Used Tools:
Physics
Capacitance is a fundamental concept in physics, defined as the ratio of the amount of charge stored in a capacitor to the potential difference between its electrodes. Let's consider an example of a parallel-plate capacitor, as shown in Figure 1. In this example, the capacitor is constructed by filling the space between two square plates with blocks of three different dielectric materials.
The capacitance of each dielectric block is determined by the equations (1), (2), and (3):
(eq.1)
(eq.2)
(eq.3)
![K subscript 1](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=96115ef05dbaf83d6ff4ef23f2322a0e&cw=20&ch=24&cb=15)
,
![K subscript 2](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=ed467bc4908d81787f65273dfeb4a0bb&cw=20&ch=24&cb=15)
and
![K subscript 3](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=322cea208fd38ca4961227488af37958&cw=20&ch=24&cb=15)
are the dielectric constants of individual dielectric materials,
![epsilon subscript 0](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=858769948b27476a6e3e32bc300e5532&cw=16&ch=24&cb=15)
is the permittivity of free space,
![A](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=eac9e7cea1ad92aeaa0abedc9e6f475c&cw=13&ch=19&cb=15)
is the capacitor plate area and
![d](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=7519f95ab3d015df65ff64ec42272a8b&cw=11&ch=19&cb=15)
is the distance between the plates. Figure 1 shows that the total capacitance is formed by connecting a capacitance of the dielectric block
![K subscript 1](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=96115ef05dbaf83d6ff4ef23f2322a0e&cw=20&ch=24&cb=15)
in parallel with a series connection of blocks
![K subscript 2](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=ed467bc4908d81787f65273dfeb4a0bb&cw=20&ch=24&cb=15)
and
![K subscript 3](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=322cea208fd38ca4961227488af37958&cw=20&ch=24&cb=15)
. 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](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=ac432be8bd5622b86f922e21cdd10474&cw=535&ch=110&cb=54)
(4)
Figure1 - parallel plate capacitor with three dielectrics
With the following parameters:
![d equals 2 m m semicolon space l equals 100 m m](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=fc7eb6c145dcab59670a7148f51c7cd7&cw=149&ch=20&cb=16)
![A equals 100 m m cross times 50 space m m equals 5000 m m squared](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=34d23baa3871dcaceaeb8d25d7bc51cb&cw=235&ch=22&cb=18)
![K subscript 1 equals 1.93 space semicolon space K subscript 2 equals 3.25 space semicolon space K subscript 3 equals 4 space](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=229f3b34efeb47364fee6487021dec22&cw=212&ch=25&cb=16)
The equivalent capacitance is: ![C equals 6.1050 cross times 10 to the power of negative 11 end exponent space F](/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=62c97e2784dc69cbb9d8c5729f43442e&cw=153&ch=23&cb=19)
Model
The following instructions show how to prescribe material to individual parts of your model and compute capacitance between two elements.
A model of the capacitor with multiple dielectrics has been created in Solidworks. The space between the parallel plates is filled by 3 different dielectric materials. The plate surface is
, while the thickness of each plate is 1mm. The thickness of the dielectric block
is the same as the distance between the plates: 2mm; the thickness of the blocks
and
is half of that:1mm (Figure 2).
The simulation is performed in the EMS Electrostatic study
. Aluminum is used as a material for the electrode plates, Teflon, polyimide, and nylon are used for dielectric
,
and
, respectively. All these materials with their electromagnetic properties can be found in the EMS material library.
Assign material
To define the material for the Dielectric 1:
- Under Materials in the EMS manager tree, right-click on Dielectric 1.
- Select Apply Material
![2](/ckfinder/userfiles/images/2%282%29.jpg)
- The Material Browser folder appears
- Under “Cables” folder, choose Teflon
- Click Apply and Close
![3](/ckfinder/userfiles/images/3.jpg)
This procedure is repeated to assign Polyimide and Nylon to Dielectrics 2 and 3, as well as to assign Aluminum to the plates.
Figure 2 - Solid works model of a capacitor with 3 dielectrics
Compute capacitance
To obtain capacitance results from EMS:
- In the EMS manger tree, Right-click on the Electrostatic study
folder.
- Select Properties
![5](/ckfinder/userfiles/images/5.jpg)
- Under General Properties, check Compute Capacitance box
- Click OK
.
Boundary Conditions
To account for the capacitance, a Floating conductor boundary condition is assigned to both plates.
To do so:
- In the EMS manager tree, Right-click on the Load/Restraint
folder.
- Select Floating Conductor
.
- Click inside the BodiesSelection
box and then select the Topplate.
- Click OK
.
For the bottom plate:
- In the EMS manger tree, Right-click on the Load/Restraint
folder.
- Select Floating Conductor
.
- Click inside the BodiesSelection
box and then select the Bottom plate.
- Click OK
.
Results
In the EMS manager tree, under Results
, open the Results Table to find the Capacitance matrix. EMS solution for total capacitance is
F (Figure 3) and it matches the theoretical result very closely.
![EMS results for Capacitance](/ckfinder/userfiles/images/EMS-results-for-Capacitance.jpg)
Figure 3 - EMS results for Capacitance
Conclusion
This application note explores the concept of capacitance through the simulation of a parallel-plate capacitor filled with three different dielectric materials in EMS. By employing the superposition principle, the study calculates the capacitance contributed by each dielectric block and the total equivalent capacitance of the system, showcasing the impact of dielectric constants on the capacitor's overall capacitance. The simulation, created in SolidWorks and analyzed in an EMS Electrostatic study, utilizes materials like aluminum for electrodes and Teflon, polyimide, and nylon for the dielectrics. The process involves assigning materials, computing capacitance, and setting boundary conditions to accurately model the capacitor's behavior.
The findings from EMS closely align with theoretical calculations, demonstrating the software's effectiveness in predicting the capacitor's performance. The total capacitance obtained from the simulation is 6.1056 x 10^-11 F, validating the theoretical approach. This study not only reinforces the fundamental principles of capacitance but also illustrates the practical application of simulation tools in understanding and analyzing complex electrical systems. Through detailed modeling and precise simulations, EMS proves to be a valuable tool in the design and analysis of capacitive systems, offering insights that bridge theoretical physics with practical engineering solutions.