Magnetic Shielding
Theory and Physics
Principles of Magnetic Shielding
Professor, how does magnetic shielding work?
By surrounding an area with a high-permeability material (such as permalloy or silicon steel), magnetic flux is diverted, attenuating the internal magnetic field. Shielding Effectiveness (SE):
Theoretical formula for a single-layer spherical shell shield:
$t$: shield thickness, $r$: shield radius, $\mu_r$: relative permeability.
So, higher permeability and greater thickness increase shielding effectiveness.
Correct. However, permalloy ($\mu_r \sim 80,000$) has a low saturation flux density ($B_s \approx 0.75$ T). Under strong magnetic fields, it saturates and its effectiveness drops, making multilayer shields (outer layer: low-carbon steel, inner layer: permalloy) effective.
Summary
- Divert magnetic flux with high-permeability materials — Basic principle
- $SE \propto \mu_r \cdot t/r$ — Permeability × thickness / radius
- Multilayer structure — For strong fields, prevent saturation with an outer layer and finish with an inner layer
Static Magnetic Field Shielding — The Mechanism of "Permalloy Diverting Magnetic Fields"
Low-frequency and DC magnetic field shielding functions via a completely different mechanism than electromagnetic shielding. High-permeability materials (permalloy, μ-metal) "suck in" magnetic flux and divert it around the interior, reducing the magnetic field inside the shield. Shielding effectiveness can be approximated by SE=1+μr×t/(2r) (spherical shell approximation), with higher permeability μr yielding greater effectiveness. However, there is a "saturation problem": when ferromagnetic materials saturate under strong external fields, their permeability drastically decreases, and shielding effectiveness is lost. In magnetic shield design for MRI and electron microscopes, multilayer shields that consider this saturation limit are used.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a time-varying magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heating.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère–Maxwell's law. Electric current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet against hair makes hair stand up due to static electricity—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the absence of magnetic monopoles. 【Everyday Example】Cutting a bar magnet in half does not create a magnet with only a N pole or only a S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used to satisfy this condition, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and edge effects can be ignored
- Isotropic assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive relations are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic flux density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2–1.4T |
| Magnetic field strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS Oersted (Oe): 1 Oe = 79.577 A/m |
| Current density $J$ | A/m² | Calculated from conductor cross-section and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
Shielding Analysis with FEM
Are there any points to note when using FEM for shield analysis?
When the shield material is thin, the aspect ratio of volume elements deteriorates. Countermeasures:
- Thin shell elements (Shell elements): Supported by JMAG, Maxwell. Treat thickness direction via shell attributes
- Impedance boundary condition: Model only the surface to reduce computational cost
- Nonlinear analysis: Essential because permalloy's $\mu_r$ strongly depends on B
It becomes an open-space problem, right?
The area outside the shield is infinite space. Model using BEM (Boundary Element Method), infinite elements, or a sufficiently large air domain. COMSOL's Infinite Element Domain is convenient.
Summary
- Thin shell elements or impedance boundary — Mesh countermeasure for thin shields
- Nonlinear analysis — Consider saturation
- Open space handling — BEM or infinite elements
Numerical Analysis of Magnetic Shielding — FEM and "Finite Element Approximation of Magnetic Reluctance"
In FEM analysis of magnetic shielding, the "modeling" of high-permeability thin plates directly affects accuracy. Realistically modeling a 1 mm thick permalloy plate can create an element size difference of over 1000 times compared to the surrounding air region. Instead, a thin plate boundary condition (SHIE element) that equivalently treats it as a "surface magnetic reluctance sheet" is used, significantly reducing the number of elements while accurately reproducing the magnetic flux diversion effect. ANSYS and COMSOL support this thin magnetic shell element as standard, enabling rapid FEM design of magnetic shields with arbitrary shapes.
Edge elements (Nedelec elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal elements
Used for scalar potential formulations. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs BEM (Boundary Element Method)
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