Eddy current

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

Eddy currents are currents induced in conductors to oppose the change in flux that generated them. It is caused 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 a current, within the body of the conductor. These circulating eddies of current create induced magnetic fields that oppose the change of the original magnetic field due to Lenz's law, causing repulsive or drag forces between the conductor and the magnet. The stronger the applied magnetic field, or the greater the electrical conductivity of the conductor, or the faster the field that the conductor is exposed to changes, then the greater the currents that are developed and the greater the opposing field.

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.

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 often be minimised with thin plates, by lamination of conductors or other details of conductor shape.

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.

When a conductor moves relative to the field generated by a source, electromotive forces (EMFs) 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 the changes in the field.

Eddy currents are created when a moving conductor experiences changes in the magnetic field generated by a stationary object, as well as when a stationary conductor encounters a varying magnetic field. 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. In many devices, 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. 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.

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 equations:

For thin sheets: $P={\frac {\pi ^{2}B_{p}^{2}d^{2}f^{2}}{6\rho D}}.$

For thin wires: $P={\frac {\pi ^{2}B_{p}^{2}d^{2}f^{2}}{12\rho D}}$

Where Penetration Depth: $D={\frac {1}{\sqrt {\pi f\mu \sigma }}}$

where: B_p - peak flux density (T), d - thickness of the sheet or diameter of the wire (m), ρ - resistivity (Ωm), σ - electrical conductivity, μ - magnetic permeabiliy D - penetration depth or skin depth (m).

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.)
• very fast changing fields - due to skin effect the above equations are not valid because the magnetic field does not penetrate the material uniformly.

Applications

Repulsive effects and levitation

In a fast varying magnetic field the induced currents, in good conductors, particularly copper and aluminium, 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, such as the US penny.

Superconductors allow perfect, lossless conduction, which creates perpetually circulating eddy currents that are equal and opposite to the external magnetic field, thus allowing magnetic levitation. For the same reason, the magnetic field inside a superconducting medium will be exactly zero, regardless of the external applied field.

Identification of metals

In coin operated vending machines, eddy currents are used to detect counterfeit coins, or slugs. The coin rolls past a stationary magnet, and eddy currents slow its speed. The strength of the eddy currents, and thus the amount of slowing, 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 | Position Sensing

Eddy currents are used in certain types of proximity sensors to observe the vibration and position of rotating shafts within their bearings. This technology was originally pioneered in the 1930s by researchers at General Electric using vacuum tube circuitry. In the late 1950s, solid-state versions were developed by Donald E. Bently at Bently Nevada Corporation. 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 modern turbomachinery. 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 are American Petroleum Institute (API) Standard 670 and ISO 7919.

Electromagnetic braking

Main article: Eddy current brake

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 examination (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 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
• Eddy current adjustable-speed drives
• Eddy-current testing
• Electric meters (Electromechanical Induction Meters)
• Eddy current brakes
• Induction heating
• Proximity sensor (Displacement sensors)
• Traffic detection systems
• Vending machines (detection of coins)
• Coating Thickness Measurements
• Sheet Resistance Measurement
• 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 Law, providing an expression for the magnetic field $H$ surrounding a current density $J$,

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

The curl is taken on both sides of the equation,

$\nabla \times \left(\nabla \times \mathbf {H} \right)=\nabla \times \mathbf {J}$,

and using a common vector calculus identity for the curl of the curl results in

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

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

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

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

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

The differential form of Faraday's law, $\nabla \times {\boldsymbol {\mathrm {E} }}=-{\frac {\partial \mathbf {B} }{\partial t}}$, provides an equivalence for the change in magnetic flux $B$ in place of the curl of the electric field, so that the equation can be simplified to

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

By definition, $\mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)$, where $M$ is the magnetization of a material, and the diffusion equation finally appears as

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

References

• Fitzgerald, A. E. (1983). Electric Machinery (4th ed. ed.). Mc-Graw-Hill, Inc. p. 20. ISBN 0-07-021145-0. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Sears, Francis Weston (1955). University Physics (2nd ed. ed.). Reading, MA: Addison-Wesley. pp. 616–618. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Stoll, R. L. (1974). The analysis of eddy currents. Oxford University Press.
• Krawczyk, Andrzej. Numerical modelling of eddy currents. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Eddy Currents and Lenz's Law (Audio slideshow from the National High Magnetic Field Laboratory)