll, R. L. (1974). The analysis of eddy currents. Oxford University Press. 
• Krawczyk, Andrzej; J. A. Tegopoulos. Numerical modelling of eddy currents. 

External links

• Eddy Currents and Lenz's Law (Audio slideshow from the National High Magnetic Field Laboratory)
• Eddy Current Separator Cogelme for non-ferrous metals separation - Info and Video in Cogelme site
• Eddy-Current Displacement Sensor Overview, Lion Precision
• Comparing Eddy-Current and Capacitive Noncontact Sensors, Lion Precision