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FasterCap and FastCap2

Sphere-plane capacitance

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[quote][i]Originally posted by Sergio[/i]
[br]Hello

I am trying to compare the results obtained using FASTCAP2 and the exact analytical solution for the capacity between an infinite plane and a sphere. I have some problems in order to reproduce the exact solution.

First, I create the plane capacitor using the cubegen.exe utility and I subtract all the edges of the cube except one. i.e using

cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr>plane.txt

I create a plane of 100*100.

Using

spheregen.exe �r1>sphere.txt

I create a sphere of radius 1m separate 3m (xh) from the plane.

I run Fastcap2 using \epsilon=1 and I obtain an error.

Warning - capacitance matrix is not strictly diagonally dominant due to row 2

CAPACITANCE MATRIX, nanofarads
                        1          2 
1%GROUP1      1      2.922    -0.6699
sphere%GROUP1 2    -0.6699     0.2225

Ok, the problem is solved if I use more panels to describe the plane i.e
cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr �n10>plane.txt

CAPACITANCE MATRIX, nanofarads
                        1          2 
1%GROUP1      1      2.144    -0.1255
sphere%GROUP1 2    -0.1255     0.1305

The results are different from the expression obtained using the image charges method.
Cimage= 1.3007e-010

I can understand that in the simple image charges method I do not take intro account the edges of the plane and other finite size effects. The problem appears when I try to refine the plane, using

cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr �n20>plane.txt

the result is

CAPACITANCE MATRIX, nanofarads
                        1          2 
1%GROUP1      1      2.172    -0.1434
sphere%GROUP1 2    -0.1434     0.1482

Total time: 0.078
Total memory allocated: 2152 kilobytes

And for different n,
n=30
CAPACITANCE MATRIX, nanofarads
                        1          2 
1%GROUP1      1      2.182    -0.1499
sphere%GROUP1 2    -0.1499     0.1548

n=40
CAPACITANCE MATRIX, nanofarads
                        1          2 
1%GROUP1      1      2.185    -0.1519
sphere%GROUP1 2    -0.1519     0.1566

and so on, what am I doing wrong? The capacity is increasing with n, why?

Thank you in advance

Sergio
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T O P I C R E V I E W
Sergio	Posted - Jan 16 2013 : 18:27:58 Hello I am trying to compare the results obtained using FASTCAP2 and the exact analytical solution for the capacity between an infinite plane and a sphere. I have some problems in order to reproduce the exact solution. First, I create the plane capacitor using the cubegen.exe utility and I subtract all the edges of the cube except one. i.e using cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr>plane.txt I create a plane of 100*100. Using spheregen.exe �r1>sphere.txt I create a sphere of radius 1m separate 3m (xh) from the plane. I run Fastcap2 using \epsilon=1 and I obtain an error. Warning - capacitance matrix is not strictly diagonally dominant due to row 2 CAPACITANCE MATRIX, nanofarads 1 2 1%GROUP1 1 2.922 -0.6699 sphere%GROUP1 2 -0.6699 0.2225 Ok, the problem is solved if I use more panels to describe the plane i.e cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr �n10>plane.txt CAPACITANCE MATRIX, nanofarads 1 2 1%GROUP1 1 2.144 -0.1255 sphere%GROUP1 2 -0.1255 0.1305 The results are different from the expression obtained using the image charges method. Cimage= 1.3007e-010 I can understand that in the simple image charges method I do not take intro account the edges of the plane and other finite size effects. The problem appears when I try to refine the plane, using cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr �n20>plane.txt the result is CAPACITANCE MATRIX, nanofarads 1 2 1%GROUP1 1 2.172 -0.1434 sphere%GROUP1 2 -0.1434 0.1482 Total time: 0.078 Total memory allocated: 2152 kilobytes And for different n, n=30 CAPACITANCE MATRIX, nanofarads 1 2 1%GROUP1 1 2.182 -0.1499 sphere%GROUP1 2 -0.1499 0.1548 n=40 CAPACITANCE MATRIX, nanofarads 1 2 1%GROUP1 1 2.185 -0.1519 sphere%GROUP1 2 -0.1519 0.1566 and so on, what am I doing wrong? The capacity is increasing with n, why? Thank you in advance Sergio
2 L A T E S T R E P L I E S (Newest First)
Sergio	Posted - Jan 18 2013 : 15:36:08 Thanks you for your complete answer I can see that all my problems appear when I use the �o option in the cubegen.exe. This option confused me and I created a 5050 plane instead of a 100100 plane. Ok Obviously I forgot the discretization of the sphere as well.. Thanks again. Sergio
Enrico	Posted - Jan 18 2013 : 10:45:47 There are two issues here. - The first one is the discretization - The second one is the distance of the sphere from the gnd plane Discretization, part 1: FastCap2 is NOT able to automatically refine the discretization. This is completely different from FasterCap, that can refine the discretization in an intelligent way until convergence of the results is reached. So with the 'basic' cubegen.exe output cubegen.exe �o �xh3 �yh50 �zh50 �t �b �pfr �pbl �pbr > plane.txt you get an almost NON-refined plane. Therefore the first results from FastCap2 are almost not meaningful. You already understood this point and started to refine the ground plane. However you made the gnd plane so large w.r.t. the sphere that the dimension of the panels composing the plan is still large if compared with the panels composing the sphere when you use the '-n20' option in cubegen.exe. You may remember from FastCap2 documentation that you should go on refining the discretization until the difference between the results converge within 1%, which is the declared solver accuracy. So you correctly go on refining the plane and you observe that the capacity increase, however the relative difference between the results is converging: even looking at the single element and at the Frobenius norm of the matrix, you see the convergent sequence 130.5 pf -> 148.2 pF -> 154.8 -> 156.6, where the relative difference is 12% -> 4.2% -> 1.1% So why the result is still not what you expect? Distance: The cubegen.exe option you used include the '-o' parameter. This means that the generated parallelepiped is centered at zero; therefore, the plane is displaced along the x-axis of 1.5 m, not 3.0 m The sphere is generated around the origin. Since I reverse-engineer from the FastCap2 results that the list file you used was like the following one: * Sphere over GND plane * C sphere.txt 1.0 0.0 0.0 0.0 C plane.txt 1.0 0.0 0.0 0.0 the actual distance of the center of the sphere from the plane is 1.5 m, not 3 m If you correct the list file to: * Sphere over GND plane * C sphere.txt 1.0 -1.5 0.0 0.0 C plane.txt 1.0 0.0 0.0 0.0 you get the following result: CAPACITANCE MATRIX, nanofarads 1 2 sphere%GROUP1 1 0.1258 -0.1176 1%GROUP2 2 -0.1176 2.147 getting closer, but still not there. Discretization, part 2: You have addressed so far only an increase in the discretization of the gnd plane. Consider however that also the sphere discretization is very crude. Let's re-create the sphere with an additional level of discretization: spheregen.exe �r1 -d2 > sphere.txt Now FastCap2 gives: CAPACITANCE MATRIX, nanofarads 1 2 sphere%GROUP1 1 0.1316 -0.123 1%GROUP2 2 -0.123 2.152 Which is quite close to the theorical value (by the way, the value I get for the problem is about 133.6 pF, unless I am mistaken) Now if you use FasterCap, you can direclty start with the non-refined gnd plane and the additionally-refined sphere to get (setting the '-d' parameter to 1 or lower, since the geometrical ratios with so large a gnd plane are high): Capacitance matrix is: Dimension 2 x 2 g1_sphere 1.32894e-010 -1.28555e-010 g2_1 -1.24708e-010 2.08375e-009 You can also see in the following picture the refinement produced by FasterCap:

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