Electret Analysis
Theory and Physics
What is an Electret?
Professor, what is an electret?
A dielectric that permanently retains electric charge. The electrical equivalent of a magnet. Used in condenser microphones, air filters, and energy harvesting.
Primarily manufactured by injecting charge via corona discharge into fluoropolymer materials (PTFE, FEP). Capable of retaining charge for decades.
Summary
- A dielectric that retains permanent charge — the electrical version of a permanent magnet
- Condenser microphones — the largest application of electrets
- Calculate electric field distribution with FEM — apply surface charge density as a Neumann BC
The Discovery of Electrets—Mototaro Eguchi Created an Electrical "Fossil" in the 1920s
An electret is a material with permanent electrical polarization, first created in 1919 by Japan's Mototaro Eguchi by applying a strong electric field to a mixture of beeswax and carnauba wax and then cooling it. This material, which could be called an "electrical fossil," revolutionized microphones in the 1960s; nearly all microphones in modern smartphones and hearing aids are of the electret type. In FEM analysis, a special formulation is required that incorporates the permanent polarization P (polarization vector) of the electret as a material constant.
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 stating that a time-varying 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. 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 $\partial D/\partial t$ becomes non-negligible and describes 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 causes static electricity, making hair stand up—electric field lines radiate from the charged sheet (charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated 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 the non-existence of magnetic monopoles. 【Everyday Example】Cutting a bar magnet in half does not 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, to satisfy this condition, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used, 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 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
Electrets in FEM
Set the surface charge density $\sigma_s$ on the electret surface as a Neumann boundary condition. Treat internal polarization as a volume charge density $\rho_p = -\nabla \cdot \mathbf{P}$.
Summary
- Set surface charge $\sigma_s$ as a Neumann BC
- Treat internal polarization as a volume charge density
FEM Formulation for Electrets—How to Incorporate Polarization Vector P
In FEM formulation for electrets, a common method is to convert the permanent polarization P into equivalent charge densities (volume charge density ρ_pol=-∇·P, surface charge density σ_pol=P·n) and add them to the Poisson equation. Commercial tools often implement material models that perform this conversion automatically, but when using custom materials, it is necessary to manually calculate ρ_pol and σ_pol and provide them as boundary conditions. If the permittivity ratio between the electret layer and the air layer is large, the mesh near the interface must be refined, otherwise the accuracy of the electric field concentration will degrade.
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 boundary problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
Nonlinearity of B-H curve is handled by the Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is typical.
Frequency Domain Analysis
Time-harmonic assumption reduces the problem to a steady-state problem. Requires complex number operations, but broadband characteristics are obtained via time-domain analysis.
Time Domain Time Step
A 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 compute 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 at a high computational cost.
Practical Guide
Practical Applications
Design of MEMS microphones, vibration power generation devices (electret-type energy harvesters).
Checklist
- [ ] Are the charge density values based on actual measurements? (PTFE: approx. 1 to 10 mC/m²)
- [ ] Is the mesh for the air gap sufficient?
- [ ] Is the calculation of electrostatic force (Maxwell stress tensor) correct?
Electret Fibers in N95 Masks—CAE of the Mechanism that Captures Viruses via Static Electricity
The filter layer of an N95 mask is made of electret non-woven fabric, where the fibers are permanently charged and capture PM2.5 and virus droplets via electrostatic attraction. During the COVID-19 pandemic, information spread that "sterilizing masks in a microwave oven removes the static electricity, making them no longer N95." This is based on the fact that the polarization of electret fibers disappears with heat. By analyzing the relationship between the electric field distribution between fibers and particle capture efficiency using FEM, the design optimization of next-generation filters is advancing.
Analogy for the Analysis Flow
Electromagnetic field analysis of a motor is similar to "tuning a guitar." You adjust the string thickness (number of coil turns) and bridge position (magnet arrangement) to bring out the most beautiful sound (efficient torque characteristics). Changing one parameter alters the overall balance—that's why parametric studies are important.
Common Pitfalls for Beginners
"Air region? Why do we need to mesh air?"—This is a question almost everyone new to electromagnetic field analysis asks. The answer is, "Because magnetic field lines spread outside the iron core as well." If the analysis domain is set right up to the iron core, magnetic flux with nowhere to go will "collide" with the wall and reflect, causing unrealistic magnetic flux concentration. Imagine a room too small where a ball keeps bouncing off the walls.
Approach to Boundary ConditionsRelated Topics
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