2D electrostatics

Conductors at different voltages set up an electric field in the insulation between them. How strong the field gets decides whether the insulation holds, and how much charge the conductors take up for a given voltage decides their capacitance. Formulas give both for a few shapes, such as a coaxial cable or a plate capacitor without its fringing field. For a real cross-section with several dielectrics, rounded and sharp edges and conductors in open space, they follow from a solution of the electrostatic field equation. The tool meshes the part and solves that equation with the finite element method, and shows the potential, the field and the place of the highest field strength.

Planar or axisymmetric

A planar model stands for a long part, such as a cable, a bus bar or a strip line, in which the field lies in the plane of the drawing. The Length gives the part's length, and the charges and the capacitance hold for that length and also per meter. An axisymmetric model stands for a part that is round about an axis, such as a bushing, an insulator or a sphere gap. The drawing is then the half-section right of its y axis, which is the axis of rotation, and the results hold for the whole part. With 3D on, the drawing shows the part revolved through half a turn, seen at an angle, with the field on its cut face on both sides of the axis.

Regions, dielectrics and metal

Every closed region of the drawing gets a number, F1, F2 and so on, and a material from the list, with its relative permittivity εr and its dielectric strength. A loop inside another on the same layer is a hole, and a loop on a layer of its own is an insert, a region of its own material. Regions that touch must lie on layers of their own. A region marked as metal is a conductor, which the field does not enter, and a hole without a condition is filled with air.

Edge, hole and interface numbers

The outline's edges get E1, E2 and so on from the lowest left corner around the part, every hole one number for its whole boundary, H1, H2 and so on from the left, and every edge between two regions I1, I2 and so on. The conditions refer to these numbers and to the metal regions' numbers, and the drawing shows them, with each edge's length on hover. The Propagation column carries a row on along the outline, with planex2d's selection propagation.

• Minimal keeps the edge alone.
• Coincident adds the edges joined to it end to end, which on a closed outline is all of it.
• Tangent adds the edges that continue it tangentially, such as the lines on either side of a fillet.
• Curvature adds the edges that continue its curvature as well.

Conductors and charges

• A voltage holds an edge, a hole or a metal region at a fixed potential, such as the core of a cable or a grounded shield at 0 V.
• A floating conductor is a metal part connected to nothing, with a given charge, usually 0. Its potential follows from the solution, such as that of a shield or a fitting that is not grounded. A metal region without a row floats with no charge.
• A surface charge puts a charge per area on an edge or an interface, such as a charged foil or an electret.

Conductors at the same voltage count as connected, so the capacitance of a comb of fingers at 5 V against one at 0 V is that of the two combs. Edges not in the table carry no charge, and the field lines run along them, as on a plane of symmetry.

Open space

With Open space on, the part sits in air that reaches to infinity, for parts such as a two-wire line or a sphere above a plate that no grounded enclosure surrounds. The tool adds an air region around the drawing, and around that a thin layer whose coordinates a map stretches out to infinity. The solution in the layer is then that of all the space outside, and the result does not depend on the size of a box, as it would with a box held at 0 V. In a planar model no charge leaves the part to infinity, and in an axisymmetric model infinity is at 0 V, so a single sphere has its capacitance to infinity.

Measurements

The Measurements group, folded below the conductors, adds values for the regions and edges you choose to the results.

• A dielectric region gives its mean and highest field strength, its breakdown margin and the energy stored in it, and a metal region its voltage and charge.
• An edge, a hole or an interface, carried on with the Propagation column, gives its length, its mean potential, the voltage ∫E·dl along it and the flux of the displacement D across it. On the edges of a conductor this flux is the conductor's charge, and along an equipotential ∫E·dl is 0.

Results

The results are the voltage and the charge of every conductor, the capacitance between the conductors at the two voltages, the energy stored in the field and the highest field strength with its place. For three or more conductors the tool also gives the capacitance matrix. For each material it gives the highest field in it and the breakdown margin, its dielectric strength divided by that field. The drawing colors the field by the potential with its equipotential lines, or by the field strength, its components, the displacement or the energy density with the list beside Results. Field lines along the field can be added, as many as the number beside Results. They start on the conductors in proportion to their charge, so they crowd where the field is strong.

At a sharp metal corner the field of an electrostatic solution grows without limit, so the highest field there depends on the mesh. The accuracy estimate in the Solver group shows this as a large change. A conductor drawn with its real radius gives the field at that radius.

Simplify outline

Drawings often carry arcs and holes as chains of short lines, which give the outline many edge numbers. With Simplify outline on, planex2d fits arcs to such chains, within a ten-thousandth of the part's size, before the part is numbered and meshed. Turn it off to analyze the outline exactly as drawn. Where the fit misses an arc, the contour checker fits arcs with a tolerance and an edge length of your choice, closes gaps and removes tiny edges, and saves the cleaned drawing for the electrostatics tool.

Units

Lengths are in millimeters, voltages in volts, field strengths in V/mm, dielectric strengths in kV/mm, charges in nC and surface charges in µC/m², the displacement in µC/m² and the energy density in J/m³. A file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab, and NGSolve with the first solve, about 23 MB the first time. Your browser keeps them in its cache, so later visits start faster.

2. Read

   planex2d reads the file, joins its closed contours into regions, simplifies the outline and numbers its regions, edges and holes.

3. Solve

   NGSolve meshes the part and solves the electrostatic field equation, again whenever a setting changes after the first solve.

4. Save

   The results are saved as a PDF report or a CSV table, made in the tab as well.

The finite element model

The mesh has elements of about a thirtieth of the part's size, finer along arcs, short edges and conductors, and it is curved to the exact outline. The hp refinement in the Solver group refines the mesh geometrically toward every vertex and edge of the part, as many levels deep as given, since the field changes fastest there, and it integrates the refined elements with a higher order. It is off by default, and 3 levels sharpen the field at sharp corners. NGSolve solves the field equation with second-order elements, set from 1 to 5 in the Solver group, and a sparse Cholesky factorization. The tool solves once for every conductor at 1 V with the others at 0 V, which gives the capacitance matrix, and combines these solutions to the voltages given. Floating conductors then take the voltages at which their charges come out as given, without an approximation.

How it is tested

The tests solve parts whose capacitance, field or potential is known exactly and compare the tool's values with those, with second-order elements and third-order elements for the cases in open space. A plate capacitor has a uniform field, which the elements represent exactly, so its cases agree to the last digit. The curved cases agree within the error of the mesh.

Supported files

Run it in Python

The same analysis takes a few lines with planex2d and NGSolve. The example is the coaxial cable sample, a core of 0.9 mm in a polyethylene dielectric of 2.95 mm with εr = 2.25. planex2d draws the section and names its two circles, and NGSolve meshes it and solves for the potential with the core at 1 V and the shield at 0 V. The field energy then gives the capacitance per meter, C = ε∫|∇V|² for 1 V. See Meshing with netgen in the docs, and the microstrip impedance example for a field solve in a trace's cross-section.

$ pip install "planex2d[occ]" ngsolve

import math

import ngsolve as ngs
import planex as px

coax = px.QuickSketch().Circle(1.475).Circle(0.45).Face()
for e in coax.edges:
    e.name = "core" if e.mass < 3 else "shield"
mesh = ngs.Mesh(px.OCCGeometry(coax).GenerateMesh(maxh=0.1))
mesh.Curve(2)
fes = ngs.H1(mesh, order=2, dirichlet="core|shield")
u, v = fes.TnT()
a = ngs.BilinearForm(2.25 * ngs.grad(u) * ngs.grad(v) * ngs.dx).Assemble()
gf = ngs.GridFunction(fes)
gf.Set(1, definedon=mesh.Boundaries("core"))
r = -a.mat * gf.vec
gf.vec.data += a.mat.Inverse(fes.FreeDofs(), inverse="sparsecholesky") * r
c = ngs.Integrate(2.25 * ngs.grad(gf) * ngs.grad(gf), mesh) * 8.8541878128
print("C", round(c, 3), "pF/m, exact", round(2 * math.pi * 2.25 * 8.8541878128 / math.log(1.475 / 0.45), 3))

It prints a capacitance of 105.439 pF per meter, against 105.439 pF from 2πε/ln(b/a).

Privacy

Limits

• The tool solves the electrostatic field with constant, isotropic permittivities. Currents, magnetic fields, losses and frequency effects are not computed.
• A planar model assumes no field along the part, and an axisymmetric model a part and a field that are round about the y axis.
• The dielectric strengths are typical values, and the breakdown margin compares them with the field. It is not a prediction of the breakdown voltage, which also depends on the gap, the thickness, the voltage form and the surface.
• The field at a sharp metal corner depends on the mesh. Draw conductors with their radii where the highest field matters.
• A volume charge in a dielectric is not part of the tool, and a file must be planar, and its text, dimensions and hatches are not read.
• A part that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.