Proximity Effect
Theory and Physics
What is the Proximity Effect?
Professor, what's the difference between the proximity effect and the skin effect?
The skin effect is the current crowding caused by "one's own magnetic field." The proximity effect is the current crowding caused by "the magnetic field from a neighboring conductor."
An external magnetic field $H_{ext}$ created by an adjacent conductor induces additional eddy currents within the conductor. Loss according to Dowell's formula:
So it becomes a problem in multi-layer windings, right?
Exactly. The external magnetic field experienced by a conductor in the $m$-th layer is proportional to $H = (m-1) \cdot n \cdot I$. Since inner layers receive a larger external magnetic field, the loss in inner layers is overwhelmingly greater than in outer layers. The AC resistance factor $F_r$ in Dowell's formula increases proportionally to $m^2$.
Summary
- Magnetic field of adjacent conductors — Additional eddy current loss
- $F_r \propto m^2$ — Loss increases with the square of the number of layers
- Multi-layer windings — Dominant in transformers and inductors
"The Neighbor's Wire Gets in the Way" — Why the Proximity Effect Complicates Coil Design
The skin effect is a phenomenon where the current distribution becomes uneven due to "the magnetic field created by its own current," while the proximity effect is the unevenness caused by "the magnetic field created by a neighboring conductor's current." When currents flow in the same direction, the current is pushed to the outer side of the conductor; when they flow in opposite directions, it concentrates on the inner side. When this occurs in transformer windings or inductor coils, the effective resistance can become several times the design value. In multi-layer coils, the effects of each layer accumulate, making it impossible to handle with analytical formulas for a single conductor. One of the main reasons FEA has become essential in modern power electronics coil design is to accurately predict this proximity effect.
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 changing magnetic field induces an electric field. Induction heating (IH) cookers 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 the absence of magnetic monopoles. 【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—they always exist as a N-S 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 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 relations 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-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
Proximity Effect Analysis with FEM
How do you capture the proximity effect with FEM?
Perform eddy current analysis by meshing each conductor individually. The magnetic interaction between conductors is automatically included.
2D cross-section analysis is efficient: If the coil's longitudinal direction is uniform, 2D provides sufficient accuracy. You can visualize the current density distribution in each conductor and confirm the current crowding due to the proximity effect.
Can the proximity effect be reduced with interleaved winding?
Yes. By alternately arranging primary and secondary windings, the MMF (Magnetomotive Force) distribution is flattened, significantly reducing the proximity effect. Comparing losses before and after interleaving with FEM clearly shows the effect.
Summary
- Individual conductor meshing — Directly calculates proximity effect
- 2D cross-section analysis — Efficiently visualizes current distribution
- Interleaved winding — Reduces proximity effect by flattening MMF
The Compromise of 2.5D Analysis — Why Approximate Calculations for Proximity Effect Survive
Attempting a strict 3D analysis of a multi-layer coil can sometimes take hours just for mesh generation. Therefore, in practice, "2.5D analysis"—that is, an approximate method that precisely analyzes the cross-sectional shape in 2D and extrapolates in the length direction using a multiplier—remains widely used. This method can evaluate current distribution including the proximity effect with high accuracy while reducing computation time to less than 1/100 of 3D. While end effects and three-dimensional magnetic field wrapping are ignored, the empirical rule that the main losses in the winding section are largely governed by the 2D cross-section provides sufficient accuracy for many practical designs. "Knowing where to approximate" is also an important skill for design engineers.
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). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
Nonlinearity of B-H curve handled by 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 wideband characteristics are obtained via time-domain analysis.
Time Domain Time Step
Time step smaller 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 has high computational cost.
Practical Guide
Countermeasures in Practice
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