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 Mutual Capacitance Between Cubes

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T O P I C    R E V I E W
teny Posted - May 16 2023 : 13:55:51
Dear Enrico,
I used FasterCap to calculate seven cubes with a spacing of 2m in the x direction. The size of the cubes is 1m1m1m. The calculation results are as follows:
g1_3 8.34737e-011 -2.51482e-011 -5.00025e-012 -2.70331e-012 -1.84663e-012 -1.43414e-012 -1.48831e-012
g2_3 -2.51225e-011 9.18812e-011 -2.36591e-011 -4.24007e-012 -2.21896e-012 -1.52494e-012 -1.43412e-012
g3_3 -4.99838e-012 -2.36466e-011 9.21686e-011 -2.35392e-011 -4.17789e-012 -2.22075e-012 -1.84705e-012
g4_3 -2.70319e-012 -4.24386e-012 -2.35205e-011 9.22219e-011 -2.35431e-011 -4.23989e-012 -2.70507e-012
g5_3 -1.84709e-012 -2.21965e-012 -4.18008e-012 -2.35267e-011 9.21672e-011 -2.36626e-011 -4.99653e-012
g6_3 -1.4344e-012 -1.52521e-012 -2.2215e-012 -4.23977e-012 -2.36522e-011 9.18832e-011 -2.51258e-011
g7_3 -1.48883e-012 -1.43396e-012 -1.8484e-012 -2.70656e-012 -4.99836e-012 -2.51258e-011 8.34625e-011

1.I have a question as to why the mutual capacitance between the first cube and the sixth cube is smaller than that between the first cube and the seventh cube. According to the formula, shouldn't the capacitance decrease as the distance increases?

* conductor | dielectric | offset
* name | constant | in space
*
C cube3.qui 3.9000000 0.0 0.0 0.0
C cube3.qui 3.9.000000 2.0 0.0 0.0
C cube3.qui 3.9.000000 4.0 0.0 0.0
C cube3.qui 3.9.000000 6.0 0.0 0.0
C cube3.qui 3.9.000000 8.0 0.0 0.0
C cube3.qui 3.9.000000 10.0 0.0 0.0
C cube3.qui 3.9.000000 12.0 0.0 0.0

I have another question to ask. When these seven cubes are placed in the same dielectric constant, will the mutual capacitance between Cube 1 and Cube 3 be affected by the dielectric constant of the middle cube? Looking at the formula, compared with the capacitance in a dielectric constant of 1, has the capacitance in a dielectric constant of 3.9 increased by 3.9 times?

Best Regards

Teny


3   L A T E S T    R E P L I E S    (Newest First)
Enrico Posted - May 21 2023 : 12:56:12
Dear Tony

1) there is mutual capacitance but the value is very small. In a numerical sense (as every simulation you perform with a computer has numerical rounding and approximations) the error is below the set threshold for accuracy. If you were to solve the problem analytically, you could get a precise value. Now, solving analytically this problem is very challenging but you can refer, for the sake of learning, to a similar problem of a series of spheres. This can be analytically solved, refer e.g. to "Dissertation About Single-Electron Devices and Circuits", C. Wasshuber, Universitšt Wien, Jan 1997, appendix A

2) this is the capacitive shielding effect of the middle cube. Again, you may play with the analytical formulae to better understand what's happening from a physical perspective.

3) if that's the result you get, it is highly inaccurate. There may be multiple reasons - depends on the software, on the solution method, how you imposed the conditions, etc.

Best,
Enrico
teny Posted - May 18 2023 : 05:24:18
Dear Enrico,

Thank you for the prompt reply.
1.' Consider that the last cube is really shielded so the interaction is very weak',How should this sentence be understood? Is the last cube too far away and there is no mutual capacitance with the first cube?

2. I conducted another experiment.
* conductor | dielectric | offset
* name | constant | in space
*
C cube3.qui 1.000000 0.0 0.0 0.0
C cube3.qui 1.000000 2.0 0.0 0.0
C cube3.qui 1.000000 4.0 0.0 0.0
The calculation results are as follows:
g1_3 8.31687e-011 -2.54972e-011 -6.31038e-012
g2_3 -2.54635e-011 9.137e-011 -2.54848e-011
g3_3 -6.31074e-012 -2.5497e-011 8.31694e-011

Then I calculated the results under this condition#65306;
* conductor | dielectric | offset
* name | constant | in space
*
C cube3.qui 1.000000 0.0 0.0 0.0
C cube3.qui 1.000000 4.0 0.0 0.0

The calculation results are as follows:
g1_3 7.36914e-011 -1.18839e-011
g2_3 -1.18838e-011 7.36914e-011
My question is why, when the spacing between two cubes is 4, inserting a cube in the middle creates such a large difference in their mutual capacitance. How can this be explained from the perspective of physical mechanisms?

3.I compared the FasterCap results with those from other commercial software. When the cubes were embedded in a dielectric constant of 3.9 environment, the mutual capacitance between the first and third cubes only increased by 1.6 times, not 3.9 times. Is this because the algorithm used in the commercial software is not accurate enough?

Best Regards

Teny
Enrico Posted - May 17 2023 : 17:54:56
Hi Tony,

1. yes it should decrease; however consider that the error that you observe is 1.43e-12 - 1.488e-12 = 5.8e-14 that, with respect to 8.347e-11, about 0.07%. FasterCap (or FastCap) usually guarantee an accuracy of 1%. You may get better accuracies of course, but in this case you already get a very good result. Consider that the last cube is really shielded so the interaction is very weak; numerical precision of the algorithms here may not calculate with high accuracy the second decimal digit of the last mutual capacitance. I find it quite reasonable.

2. I'm not sure I get your question. However if you embed all your cubes in a dielectric medium with relative permittivity of 3.9, yes you get an increase in capacitance of 3.9.

Best,
Enrico

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