Test3d22.dat, 22nd 'test' file for CPO3DS

Spherical capacitor with dielectric in gap

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

The inner and outer conducting spheres have radii of 0.5 and 1mm respectively, and voltages of 1 and 0 respectively.

The dielectric between them has a dielectric constant of K = 2 and inner and has outer radii of 0.6 and 0.9.

The number of segments is 288.

The inaccuracy used for evaluation of the fields at the dielectric interfaces is 1E-5 (but see below).

The theoretical capacitance of a sphere of radius 1m is C = 4*pi*epsilon0 = 1.11265003E-10 farad = C0, say.

If the dielectric were to completely fill the gap the capacitance of the present system would be K*1E-3*C0.

It can be shown that for radii r1 and r2 of the conducting spheres and radii s1 and s2 of the dielectric interfaces, the capacitance is C0/( (1/r1 - 1/s1) + (1/s1 - 1/s2)/K +(1/s2 - 1/r2) ).

In the present case this gives 1.384615*1E-3*C0 = 1.540592E-13 farad.

4 reflection planes are used here, so the charge given by the program is 1/16 of the full charge.

Therefore for a voltage difference of 1V, the program charge should be 9.62870E-15 coulomb.

In fact the cumulative charge for the inner conducting sphere given by the program is 9.53202E-15 -this is the cumulative charge for segment 48 in the output file, in the 3rd (last, be careful) listing of charges. The error is therefore 1.0%.

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

When the dielectric is removed the capacitance is given with an error of 0.005% and in a computing time of 1.6s (as opposed to the time of 35s for the dielectric calculation).

This illustrates a general observation in the examples which we have tested and which involve curved electrodes: the dielectric calculations do not have the very high accuracy of the normal non-dielectric calculations.

Also, dielectric calculations with CPO3D are always much slower than the analogous non-dielectric calculations.

Furthermore, CPO3D/Di has been optimised to give the most accurate fields inside dielectrics (in preparation for the magnetic version) rather than being optimised for capacitance calculations.

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.

Note also that this inaccuracy is only loosely correlated with the obtained inaccuracy.

Here are the results of using different numbers of segments N of the inner sphere, dielectric constants K and 'dielectric interface' inaccuracies.

 N K inaccuracy time (sec) Error, % 48 2 1E-7 55.4 1.00 48 2 1E-6 42.2 1.00 48 2 1E-5 35.1 1.00 48 2 1E-4 29.5 0.99 48 2 1E-3 12.8 0.95 48 2 1E-2 6.4 1.07 48 2 1E-1 5.7 0.75 24 2 1E-5 10.3 1.94 96 2 1E-5 122 0.51 192 2 1E-5 451 0.27 48 10 1E-5 55.4 1.63

In this example the inaccuracy obtained is relatively insensitive to the requested inaccuracy of the dielectric evaluation, but is clearly improved by using more segments.