eee 431 computational methods in electrodynamics
DESCRIPTION
EEE 431 Computational Methods in Electrodynamics. Lecture 17 By Dr. Rasime Uyguroglu [email protected]. Charged Conducting Plate. Moment Method Solution. Charged Conducting Plate/ MoM Solution. - PowerPoint PPT PresentationTRANSCRIPT
3
Charged Conducting Plate/ MoM Solution
Consider a square conducting plate 2a meters on a side lying on the z=0 plane with center at the origin.
x
y
z
2a
2a
2b
2b
sn
4
Charged Conducting Plate/ MoM Solution
Let represent the surface charge density on the plate.
Assume that the plate has zero thickness.
( , )y x
5
Charged Conducting Plate/ MoM Solution
Then, V(x,y,z):
Where;
0
1 ( ', ', ')( , , ) ' '
4
a a
a a
x y zV x y z dx dy
R
1/ 22 2 2( ') ( ')R x x y y z
6
Charged Conducting Plate/ MoM Solution
Integral Equation: When
This is the integral equation for
, , 0, ( , , ) ( .)x a y a z V x y z V const
0 2 2
( ', ')4 ' '
( ') ( ')
a a
a a
x yV dx dy
x x y y
7
Charged Conducting Plate/ MoM Solution
Method of Moment Solution: Consider that the plate is divided into N
square subsections. Define:
And let:
1
0n
nm
on Sf
on all other S
1
( , )N
n nn
x y f
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Charged Conducting Plate/ MoM Solution
Substituting this into the integral equation and satisfying the resultant equation at the midpoint of each , we get:
( , )m mx y
mS
1
, 1, 2,3,...,N
m mn nn
V A m N
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Charged Conducting Plate/ MoM Solution
Where:
is the potential at the center of
due to a uniform charge density of unit amplitude over
1/ 22 2
0
1' '
4 ' 'n n
mn
x ym m
A dx dyx x y y
mnAmS
nS
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Charged Conducting Plate/ MoM Solution
Let : denote the side length of each
the potential at the center of
due to the unit charge density over its own surface.
22
ab
N nS
nnA nS
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Charged Conducting Plate/ MoM Solution
So,
2 20
0
0
1
4
2ln(1 2)
2(0.8814)
b b
nn
b b
A dx dyx y
b
b
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Charged Conducting Plate/ MoM Solution
The potential at the center of can simply be evaluated by treating the charge over as if it were a point charge, so,
mS
nS
2
1/ 22 20 0
4 ( ) ( )
nmn
mn m n m n
S bA m n
R x x y y
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Charged Conducting Plate/ MoM Solution
The capacitance:
1 1
1( , )
1 1( )
a a
a a
a a a aN N
n n n nn na a a a
qC dx dy x y
V V
C dx dy f f dxdyV V
1
1
1 1 1
1 1
1 N
n n mn nn mn
N N
n mn mnm m
C S A SV
A V A V V A
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Charged Conducting Plate/ MoM Solution
The capacitance (Cont.):
Number of sub areas
C/2a
Approx.
C/2a
Exact
1 31.5 31.5
9 37.3 36.8
16 38.2 37.7
36 39.2 38.5
mnA mnA
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Charged Conducting Plate/ MoM Solution
Harrington, Field Computation by Moment Methods
The charge distribution along the width of the plate
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Moment Method/ Review
Consider the operator equation:
Linear Operator. Known function, source. Unknown function. The problem is to find g from f.
...........................................................(1)Lf g
:L:g:f
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Moment Method/ Review
Let f be represented by a set of functions
scalar to be determined (unknown expansion coefficients.
expansion functions or basis functions.
1
.................................................(2)N
i ii
f f
1 2 3,..., ,f f f
i
if
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Moment Method/ Review
Now, substitute (2) into (1):
Since L is linear: 1
N
i ii
L f g
1
.....................................(3)N
i ii
Lf g
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Moment Method/ Review
Now define a set of testing functions or weighting functions
Define the inner product (usually an integral). Then take the inner product of (3) with each and use the linearity of the inner product:
1 2, ,..., Nw w w
1
, , , 1, 2,....., .......(4)N
i i j ji
Lf w g w j M
jw
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Moment Method/ Review
It is common practice to select M=N, but this is not necessary.
For M=N, (4) can be written as:
................(5)A g
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Moment Method/ Review
Or,
1 1 1 2 1
2 1 2 2 2
1 2
, , ... ,
, , ... ,
. . ... .
. . ... .
, , ... ,
N
N
N N N N
w Lf w Lf w Lf
w Lf w Lf w Lf
A
w Lf w Lf w Lf
25
Moment Method/ Review
If is nonsingular, its inverse exists and .
Let
A
1A g
1 2
1
1
...... (1 )
................................(6)
N
N
i ii
f f f f XN
f f f
f f A g
26
Moment Method/ Review
The solution (6) may be either approximate or exact, depending upon on the choice of expansion and testing functions.
27
Moment Method/ Review
Summary: 1)Expand the unknown in a series of basis
functions. 2) Determine a suitable inner product and
define a set of weighting functions. 3) Take the inner products and form the matrix
equation. 4)Solve the matrix equation for the unknown.
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Moment Method/ Review
Inner Product:
Where:
, ,
, , ,
, 0
, 0
u v v u
u v w u w v w
u u if u
u u if u
, , :
, :
u v w functions
real numbers
30
Moment Method/ Review
If u and v are complex:
*, ,
, , ,
, 0
, 0
u v v u
u v w u w v w
u u if u
u u if u
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Moment Method/ Review
Example: Find the inner product of u(x)=1-x and
v(x)=2x in the interval (0,1). Solution: In this case u and v are real functions.