Insulation Design
Theory and Physics
Approach to Insulation Design
Professor, is insulation design essentially about "keeping the electric field below the dielectric strength"?
Exactly. The design flow is:
1. Determine electrode shape and insulation configuration
2. Calculate electric field distribution using FEM
3. Verify that the maximum electric field $E_{max}$ is below the dielectric strength $E_b$ of each material
4. Evaluate the safety factor $SF = E_b / E_{max}$
Electric Field Mitigation Techniques
| Technique | Principle | Application Example |
|---|---|---|
| Fillet (Rounding) | Increase the curvature radius of edges to mitigate electric field concentration | High-voltage electrodes |
| Corona Ring | Enlarge the equipotential surface to homogenize the electric field | Transmission line insulators |
| Stress Cone | Push out the electric field using high-permittivity material | Cable terminations |
| Shield Electrode | Shield the electric field with a grounded electrode | GIS (Gas Insulated Switchgear) |
| Graded Insulation | Gradually change the $\varepsilon_r$ (relative permittivity) | Bushings |
Summary
- $E_{max} < E_b / SF$ — Fundamental condition for insulation design
- Electric Field Mitigation — Fillet, corona ring, stress cone
- Predict electric field distribution with FEM — Optimization of design
The Dawn of Insulation Engineering—The History of Cable Insulation and Gutta-Percha (1850s)
The engineering use of electrical insulating materials began in the 1850s during the era of submarine cable laying. When the Dover Strait submarine cable connecting England and France was laid in 1851, a natural rubber from Malaysia called "Gutta-percha" was used as the insulating material. However, the first transatlantic cable (1858) failed due to insulation breakdown after just three weeks—insulation engineering at the time relied solely on empirical rules. Subsequently, the electric line of force theory, which evolved from Coulomb's law (1785), was systematized by Maxwell (1873), establishing the foundation for modern electric field analysis. Today's FEM electric field analysis solves Maxwell's equations through discretization, with theories from 170 years ago forming the mathematical basis for cutting-edge insulation design tools.
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 lines of force 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 a N pole or only a S pole—N and S poles always exist as a pair. This means magnetic lines of force 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
FEM Flow for Insulation Design
1. CAD Model Construction (Electrodes + Insulators + Surrounding Space)
2. Material Settings ($\varepsilon_r$ for each region)
3. Boundary Conditions (Electrode potential, ground, symmetry plane)
4. Mesh (Refine areas of electric field concentration)
5. Solve (Laplace/Poisson equation)
6. Postprocessing ($E_{max}$, safety factor map)
How does mesh coarseness affect the results?
The electric field has high mesh dependency (derivative of potential). Place at least 4–6 layers of elements at electrode edges. Second-order elements are recommended.
Summary
- Mesh quality in electric field concentration areas is key
- Ensure electric field accuracy with second-order elements
- Visualize design margin with safety factor maps
FEM Analysis of Solid Insulation—Mesh Refinement and Convergence Verification of Electric Field Concentration Factor
In FEM electric field analysis of solid insulating materials (epoxy, XLPE, ceramic), electric field concentration occurs at shape corners, edges, and electrode ends. Mesh refinement in these concentrated areas is crucial for accuracy. Practical procedure for verifying electric field convergence: ① Analyze the target area with three mesh levels (coarse, medium, fine) and judge convergence when the change rate of the maximum electric field value is below 1%. ② Set the minimum mesh size to R/20 or less relative to the edge curvature radius R (e.g., mesh ≤5um for R=0.1mm). ③ At interfaces where the relative permittivity ratio (er) differs significantly (e.g., air er=1 vs. epoxy er=4), the electric field changes abruptly due to discontinuity in normal electric flux density, so ensure equally dense meshes on both sides of the interface. Underestimating the electric field concentration factor Kt by more than 3% risks insufficient design safety factor in actual equipment.
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 formulation. 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)
Handles nonlinearity of B-H curve using Newton-Raphson method. Residual criterion:
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