2. - 1 -
Abstract
The purpose of this study was to analyze the electric field effects of various capacitor
designs. These included a spherical capacitor, a cylindrical capacitor, and a GIS dead-end (a
cylindrical capacitor with a hemispherical end). We analyzed both the electric potential and the
electric field produced within the capacitors and compared results from a FE computer
simulation to the theoretical values based on ideal cases.
3. - 2 -
Contents
Introduction ..................................................................................................................................................3
Results...........................................................................................................................................................4
Spherical Capacitor .......................................................................................................................................5
Cylindrical Capacitor .....................................................................................................................................7
GIS Dead-End Capacitor................................................................................................................................9
Name Plate Rating from Dead-end Dielectric Strength..............................................................................11
Conclusion...................................................................................................................................................13
Reference....................................................................................................................................................14
Appendix A (Additions for Spherical)..........................................................................................................15
Appendix B (Additions for Cylindrical)........................................................................................................16
Appendix C (Additions for GIS)....................................................................................................................17
4. - 3 -
Introduction
This report includes analyses from three different capacitors including a description of
the capacitors behavior, figures of the electric potential field within each capacitor, and a
MATLAB plot comparing the data generated from FEMM 4.2 to the theoretical expectations.
Theory would expect that the V field within a long, cylindrical capacitor or a spherical capacitor
to linearly decrease radially outward. Our finite element data shows slight deviations from
these expected results, most notably half way between the positive and negative voltage
terminals of the capacitors.
The theoretical equations used in this report are as follows:
For spherical capacitors:
(1) 𝑉(𝑟) =
𝑉0(
1
𝑟
−
1
𝑏
)
(
1
𝑎
−
1
𝑏
)
(2) 𝐸 =
𝑉0
𝑏−𝑎
(
𝑎𝑏
𝑟2) ȃr
For cylindrical capacitors:
(3) 𝑉𝑟 =
𝑉0
ln( 𝑏
𝑎⁄ )
ln( 𝑏
𝑟⁄ )
(4) 𝐸 =
𝑉0
ln( 𝑏
𝑎⁄ )
∗ 1
𝑟⁄ ȃr
5. - 4 -
Results
The following sections reviews the results of this study.
6. - 5 -
Spherical Capacitor
The simulated spherical capacitor behaved nearly identical to the theoretical model. In
this experiment, the inner spherical conductor contains a constant voltage (5000 v) while the
outer spherical conductor is grounded. Theory would expect the V-field between the capacitors
to be inversely proportionate to the radius as described by equation (1).
Figure 1. FEMM model of axisymmetric spherical capacitor. V-field is shown in color with equipotential
lines. The red line is the path along which data was compared.
7. - 6 -
The theoretical and analytical data were almost perfectly in sync:
Figure 2. Theoretical vs FEMM results of spherical capacitor.
The two values overlapped nearly perfectly showing that the FEMM software accurately
depicts the V-field.
8. - 7 -
Cylindrical Capacitor
The second phase examined a cylindrical capacitor. The analyses occurred at the center
of the capacitor, far from the fringe effects occurring at the edges. Again, the inner cylinder
represents a constant voltage (5000 v) while the outer sheath is grounded.
Figure 3. FEMM model of axisymmetric cylindrical capacitor. V-field is shown in color with equipotential
lines. Data was taken along a radial path at the center of the capacitor.
9. - 8 -
Again, the theoretical and analytical data were very close:
Figure 4. Theoretical vs FEMM results for cylindrical capacitor.
The comparison of the two solutions shows minor differences between the theoretical
and analytical values for V-field. This likely could be due to the theoretical assumption of an
infinitely long cylindrical capacitor whereas the simulated capacitor was finite in length.
10. - 9 -
GIS Dead-End Capacitor
The GIS dead-end capacitor behaved very similar to the spherical and cylindrical
capacitors (as can be expected). In this study, the V-field was analyzed along two different
paths. Along the z-axis from the tip of the hemispherical end and radially outward near the
center of the cylinder. The theoretical equation for a spherical capacitor was used for the first
path while the theoretical equation for a cylindrical capacitor was used for the second.
Figure 5. FEMM model of axisymmetric GIS dead-end. V-field is shown in color with equipotential lines.
11. - 10 -
The theoretical and analytical models matched up near perfectly for both the spherical
and cylindrical approximations:
Figure 6. Theoretical vs analytical V-field values. Both the spherical and cylindrical approximations nearly
matched the FEMM simulation.
The comparison for the GIS dead-end was near perfect. This shows that the spherical
and cylindrical approximations can accurately deduce the V-field even when the shape of the
conductor is dissimilar to the ideal representation.
12. - 11 -
Name Plate Rating from Dead-end Dielectric Strength
In the final section of the results, the GIS dead-end E-field is evaluated to find the
maximum safe operating voltage for the dielectric. Our dielectric is capable of operating in
electric fields up to E* = 7.35x106 V/m. Due to the proportionality of E to voltage, the ratios of
E/v are equal. This yields:
(5) 𝑉∗
= (
𝐸∗
𝐸 𝑀𝐴𝑋
)𝑉1
Figure 7. Table of values, equations, and descriptions for elements used in determining the
maximum safe operating voltage and the rated center-pin-to-sheath voltage.
13. - 12 -
The max E-field was found at the tip of the hemisphere in the GIS using the FEMM
software. This yielded:
EMAX = 1.9389x105 V/m
Using equation (5) with the values of EMAX, E*, and V1 = 5000 V, we can calculate V*.
V* = 189.540 kV
Using a safety factor of 25% we can calculate the maximum operational voltage, VOPERATE
as:
VOPERATE = 142.156 kV
and:
VRATED,RMS = 100.519 kV
14. - 13 -
Conclusion
In conclusion, the similar results between the analytical FEMM software and the
theoretical equations served to validate each other and prove that both were acceptable
methods for determining the V-field strength inside of various axisymmetric capacitors. The GIS
dead-end took it a step further and showed that even with large dissimilarities to the ideal
shape the theoretical approximations yield accurate results. The analysis of the maximum E-
field allowed us to determine the exact value for breakdown voltage, at which point the
dielectric fails. Finally, with a safety factor of 25%, the Voperate came out to be 142.2 kV.
15. - 14 -
Reference
P. Pedrow. EE331. Class Lecture, Topic: "Dielectrics and Capacitors." SH 7, Doctorate of
Engineering, Washington State University, Pullman, Washington, Apr. 22, 2015.
16. - 15 -
Appendix A (Additions for Spherical)
Below is the MATLAB code used to plot the theoretical vs analytical V-field to radius:
function V = sphere_potential(r)
if r > 0.05 && r < 0.1
V = 5000*((1/r)-10)/10;
else
V = 0;
end
r = 0 : 0.00333 : 0.5;
V = zeros(length(r));
for jr = 1 : length(r)
V(jr) = shpere_potential(r(jr));
end
fileID = fopen('voltage_data1.txt','r');
spec = '%f %f';
sizeV_femm = [2 Inf];
V_femm = fscanf(fileID,spec,sizeV_femm)';
plot(r,V,'g',V_femm(1:150,1),V_femm(1:150,2),'--r')
title('Potential (V) vs Radius(r)')
legend('Theoretical','FEMM Simulation')
xlabel('r')
ylabel('V')
17. - 16 -
Appendix B (Additions for Cylindrical)
MATLAB code:
function V = cylinder_potential(r)
if r > 0.05 && r < 0.1
V = 5000*log(0.1/r)/log(2);
else
V = 0;
end
r = 0 : 0.00333 : 0.5;
V = zeros(length(r));
for jr = 1 : length(r)
V(jr) = cylinder_potential(r(jr));
end
fileID = fopen('voltage_data2.txt','r');
spec = '%f %f';
sizeV_femm = [2 Inf];
V_femm = fscanf(fileID,spec,sizeV_femm)';
plot(r,V,'g',V_femm(1:150,1),V_femm(1:150,2),'--r')
title('Potential (V) vs Radius(r)')
legend('Theoretical','FEMM Simulation')
xlabel('r')
ylabel('V')
18. - 17 -
Appendix C (Additions for GIS)
MATLAB code:
function V = cylinder_potential(r)
if r > 0.05 && r < 0.1
V = 5000*log(0.1/r)/log(2);
else
V = 0;
End
function V = sphere_potential(r)
if r > 0.05 && r < 0.1
V = 5000*((1/r)-10)/10;
else
V = 0;
End
r = 0 : 0.0001 : 0.15;
V_sphere = zeros(length(r));
V_cyl = zeros(length(r));
for jr = 1 : length(r)
V_sphere(jr) = sphere_potential(r(jr));
end
for ir = 1 : length(r)
V_cyl(ir) = cylinder_potential(r(ir));
end
file1ID = fopen('voltage_data3_horizontal.txt','r');
file2ID = fopen('voltage_data3.txt','r');
spec = '%f %f';
sizeV_femm_sphere = [2 Inf];
sizeV_femm_cyl = [2 Inf];
V_femm_sphere = fscanf(file2ID,spec,sizeV_femm_sphere)';
V_femm_cyl = fscanf(file1ID,spec,sizeV_femm_cyl)';
plot(r,V_sphere,'c',V_femm_sphere(1:150,1),V_femm_sphere(1:150,2),'--
r',r,V_cyl,'g',V_femm_cyl(1:150,1),V_femm_cyl(1:150,2),'b--')
title('Potential (V) vs Radius(r)')
legend('Theoretical Sphere','FEMM Simulation Sphere', 'Theoretical Cylincer',
'FEMM Cylinder')
xlabel('r')
ylabel('V')
