Test3d23.dat, 23rd 'test' file for CPO3DS

Parallel plate capacitor with dielectric in gap

Care must be taken in setting up dielectric simulations, so it is very important to read the guidance.

A circular disc of radius 0.5mm is at z = -0.1mm, at -0.5V. A similar disc is at z = +0.1mm, at +0.5V. The number of segments in each of these discs is 96.

The textbook formula for the capacitance of this parallel plate capacitor is K*(A/d)*epsilon0, where K is the dielectric constant of the medium between the plates and A and d are the area and spacing. In the present case this gives 3.477031E-14 farad.

However in practice this is an under-estimate (see also the remarks at the end), because it ignores the extra charges on the inside surfaces caused by the fringe fields at the edges and it also ignores the charges on the outer surfaces (which are significant near the edges).

Therefore in this benchmark test we have added surrounds of radius 1mm to the inner discs. We have also added electrodes at z = +/-0.2mm to create thick plates. In this test we shall only be concerned with the charges on the inner discs.

A dielectric medium (K = 2) has been added between z = -0.9999 and +0.9999mm (the distances from the conducting surfaces are then almost at the minimum that the program will allow). The total number of segments is 608.

The inaccuracy used for evaluation of the fields at the dielectric interfaces is 0.001 (but see below).

In a subsidiary test the dielectric was removed and the cumulative charge on the inner disc was found to be 4.3352319E-15 -this is the cumulative charge for segment 96 in the output file, in the 3rd (last) listing of charges. This is 1/8 of the full charge because the x, y and x=y reflection planes have been used. It is 0.25% smaller than the texbook result 4.346288E-15.

When the dielectric medium is included the effective value of K is (d1 + d2)/(d1 + d2/K), where d1 and d2 are the depths of the combined air gaps and the dielectric respectively. In the present case d1 = 0.0002, d2 = 0.1998, K = 2, so the effective constant is 1.998 and the expected charge is 8.683892E-15. The program gives 8.6611945E-15, which is too small by 0.26%.

When K = 10 the error is approximately the same, as it is when the z values of the dielectric interfaces are changed to +/-0.05.

Higher accuracy can of course be obtained by using more segments.

The computing times for the dielectric calculations are typically approximately 10 times longer than those for analogous non-dielectric calculations.

Finally we consider the capacitance of the complete system, of radius 1mm and with thick electrodes, without the dielectric medium. The cumulative charge for all the segments at -0.5V is found to be 2.4363385E-14. The texbook formula (allowing for reflection planes) gives 1.7385E-14, which is smaller by 29%! The textbook correction for the fringing fields (which seems to take into account only the additional charges on the inner faces, neglecting those at the edges themselves and on the outer faces) is to add d/pi to the radius, but this improves the result by only 8%.

The inaccuracy used for evaluation of the fields at the dielectric interfaces can be selected by the user (at the bottom of the 'tracing accuracy' sheet). The default value is 0.001.

Note that this inaccuracy is not the same as the inaccuracy used in calculating the surface charges of the conducting electrodes, which is always 1.E-7 and which cannot usually be changed by the user.

Here are the results of using different 'dielectric interface' inaccuracies, still using only 608 segments:

 inaccuracy time (sec) error, % 0.1 27 -33 0.01 29 -26 0.003 43 -0.27 0.001 43 -0.26 0.0001 130 -0.23 0.00001 168 -0.22

The default value in CPO3D for this inaccuracy is 0.001, but it might sometimes be better to change this to 0.0001 (at the bottom of the 'tracing accuracy' sheet).