Xmpl2d20, 20th 'example' data file for CPO2D

Field and potential penetration through a flat mesh of parallel round wires.

One of a set of files:

test2d18.dat: Field penetration through a flat mesh of flat strips.

xmpl2d19.dat: Field penetration through a cylindrical mesh of round wires.

xmpl2d20.dat: Field penetration through a flat mesh of round wires.

xmpl3d23.dat: Field and potential penetration through a flat mesh of crossed round wires.

xmpl3d29.dat: Field and potential penetration through a flat mesh of crossed flat wires.

For further details see the papers:

Short and long range penetration of fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward Nucl. Instr. Meth. A427, 363-367 (1999).

Penetration of electrostatic fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward, Rev. Sci. Instrum. 69, 2000-2006 (1998).

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

The mesh is composed of 20 parallel wires (in planar symmetry) of radius 0.0125, spaced by 1.0. The mesh is at potential 0 and is enclosed by a box that has end potentials of 0 at z=-10 and +10 at z=+10 (but see below). The sides of the box at z>0 and z>0 are given uniform potential gradients.

The following quantities are obtained:

(1) v1 = the average potential at z = -r, where r is the radius of the wires

(2) v2 = the average potential at z = +r

(3) vc = the potential at the centre of the mesh (that is, at x=0, z=0)

(4) ec = the field at the centre of the mesh.

All these values are extrapolated to an infinite number of segments (see below).

For further details see the paper by F H Read, N J Bowring, P D Bullivant and R R A Ward, submitted to Rev Sci Instrum, to appear (hopefully) in 1998.

The mesh is at zero potential and fields are created on both sides of it by means of continuous end plates at a different potential. An extrapolation technique is used (see below) to simulate a mesh that is infinite in extent.

To simulate end plates at infinity, vl is applied to the plate at z=-10, 10+vr is applied to the plate at z=10, and va (=average of vl and vr) is applied to the mesh surround at z=0.

The resulting values of the potential v1 averaged over z=-r, and v2 averaged over z=r, are noted and then vl and vr are adjusted iteratively until v1=vl and v2=vr. The effective fields are then 0 and 1V/mm, as required.

vc and ec are the potential and field at the origin.

Results:

(The initial values of vl and vr are taken from a previous, less accurate study)

 r nseg vr vl va v1 v2 vc ec 0.0125 336 0.415 0.403 0.409 0.415618 0.403654 0.515885 -0.499872 0.0125 672 0.415 0.403 0.409 0.415477 0.403514 0.515732 -0.499869 0.0125 996 0.415 0.403 0.409 0.415446 0.403482 0.515697 -0.499865 0.0125 inf 0.415 0.403 0.409 0.40944 0.00598 0.51568 -0.49987 0.0250 336 0.317 0.294 0.306 0.317979 0.294992 0.406112 -0.498199 0.0250 672 0.317 0.294 0.306 0.317844 0.294861 0.405966 -0.498193 0.0250 inf 0.317 0.294 0.306 0.31780 0.29482 0.40592 -0.49819 0.0250 inf 0.317 0.294 0.306 0.30631 0.01149 0.40592 -0.49819 0.050 336 0.229 0.187 0.208 0.230638 0.188409 0.296401 -0.490302 0.050 672 0.229 0.187 0.208 0.230513 0.188296 0.296265 -0.490279 0.050 996 0.229 0.187 0.208 0.230486 0.188271 0.296237 -0.490272 0.050 inf 0.229 0.187 0.208 0.23047 0.18825 0.29622 -0.49027 0.050 inf 0.229 0.187 0.208 0.20936 0.02110 0.29622 -0.49027 0.100 336 0.088 0.158 0.123 0.159834 0.089805 0.186737 -0.457138 0.100 672 0.088 0.158 0.123 0.159704 0.089719 0.186616 -0.457062 0.100 996 0.088 0.158 0.123 0.159676 0.089699 0.186590 -0.457042 0.100 inf 0.088 0.158 0.123 0.15965 0.08968 0.18657 -0.45703 0.100 inf 0.088 0.158 0.123 0.12467 0.03499 0.18657 -0.45703

Summary of final results for potentials:

 r vm vd vc ec 0.0125 0.40944 0.00598 0.51568 -0.49987 0.0250 0.30631 0.01149 0.40592 -0.49819 0.050 0.20936 0.02110 0.29622 -0.49027 0.100 0.12467 0.03499 0.18657 -0.45703

Investigation of average and maximum values of field, and fourier components of potential:

e = exp(-2*pi*z)

r = 0.0125:

 z av*e max*e f1*e f2*e**2 f3*e**3 0 1.269 10.61 0.1513 -0.0716 0.045 -0.2 1.01894 1.39279 0.1588 -0.0792 0.053 -0.4 0.99948 1.08368 0.1588 -0.0790 0.052 -0.6 0.99766 1.01371 0.1587 -0.0773 0.016 -0.8 0.99644 1.02022 0.1583 -0.0600 -1.248 -1 0.99549 1.1042 0.1574 0.1128 -45.72

average should be 2*pi*f1 at large z.

Taking f1*e=0.1588, 2*pi*f1*e=0.9978, which is consistent with the values found for av*e.

Take f2*e**2=-0.079, f3*e**3=0.053, r = 0.025:

 z av*e max*e f1*e f2*e**2 f3*e**3 0.000E+00 9.926E-01 5.299E+00 1.437E-01 -6.402E-02 3.752E-02 -2.000E-01 2.876E-01 3.929E-01 1.576E-01 -7.783E-02 5.122E-02 -4.000E-01 8.031E-02 8.792E-02 1.576E-01 -7.761E-02 4.991E-02 -6.000E-01 2.282E-02 2.405E-02 1.574E-01 -7.522E-02 4.806E-04 -8.000E-01 6.498E-03 7.206E-03 1.569E-01 -4.975E-02 -1.837E+00 -1.000E+00 1.893E-03 2.500E-03 1.554E-01 2.225E-01 -6.965E+01

Take f1*e=0.1576, f2*e**2=-0.078, f3*e**3=0.051

r = 0.05:

 z av*e max*e f1*e f2*e**2 f3*e**3 0 0.7851 2.868 0.12815 -0.0487 0.02287 -0.2 0.97467 1.31725 0.15235 -0.0724 0.04573 -0.4 0.95788 1.05565 0.15227 -0.0721 0.04408 -0.6 0.95689 1.04234 0.15205 -0.0690 -0.02027 -0.8 0.96611 1.18192 0.15138 -0.0343 -2.525 -1 1.10527 1.71413 0.14913 0.357 -101.675

Take f1*e=0.1523, f2*e**2=-0.072, f3*e**3=0.046

Summary of fourier coefficients:

 r a10 a20 a30 0.0125 0.1588 -0.079 0.053 0.025 0.1576 -0.078 0.051 0.050 0.1523 -0.072 0.046

Compare with 3D results (xmpl3d23.dat):

 r a10 a20 a30 a11 0.0125 0.070 -0.037 0.025 0.028 0.025 0.067 -0.035 0.024 0.033 0.050 0.061 -0.031 0.020 0.040

Appendix A

Program used to deduce v1, v2, vc and ec from the results in the output data.File temp20a.dat:

-see first part of program 16.

Appendix B

Program used to derive fourier components of the potential, and average and maximum values of the field, from the results in the output data file (temp20a.dat), when the above file has the lines:

-see second part of program 16.