ИЭ / 6 семестр (англ) / Лаба / Capacitor

Peter the Great St. Petersburg Polytechnic University Institute of Energy and Transport Systems. Department of Theoretical Electrical Engineering and Electromechanics.

Project 1

Plane Capacitor

Student:

gr. 3231302/90201

Supervisor:

Dubitsky S.D.

St. Petersburg

2022

1. Task description and given data

Table 1. Personal data

2. Sketch of the geometry model with no mesh

1. X-axis is equipotential: U=0 (zero Dirichlet condition) as we consider that both capacitor plates have the same charge, but with the opposite sign, so x-axis is the middle between them and its voltage is zero.

2. Y-axis is normal to the equipotential lines, so (zero Neuman condition).

3. The artificial outer boundary (there’s no field here, so U=0).

Figure 1. Sketch of the geometry model with no mesh

Figure 2. Model of ¼ of the plane capacitor

3. Dependency of maximal field E_max on the mesh spacing on the rounded end of the conductor

In order to study the influence of the number of mesh nodes to the maximal field intensity (E_max), let’s add the mesh and then it’s necessary to control its step manually.

Table 2. Maximal field E_max

Mesh density (S)	Value of Emax, V/m	Difference, %
0,8	44295,8
0,4	46659,8	5,34
0,2	49099,2	5,23
0,1	49287,7	0,38
0,05	49951,5	1,35
0,02	50250,4	0,6

Figure 3. Dependency of maximal field E_max on the mesh density

That figure shows us that with the growth of mesh density, the value of E_max grows too. The discrepancy between the last two values equals 0,6%.

3. Pictures with the mesh and E-field around the rounded end of the conductor

Figure 4. Sketch of rounded end of the conductor with default mesh spacing S=0.8

Figure 5. Picture of field with default mesh spacing S=0.8

Figure 6. Sketch of rounded end of the conductor with final mesh spacing S=0.02

Figure 7. Picture of field with final mesh spacing S=0.02

As we can see, maximum of field intensity is at the rounded end of the conductor.

5. Dependency of E_max on the radius R of the artificial outer boundary

With our optimal mesh spacing at the rounded end of the conductor (S=0.02) we will increase radius R of the artificial outer boundary from default (R=500 mm) to such a value that the E_max will converge.

Table 3. Maximal field E_max

Radius, mm	Value of Emax, V/m	Difference, %
100	50527,1	–
200	50305,3	0,44
500	50250,4	0,11

Figure 8. Dependency of maximal field E_max on radius of the outer boundary

That figure shows us that with the growth of radius of outer boundary, the value of E_max decreases. The discrepancy between the last two values is about 0,11%.

6. Capacitance calculations

• By a simple formula:

Some values which we need to use in this formula:

– dielectric permittivity; F/m;

d = 9 mm – distance between plates; w = 120 mm – width;

Z_L = 195 mm – axial length;

• In the «Quick Field» (using charge and using energy):

This problem we will solve in the «Quick Field» program, but firstly we should draw a contour for it.

Figure 9. Contour for finding capacitance with results of Capacitance Wizard

Table 4. The results for the capacitance by different methods

By a simple formula	By charge and voltage	By energy and voltage

According to our calculations, we see that the final results for the capacity differ by 2.3%.

7. Maximal field intensity E_max and the field in the middle E_c.

• For the field in the middle E_c

Figure 10. The value of the field in the middle E_c.

So finally, we got the value of E_c = 33333 V/m.

With the theoretical calculation, we get .

As we can see, the values match.

• For the maximal field intensity E_max

According to the tables 2 and 3 «Maximal field E_max» we took mesh density S=0.02 and radius R = 500 mm, so we get E_max = 50250,4 V/m.

Figure 11. XY-plot of E for S=0,02 and R=500 mm

Conclusion

I created a geometry model of the plane capacitor according to my personal data and set the necessary properties and boundary conditions.

Were received the dependency of the maximal field intensity E_max x on the mesh spacing on the rounded end of the conductor (with the growth of mesh density, the value of E_max grows too); and dependency of maximal field E_max on radius of the outer boundary (with the growth of radius of outer boundary, the value of E_max decreases).

I calculated the capacitance by three methods and all results are quite similar (final results for the capacity differ by 2.3%).

I found the values of the field in the center E_c = 33333 V/m and the maximal field E_max=50250,4 V/m.