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Computing the Effect of Fringing Fields on Capacitance
Introduction
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, which is 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 on the computed 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 field lines 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 and electric field lines for the case of the Zero Charge boundary condition.
Figure 3: The electric field norm and electric field lines 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.
Application Library path: ACDC_Module/Introductory_Electrostatics/capacitor_fringing_fields
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select AC/DC > Electric Fields and Currents > Electrostatics (es).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
r_air
15[cm]
0.15 m
Radius, air domain
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose cm.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 10.
4
In the Height text field, type 0.5.
5
Locate the Position section. In the z text field, type -2.
6
Click  Build Selected.
Mirror 1 (mir1)
1
In the Geometry toolbar, click  Transforms and choose Mirror.
2
Select the object cyl1 only.
3
In the Settings window for Mirror, locate the Input section.
4
Select the Keep input objects checkbox.
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Click  Build Selected.
Sphere 1 (sph1)
1
In the Geometry toolbar, click  Sphere.
2
In the Settings window for Sphere, locate the Size section.
3
In the Radius text field, type r_air.
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Click  Build Selected.
5
Click the  Wireframe Rendering button in the Graphics toolbar.
6
Click the  Zoom Extents button in the Graphics 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 Floating Potential boundary condition.
Exterior
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Exterior in the Label text field.
3
Locate the Input Entities section. Select the All domains checkbox.
4
Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries.
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 Electrostatics interface, then add a Hide node.
Hide for Physics 1
1
In the Model Builder window, right-click View 1 and choose Hide for Physics.
2
In the Settings window for Hide for Physics, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 2 only.
The default boundary condition is Zero Charge, which is applied to all exterior boundaries. Add two Terminal features for the electrodes. One connected to the source, and one connected to ground.
Electrostatics (es)
Domain Terminal 1
1
In the Physics toolbar, click  Domains and choose Domain Terminal.
2
Select Domain 2 only.
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In the Settings window for Domain Terminal, locate the Terminal section.
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From the Terminal type list, choose Voltage.
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In the V0 text field, type 0.
Domain Terminal 2
1
In the Physics toolbar, click  Domains and choose Domain Terminal.
2
Select Domain 3 only.
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In the Settings window for Domain Terminal, locate the Terminal section.
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From the Terminal type list, choose Voltage.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarse.
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Click  Build All.
Study 1
The stationary study should be solved for different values of the radius of the air domain. Set the range of possible values by using a parametric sweep.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
r_air (Radius, air domain)
range(15,6,39)
cm
5
In the Study toolbar, click  Compute.
Results
Insulating Exterior
The second default plot shows the electric field. Modify it to get a better view of the field shape in the x,z-plane.
1
In the Model Builder window, under Results click Electric Field (es).
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In the Settings window for 3D Plot Group, type Insulating Exterior in the Label text field.
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Locate the Data section. From the Parameter value (r_air (cm)) list, choose 21.
Multislice 1
1
In the Model Builder window, expand the Insulating Exterior node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Multiplane Data section.
3
Find the x-planes subsection. From the Entry method list, choose Number of planes.
4
In the Planes text field, type 0.
Streamline Multislice 1
1
In the Model Builder window, click Streamline Multislice 1.
2
In the Settings window for Streamline Multislice, locate the Multiplane Data section.
3
Find the x-planes subsection. From the Entry method list, choose Number of planes.
4
In the Planes text field, type 0.
Transparency 1
1
In the Model Builder window, expand the Streamline Multislice 1 node, then click Transparency 1.
2
In the Settings window for Transparency, locate the Transparency section.
3
Find the Transparency subsection. In the Transparency text field, type 0.8.
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Click the  Go to XZ View button in the Graphics toolbar.
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Click the  Zoom Extents button in the Graphics toolbar.
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In the Insulating Exterior toolbar, click  Plot.
Compare the resulting plot with Figure 2.
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Click the  Go to Default View button in the Graphics toolbar.
Electrostatics (es)
Next, apply a Floating Potential boundary condition to the exterior. This condition overrides the default Zero Charge condition.
Floating Potential 1
1
In the Physics toolbar, click  Boundaries and choose Floating Potential.
2
In the Settings window for Floating Potential, locate the Boundary Selection section.
3
From the Selection list, choose Exterior.
Add a new study to keep the result separate from the previous one.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
r_air (Radius, air domain)
range(15,6,39)
cm
5
In the Model Builder window, click Study 2.
6
In the Settings window for Study, locate the Study Settings section.
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Clear the Generate default plots checkbox.
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In the Study toolbar, click  Compute.
Results
Conducting Exterior
1
In the Model Builder window, right-click Insulating Exterior and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Conducting Exterior in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2/Parametric Solutions 2 (sol9).
4
Click the  Go to XZ View button in the Graphics toolbar.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
In the Conducting Exterior toolbar, click  Plot.
The resulting plot should look like Figure 3.
7
Click the  Go to Default View button in the Graphics toolbar.
Join 1
1
In the Results toolbar, click  More Datasets and choose Join.
2
In the Settings window for Join, locate the Data 1 section.
3
From the Data list, choose Study 1/Parametric Solutions 1 (sol2).
4
Locate the Data 2 section. From the Data list, choose Study 2/Parametric Solutions 2 (sol9).
5
Locate the Combination section. From the Method list, choose General.
6
In the Expression text field, type (data1+data2)/2.
1D Plot Group 4
In the Results toolbar, click  1D Plot Group.
Global 1
1
Right-click 1D Plot Group 4 and choose Global.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Electrostatics > Terminals > Maxwell capacitance - F > es.C22 - Maxwell capacitance, 22-component.
5
Click Replace Expression in the upper-right corner of the x-Axis Data section. From the menu, choose Global definitions > Parameters > r_air - Radius, air domain - m.
6
Click to expand the Legends section. From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
Insulating Exterior
8
In the 1D Plot Group 4 toolbar, click  Plot.
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 2/Parametric Solutions 2 (sol9).
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Conducting Exterior
6
In the 1D Plot Group 4 toolbar, click  Plot.
Global 3
1
Right-click Global 2 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Join 1.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Average
5
In the 1D Plot Group 4 toolbar, click  Plot.
This should reproduce Figure 4.
Study 1
Optionally, to allow recomputing Study 1, you can disable the Floating Potential boundary condition for that particular study as follows.
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step checkbox.
4
In the tree, select Component 1 (comp1) > Electrostatics (es) > Floating Potential 1.
5
Click  Disable.