Eddy Current Loss
Theory and Physics
Fundamentals of Eddy Current Loss
Professor, how do you calculate eddy current loss?
It's the Joule loss caused by eddy currents induced in a conductor within an alternating magnetic field. For a single thin sheet:
$d$: sheet thickness, $f$: frequency, $B_m$: magnetic flux density amplitude, $\rho$: resistivity. Since it's proportional to $d^2$, laminating to reduce sheet thickness is fundamental for loss reduction.
That's the reason for laminating electrical steel sheets.
Eddy current loss decreases as thickness is reduced: 0.5 mm → 0.35 mm → 0.2 mm. However, the number of laminations increases and the space factor (volume ratio of iron) decreases. For high-frequency applications, amorphous alloys (25 μm) and nanocrystalline alloys (18 μm) are also used.
Summary
- $P_e \propto d^2 f^2 B_m^2$ — Proportional to the square of thickness and frequency
- Laminated Core — Reduces thickness and interrupts eddy current paths
- Amorphous Alloy — Significantly reduces high-frequency loss with ultra-thin sheets
The Secret of 0.35mm Silicon Steel Sheet—The Thickness-Squared Law Governs Core Design
Have you ever wondered why transformer cores are made of hundreds of thin laminated sheets? Since eddy current loss is proportional to the square of the thickness d, slicing a single iron sheet (35mm thick) into 1/100th thickness (0.35mm) and laminating them reduces the loss per sheet to 1/10000. Even with 100 sheets laminated, the total loss is 1/100 of the original. Modern hybrid vehicle motors use high-strength non-oriented silicon steel sheets with thicknesses of 0.20mm or 0.15mm, with thickness uniformity within ±5μm. Balancing rolling technology and magnetic properties is where material manufacturers show their skill, and this "thinness competition" is still ongoing.
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 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 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 hair with a plastic sheet creates static electricity, making hair stand up—electric field lines radiate from the charged sheet (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 $\nabla \cdot \mathbf{B} = 0$: States 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 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 edge 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 Systems
| 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 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
Eddy Current Loss Calculation with FEM
How do you calculate eddy current loss with FEM?
There are two approaches.
1. Direct Method — Solve the eddy current equation and integrate loss via $P = \int \mathbf{J}^2/\sigma \, dV$. Requires meshing each steel sheet, leading to high computational cost.
2. Iron Loss Formula Method — Obtain magnetic flux density distribution via FEM, then calculate via post-processing using iron loss formulas (e.g., modified Steinmetz equation).
Term 1: Hysteresis loss, Term 2: Classical eddy current loss, Term 3: Anomalous eddy current loss.
Which method is used in JMAG?
The iron loss formula method is standard in JMAG. It enables high-precision iron loss prediction without meshing each sheet individually. It can also account for iron loss increase due to harmonic magnetic fields.
Summary
- Direct Method — Accurate but computationally expensive
- Iron Loss Formula Method — Practical. Fast calculation via post-processing
- Harmonic Consideration — Harmonic iron loss can become dominant in PWM drives
Amorphous Metal—The "Ultra-Low Loss" Core Born from a Cooling Rate of 10⁶℃/s
You may have encountered the material "amorphous alloy" in numerical analysis of eddy current loss. This is a non-crystalline metal obtained by spraying molten metal onto a roll and rapidly quenching it at a rate of one million degrees Celsius per second. The lack of crystal structure makes magnetic domain movement extremely smooth. Its thickness is also 25μm, over 10 times thinner than silicon steel sheets. Using it for distribution transformer cores can theoretically reduce iron loss by about 70%, leading major Japanese transformer manufacturers to start mass production in the 2010s. However, it has drawbacks: "brittle," "difficult to cut," and "expensive." Care is also needed when handling its material constants in analysis.
Edge Element (Nedelec Element)
An element specialized for electromagnetic field analysis. Automatically guarantees continuity of tangential components and eliminates spurious modes. Standard for 3D high-frequency analysis.
Nodal Element
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 typical.
Frequency Domain Analysis
Reduces 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 less than 1/20 of the highest frequency component is required. Implicit time integration allows larger steps but accuracy must be considered.
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 includes all frequency components at...
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