EERI 313 – Practical 1
Rian Engelbrecht
37252623
Contents
Introduction ........................................................................................................................... 2
Shape .................................................................................................................................... 2
Calculation ............................................................................................................................ 4
two triangles ...................................................................................................................... 4
Trapezoids .......................................................................................................................... 4
rectangle............................................................................................................................ 4
Semi-circle ........................................................................................................................ 4
Total equivalent capacitance .............................................................................................. 4
Total stored energy ............................................................................................................. 4
Analytical voltage calculation ............................................................................................. 5
First iteration .................................................................................................................. 5
Second iteration ............................................................................................................. 5
Third iteration.................................................................................................................. 5
FEMM simulation ................................................................................................................... 6
FDM (excel) ............................................................................................................................ 7
Comparison........................................................................................................................... 8
References ............................................................................................................................ 8
Introduction
In this practical we must design a unique capacitor shape. We will be using computer
aided simulation (femm), finite difference method (FDM) and analytical calculations to
compare the different methods comparison. We must then look at how each method
complements each other.
The reason we do this comparison is to understand and concept of capacitor and how
the design will influence the end results. We must also search and learn about
capacitors like super capacitors in electric vehicles. Super capacitors use Barium
titanate(BaTiO3) and Barium zirconium oxide (BaZrO3) as their dielectric.
Shape
• The first part of the shape of the capacitor will exist out of two triangles.
Figure 1: Part one of capacitor.
• The second part will exist out of two trapezoids where one is on top of the other
on the bottom their long side points to each other it has 7.2mm from top to
bottom.
Figure 2: Part two of capacitor.
• The third part will be made from a rectangle. It is 5.1mm from top to bottom.
Figure 3: Part three of capacitor.
• The final part will be made from a half circle.
Figure 4: Part four of capacitor.
Calculation
For the calculation I used the length of the capacitor as 1 meter
two triangles
The capacitors between the two parts will be zero because there is no voltage drop between the
top of the triangle and the bottom part.
C1=0
Trapezoids
𝜖0 ∗ 𝐴
𝐶2 =
𝑑
𝜖0 ∗ 12.5 ∗ 10−3
𝐶2 =
8.5 ∗ 10−3
𝐶2 = 1.302 ∗ 10−11
rectangle
𝜖0 ∗ 𝐴
𝐶3 =
𝑑
𝜖0 ∗ 9.5 ∗ 10−3
𝐶3 =
6.5 ∗ 10−3
𝐶3 = 1.29 ∗ 10−11
Semi-circle
The capacitors between the two parts will be zero because there is no voltage drop between the
top of the semi-circle and the bottom part.
Total equivalent capacitance
1 1
𝐶𝑒𝑞 = ( + )−1
𝐶2 𝐶3
𝐶𝑒𝑞 = 6.49 ∗ 10−12
Total stored energy
𝑊 = 0.5𝐶𝑉 2
𝑊 = 3.245 ∗ 10−12
Analytical voltage calculation
A
B C
First iteration
1+1+1+0
𝑉𝑎1 = = 0.75V
4
0 + 0.75
𝑉𝑏1 = = 0.375𝑉
2
0.375 ∗ 2 + 0 + 1
𝑉𝑐1 = = 0.44𝑉
4
0.875+0
𝑉𝑑1 = = 0.4375V
2
0.4375
𝑉𝑒1 = = 0.1093V
4
Second iteration
1 + 1 + 1 + 0.375
𝑉𝑎2 = = 0.84𝑉
4
0.44 + 0.84
𝑉𝑏2 = = 0.64𝑉
2
0.64 ∗ 2 + 0.44 + 1
𝑉𝑐2 = = 0.68𝑉
4
0.875+0.11
𝑉𝑑2 = = 0.395V
2
0.4375
𝑉𝑒2 = = 0.099V
4
Third iteration
1+1+1+0.64
𝑉𝑎3 = = 0.91V
4
0.91+0.68
𝑉𝑏3 = = 0.795V
2
0.795 ∗ 2 + 0.68 + 1
𝑉𝑐3 = = 0.82𝑉
4
0.82 + 0.099
𝑉𝑑3 = = 0.458𝑉
2
0.4375
𝑉𝑒3 = = 0.115𝑉
4
FEMM simulation
The first thing I did for the simulation was to create the design. I the put the material properties
and boundaries in the design. The voltage of 1 volt was put on the top and a voltage of 0 volts
was put at the semi-circle. I put an electric permittivity inside the design.
After the design was finished, I simulated the design. I then use the display option to display the
equipotential lines.
FDM (excel)
In this figure of the simulation there was a voltage of 1 Volts put on the triangular and the flat
space between the triangular. For the low voltage of 0 Volts was put on the semi-circle.
Everything else is dielectric.
Using the FDM I was able to calculate the voltages throughout the dielectric and with this I was
able to get the electrostatic fields in the capacitor.
The calculation is used to calculate each node voltage:
𝐴4 + 𝐶4 + 𝐵3 + 𝐵5
𝐵4 =
4
Comparison
The result of the FEMM simulation and that of the FDM excel gave a very close result to each
other. The result where very close to each other.
The Va3 of 0.91V of the calculation is close to the 9.5 volts that we had by the simulation.
The Vb3 of 0.795V of the calculation is close to the 8.1 volts that we had by the simulation.
The Vc3 of 0.82V of the calculation is close to the 8.1 volts that we had by the simulation.
The Vd3 of 0.458V of the calculation is close to the 0.45 volts that we had by the simulation.
The Ve3 of 0.115V of the calculation is close to the 0.1 volts that we had by the simulation.
By looking at the results we can see that the FEMM simulation, FDM excel, and analytical
calculations give an accurate result that is close to each other. The analytical calculations can
be made more accurate if we put in more iteration.
References
[ “Electronics Tutorials Logo,” [Online]. Available: https://www.electronics-
1 tutorials.ws/capacitor/cap_1.html#:~:text=The%20generalised%20equation%20for%20the,
] 10%2D12%20Farads%20per%20metre..
[ D. j. Griffiths, Introductions to electrodynamics.
2
]
[ S. Jain, “Hi-k dielectric ceramic material for supercapacitors,” 14 september 2015. [Online].
3 Available: https://www.hilarispublisher.com/proceedings/hik-dielectric-ceramic-material-
] for-supercapacitors-
4036.html#:~:text=Barium%20titanate%20(BaTiO3)%20and%20barium,cost%20of%20the%
20ceramic%20material..
