Eddy current

For the comic-book series, seeEddy Current (comics).For the Australian rock band, seeEddy Current Suppression Ring.

Inelectromagnetism,aneddy current(also calledFoucault's current) is a loop ofelectric currentinduced withinconductorsby a changingmagnetic fieldin the conductor according toFaraday's law of inductionor by the relative motion of a conductor in a magnetic field. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can beinducedwithin nearby stationary conductors by a time-varying magnetic field created by an ACelectromagnetortransformer,for example, or by relative motion between amagnetand a nearby conductor. The magnitude of the current in a given loop is proportional to the strength of the magnetic field, the area of the loop, and the rate of change offlux,and inversely proportional to theresistivityof the material. When graphed, these circular currents within a piece of metal look vaguely likeeddiesor whirlpools in a liquid.

ByLenz's law,an eddy current creates a magnetic field that opposes the change in the magnetic field that created it, and thus eddy currents react back on the source of the magnetic field. For example, a nearby conductive surface will exert a drag force on a moving magnet that opposes its motion, due to eddy currents induced in the surface by the moving magnetic field. This effect is employed ineddy current brakeswhich are used to stop rotating power tools quickly when they are turned off. The current flowing through the resistance of the conductor also dissipates energy asheatin the material. Thus eddy currents are a cause of energy loss in alternating current (AC)inductors,transformers,electric motorsandgenerators,and other AC machinery, requiring special construction such aslaminated magnetic coresorferrite coresto minimize them. Eddy currents are also used to heat objects ininduction heatingfurnaces and equipment, and to detect cracks and flaws in metal parts usingeddy-current testinginstruments.

Origin of term

The termeddy currentcomes from analogous currents seen inwaterinfluid dynamics,causing localised areas of turbulence known aseddiesgiving rise to persistent vortices. Somewhat analogously, eddy currents can take time to build up and can persist for very long times in conductors due to their inductance.

History

The first person to observe eddy currents wasFrançois Arago(1786–1853), the President of the Council of Ministers of the 2nd French Republic during the brief period 10th May to June 24, 1848 (equivalent to the current position of the French Prime Minister), who was also a mathematician, physicist and astronomer. In 1824 he observed what has been called rotatory magnetism, and that most conductive bodies could be magnetized; these discoveries were completed and explained byMichael Faraday(1791–1867).

In 1834,Emil LenzstatedLenz's law,which says that the direction of induced current flow in an object will be such that its magnetic field will oppose the change of magnetic flux that caused the current flow. Eddy currents produce a secondary field that cancels a part of the external field and causes some of the external flux to avoid the conductor.

French physicistLéon 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. The first use of eddy current for non-destructive testing occurred in 1879 whenDavid E. Hughesused the principles to conduct metallurgical sorting tests.

Explanation

Eddy currents(I,red)induced in a conductive metal plate(C)as it moves to the right under a magnet(N).The magnetic field(B,green)is directed down through the plate. The Lorentz force of the magnetic field on the electrons in the metal induces a sideways current under the magnet. The magnetic field, acting on the sideways moving electrons, creates a Lorentz force opposite to the velocity of the sheet, which acts as a drag force on the sheet. Theblue arrowsare counter magnetic fields generated by the circular motion of the charges.

Forces on an electron in the metal sheet under the magnet, explaining where the drag force on the sheet comes from. The red dote₁shows a conduction electron in the sheet right after it has undergone a collision with an atom, ande₂shows the same electron after it has been accelerated by the magnetic field. On average ate₁the electron has the same velocity as the sheet (v,black arrow) in the+xdirection. The magnetic field (B,green arrow) of the magnet's North pole N is directed down in the−ydirection. The magnetic field exerts aLorentz forceon the electron(pink arrow)ofF₁= −e(v×B),whereeis theelectron's charge.Since the electron has a negative charge, from theright hand rulethis is directed in the+zdirection. Ate₂this force gives the electron a component of velocity in the sideways direction (v₂,black arrow) The magnetic field acting on this sideways velocity, then exerts a Lorentz force on the particle ofF₂= −e(v₂×B).From the right hand rule, this is directed in the−xdirection, opposite to the velocityvof the metal sheet. This force accelerates the electron giving it a component of velocity opposite to the sheet. Collisions of these electrons with the atoms of the sheet exert a drag force on the sheet.

Eddy current brake. The North magnetic pole piece(top)in this drawing is shown further away from the disk than the South; this is just to leave room to show the currents. In an actual eddy current brake the pole pieces are positioned as close to the disk as possible.

A magnet induces circularelectric currentsin a metal sheet moving through its magnetic field. See the diagram at right. It shows a metal sheet(C)moving to the right with velocityvunder a stationary magnet. The magnetic field(B,green arrows)of the magnet's north pole N passes down through the sheet. Since the metal is moving, themagnetic fluxthrough a given area of the sheet is changing. In the part of the sheet moving under the leading edge of the magnet(left side)the magnetic field through a given point on the sheet is increasing as it gets nearer the magnet,⁠dB/dt⁠> 0.FromFaraday's law of induction,this creates a circularelectric fieldin the sheet in a counterclockwise direction around the magnetic field lines. This field induces a counterclockwise flow of electric current(I,red),in the sheet. This is the eddy current. In the part of the sheet under the trailing edge of the magnet(right side)the magnetic field through a given point on the sheet is decreasing as it is moving further away from the magnet,⁠dB/dt⁠< 0,inducing a second eddy current in a clockwise direction in the sheet.

Another equivalent way to understand the current is to see that the freecharge carriers(electrons) in the metal sheet are moving with the sheet to the right, so the magnetic field exerts a sideways force on them due to theLorentz force.Since the velocityvof the charges is to the right and the magnetic fieldBis directed down, from theright hand rulethe Lorentz force on positive chargesF=q(v×B)is toward the rear of the diagram (to the left when facing in the direction of motionv). This causes a currentItoward the rear under the magnet, which circles around through parts of the sheet outside the magnetic field, clockwise to the rightand counterclockwise to the left, to the front of the magnet again. The mobilecharge carriersin the metal, theelectrons,actually have a negative charge (q< 0) so their motion is opposite in direction to theconventional currentshown.

The magnetic field of the magnet, acting on the electrons moving sideways under the magnet, then exerts a Lorentz force directed to the rear, opposite to the velocity of the metal sheet. The electrons, in collisions with the metal lattice atoms, transfer this force to the sheet, exerting a drag force on the sheet proportional to its velocity. Thekinetic energywhich is consumed overcoming this drag force is dissipated as heat by the currents flowing through theresistanceof the metal, so the metal gets warm under the magnet.

Due toAmpère's circuital laweach of the circular currents in the sheet creates a counter magnetic field (blue arrows). Another way to understand the drag force is to see that due toLenz's lawthe counterfields oppose the change in magnetic field through the sheet. At the leading edge of the magnet(left side)by theright hand rulethe counterclockwise current creates a magnetic field pointed up, opposing the magnet's field, causing a repulsive force between the sheet and the leading edge of the magnet. In contrast, at the trailing edge(right side),the clockwise current causes a magnetic field pointed down, in the same direction as the magnet's field, creating an attractive force between the sheet and the trailing edge of the magnet. Both of these forces oppose the motion of the sheet.

Properties

Eddy currents in conductors of non-zeroresistivitygenerate heat as well as electromagnetic forces. The heat can be used forinduction heating.The electromagnetic forces can be used for levitation, creating movement, or to give a strongbrakingeffect. Eddy currents can also have undesirable effects, for instance power loss intransformers.In this application, they are minimized with thin plates, bylaminationof conductors or other details of conductor shape.

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

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.

(left)Eddy currents(I,red)within a solid iron transformer core.(right)Making the core out of thinlaminationsparallel to the field(B,green)with insulation(C)between them reduces the eddy currents. Although the field and currents are shown in one direction, they actually reverse direction with the alternating current in the transformer winding.

Eddy currents generate resistive losses that transform some forms of energy, such as kinetic energy, into heat. ThisJoule heatingreduces efficiency of iron-coretransformersandelectric motorsand other devices that use changing magnetic fields. Eddy currents are minimized in these devices by selectingmagnetic corematerials that have low electrical conductivity (e.g.,ferritesor iron powder mixed withresin) or by using thin sheets of magnetic material, known aslaminations.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 theHall 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 aseddy current brakes.During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. This eddy current is formed by the movement of the wheels. So, byLenz's law,the magnetic field formed by the eddy current will oppose its cause. Thus the wheel will face a force opposing the initial movement of the wheel. 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 heatingmakes use of eddy currents to provide heating of metal objects.

Power dissipation of eddy currents

Under certain assumptions (uniform material, uniform magnetic field, noskin effect,etc.) the power lost due to eddy currents per unit mass for a thin sheet or wire can be calculated from the following equation: $$P={\frac {\pi ^{2}{B_{\text{p}}}^{2}d^{2}f^{2}}{6k\rho D}},$$ where

• Pis the power lost per unit mass (W/kg),
• B_pis the peak magnetic field (T),
• dis the thickness of the sheet or diameter of the wire (m),
• fis the frequency (Hz),
• kis a constant equal to 1 for a thin sheet and 2 for a thin wire,
• ρis theresistivityof the material (Ω m), and
• Dis thedensityof the material (kg/m³).

This equation is valid only under the so-called quasi-static conditions, where the frequency of magnetisation does not result in theskin effect;that is, the electromagnetic wave fully penetrates the material.

Skin effect

Main article:Skin effect

In very fast-changing fields, the magnetic field does not penetrate completely into the interior of the material. Thisskin effectrenders the above equation invalid. 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 for a good conductor can be calculated from the following equation: $$\delta ={\frac {1}{\sqrt {\pi f\mu \sigma }}},$$ whereδis the penetration depth (m),fis the frequency (Hz),μis themagnetic permeabilityof the material (H/m), andσis theelectrical conductivityof the material (S/m).

Diffusion equation

The derivation of a useful equation for modelling the effect of eddy currents in a material starts with the differential, magnetostatic form ofAmpère's Law,providing an expression for themagnetizing fieldHsurrounding a current densityJ: $$\nabla \times \mathbf {H} =\mathbf {J}.$$

Taking thecurlon both sides of this equation and then using a common vector calculus identity for thecurl of the curlresults in $$\nabla \left(\nabla \cdot \mathbf {H} \right)-\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J}.$$

FromGauss's law for magnetism,∇ ⋅H= 0,so $$-\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J}.$$

UsingOhm's law,J=σE,which relates current densityJto electric fieldEin terms of a material's conductivityσ,and assuming isotropic homogeneous conductivity, the equation can be written as $$-\nabla ^{2}\mathbf {H} =\sigma \nabla \times \mathbf {E}.$$

Using the differential form ofFaraday's law,∇ ×E= −⁠∂B/∂t⁠,this gives $$\nabla ^{2}\mathbf {H} =\sigma {\frac {\partial \mathbf {B} }{\partial t}}.$$

By definition,B=μ₀(H+M),whereMis themagnetizationof the material andμ₀is thevacuum permeability.The diffusion equation therefore is $$\nabla ^{2}\mathbf {H} =\mu _{0}\sigma \left({\frac {\partial \mathbf {M} }{\partial t}}+{\frac {\partial \mathbf {H} }{\partial t}}\right).$$

Applications

Electromagnetic braking

Main article:Eddy current brake

Demonstration of Waltenhofen's pendulum, precursor of eddy current brakes. The formation and suppression of eddy currents is here demonstrated by means of this pendulum, a metal plate oscillating between the pole pieces of a strong electromagnet. As soon as a sufficiently strong magnetic field has been switched on, the pendulum is stopped on entering the field.

Eddy current brakesuse the drag force created by eddy currents as abraketo slow or stop moving objects. Since there is no contact with a brake shoe or drum, there is no mechanical wear. However, an eddy current brake cannot provide a "holding" torque and so may be used in combination with mechanical brakes, for example, on overhead cranes. Another application is on some roller coasters, where heavycopperplates extending from the car are moved between pairs of very strong permanent magnets.Electrical resistancewithin 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. Using electromagnets, as opposed to permanent magnets, the strength of the magnetic field can be adjusted and so the magnitude of braking effect changed.

Repulsive effects and levitation

Main article:electrodynamic suspension

A cross section through a linear motor placed above a thick aluminium slab. As thelinear induction motor's field pattern sweeps to the left, eddy currents are left behind in the metal and this causes the field lines to lean.

In a varying magnetic field, the induced currents exhibit diamagnetic-like repulsion effects. A conductive object will experience a repulsion force. This can lift objects against gravity, though with continual power input to replace the energy dissipated by the eddy currents. An example application is separation ofaluminum cansfrom other metals in aneddy current separator.Ferrous metals cling to the magnet, and aluminum (and other non-ferrous conductors) are forced away from the magnet; this can separate a waste stream into ferrous and non-ferrous scrap metal.

With a very strong handheld magnet, such as those made fromneodymium,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, such as the USpenny.Another example involves dropping a strong magnet down a tube of copper– the magnet falls at a dramatically slow pace.

In a perfect conductor with noresistance,surface eddy currents exactly cancel the field inside the conductor, so no magnetic field penetrates the conductor. Since no energy is lost in resistance, eddy currents created when a magnet is brought near the conductor persist even after the magnet is stationary, and can exactly balance the force of gravity, allowingmagnetic levitation.Superconductors also exhibit a separate inherentlyquantum mechanicalphenomenon called theMeissner effectin which any magnetic field lines present in the material when it becomes superconducting are expelled, thus the magnetic field in a superconductor is always zero.

Usingelectromagnetswith electronic switching comparable toelectronic speed controlit is possible to generate electromagnetic fields moving in an arbitrary direction. As described in the section above about eddy current brakes, a non-ferromagnetic conductor surface tends to rest within this moving field. When however this field is moving, a vehicle can be levitated and propelled. This is comparable to amaglevbut is not bound to a rail.

Identification of metals

In some coin-operatedvending machines,eddy currents are used to detect counterfeit coins, orslugs.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 and position sensing

Eddy currents are used in certain types ofproximity sensorsto observe the vibration and position of rotating shafts within their bearings. This technology was originally pioneered in the 1930s by researchers atGeneral Electricusing vacuum tube circuitry. In the late 1950s, solid-state versions were developed byDonald E. BentlyatBently NevadaCorporation. These sensors are extremely sensitive to very small displacements making them well suited to observe the minute vibrations (on the order of several thousandths of an inch) in modernturbomachinery.A typical proximity sensor used for vibration monitoring has a scale factor of 200 mV/mil.Widespread use of such sensors in turbomachinery has led to development of industry standards that prescribe their use and application. Examples of such standards areAmerican Petroleum Institute(API) Standard 670 andISO7919.

A Ferraris acceleration sensor, also called aFerraris sensor,is a contactless sensor that uses eddy currents to measure relative acceleration.

Structural testing

Eddy current techniques are commonly used for thenondestructive examination(NDE) and condition monitoring of a large variety of metallic structures, includingheat exchangertubes, aircraft fuselage, and aircraft structural components.

Skin effects

Eddy currents are the root cause of theskin effectin conductors carryingalternating current.

Lamination of magnetic cores in transformers greatly improves the efficiency by minimising eddy currents

Similarly, in magnetic materials of finite conductivity, eddy currents cause the confinement of the majority of the magnetic fields to only a coupleskin depthsof the surface of the material. This effect limits theflux linkageininductorsandtransformershavingmagnetic cores.

E-I transformer laminations showing flux paths. The effect of the gap where the laminations are butted together can be mitigated by alternating pairs of E laminations with pairs of I laminations, providing a path for the magnetic flux around the gap.

Other applications

• Rock climbing auto belays
• Zip line brakes
• Free fall devices
• Metal detectors
• Conductivity meters for non-magnetic metals
• Eddy current adjustable-speed drives
• Eddy-current testing
• Eddy current brake
• Electricity meters(electromechanical induction meters)
• Induction heating
• Cooking (induction cooking)
• Proximity sensor(displacement sensors)
• Vending machines(detection of coins)
• Coating thickness measurements
• Sheet resistance measurement
• Eddy current separatorfor metal separation
• Mechanicalspeedometers
• Safety hazard and defect detection applications
• Magnetic damping

References

Online citations

General references

• Fitzgerald, A. E.; Kingsley, Charles Jr.; Umans, Stephen D. (1983).Electric Machinery(4th ed.). Mc-Graw-Hill, Inc. p.20.ISBN978-0-07-021145-2.
• Sears, Francis Weston; Zemansky, Mark W. (1955).University Physics(2nd ed.). Addison-Wesley. pp.616–618.

Further reading

• Stoll, R. L. (1974).The Analysis of Eddy Currents.Oxford University Press.
• Reitz, J. R. (1970). Forces on Moving Magnets due to Eddy Currents. Journal of Applied Physics 41, 2067-2071.https://doi.org/10.1063/1.1659166
• Krawczyk, Andrzej; J. A. Tegopoulos.Numerical Modelling of Eddy Currents.

External links

• Eddy Current Separator Cogelme for non-ferrous metals separation– Information and video in Cogelme site