Capacitance not being calculated correctly for ideal case, in center

[uladkasach]

Capacitance not being calculated correctly for ideal case, in center

[uladkasach]

Discovered the Doubling Nz between plates halves C -> doubles V. This should not be occurring.

[uladkasach]

Voltage responds properly to doubling Nz. When Nz is doubled, voltage is halved at each step. The "discovered doubling Nz results in halfs C" may be a product of the electric field being disproportionately high near the plate.

For example:

• for case of 500 elements in z direction between, max E_z value is 8E23
  
• while for case of 125 elements in z direction between, max E_z value is 5E22.
  

Perhaps this is a factor of currently setting the plate width to be zero, resulting in the plate only taking up one Z element point.

[uladkasach]

Adding thickness did not make an effect on the exponential increase. This may be a case of the plates not having enough area.

Better illustration of exponential increase, the Y axis is logarithmic:

[uladkasach]

Adding plate area does not fix the problem either; at a ratio of 1:100,000, d:sqrt(A) the same problem exists.

The problem is that as Nz increases, the number of elements in the extreme range as r - r' goes to zero increases - impacting the summation V = integral E dot dl.

[uladkasach]

This is a product of discretization/numerics. Because the model requires the plate to have some thickness ( a minimum of Dz = Lz/Nz ) and the steps at which E is calculated are at intervals of Dz away - When z ~ (r_z - r'_z) the Electric field explodes. By idea the model should be doing well in the center but breaking down towards the boundaries - like in fluid mechanics models.

There are two options to consider at this point:

• Increase the step size in z direction when calculating potential
  • e.g., set plate_thickness = 0, double Nz, double steps in z per step
  • this could keep z >> (r_z - r'_z)
• Explicitly calculate V
  • since we know we are dealing with finite objects V = int rho(r') * dtau / |r - r'| applies
  • V should be less sensitive to the discretization
    • as we get closer to a charge the position we are at should "see" less charges
    • the "blow up" should be much smaller
      - E proportional to (r - r')/|(r - r')|^3
      - V proportional to 1/|(r - r')|

[uladkasach]

Option of increasing step size in z direction does not help. When both Nz and Z/step are doubled, the physical distance from plate to dl is constant. If they are not both doubled, then essentially we are simply manipulating Nz on its own.

[uladkasach]

For setup of plate_lx = plate_ly = 4.9m and plate_seperation = 0.005, C = e_0*A/d = 4.24977E-8

N_xy	Nz	N_plate_thickness_1	C_1	N_plate_thickness_2	C_2
100	100	1	6.9251 E-11	3	6.945 E-11
100	200	1	3.4188 E-11	3	3.439 E-11
100	400	1	1.6989 E-11	n/a	n/a
100	800	1	8.46 E-12	n/a	n/a
250	100	1	4.218 E-10	n/a	n/a
250	200	1	2.08 E-10	n/a	n/a
250	400	1	1.~ E-10	n/a	n/a

We can note that as Nz doubles, C is reduced by half.

We can also note that Nxy affects C.

More data at https://github.com/uladkasach/Finite-Capacitance/blob/master/z_dev_notes/issue_1/data_table.md

[uladkasach]

Data at above mentioned table shows that this method will actually create accurate results, but only when the N_xy is large enough (~8k). At this scale, issue #2 shows up.

[uladkasach]

Follow up demonstrating that Coefficient of Variance decreases as Nxy increases: