Skin Effect
Theory and Physics
What is the Skin Effect?
Professor, the skin effect is the phenomenon where current concentrates near the surface at high frequencies, right?
Correct. When alternating current flows through a conductor, the eddy currents induced inside the conductor cancel the current in the central region, causing the current to concentrate near the surface. Skin depth:
$f$: Frequency, $\mu$: Permeability, $\sigma$: Electrical Conductivity.
Specifically, how much is it for a copper wire?
| Frequency | Copper $\delta$ | Iron $\delta$ |
|---|---|---|
| 50 Hz | 9.4 mm | 0.65 mm |
| 1 kHz | 2.1 mm | 0.15 mm |
| 100 kHz | 0.21 mm | 0.015 mm |
| 1 MHz | 0.066 mm | — |
Iron has a very small skin depth due to its large $\mu_r$.
Summary
- $\delta = \sqrt{2/(\omega\mu\sigma)}$ — Determined by frequency and material
- Conductor radius > $\delta$ — Skin effect becomes significant
- Increase in AC resistance — Reduction in effective cross-sectional area
"Current only flows in the skin of the wire" — What surprises busbar designers first
Current that flows uniformly across the entire cross-section under DC conditions concentrates only near the surface as frequency increases—this is the skin effect. For 50Hz commercial power, the skin depth of copper is about 9mm. Even a 10mm diameter copper rod "seems to use almost the entire cross-section," but at 1kHz, the skin depth shrinks to about 2.1mm, and the center of a thick rod becomes "just dead weight." If you design a busbar for a power conversion device without knowing this, you'll suffer from higher resistance and heat generation than calculated. In practice, just remembering the √f rule that "skin depth halves when frequency quadruples" makes initial cross-sectional shape considerations much faster.
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) generates voltage in a nearby coil by rotating a magnet—a direct application of this law 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 heat.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère–Maxwell's law. 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 on hair makes hair stand up due to static electricity—electric field lines radiate from the charged sheet (electric charge), 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 that magnetic monopoles do not exist. 【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 the 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 do not depend on 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 end effects can be ignored
- Isotropic assumption: Direction-specific property definitions needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- Non-applicable cases: Additional constitutive laws needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit System
| 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 Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area 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
Eddy Current FEM
How do you solve the skin effect with FEM?
Frequency-domain eddy current equation:
$j\omega\sigma\mathbf{A}$ is the eddy current term. Complex analysis solves for amplitude and phase simultaneously.
Are there any mesh considerations?
A minimum of 3-4 mesh layers are required within the $\delta$ range from the conductor surface. The element size must be less than $\delta$ to resolve the current distribution. JMAG and COMSOL have automatic meshing functions based on skin depth.
Summary
- $j\omega\sigma\mathbf{A}$ term — Frequency-domain representation of eddy currents
- Mesh — At least 3-4 layers within the skin depth
- Complex solution — Obtain amplitude and phase simultaneously
The "3-Layer Rule" for Mesh and Skin Depth — A Pitfall for FEA Beginners
A common failure in numerical analysis of the skin effect is "mesh being too coarse relative to the skin depth." If you don't place at least 3 layers of mesh elements in the skin depth δ region, you cannot accurately reproduce the current density distribution. For example, when analyzing copper at 10kHz (δ≈0.66mm), a mesh size of 0.22mm or less is needed from the conductor surface. However, making the entire shape that fine explodes the element count, so in practice, "Boundary layer mesh" that becomes exponentially finer towards the surface is used. This mesh generation can be said to determine the success or failure of skin effect analysis, and it's also where the superiority of commercial tools' automatic meshing functions is tested.
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)
FEM: Handles nonlinear materials and non-homogeneous media. BEM: Naturally handles infinite domains (open boundary problems). Hybri
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