Induction Heating
Theory and Physics
Principles of Induction Heating
Professor, what's the difference between induction heating and high-frequency hardening?
The principle is the same (Joule heating by eddy currents), but the applications differ. Induction heating is used for a wide range of purposes such as melting, forging heating, brazing, and cooking (IH cooking heaters). Heat generation:
IH cooking heaters use the same principle, right?
Yes. They induce eddy currents in the pot bottom at 20–100 kHz. Aluminum pots have high conductivity but $\mu_r \approx 1$, making them difficult to heat. Iron pots have a large $\mu_r$ and can be heated efficiently. All-metal compatible IH heaters use higher frequencies to work with aluminum as well.
Summary
- Joule heating by eddy currents — $Q = J^2/\sigma$
- More efficient for magnetic materials — Larger $\mu_r$ leads to smaller $\delta$ and current concentration
- Non-contact, rapid heating — Energy efficiency 80–90%
Solving the Mystery of IH Stoves: "Why is the pot bottom hot but the unit itself isn't?"
If you've ever used an IH (induction heating) cooking heater, you might have wondered, "Why does the pot bottom get so hot, yet the appliance body remains cool enough to touch with bare hands?" The answer is because eddy currents are generated "only in the pot bottom." The alternating magnetic flux created by the IH's heating coil passes through the electrically conductive pot bottom (iron or stainless steel), causing eddy currents to flow within the pot bottom and generate Joule heat. Glass or wood do not conduct electricity, so no eddy currents flow and they are not heated. This selective heating is the essence of IH, achieving thermal efficiency over 90%. CAE induction heating analysis is precisely the technology that predicts "which material heats up and by how much."
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 a voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a temporally changing magnetic field induces an electric field. IH cooking heaters also use the same principle, where the changing high-frequency 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. Electric 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 makes hair stand up due to static electricity—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design uses this law to calculate the electric field distribution between electrodes. 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 the 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 are needed for anisotropic materials (e.g., silicon steel sheet rolling direction).
- 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: 0.2–1.4T |
| 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
Electromagnetic-Thermal Coupled Analysis
How do you set up a simulation for induction heating?
It's a coupling of electromagnetic fields and heat conduction. The heat generation distribution is obtained from frequency-domain eddy current analysis and passed to thermal analysis.
Since material properties ($\mu$, $\sigma$, $k$, $c_p$) are all temperature-dependent, the weak coupling method, which alternately calculates electromagnetic fields and heat, is standard.
Do you consider convection in melting simulations?
To handle stirring by Lorentz force in molten metal (electromagnetic stirring), three-way coupling of electromagnetic-thermal-fluid is required. This is where COMSOL Multiphysics excels.
Summary
- Electromagnetic-Thermal Coupling — Using $Q_{eddy}$ as a heat source
- Temperature dependence of materials — $\mu(T)$, $\sigma(T)$ are particularly important
- Electromagnetic-Thermal-Fluid — Necessary for melting simulation
The "Nonlinear Loop" of Induction Heating Analysis—Magnetization and Temperature Interfere with Each Other
Induction heating numerical analysis is difficult because electromagnetic fields, heat, and material properties interfere with each other in a three-way struggle. As temperature rises, electrical resistivity increases, changing the eddy current distribution. When iron exceeds the Curie temperature (770°C for pure iron) where it transitions from ferromagnetic to paramagnetic, permeability changes drastically, completely altering the magnetic flux distribution. On the other hand, if the eddy current distribution changes, the heat generation pattern also changes, which in turn changes the temperature distribution. To properly solve this three-way coupling of "electromagnetic ⇔ thermal ⇔ nonlinear material," iterative convergence calculation is essential. Many troubles like "analysis not converging" or "temperature diverging" stem from how these nonlinear material properties are handled.
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 region problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
Nonlinearity of B-H curve is handled by Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is typical.
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
Time step smaller 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 can reproduce transient phenomena containing all frequency components, but computational cost is high.
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