Dielectric Breakdown Analysis
Theory and Physics
Dielectric Breakdown
Professor, is dielectric breakdown when the electric field is too strong and the insulator breaks down?
Yes. When the electric field strength exceeds the dielectric strength (breakdown field strength), insulation breaks down and discharge occurs.
Dielectric Strength Reference
| Material | Dielectric Strength [kV/mm] |
|---|---|
| Air (1atm) | 3.0 |
| SF₆ (0.1MPa) | 8.9 |
| Transformer Oil | 10–20 |
| Epoxy Resin | 20–30 |
| Polyethylene | 20–50 |
| SiO₂ (thin film) | 500–1000 |
SiO₂ thin film is orders of magnitude stronger!
Thin films have high strength due to fewer defects. However, at thicknesses of a few nm, leakage current flows due to the quantum tunneling effect.
Paschen's Law
The breakdown voltage of a gas is a function of the product $pd$ of pressure $p$ and electrode gap distance $d$:
$A, B$: gas constants, $\gamma$: secondary electron emission coefficient. There is a minimum value (Paschen minimum) for $pd$.
Summary
- Breakdown when Electric Field > Dielectric Strength — varies by material
- Paschen's Law — breakdown voltage for gases
- Calculate electric field distribution with FEM → Compare with dielectric strength — basic flow of insulation design
Townsend Avalanche—Understanding Lightning's "Avalanche" at the Quantum Level
The Townsend electron avalanche theory (early 1900s), fundamental to gas insulation breakdown theory, describes the process where one electron accelerates and collides with a neutral molecule → new electrons and ions are generated → these further accelerate → electron number increases exponentially. The exponential growth coefficient (Townsend's first ionization coefficient $\alpha$) depends on the electric field and gas pressure, giving rise to the pressure and gap dependence of the breakdown voltage. Lightning is a Townsend avalanche on a massive scale.
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 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 a plastic sheet on hair makes hair stand up due to static electricity—electric field lines radiate outward 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】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 in pairs. 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 do not depend on 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
- Isotropy 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 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
Dielectric Breakdown FEM
FEM does not directly calculate "breakdown"; rather, it calculates the electric field distribution and compares it with the dielectric strength.
1. Obtain electric field $E$ via electrostatic field analysis
2. Check if $E < E_{breakdown}$ throughout the entire domain
3. Evaluate safety factor $SF = E_{breakdown} / E_{max}$
How much safety factor is needed?
| Application | Recommended Safety Factor |
|---|---|
| Power Equipment (IEC Standards) | 2.0–3.0 |
| Automotive (AEC-Q Standards) | 2.0 or higher |
| Aerospace | 3.0 or higher |
| Consumer Products | 1.5–2.0 |
Summary
- FEM Electric Field Analysis + Dielectric Strength Comparison — breakdown judgment
- Safety Factor 2–3 is standard — varies by application field
Paschen's Law—Discharge Voltage Determined by "Product of Pressure and Electrode Gap"
The breakdown voltage of a gas follows the "Paschen curve" (discovered in 1889), uniquely determined by the product of gas pressure $p$ and electrode gap $d$ (the $pd$ product). A minimum breakdown voltage (Paschen minimum) exists; for air, it's about 330V at approximately 1cm electrode gap and 1 atm. The breakdown voltage increases for gaps narrower or wider than this. Understanding this law helps answer questions like "Why is insulation more difficult at low pressure?" in equipment design for high altitudes (low pressure) or vacuum insulation design.
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)
Magnetic saturation
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