Eddy current

The term eddy current comes from analogous currents seen in water in fluid dynamics, causing localised areas of turbulence known as eddies giving 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.

The first person to observe eddy currents 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 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 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 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. The first use of eddy current for non-destructive testing occurred in 1879 when David E. Hughes used the principles to conduct metallurgical sorting tests.

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. The blue arrows are 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 dot e₁ shows a conduction electron in the sheet right after it has undergone a collision with an atom, and e₂ shows the same electron after it has been accelerated by the magnetic field. On average at e₁ the electron has the same velocity as the sheet (v, black arrow) in the +x direction. The magnetic field (B, green arrow) of the magnet's North pole N is directed down in the −y direction. The magnetic field exerts a Lorentz force on the electron (pink arrow) of F₁ = −e(v × B), where e is the electron's charge. Since the electron has a negative charge, from the right hand rule this is directed in the +z direction. At e₂ this 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 of F₂ = −e(v₂ × B). From the right hand rule, this is directed in the −x direction, opposite to the velocity v of 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 circular electric currents in a metal sheet moving through its magnetic field. See the diagram at right. It shows a metal sheet (C) moving to the right under 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, the magnetic flux through 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. From Faraday's law of induction, this creates a circular electric field in 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 free charge 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 the Lorentz force. Since the velocity v of the charges is to the right and the magnetic field B is directed down, from the right hand rule the Lorentz force on positive charges F = q(v × B) is toward the rear of the diagram (to the left when facing in the direction of motion v). This causes a current I toward the rear under the magnet, which circles around through parts of the sheet outside the magnetic field, clockwise to the right and counterclockwise to the left, to the front of the magnet again. The mobile charge carriers in the metal, the electrons, actually have a negative charge (q < 0) so their motion is opposite in direction to the conventional current shown.

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. The kinetic energy which is consumed overcoming this drag force is dissipated as heat by the currents flowing through the resistance of the metal, so the metal gets warm under the magnet.

Due to Ampere's circuital law each 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 to Lenz's law the counterfields oppose the change in magnetic field through the sheet. At the leading edge of the magnet (left side) by the right hand rule the 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.

Eddy currents in conductors of non-zero resistivity generate heat as well as electromagnetic forces. The heat can be used 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 minimized 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.[1] The latter can be used for non-destructive testing of materials for geometry features, like micro-cracks.[2] A similar effect is the proximity effect, which is caused by externally induced eddy currents.[3]

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 thin laminations parallel 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. This Joule heating reduces efficiency of iron-core transformers and electric motors and other devices that use changing magnetic fields. Eddy currents are minimized in these devices by selecting magnetic core materials that have low electrical conductivity (e.g., ferrites) 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. This eddy current is formed by the movement of the wheels. So, by Lenz'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 heating makes use of eddy currents to provide heating of metal objects.

Power dissipation of eddy currents

Under certain assumptions (uniform material, uniform magnetic field, no skin 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:[4]

$${\displaystyle P={\frac {\pi ^{2}{B_{\text{p}}}^{2}d^{2}f^{2}}{6k\rho D}},}$$

where

• P is the power lost per unit mass (W/kg),
• B_p is the peak magnetic field (T),
• d is the thickness of the sheet or diameter of the wire (m),
• f is the frequency (Hz),
• k is a constant equal to 1 for a thin sheet and 2 for a thin wire,
• ρ is the resistivity of the material (Ω m), and
• D is the density of 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 the skin effect; that is, the electromagnetic wave fully penetrates the material.

Skin effect

In very fast-changing fields, the magnetic field does not penetrate completely into the interior of the material. This skin effect renders 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:[5]

$${\displaystyle \delta ={\frac {1}{\sqrt {\pi f\mu \sigma }}},}$$

where δ is the penetration depth (m), f is the frequency (Hz), μ is the magnetic permeability of the material (H/m), and σ is the electrical conductivity of 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 of Ampère's Law,[6] providing an expression for the magnetizing field H surrounding a current density J:

$${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} .}$$

Taking the curl on both sides of this equation and then using a common vector calculus identity for the curl of the curl results in

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

From Gauss's law for magnetism, ∇ ⋅ H = 0, so

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

Using Ohm's law, J = σE, which relates current density J to electric field E in terms of a material's conductivity σ, and assuming isotropic homogeneous conductivity, the equation can be written as

$${\displaystyle -\nabla ^{2}\mathbf {H} =\sigma \nabla \times \mathbf {E} .}$$

Using the differential form of Faraday's law, ∇ × E = −∂B/∂t, this gives

$${\displaystyle \nabla ^{2}\mathbf {H} =\sigma {\frac {\partial \mathbf {B} }{\partial t}}.}$$

By definition, B = μ₀(H + M), where M is the magnetization of the material and μ₀ is the vacuum permeability. The diffusion equation therefore is

$${\displaystyle \nabla ^{2}\mathbf {H} =\mu _{0}\sigma \left({\frac {\partial \mathbf {M} }{\partial t}}+{\frac {\partial \mathbf {H} }{\partial t}}\right).}$$

Online citations

General references

• Fitzgerald, A. E.; Kingsley, Charles Jr.; Umans, Stephen D. (1983). Electric Machinery (4th ed.). Mc-Graw-Hill, Inc. p. 20. ISBN 978-0-07-021145-2.
• Sears, Francis Weston; Zemansky, Mark W. (1955). University Physics (2nd ed.). Addison-Wesley. pp. 616–618.

• Stoll, R. L. (1974). The Analysis of Eddy Currents. Oxford University Press.
• Krawczyk, Andrzej; J. A. Tegopoulos. Numerical Modelling of Eddy Currents.

Wikimedia Commons has media related to Eddy currents.

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