Impedance Analysis (Eddy Current)
Theory and Physics
Impedance and Eddy Current
Professor, could you explain the effect of eddy currents on coil impedance?
When a conductor is near a coil, the reaction magnetic field from eddy currents changes the coil's self-inductance and resistance.
Without conductor: $Z_0 = R_0 + j\omega L_0$
With conductor: $Z = (R_0 + \Delta R) + j\omega(L_0 - \Delta L)$
Eddy currents decrease inductance and increase equivalent resistance.
This is the operating principle of ECT sensors, right?
Correct. Plotting $\Delta R$ and $\Delta(\omega L)$ on the impedance plane allows separation of the effects of the conductor's conductivity, permeability, thickness, and defects.
Summary
- Reaction of Eddy Currents — L decreases, R increases
- Impedance Plane — Separate material properties using $\Delta R$ vs $\Delta(\omega L)$
- Frequency Dependence — Low frequency provides deep information, high frequency provides surface information
Complex Representation of Impedance—Physical Meaning of the Real Part (Resistance) and Imaginary Part (Reactance)
The impedance of a conductor carrying eddy currents is a complex number that changes with frequency. The real part (resistance component) represents eddy current loss, and the imaginary part (reactance component) represents magnetic energy storage. This frequency dependence is the cause of "AC resistance increase due to skin effect." Compared to DC resistance, the resistance of a copper round bar can increase several times at 1000 Hz. Accurately calculating this complex impedance with FEM is a central challenge in eddy current analysis.
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'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 (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 that magnetic monopoles do not exist. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only an N pole or only an 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: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed.
- Non-applicable cases: Additional constitutive laws 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: 0.2〜1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS 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
Impedance Calculation with FEM
How do you calculate coil impedance with FEM?
Calculate impedance from the eddy current FEM solution $\mathbf{A}$:
$\mathbf{J}_0$: Applied current density. In COMSOL, automatically calculated via Coded Impedance Calculation. In Maxwell, output as impedance matrix.
How about frequency sweeps?
Use a frequency domain solver to sweep logarithmically (e.g., 100 Hz〜10 MHz, 10 points/decade). Calculate Z(f) at each frequency and plot the impedance trajectory. This evaluates sensor sensitivity characteristics.
Summary
- A-J-integral — Post-process impedance from FEM solution
- Frequency Sweep — Evaluate sensitivity characteristics
- Impedance Trajectory — Separate material/defect parameters
FEM Calculation of Eddy Current Impedance—Advantages of Vector Potential Formulation
In FEM analysis of eddy currents, the formulation using magnetic vector potential A (∇×A=B) as the unknown is standard. Using A automatically satisfies the solenoidal condition (∇·B=0), eliminating the need to explicitly impose the normal continuity condition of magnetic flux density B as a boundary condition. To calculate impedance with this formulation, the ratio of voltage line integral to current area integral is required, so accurately modeling the coil geometry is key to analysis precision.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee tangential component continuity and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulation. 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 region problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
Handle B-H curve nonlinearity with Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is common.
Frequency Domain Analysis
Reduced to a steady-state problem via time-harmonic assumption. Requires complex number operations, but broadband 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 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 captures all frequency components...
Related Topics
なった
詳しく
報告