COMSOL 6.4 - Thin Low Permittivity Gap Comparison

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Thin Low Permittivity Gap Comparison

The thin low permittivity gap boundary condition is meant to approximate a thin layer of material with low relative permittivity compared to its surroundings, and is available for electrostatic field modeling. This example compares the thin low permittivity gap boundary condition to a full-fidelity model and discusses the range of applicability of this boundary condition.

Figure 1: A two-dimensional parallel plate capacitor with a high dielectric between the plates. A thin circular gap in the high dielectric distorts the electric field.

Model Definition

The situation being modeled is shown in Figure 1. Two parallel plates have a high dielectric material (relative permittivity εr = 20) in between them, and have a voltage difference applied to them, forming a capacitor. Inside of this dielectric material, there is a 1 mm circular gap with lower permittivity (εr = 1) and an outer diameter of 1 cm.

The thin gap is modeled in two different ways: first by using a full fidelity model that includes the thickness of the wall, and then by using the thin low permittivity gap boundary condition. The two models are separate, but are modeled simultaneously for comparison.

The location of the thin low permittivity gap condition is at the centerline, midway between the inner and outer radii of the gap in the full fidelity model. Note that when using the thin low permittivity gap condition, the total volume of the surrounding material is slightly larger, since the thickness of the wall is not being explicitly modeled.

Results and Discussion

The electric field and isolines of the voltage are plotted in Figure 2. The field lines can be observed to deform around the inclusion. The solution for the model using the thin low permittivity gap condition agrees well with the one in the full fidelity model.

The thin low permittivity gap boundary condition can be used in cases where the thickness of the boundary being approximated is much smaller than the characteristic size of the model domain, and when the relative permittivity of the gap region is lower than the surrounding medium. When this boundary condition is used, the number of mesh elements is much smaller, saving solution time and memory.

Figure 2: Isolines of the voltage field and streamlines of the electric field are plotted. The lines are colored according to the strength of the electric field, and the background grayscale plot is of the electric field norm. The full fidelity (left) and thin low permittivity gap (right) solutions are almost identical.

ACDC_Module/Introductory_Electrostatics/thin_low_permittivity_gap_comparison

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select > > .

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4

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In the tree, select > .

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In the window, under click .

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In the window for , locate the section.

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From the list, choose .

Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
er_a	1	1	Relative permittivity, free space and gap
er_b	20	20	Relative permittivity, dielectric
V0	1[V]	1 V	Applied voltage

Rectangle 1 (r1)

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type 8.

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In the text field, type 10.

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Locate the section. In the text field, type 0.1.

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In the text field, type -5.

Rectangle 2 (r2)

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type 3.5.

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Locate the section. In the text field, type 0.1.

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In the text field, type 2.

Fillet 1 (fil1)

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In the toolbar, click

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On the object , select Points 2 and 3 only.

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In the window for , locate the section.

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In the text field, type 0.5.

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In the toolbar, click

and choose .

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Select the object only.

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In the window for , locate the section.

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In the text field, type -5.

Rectangle 3 (r3)

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type 3.

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In the text field, type 4.

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Locate the section. In the text field, type 0.1.

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In the text field, type -2.

Mirror 1 (mir1)

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In the toolbar, click

and choose .

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Click in the window and then press Ctrl+A to select all objects.

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In the window for , locate the section.

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Select the checkbox.

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button in the toolbar.

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type 0.95.

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In the text field, type 180.

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Locate the section. In the text field, type 0.1.

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Locate the section. In the text field, type -90.

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type 180.

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Locate the section. In the text field, type -0.1.

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Locate the section. In the text field, type 90.

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Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (cm)
Layer 1	1[mm]

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The geometry on the left side describes the full fidelity model. The geometry on the right side replaces the thin layer with a boundary in order to use the feature.

Create a set of selections before setting up the physics. First, create a selection for the high dielectric material domain between two parallel plates.

High dielectric

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In the toolbar, click

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Select Domains 3, 7, 10, and 11 only.

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In the window for , type High dielectric in the text field.

Add a selection for the thin low permittivity gap boundaries.

Thin low permittivity gap

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Select Boundaries 44 and 45 only.

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In the text field, type Thin low permittivity gap.

Add a selection for the model domain.

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In the toolbar, click

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Select Domains 1, 3, 5–8, 10, and 11 only.

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In the window for , type Model domain in the text field.

Now, add the physics features needed for the two sides of the model.

Electrostatics (es)

Charge Conservation in Solids 1

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In the toolbar, click

and choose .

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Select Domains 3, 7, 10, and 11 only.

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In the window, click .

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In the window for , locate the section.

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From the list, choose .

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In the toolbar, click

and choose .

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Select Boundaries 23, 25, 46, and 47 only.

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In the toolbar, click

and choose .

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Select Boundaries 4, 6, 36, and 37 only.

Boundary Terminal 1

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In the toolbar, click

and choose .

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Select Boundaries 30, 32, 48, and 49 only.

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In the window for , locate the section.

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From the list, choose .

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In the V0 text field, type V0.

Boundary Terminal 2

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In the toolbar, click

and choose .

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Select Boundaries 7, 8, 38, and 39 only.

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In the window for , locate the section.

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From the list, choose .

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In the V0 text field, type V0.

Thin Low Permittivity Gap 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. In the ds text field, type 1[mm].

Material 1 (mat1)

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In the window, under right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	er_b	1	Basic

Material 2 (mat2)

1

Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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From the list, choose .

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Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	er_a	1	Basic

In the window, under right-click and choose .

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In the window, click .

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In the window for , locate the section.

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Clear the checkbox.

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In the toolbar, click

.

Begin the result analysis by suppressing the domain of the wall of the inclusion, which is not of interest.

In the window, expand the node.

Study 1/Solution 1 (sol1)

In the window, expand the > node, then click .

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Select Domains 1, 3, 7, 8, 10, and 11 only.

Create a custom plot to show the direction and norm of the electric field.

Electric Field (es)

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In the toolbar, click

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In the window for , type Electric Field (es) in the text field.

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Right-click and choose .

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In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

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Locate the section. From the list, choose .

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Clear the checkbox.

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From the list, choose .

Electric Field (es)

Next, add a contour plot showing the electric potential.

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In the window, right-click and choose .

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In the window for , locate the section.

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In the text field, type 21.

Color Expression 1

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Right-click and choose .

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In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

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Locate the section. Clear the checkbox.

Electric Field (es)

Then, add a streamline plot of the electric field.

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In the window, right-click and choose .

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Select Boundaries 7, 30, 38, 39, 48, and 49 only.

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In the window for , locate the section.

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In the text field, type 30.

Color Expression 1

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Right-click and choose .

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In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

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button in the toolbar.

Compare the plot with Figure 2.

Finish the result analysis by evaluating the capacitance of the system.

Global Evaluation 1

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In the toolbar, click

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In the window for , locate the section.

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In the table, enter the following settings:


Expression	Unit	Description
es.Q0_1/V0	F

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Global Evaluation 2

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In the toolbar, click

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In the window for , locate the section.

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In the table, enter the following settings:


Expression	Unit	Description
es.Q0_2/V0	F

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, then choose .

The capacitance should be about 120 pF in both cases.