AC Resistance
AC Resistance: Theoretical Foundations
What is AC Resistance?
Professor, why is AC resistance larger than DC resistance?
Due to the skin effect and proximity effect, the current distribution becomes non-uniform across the conductor cross-section, reducing the effective cross-sectional area.
$F_r$: AC resistance factor. Approximate formula for round wire conductors (simplified version of Dowell's formula):
$\xi = d/\delta$ (conductor diameter / skin depth), $m$: number of layers.
It increases rapidly when there are many layers, doesn't it?
Losses increase sharply from the second layer onward due to the proximity effect. For $m=5$ layers, $F_r$ can become 10 to 100 times larger.
Summary
- $R_{AC} = R_{DC} \cdot F_r$ — Evaluated using the AC resistance factor
- Skin effect + Proximity effect — The two effects are superimposed
- Effect of layer count $m$ — Increases sharply with multi-layer winding
History of the Skin Effect — "Expulsion of Current" Discovered by Lord Kelvin in 1887
The skin effect, where AC current concentrates near the conductor surface, was theoretically predicted by William Thomson (later Lord Kelvin) in 1887. Behind the simple formula for skin depth δ=√(2/ωμσ) lies the physics that the diffusion term in Maxwell's equations becomes dominant at high frequencies. Kelvin's involvement in the design of submarine telegraph cables was motivated by the awareness that this skin effect would cause signal attenuation — the fundamental equations of CAE originated from practical problems over 150 years ago.
Computational Methods for AC Resistance
AC Resistance Calculation with FEM
How do you find AC resistance using FEM?
Obtain the current density distribution $\mathbf{J}(x,y)$ from eddy current analysis and calculate the equivalent resistance from the loss:
Modeling each individual strand provides high accuracy, but computational cost becomes enormous for windings with hundreds of turns.
How do you use homogenized winding models?
Features like JMAG's FEM Coil or COMSOL's Homogenized Multi-Turn Coil treat the winding region as an equivalent continuum, allowing AC loss calculation without modeling individual strands. Accuracy is around 90% of the individual model.
Summary
- $R_{AC} = P/I^2$ — Calculated inversely from loss
- Individual strand model — High accuracy but high cost
- Homogenized model — Practical approximation method
FEM Formulation of the Proximity Effect — Non-uniform Current Distribution Caused by Adjacent Conductors
When another conductor is placed adjacent to a conductor carrying AC current, the "proximity effect" occurs where the magnetic field influences and creates an asymmetric current distribution. This effect can be quantitatively evaluated by solving for the complex current density in 2D FEM, but sufficiently fine meshing between conductors is necessary to accurately represent the interaction of parallel conductors. For multi-layer coils, the proximity effect of 10 or more winding layers must be handled in a single FEM analysis, and comparison/verification with Dowell's method is used for reliability confirmation in practice.