Capacitance Analysis
Theory and Physics
Overview
Professor, how is capacitance calculated using FEM?
Capacitance is the ratio of charge to voltage between conductors.
In FEM, we solve Poisson's equation for electric potential and calculate the charge on conductor surfaces by integrating using Gauss's law.
Typical Analytical Solutions
Are there any shapes we can compare with theoretical formulas?
| Structure | Capacitance |
|---|---|
| Parallel Plates | $C = \varepsilon A / d$ |
| Coaxial Cylinders | $C = 2\pi\varepsilon L / \ln(b/a)$ |
| Concentric Spherical Shells | $C = 4\pi\varepsilon ab / (b-a)$ |
| Isolated Spherical Conductor | $C = 4\pi\varepsilon_0 a$ |
The golden rule is to verify the validity of FEM results with these before moving on to real-world problems.
Capacitance Matrix
How is it handled for multi-conductor systems?
The capacitive relationship between $n$ conductors is described by the capacitance matrix $[C]$.
The diagonal components $C_{ii}$ are self-capacitance, and the off-diagonal components $C_{ij}$ are mutual capacitance, which take negative values. This capacitance matrix is essential for evaluating crosstalk between PCB traces. Ansys Q3D can extract it automatically.
The Leyden Jar—Humanity's First Capacitor Was Born in the Netherlands in 1745
The Leyden Jar is humanity's first capacitor that embodies the concept of capacitance (capacitance). It stores static electricity inside a glass jar covered with metal foil on the inside and outside. In 1746, Franklin used this to conduct his kite experiment, proving that lightning is electricity. The essence of modern capacitance theory $C = Q/V$—"how much charge can be stored for a given potential difference"—was inspired by observing this jar.
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 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. In capacitor (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 $\nabla \cdot \mathbf{B} = 0$: Represents the absence of magnetic monopoles. 【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 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
- Isotropic assumption: Direction-specific property definitions are needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- 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 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-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
Details of Numerical Methods
How is the capacitance matrix calculated using FEM?
For a system with $n$ conductors, we solve Poisson's equation by applying 1V to conductor $j$ and 0V to the others. The charge $Q_i$ on each conductor surface directly becomes $C_{ij}$. This is repeated for all conductors.
Doing $n$ analyses sounds tough.
We can reduce the number of runs by utilizing the symmetry $C_{ij} = C_{ji}$. In Ansys Q3D, this operation is fully automated.
Energy Method
Are there methods other than charge integration?
There is the energy method, which calculates capacitance from electrostatic energy.
This can be calculated using COMSOL's "Volume Integration." Calculate using both the charge method and the energy method; if the difference is within 1%, it's reliable. A large difference indicates the mesh is too coarse.
Structures with large fringing fields are prone to insufficient meshing, right?
Exactly. When the trace width and gap are comparable, fringing capacitance can account for 30-50% of the total. The parallel plate approximation $C = \varepsilon A/d$ alone is insufficient.
Why the "Parallel Plate Capacitor" Never Disappears from Textbooks
The formula for the parallel plate capacitor $C = \varepsilon_0 \varepsilon_r A / d$ always appears as a fundamental of capacitance analysis. You might think, "Such a simple shape doesn't exist in reality," but semiconductor gate oxide layers, PCB power planes, and interlayer laminations in flexible boards are essentially parallel plates. In actual design, edge effects (fringing effects) cause 10-30% error, making correction by CAE solvers essential. The textbook formula is merely a starting point.
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)
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