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Computing capacitance of a multi-material capacitor Application


Physics

The capacitance is defined as the ratio of the amount of charge stored in the capacitor and the potential difference between the electrodes.
The example of a parallel-plate capacitor, in Figure1, is constructed by filling the space between two square plates with blocks of three dielectric materials.

The capacitance of each dielectric block is given by equations (1), (2) and (3):

 C subscript 1 equals space fraction numerator K subscript 1 space epsilon subscript 0 space space A ? 2 space space space over denominator d end fraction space   (eq.1)

 C subscript 2 equals fraction numerator space K subscript 2 space epsilon subscript 0 space space A divided by 2 over denominator d divided by 2 end fraction space space space space space space space (eq.2)
   C subscript 3 equals fraction numerator K subscript 3 space epsilon subscript 0 space A divided by 2 over denominator d divided by 2 end fraction space space space        (eq.3)

K subscript 1,K subscript 2 and K subscript 3are the dielectric constants of individual dielectric materials, epsilon subscript 0 is the permittivity of free space,A  is the capacitor plate area and d is distance betweenthe plates. Figure1 shows that the total capacitance is formed by connecting a capacitance of the dielectric block K subscript 1 in parallel with a series connection of blocks K subscript 2and K subscript 3. Therefore, the equivalent capacitance is given by equation (4) as: 
 
C equals C subscript 1 plus left parenthesis 1 over C subscript 2 space plus space 1 over C subscript 3 space right parenthesis to the power of negative 1 end exponent equals C subscript 1 plus space fraction numerator C subscript 2 space C subscript 3 over denominator C subscript 2 plus space C subscript 3 space end fraction equals space fraction numerator K subscript 1 space epsilon subscript 0 space A divided by 2 over denominator d end fraction plus fraction numerator epsilon subscript 0 space A over denominator d end fraction space left parenthesis fraction numerator K subscript 2 space K subscript 3 over denominator K subscript 2 plus K subscript 3 end fraction right parenthesis
space space equals space space fraction numerator epsilon subscript 0 space A over denominator d end fraction left parenthesis K subscript 1 over 2 space plus fraction numerator K subscript 2 space K subscript 3 over denominator K subscript 2 plus K subscript 3 end fraction right parenthesis space space space space space space space space space space                                                   (4)

parallel-plate-capacitor-with-three-dielectrics
Figure1 - parallel plate capacitor with three dielectrics


With the following parameters: 
d equals 2 m m semicolon space l equals 100 m m
A equals 100 m m cross times 50 space m m equals 5000 m m squared

K subscript 1 equals 1.93 space semicolon space K subscript 2 equals 3.25 space semicolon space K subscript 3 equals 4 space
The equivalent capacitance is: C equals 6.1050 cross times 10 to the power of negative 11 end exponent space F

Model

The following instructions show how to prescribe material to individual parts of your model and compute capacitance between two elements.
Model of the capacitor with multipledielectrics has been created in Solidworks. The space betweenthe parallel plates is filled by 3 different dielectric materials.Plate surface is 50 cross times 100 m m squared, whilethe thickness of each plate is 1mm.Thickness of dielectric block K subscript 1is the same as the distance between the plates: 2mm; thickness of blocks K subscript 2and K subscript 3 is half of that:1mm (Figure 2).

The simulation is performed in the EMS Electrostatic study 1 . Aluminum is used as a material for the electrode plates, Teflon, polyimide and nylon are used for dielectricK subscript 1,K subscript 2and K subscript 3 , respectively. All these materials with their electromagnetic properties can be found in the EMS material library. 

Assign material

To define material for the Dielectric 1:
  1. Under Materials in the EMS manager tree, right click on the Dielectric 1.
  2. Select Apply Material2
  3. The Material Browser folder appears
  4. Under “Cables” folder, choose Teflon
  5. Click Apply and Close 3
This procedure is repeated to assign Polyimide and Nylon to Dielectrics 2 and 3, as well as to assign Aluminum to the plates.

Solid works model of a capacitor with 3 dielectrics
Figure 2 - Solid works model of a capacitor with 3 dielectrics

Compute capacitance

To obtain capacitance results from EMS:

  1. In the EMS manger tree, Right-click on the Electrostatic study 1 folder.
  2. Select Properties5
  3. Under General Properties, check Compute Capacitance box
  4. Click OK6.

Boundary Conditions 

To account for the capacitance, a Floating conductor boundary condition is assigned to both plates.
To do so:             
  1. In the EMS manger tree, Right-click on the Load/Restraint 7folder.
  2. Select Floating Conductor8 .
  3. Click inside the BodiesSelection 9box and then select the Topplate.
  4. Click OK6.
For the bottom plate:
  1. In the EMS manger tree, Right-click on the Load/Restraint 7folder.
  2. Select Floating Conductor8 .
  3. Click inside the BodiesSelection9 box and then select the Bottom plate.
  4. Click OK6.

Results

In the EMS manger tree, under Results 10, open the Results Table to find Capacitance matrix. EMS solution for total capacitance is6.1056 cross times 10 to the power of negative 11 end exponent F (Figure 3) and it matches the theoretical result very closely.

EMS results for Capacitance
Figure 3 - EMS results for Capacitance