AC Magnetic Field Response of a Conducting Plate
AC Magnetic Field Response of a Conducting Plate: Theoretical Foundations
Eddy Currents in a Conducting Plate
Professor, what happens when an AC magnetic field is applied to a metal plate?
According to Faraday's law, eddy currents are induced, flowing in a direction that shields the external magnetic field. Magnetic field decay in an infinite flat plate:
$z$: depth from the surface. The amplitude decays exponentially as $e^{-z/\delta}$, and the phase also advances.
So the amplitude reduces to $1/e$ at the skin depth $\delta$.
It decays to about 5% at a depth of 3$\delta$. The magnetic shielding effect (shielding) by a conducting plate is based on this principle. Shielding effectiveness:
$t$: plate thickness.
Summary
- Exponential Decay — $H \propto e^{-z/\delta}$
- Shielding Effect — $SE \approx 8.7 \cdot t/\delta$ dB
- Heating by Eddy Currents — Principle of induction heating
Electromagnetic Induction in a Conducting Plate—The Physics of Penetration Depth Predicted by Maxwell's Equations
When a time-varying magnetic field penetrates a conducting plate, eddy currents are induced within the plate in a direction that opposes the original magnetic field change (Lenz's law). This shielding effect causes the magnetic field to decay exponentially inside the plate, with a characteristic length represented by the skin depth δ=√(2/ωμσ). The property that δ is larger at lower frequencies (magnetic field penetrates deeper) and smaller at higher frequencies (only the surface changes) is the basis for the practical understanding that "higher frequencies can be shielded with thinner plates" in electromagnetic shielding.
Computational Methods for AC Magnetic Field Response of a Conducting Plate
Solution with FEM
How do you solve the eddy current problem in a conducting plate using FEM?
A-φ method in the frequency domain:
Sufficient mesh division in the thickness direction is required (at least 3 layers within $\delta$). For thin plates, impedance boundary conditions can be used as an alternative.
Should I use the time domain or the frequency domain?
The frequency domain is efficient for sinusoidal excitation. Use the time domain for nonlinear materials (ferromagnetics) or non-sinusoidal waveforms. Transient responses can also be calculated in the time domain.
Summary
- Frequency Domain — Efficient for sine waves. Complex solution.
- Time Domain — Handles nonlinearity and non-sinusoidal waves.
- Impedance Boundary — Approximation method for thin plates.
Analysis of AC Magnetic Fields in Conducting Plates—Applicability Limits of the Thin Plate (Sheet) Approximation
When analyzing the AC magnetic field response of a conducting plate, if the plate thickness is sufficiently smaller than the skin depth, the "thin plate (sheet) approximation" can be used to reduce the 3D model to a 2D problem. This approximation can reduce computational cost to less than 1/100th, but errors increase rapidly when the thickness/δ ratio exceeds 0.1. For intermediate thicknesses, full 3D FEM is required, and numerically verifying "when the sheet approximation can be used" becomes the first step before analysis.