AC Resistance
Theory and Physics
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.
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 changing magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where a high-frequency changing magnetic field induces 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 against hair causes static electricity, making hair stand up — electric field lines radiate from the charged sheet (electric charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated 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】Even if you cut a bar magnet in half, you cannot create a magnet with only a N pole or only a S pole — a N pole and S pole 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-sectional analysis): Effective when current direction is uniform and end effects can be ignored
- Isotropic assumption: Direction-specific property definitions are needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- 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 to 1.4T |
| Magnetic field strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS unit 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
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.
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 inhomogeneous media. BEM: Naturally handles infinite domains (open region problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic saturation)
Nonlinearity of B-H curve is handled by the Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is common.
Frequency Domain Analysis
Reduced to a steady-state problem by assuming time-harmonic conditions. Requires complex number operations, but broadband characteristics are obtained via time-domain analysis.
Time Domain Time Step
A time step less than 1/20 of the highest frequency component is required. Implicit time integration allows larger steps but requires attention to accuracy.
Choosing Between Frequency Domain and Time Domain
Frequency domain analysis is like "tuning a radio to a specific frequency" — it can efficiently calculate the response at a single frequency. Time domain analysis is like "recording all channels simultaneously" — it can reproduce transient phenomena containing all frequency components, but computational cost is high.
Related Topics
Electromagnetic Analysis近接効果用語集表皮効果 — CAE用語解説Electromagnetic Analysis表皮効果Electromagnetic巻線損失解析（交流銅損）用語集Skin Depth Term用語集Penetration Depth
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