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Computing Capacitance
Introduction
A capacitor, in its simplest form, is a two terminal electrical device that stores electric energy when a voltage is applied across the terminals. The stored electric energy is proportional to the applied voltage squared and is determined by the capacitance of the device. This example introduces a model of a simple capacitor. The electric field and device capacitance are solved for under electrostatic conditions.
Figure 1: A simple capacitor composed of a disk of dielectric with metal plates on either side, and lead wires.
Model Definition
The capacitor being modeled is shown in Figure 1. Two metal disks, with leads, are separated by a disk of dielectric material. Since there can be significant fringing fields around the capacitor plates, an air volume is included in the model. The size of this air volume truncates the modeling space. In actuality, the fringing electric fields extend to infinity, but drop off in proportion to the inverse cube of the distance. They rapidly become small enough to be considered numerically insignificant. Here, it is assumed that the air volume is large enough to accurately capture the fringing fields. This could be checked by increasing the air volume size and comparing the results.
Under the assumption of electrostatic conditions, the entire surface of each electrode must be at the same potential, otherwise current would flow through these conductors. The air and dielectric are assumed to be perfect insulators. The quantity of interest is the electric potential in the air and in the dielectric — it is not necessary to solve for the potential in the electrode, because it is constant. There are two approaches that can be followed:
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apply the Terminal domain feature on the electrodes, or
•
remove the electrode domains from the selection of the physics interface and apply appropriate boundary conditions (such as Ground or boundary Terminal) at the interface with the air or dielectric.
This model serves as an example of the second approach.
Results and Discussion
The electric field is plotted in Figure 2. The field is relatively uniform between the plates, but some strong variations are seen at the edges of the plates. The fringing fields are observed to extend a small distance away from the capacitor. The capacitance of the device is evaluated to be approximately 43 pF. In Figure 3, a slice plot of the electric potential shows that there is an equipotential surface exactly midway between the capacitor plates.
Figure 2: The electric field strength in the dielectric and air domain surrounding the capacitor.
Figure 3: Contours of the electric potential on a slice through the center of the capacitor.
Application Library path: ACDC_Module/Introductory_Electrostatics/capacitor_dc
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).
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Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
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Click  Done.
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.
First, create a cylinder for the model domain.
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 20.
4
In the Height text field, type 20.
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Click  Build Selected.
Choose wireframe rendering to get a better view of the interior parts.
6
Click the  Wireframe Rendering button in the Graphics toolbar.
Then, add a cylinder for the disc of dielectric with the two metal plates.
Cylinder 2 (cyl2)
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 4.
5
Locate the Position section. In the z text field, type 8.
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Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (cm)
Layer 1
5[mm]
7
Clear the Layers on side checkbox.
8
Select the Layers on bottom checkbox.
9
Select the Layers on top checkbox.
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Click  Build Selected.
Finish the geometry by adding two cylinders for the leads.
Cylinder 3 (cyl3)
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 0.75.
4
In the Height text field, type 8.
Cylinder 4 (cyl4)
1
Right-click Cylinder 3 (cyl3) and choose Duplicate.
2
In the Settings window for Cylinder, locate the Position section.
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In the z text field, type 12.
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Click  Build All Objects.
The result should look like the image above.
Definitions
The model is composed of a disc of dielectric material with metal plates on either side and two lead wires. Create a set of selections to use when setting up the physics. First, create a selection for the metal domains.
Metal
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Metal in the Label text field.
3
Select Domains 2 and 4–6 only.
Add a Complement selection for the insulators.
Insulators
1
In the Definitions toolbar, click  Complement.
2
In the Settings window for Complement, type Insulators in the Label text field.
3
Locate the Input Entities section. Under Selections to invert, click  Add.
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In the Add dialog, select Metal in the Selections to invert list.
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Click OK.
Add selections for the ground boundaries and for the terminal boundaries.
Ground
1
In the Definitions toolbar, click  Explicit.
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In the Settings window for Explicit, type Ground in the Label text field.
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Select Domains 2 and 5 only.
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Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries.
Terminal
1
In the Definitions toolbar, click  Explicit.
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In the Settings window for Explicit, type Terminal in the Label text field.
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Select Domains 4 and 6 only.
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Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries.
To get a better view, hide some of the boundaries. 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.
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From the Geometric entity level list, choose Boundary.
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Select Boundaries 1, 4, and 23 only.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
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In the Settings window for Electrostatics, locate the Domain Selection section.
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From the Selection list, choose Insulators.
Charge Conservation in Solids 1
1
In the Physics toolbar, click  Domains and choose Charge Conservation in Solids.
2
Select Domain 3 only.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
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In the Settings window for Ground, locate the Boundary Selection section.
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From the Selection list, choose Ground.
Boundary Terminal 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Terminal.
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In the Settings window for Boundary Terminal, locate the Boundary Selection section.
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From the Selection list, choose Terminal.
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Locate the Terminal section. From the Terminal type list, choose Voltage.
Next, assign material properties to the model. Begin by specifying the dielectric disc with glass (quartz).
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
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In the tree, select Built-in > Glass (quartz).
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Click the Add to Component button in the window toolbar.
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In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Glass (quartz) (mat1)
Select Domain 3 only.
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Study 1
1
In the Model Builder window, click Study 1.
2
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.
Add a solution and define a Selection for the metal parts. Domains that are excluded from the selection will be hidden in the corresponding plots.
Results
Study 1/Solution 1 (2) (sol1)
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In the Model Builder window, expand the Results node.
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Right-click Results > Datasets and choose Solution.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
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From the Selection list, choose Metal.
3D Plot Group 1
In the Results toolbar, click  3D Plot Group.
The metal parts can be visualized by choosing Uniform for the coloring type. In this case the purpose of the plot is not to show a quantity, but to show a shape. The variable that the plot is based on is of no importance. Create a surface plot for this purpose.
Surface 1
1
Right-click 3D Plot Group 1 and choose Surface.
2
In the Settings window for Surface, locate the Data section.
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From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
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Locate the Coloring and Style section. From the Coloring list, choose Uniform.
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From the Color list, choose Gray.
3D Plot Group 1
Add a slice plot for the norm of the electric field and an arrow plot for the electric field.
Slice 1
1
In the Model Builder window, right-click 3D Plot Group 1 and choose Slice.
2
In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Electrostatics > Electric > es.normE - Electric field norm - V/m.
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Locate the Plane Data section. In the Planes text field, type 1.
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Locate the Coloring and Style section. From the Color table list, choose RainbowLight.
Arrow Volume 1
1
Right-click 3D Plot Group 1 and choose Arrow Volume.
2
In the Settings window for Arrow Volume, locate the Arrow Positioning section.
3
Find the x grid points subsection. In the Points text field, type 1.
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Find the y grid points subsection. In the Points text field, type 24.
5
Find the z grid points subsection. In the Points text field, type 11.
6
Locate the Coloring and Style section. From the Arrow length list, choose Logarithmic.
Color Expression 1
1
Right-click Arrow Volume 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Coloring and Style section.
3
Clear the Color legend checkbox.
4
Click the  Go to YZ View button in the Graphics toolbar.
5
Click the  Zoom Extents button in the Graphics toolbar.
The plot should look like the one in Figure 2.
Create a Cut Plane that may serve as a basis for 2D plots.
Cut Plane 1
1
In the Results toolbar, click  Cut Plane.
2
In the Settings window for Cut Plane, click  Plot.
3
Click the  Go to Default View button in the Graphics toolbar.
On this dataset, base a 2D Plot Group with contour plots for the electric potential.
2D Plot Group 2
In the Results toolbar, click  2D Plot Group.
Contour 1
1
Right-click 2D Plot Group 2 and choose Contour.
2
In the Settings window for Contour, locate the Levels section.
3
From the Entry method list, choose Levels.
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Click  Range.
5
In the Range dialog, type 0.1 in the Start text field.
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In the Step text field, type 0.1.
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In the Stop text field, type 0.9.
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Click Replace.
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In the Settings window for Contour, locate the Coloring and Style section.
10
From the Contour type list, choose Filled.
11
From the Color table list, choose RainbowLight.
Contour 2
1
In the Model Builder window, right-click 2D Plot Group 2 and choose Contour.
2
In the Settings window for Contour, click to expand the Title section.
3
From the Title type list, choose None.
4
Locate the Levels section. From the Entry method list, choose Levels.
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Click  Range.
6
In the Range dialog, type 0 in the Start text field.
7
In the Step text field, type 0.1.
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In the Stop text field, type 1.
9
Click Replace.
10
In the Settings window for Contour, locate the Coloring and Style section.
11
Select the Level labels checkbox.
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From the Coloring list, choose Uniform.
13
From the Color list, choose Black.
14
Clear the Color legend checkbox.
15
In the 2D Plot Group 2 toolbar, click  Plot.
16
Click the  Zoom Extents button in the Graphics toolbar.
Compare the resulting plot with Figure 3.
The Terminal feature automatically computes the capacitance. The computed value is available in a global variable and can be accessed using a Global Evaluation.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1) > Electrostatics > Terminals > es.C11 - Maxwell capacitance - F.
3
Click  Evaluate.
Table 1
1
Go to the Table 1 window.
The computed capacitance should be around 43 pF.