Eddy current

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.

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).

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

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

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

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

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

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

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

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

Eddy current brakes use the drag force created by eddy currents as a brake to 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 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. Using electromagnets, as opposed to permanent magnets, the strength of the magnetic field can be adjusted and so the magnitude of braking effect changed.

A cross section through a linear motor placed above a thick aluminium slab. As the linear 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 of aluminum cans from other metals in an eddy 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 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. Another example involves dropping a strong magnet down a tube of copper[7] – the magnet falls at a dramatically slow pace.

In a perfect conductor with no resistance (a superconductor), 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, allowing magnetic levitation. Superconductors also exhibit a separate inherently quantum mechanical phenomenon called the Meissner effect in 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.

Using electromagnets with electronic switching comparable to electronic speed control it 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 a maglev but is not bound to a rail.[8]

In some 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 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.

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.

A Ferraris acceleration sensor, also called a Ferraris sensor, is a contactless sensor that uses eddy currents to measure relative acceleration.[9][10][11]

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.

Eddy currents are the root cause of the skin effect in conductors carrying AC 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 couple skin depths of the surface of the material. This effect limits the flux linkage in inductors and transformers having magnetic 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.

• Rock climbing auto belays[12]
• Zip line brakes[13]
• Free fall devices[14]
• Metal detectors
• Conductivity meters for non-magnetic metals[15][16]
• Eddy current adjustable-speed drives
• Eddy-current testing
• Electric meters (electromechanical induction meters)
• Induction heating
• Proximity sensor (displacement sensors)
• Vending machines (detection of coins)
• Coating thickness measurements[17]
• Sheet resistance measurement[18]
• Eddy current separator for metal separation[19]
• Mechanical speedometers
• Safety hazard and defect detection applications