Computing the Effect of Fringing Fields on Capacitance

A typical capacitor is composed of two conductive objects with a dielectric in between. Applying a potential difference between these objects results in an electric field. This electric field exists not just directly between the conductive objects, but extends some distance away, a phenomenon known as a fringing field. To accurately predict the capacitance of a capacitor, the domain used to model the fringing field must be sufficiently large, and the appropriate boundary conditions must be used. This example models a parallel plate capacitor in air and studies the size of the air domain. The choice of boundary condition is addressed as well.

Figure 1: A simple capacitor consisting of two metal discs in an air domain.

Model Definition

Figure 1 shows the capacitor consisting of two metal discs in a spherical volume of air. The size of the sphere truncates the modeling space. This model studies the size of this air domain and its effect upon the capacitance.

One of the plates is specified as ground, with the electric potential 0 V. The other plate has an applied potential of 1 V.

The air sphere boundary can be thought of as one of two different physical situations: It can be treated as a perfectly insulating surface, across which charge cannot redistribute itself, or as a perfectly conducting surface, over which the potential does not vary.

A perfectly insulating surface is modeled using the Zero Charge boundary condition. This boundary condition implies that the electric field lines are tangential to the boundary.

A perfectly conducting surface can be modeled with the Floating Potential boundary condition. This boundary condition fixes the electric potential on the surface of the sphere to a constant value that is computed during the solution. The boundary condition also implies that the electric field lines are perpendicular to the boundary.

When studying convergence of results with respect to the surrounding domain size, it is important to fix the element size. In this model, the element size is fixed as the domain size is varied.

Results and Discussion

Figure 2 and Figure 3 plot the electric field for the cases where the air sphere boundary is treated as perfectly insulating and perfectly conducting, respectively. The fields terminate differently on the boundaries of the air sphere.

Figure 4 compares the capacitance values of the device with respect to air sphere radius for the two boundary conditions. The figure also plots the average of the two values. Notice that all three capacitance calculations converge to the same value as the radius grows. In practice, it is often sufficient to model a small air sphere with the electric insulation and floating potential boundary conditions and take the average of the two.

Figure 2: The electric field norm (multislices) and electric field (arrows) for the case of the Zero Charge boundary condition.

Figure 3: The electric field norm (multislices) and electric field (arrows) for the case of the Floating Potential boundary condition.

Figure 4: Convergence of the device capacitance as the size of the surrounding air sphere is increased. Electric insulation and fixed potential boundary conditions converge to the same result. The average of the two is also plotted.

ACDC_Module/Capacitive_Devices/capacitor_fringing_fields

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r_air	15[cm]	0.15 m	Radius, air domain

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 10.

In the text field, type 0.5.

Locate the section. In the text field, type -2.

Click

Mirror 1 (mir1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Select the check box.

Click

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type r_air.

Click

Click the

button in the toolbar.

Click the

button in the toolbar.

The geometry describes two metal discs in an air domain.

Definitions

Create a selection for the exterior boundaries. Later, this will be used for the boundary condition.

Exterior

In the toolbar, click

In the window for , type Exterior in the text field.

Locate the section. Select the check box.

Locate the section. From the list, choose .

Hide one boundary to get a better view of the interior parts when setting up the physics and reviewing the mesh. Begin by selecting the interface, then add a node.

Hide for Physics 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 2 only.

The default boundary condition is , which is applied to all exterior boundaries. Add two features for the electrodes. One connected to the source, and one connected to ground.

Electrostatics (es)

Terminal 1