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Electric Sensor
Introduction
This example illustrates — in a simplified electrostatic setting — electrical impedance tomography (EIT), a method used for imaging of the distribution of electric permittivity in an object. The tomography is performed by measuring currents and voltages at the object’s surface.
An application of this technique is medical diagnosis. Due to the different electrical properties of various organs and parts of the body, it is possible to obtain information on their position and movement in a noninvasive way.
This model shows how to determine from the outside the shape and the placement of small objects with different material properties inside a box. Applying a potential difference on the boundaries of the box creates a surface charge density that varies depending on the permittivity distribution inside the box.
Model Definition
This model solves Gauss’ law with ρ = 0:
The box contains air with εr equal to 1. The different objects are made of materials with different values of the relative permittivity, εr: 1, 2, and 3.
To get a voltage difference, a ground condition (V = 0) is set on the bottom while the condition V = 1 is applied on the top of the box. On the side, the boundary condition used is electric insulation: n · D = 0.
Results and Discussion
As seen in Figure 1, the surface charge density is higher in correspondence of materials with higher permittivity, as expected. An imaging of the figures inside the box is reproduced in the surface charge density plot.
Figure 1: Surface charge density (boundary), electric field (streamline density), and electric potential (streamline color).
Inside the geometry, the streamlines show how the electric field varies. The electric field is lower in media with larger value of the permittivity.
Application Library path: COMSOL_Multiphysics/Electromagnetics/electric_sensor
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.
Geometry 1
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 0.1.
4
Click  Go to Plane Geometry.
Work Plane 1 (wp1) > Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 0.5.
4
In the Height text field, type 2.
5
Locate the Position section. In the xw text field, type -1.
6
In the yw text field, type 0.5.
7
Click  Build Selected.
Work Plane 1 (wp1) > Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 1.5.
4
In the Height text field, type 0.25.
5
Locate the Position section. In the xw text field, type -1.5.
6
In the yw text field, type 1.
7
Click  Build Selected.
Work Plane 1 (wp1) > Rectangle 3 (r3)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 1.5.
4
In the Height text field, type 0.25.
5
Locate the Position section. In the xw text field, type -1.5.
6
In the yw text field, type 1.75.
7
Click  Build Selected.
Work Plane 1 (wp1) > Union 1 (uni1)
1
In the Work Plane toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Ellipse 1 (e1)
1
In the Work Plane toolbar, click  Ellipse.
2
In the Settings window for Ellipse, locate the Size and Shape section.
3
In the a-semiaxis text field, type 0.5.
4
Locate the Position section. In the xw text field, type 1.5.
5
In the yw text field, type 1.5.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Ellipse 2 (e2)
1
In the Work Plane toolbar, click  Ellipse.
2
In the Settings window for Ellipse, locate the Size and Shape section.
3
In the b-semiaxis text field, type 0.5.
4
Locate the Position section. In the xw text field, type 1.5.
5
In the yw text field, type 1.5.
6
Click  Build Selected.
Work Plane 1 (wp1) > Compose 1 (co1)
1
In the Work Plane toolbar, click  Booleans and Partitions and choose Compose.
2
Select the objects e1 and e2 only.
3
In the Settings window for Compose, locate the Compose section.
4
Clear the Keep interior boundaries checkbox.
5
In the Set formula text field, type e1+e2.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Extrude 1 (ext1)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
0.8
4
Click  Build Selected.
5
Click the  Zoom Extents button in the Graphics toolbar.
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 5.
4
In the Depth text field, type 3.
5
Locate the Position section. In the x text field, type -2.
6
Click  Build All Objects.
7
Click the  Zoom Extents button in the Graphics toolbar.
8
Click the  Transparency button in the Graphics toolbar.
This completes the model geometry.
Electrostatics (es)
Charge Conservation in Fluids 1
1
In the Physics toolbar, click  Domains and choose Charge Conservation in Fluids.
2
Select Domains 2 and 3 only.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 3 only.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundary 4 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type 1.
Materials
Material 1 (mat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
Select Domain 2 only.
3
In the Settings window for Material, locate the Material Contents section.
4
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
2
1
Basic
Material 2 (mat2)
1
Right-click Materials and choose Blank Material.
2
Select Domain 3 only.
3
In the Settings window for Material, locate the Material Contents section.
4
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
3
1
Basic
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.
3
Clear the Generate default plots checkbox.
4
In the Study toolbar, click  Compute.
To reproduce the plot shown in Figure 1, begin by suppressing some boundaries so that the inside of the box becomes visible.
Results
In the Model Builder window, expand the Results node.
Study 1/Solution 1 (sol1)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution 1 (sol1).
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 Boundary.
4
From the Selection list, choose All boundaries.
5
Select Boundaries 3–5 and 38 only.
6
Click the  Transparency button in the Graphics toolbar.
3D Plot Group 1
In the Results toolbar, click  3D Plot Group.
Surface 1
1
Right-click 3D Plot Group 1 and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Electrostatics > Currents and charge > es.nD - Surface charge density - C/m².
3
Locate the Expression section. In the Unit field, type pC/m^2.
4
Locate the Coloring and Style section. From the Color table list, choose Cyclic.
5
In the 3D Plot Group 1 toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Streamline 1
1
In the Model Builder window, right-click 3D Plot Group 1 and choose Streamline.
2
In the Settings window for Streamline, locate the Streamline Positioning section.
3
From the Positioning list, choose Magnitude controlled.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Type list, choose Tube.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
In the Settings window for Color Expression, 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.
3
Click to expand the Title section. From the Title type list, choose Automatic.
Compare the resulting plot with that in Figure 1.