Eddy currents are circulating electric currents induced in conductors due to a changing magnetic field. These currents generate resistive heating, leading to energy dissipation. Understanding the power loss due to eddy currents is crucial in applications such as electrical machines, transformers, induction heating, and electromagnetic shielding. This article derives the formula for power dissipation per unit mass due to eddy currents and discusses its significance. (You can check the Youtube video on eddy current simulation)
Theoretical Background
Faraday’s Law of Induction
According to Faraday’s Law, the induced electromotive force (EMF) in a conductor exposed to a changing magnetic field is:
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
where:
- $\mathcal{E}$ is the induced EMF,
- $\Phi_B$ is the magnetic flux.
Ohm’s Law in Conductive Media
In a conductor, the induced electric field E leads to the formation of eddy currents J, governed by Ohm’s Law:
$$
\mathbf{J} = \sigma \mathbf{E}
$$
where:
- $\sigma$ is the electrical conductivity ($\sigma = \frac{1}{\rho}$, where $\rho$ is resistivity),
- $\mathbf{E}$ is the electric field.
Induced Eddy Currents and Lorentz Forces
The induced eddy currents create opposing magnetic fields that resist the original field changes, as explained by Lenz’s Law. The interaction of eddy currents with the applied magnetic field produces Lorentz forces, which influence mechanical and thermal behavior.
Derivation of Power Dissipation Due to Eddy Currents
To derive the formula for power dissipation, we consider a thin conductive sheet or wire of thickness ( d ), subjected to a time-varying peak magnetic field ( $B_p$ ), oscillating at frequency ( f ).
- Induced EMF per unit length:
Using Faraday’s Law, the time-dependent magnetic flux through a loop of dimension ( d ) is: $$
\Phi_B = B_p A = B_p d^2
$$ Differentiating with respect to time: $$
\mathcal{E} = -\frac{d}{dt} (B_p d^2) = -d^2 \frac{dB_p}{dt}
$$ Assuming sinusoidal variation of ( B_p ), we express: $$
B_p = B_{p0} \cos(2\pi f t)
$$ Thus, $$
\frac{dB_p}{dt} = -2\pi f B_{p0} \sin(2\pi f t)
$$ Hence, the induced EMF becomes: $$
\mathcal{E} = 2\pi f B_p d^2
$$ - Induced Eddy Current Density:
From Ohm’s Law: $$
J = \frac{\mathcal{E}}{\rho d} = \frac{2\pi f B_p d}{\rho}
$$ - Power Dissipation per Unit Volume:
The power dissipated per unit volume due to Joule heating is: $$
P_v = J^2 \rho
$$ Substituting ( J ): $$
P_v = \left(\frac{2\pi f B_p d}{\rho}\right)^2 \rho
$$ Simplifying, $$
P_v = \frac{4\pi^2 f^2 B_p^2 d^2}{\rho}
$$ - Power Dissipation per Unit Mass:
The total power dissipation per unit mass is obtained by dividing by the material density ( D ): $$
P = \frac{P_v}{D} = \frac{4\pi^2 f^2 B_p^2 d^2}{\rho D}
$$ - Final Expression Including k Factor:
A geometric correction factor ( k ) is introduced:
- ( k = 1 ) for a thin sheet,
- ( k = 2 ) for a thin wire. Thus, the final expression for power loss per unit mass due to eddy currents is: $$
P = \frac{\pi^2 B_p^2 d^2 f^2}{6k\rho D}
$$
Discussion and Implications
- Dependence on Frequency ($f^2$):
- Power dissipation increases quadratically with frequency.
- At high frequencies, the skin effect limits current penetration, modifying this relationship.
- Material Influence ($\rho$ and $D$):
- Low resistivity ($\rho$) materials experience higher eddy currents, leading to increased power loss.
- Higher density (D) materials help reduce power dissipation per unit mass.
- Thickness Effect ($d^2$):
- Thicker conductors have higher eddy current losses.
- This is critical in transformer cores, where laminated structures are used to minimize losses.
- Engineering Applications:
- Electromagnetic Braking: Lorentz forces generated by eddy currents oppose motion, leading to braking effects.
- Transformer Core Design: Laminated iron cores reduce ($d^2$ ), minimizing eddy current losses.
- Induction Heating: Efficient heating is achieved by maximizing eddy current generation.
The derived formula for eddy current power dissipation per unit mass highlights key dependencies on frequency, material properties, and conductor dimensions. Engineers can leverage this understanding to optimize electromagnetic systems by reducing eddy current losses through lamination, material selection, and frequency control.
Best Tool for Eddy Current simulation and study Power Loss etc.? Maybe COMSOL Multiphysics is a good choice. check their official website for more info.
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