3.4 project 1, numerical analysis of charged conducting...

Lesson 10Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu

3.4 Project 1, Numerical Analysis of Charged Conducting Plates

Two parallel rectangular conducting plates with dimension 2Lx by 2Ly areseparated by 2d as shown in the figure. Assuming the plates are perfectconductors and Lx=Ly=L=10 mm,(1) Numerically calculate the capacitance C of the two plates with d=0.1L to 1L(d/L=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) and graphically present thecapacitance C numerically calculated and the C calculated by the formula ofvs. d/L.

d

SC !=

(2) Using the charge distribution calculated in (1) ford/L=0.5, calculate the potential distribution on thecentral cutting plane of y=0, for -3L <x< 3L, -2d < z <2d and graphically present the potential distribution. Itis assumed that the origin of the coordinate is locatedat the center of the two plates.

(3) Using the potential distribution,numerically calculate the E fielddistribution on the same plane using finitedifference scheme.


Letʼs consider one conducting plate with certain chargedistribution first.

Numerical Analysis of Charged Conducting Plates


We divide the plate into many small pieces so that the chargedistribution over each piece can be considered as uniform butwith different value from other pieces.



The electric potential at point P(x,y,z)contributed by cell i can be expressed as

iR

! !"

##=

iS

i ydxdzyxiR 4

1),,(

$%&'

),,( zyxP

where 222 )()()( iiii zzyyxxR !"+!"+!"=

y

x



The electric potential at point P(x,y,z)contributed by cells i, j, k, can be expressed as

iR

! !! ! ! !"" "

##+##+##=

ki j S

k

S S

ji ydxdydxdydxdzyxkji R 4

1

R 4

1

R 4

1),,(

$%&

$%&

$%&'

),,( zyxP

222 )()()( iiii zzyyxxR !"+!"+!"=

jR kR

222 )()()( jjjj zzyyxxR !"+!"+!"=

222 )()()( kkkk zzyyxxR !"+!"+!"=

y

x



Assuming there are N cells on the plate, theelectric potential at point P(x,y,z) contributed byall the cells can be expressed as

iR

! " "= #

$$=N

i S

i

i

ydxdzyx1 iR 4

1),,(

%&'(

),,( zyxP222 )()()( iiii zzyyxxR !"+!"+!"=

y

x



If there are total 2 conducting plates with 2 N number of cells, theelectric potential at point P(x,y,z) contributed by all the cells can beexpressed as

iR

! " "= #

$=N

i S

i

i

sdzyx2

1 i R4

1),,(

%&'(

),,( zyxP222 )()()( iiii zzyyxxR !"+!"+!"=



Consider the two square conducting plates of L by L mm2. Letrepresent the surface charge density on the plates (unknown in thismoment). The electrostatic potential at any point in space is

i!

The boundary condition for the capacitor problem is

surface top theon anywhere 2/V=!

! " "= #

$=N

i S i

i

i

sdR

zyx2

1 4

1),,(

%&'(



The integral equation for the problem is therefore

We satisfying the resultant equation at the midpoint ofeach cell, say for m-th cell, the midpoint is

By doing this, we obtain a set of 2N by 2Nlinear equations:

2/)()()( 4

12

1222

Vsdzzyyxx

N

i S iii

i

i

=!!"+!"+!"

# $ $= % &'(

),,(mmmzyx

where (x,y,z) is any point on the surface of the conducting plates.

2/)()()( 4

12

1222

Vsdzzyyxx

N

i S imimim

i

i

=!!"+!"+!"

# $ $= % &'(

m = 1, 2, 3, … 2N.



2/)()()( 4

12

12

1

2

1

2

1

Vsdzzyyxx

N

n S nnn

n

n

=!!"+!"+!"

# $ $= % &'(

2/)()()( 4

12

1222

Vsdzzyyxx

N

n S nNnNnN

n

n

=!!"+!"+!"

# $ $= % &'(

2/)()()( 4

12

12

2

2

2

2

2

Vsdzzyyxx

N

n S nNnNnN

n

n

!=""!+"!+"!

# $ $= % &'(

2/)()()( 4

12

12

1

2

1

2

1

Vsdzzyyxx

N

n S nNnNnN

n

n

!=""!+"!+"!

# $ $= % +++&'(

:

::

:



In a matrix form, there is

The matrix elements can beexpressed as

! !+

"

+

" #"+#"+#"##=

ax

ax

ay

ay nmnmnm

nnmn

n

n

n

nzzyyxx

ydxdl222 )()()(4

1

$%

The corresponding approximate capacitance of the two plates is

!=

"#N

n

nnS

VC

1

1$

Note that is the potential at the center of due to a uniform chargedensity of unit amplitude over . The solution to the equation gives the .

nS!

mS!

m!

mnl

[ ]{ }

!!!

"

!!!

#

$

!!!

%

!!!

&

'

(

(=

2/

2/

2/

2/

V

V

V

V

lnmn

M

M

)

a a

a

a

cell n

x

y

z

(xn, yn, zn)

nn ydxdsd !"!=! If



mnl

)8814.0(2

)21ln(2

4

122 !"!"!"

aa

yxdydxl

a

a

a

a

nn =+=+

= # #$ $

For numerical results, the of the matrix elements must be evaluated.The potential at the center of due to unit charge density over itsown surface is

nS!

!!!!

"

#

$$$$

%

&

''''

(

)

****

+

,

-

''''

(

)

****

+

,

+=

!!"

#

$$%

&'(

)*+

,+'

(

)*+

,+=

'''

(

)

***

+

,

+=

'''

(

)

***

+

,

+..=+

=

/ /

//////--

2

arctan

tanln42

arctan

tanln1

2ln

42ln

1

sincos

1

44

1

4

1

2

arctan

arctan

0

tanarg

0

2

tanarg

sin

0

2

tanarg

cos

0

tanarg

022

a

b

ba

b

a

xtgb

xtgad

bd

a

drddrdyx

dydxl

a

ba

ba

b

a

b

b

a

b

a

a

b

b

b

a

a

mn

0

01

0

012

22

201

220101

00

2

0

2

For the special case where a=b and m=n, there is



Another option is to integrate directly w.r.t. y and then x (without going topolar coordinate) in such a way as to find a function that satisfies


for some parameter z. Then the matrix element to compute is directlyobtained as



By using several changes of variables and part integration, it possible tofind an exact expression for this function:

Verify it (by performing a double derivative w.r.t. x and y)!


The potential at the center of due to unit charge over can besimilarly evaluated, but the formula is complicated. For most purposes it issufficiently accurate to treat the charge on as if it were a point charge,and use

mS!

nS!

nS!

nmzzyyxx

a

R

Sl

nmnmnmmn

nmn !

"+"+"=

#$ ,

)()()(4 222

2

%&%&

It this point, we have discussed all the details of how to analyze a staticelectric field problem consisting of two conducting plates. The sameprinciple can be applied to other static electric field problems consisting ofany number of conducting objects of any shapes.

Please pay attention to (1) How the problem is approximated? and (2) Howthe integral equation is converted into a matrix equation?


3.4 project 1, numerical analysis of charged conducting...

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