## Origin or Cause of eddy currents

When time varying magnetic flux flows in the Ferromagnetic core, an emf is induced in the core in accordance with the Faraday’s law.

$\oint&space;E.dl&space;=&space;-d\Phi&space;/dt$

Φ = magnetic flux = $\int&space;_{_s}B.ds$

where

B is magnetic flux density,

S is the surface though which flux flows.

According to Lenz’s law this EMF is induced in a direction opposing the original field producing it. This induced emf will produce local currents in the conducting core normal to the magnetic flux these currents are called eddy currents.

## Skin effect

The field produced by this time varying eddy currents try to nullify the fields inside the core. Hence these currents flow within the skin depth of core near the surface. This is termed as skin effect. The magnitude of Ac current density inside the core can be approximated as

|J| = Jo*e (-Z*α)

Where the direction of energy propagation is thought to be in Z direction,

α is the attenuation constant = $\sqrt{}$(ω*µo*µr*σ/2) and has the units of m-1.

ω = 2*pi*f

where f is the frequency of alternating current,

µr is the relative magnetic permeability of core,

σ is the conductivity of core,

µo is the permeability of vacuum = 4*∏*10-7 H⋅m−1

The skin depth is defined as the distance within which the fields reduce to 1/e of its maximum value. Skin depth is the characteristic of the material which also depends on the frequency of the time varying magnetic field. The skin depth is denoted by δ and is given as

Skin depth δ = 1/ α = $\sqrt{}$ (2/( ω*µo*µr*σ))

The resistance of the core for alternating currents will be different from the DC resistance (L/(σ*A)) where L is the length of core and A is the area of the core. The effective resistance due to skin effect for long cylindrical conductor such as a wire, having a diameter D large compared to δ, has a resistance approximately that of a hollow tube with wall thickness δ carrying direct current is given as

R = L/ (σ*∏*D* δ)

Due to finite resistivity of core these eddy currents leads to power dissipation in the core.

Quantification of Eddy current losses

According to Steinmetz’s formula the eddy current losses in a transformer

Pe = Kef2Kf2Bm2     watts

Where Ke = Eddy Current Constant.

Kf = form Constant.

The total core losses in transformer is given as Pc = Ph + Pe

= Kh*f*Bmn+ Kef2Kf2Bm2

Where

f is the frequency of the external magnetic field,

B is flux density,

Kh, Ke and n are the coefficients, which depend on the lamination material, thickness, conductivity, as well as other factors. However, this formula is only applicable under the assumption that the maximum magnetic flux density of 1.0 Tesla is not exceeded and the hysteresis loop is under the static situation.

Eddy current losses increases with increasing frequency and increases with increasing conductivity.

Remedies to reduce eddy current losses

In transformers eddy current loss is undesirable and can be reduced by using the core materials that have high permeability but low conductivity. Soft ferrites are such materials.

For low frequency high power applications laminated cores out of stacked ferromagnetic sheets each electrically insulated from its neighbors by thin oxide coatings (insulator). The insulating coatings parallel to the direction of magnetic flux so that eddy currents normal to the flux is restricted to the laminated sheets. The eddy current power loss decreases as the number of lamination’s increases.