Skin Effect
Skin Effect: Theoretical Foundations
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.
Computational Methods for Skin Effect
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.