DEMO_XY / gallery 16: Electrostatic quadrupole. Verion 2, defined in radial coordinates. Mesh size: 401 x 401 nodes.

Running program

Windows:
DEMO_XY.exe 16

Linux:
./DEMO_XY.run 16

Problem definition

Problem drafted in books about electron optics (multiple symmetry lenses).

Numeric meshes

Each pixel represents one mesh point.

colour white - vacuum
colour red   - casing     1 (U = +0.0 [V])
colour green - electrodes 2 (U = +1.0 [V])
colour blue  - electrodes 3 (U = -1.0 [V])

Visualization of mesh on picture.

Model parameters

mesh size:

number of rows     ==  401
number of columns  ==  401

x ==  -200.0 ... +200.0 [mm]
y ==  -200.0 ... +200.0 [mm]

number of iterations

>>> SUCCESS - solution V(x,y) has been found
after 27264 iterations

pi constant:
M_PI ==  +3.14...


electrode type 1 (casing, red)
boundary ==  +1
voltage  ==  +0.0   [V]
x_center ==  +0.0   [mm]
y_center ==  +0.0   [mm]
radius   >=  +190.0 [mm]


electrodes type 2 (2 elements, green)
boundary   ==  +2
voltage    ==  +1.0  [V]
radius     <=  +20.0 [mm]
N          ==  +4 /* quadrupole */
alpha_zero ==  +0.25 * M_PI [rad] /* angle = +45 degrees */
R          ==  +50.0 [mm] 
coordinates of centres of electrodes type 2:
1/2	
	n = +0	
	x_1 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_1 = R * sin(alpha_zero + 2 * n * M_PI / N);
2/2
	n = +2	
	x_3 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_3 = R * sin(alpha_zero + 2 * n * M_PI / N);

	
electrodes type 3 (2 elements, blue)
boundary   ==  +3
voltage    ==  -1.0  [V]
radius     <=  +20.0 [mm]
N          ==  +4 /* quadrupole */
alpha_zero ==  +0.25 * M_PI [rad] /* angle = +45 degrees */
R          ==  +50.0 [mm] 
coordinates of centres of electrodes type 3:
1/2
	n = +1	
	x_2 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_2 = R * sin(alpha_zero + 2 * n * M_PI / N);
2/2	
	n = +3	
	x_4 = R * cos(alpha_zero + 2 * n * M_PI / N);
	y_4 = R * sin(alpha_zero + 2 * n * M_PI / N);


computation accuracy:
eps ==  1.0E-9 [V]

Computation results

Equipotential lines

Mapping of electrostatic potential [V] to colours

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of component x of electric force (E_x) [V/mm] to colours

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of component y of electric force (E_y) [V/mm] to colours

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of electric force (E) [V/mm] to colours

algorihm of computation:
E = sqrt(E_x*E_x + E_y*E_y)

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

Mapping of square of electric force (E*E == E2) [V2/mm2] to colours

algorithm of computation:
E*E = E_x*E_x + E_y*E_y

Important: determinig of exact values of E near electrode surface is generally problematic. These maps have generally illustrative character.

Colormap: grayscale

Colormap: grayscale inverted

Colormap: hot-to-cold

Colormap: hot-to-cold inverted

Colormap: jet

Colormap: jet inverted

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