Download The Method of Moments

The Method of Moments
Basic Concepts of the Method of Moments
The method of moment (MoM) is a numerical procedure for solving linear
operator equation by transforming it into a system of simultaneous linear algebraic
equation, referred to as matrix equation. Many problems in electromagnetics can be cast
in the form of integral equations. An integral equation is one in which the unknown
function appears in the integrand. The MoM provides a way to solve such integral
equations. Instead of discussing the general procedure for MoM, here we introduce the
basic concepts involved in MoM solution through some examples, both for static as well
as time harmonic electromagnetic fields.
To start with, let us consider the an example of determining the electrostatic
potential due to an isolated charged conducting plate, 2a meters on a side and lying on
the z  0 plane with its center at the origin as shown in figure 8.5.
Fig. 8.5 A square conducting charged plate
The plate is assumed to have zero thickness. Let  ( x, y ) represent the surface charge
density on the plate. Further, the boundary condition on the potential is V  V0 =constant
on the plate. For the charged conducting plate, the potential V at any point ( x, y, z ) can be
written as:
a a
V ( x, y , z ) 
a a
 ( x' , y ' )dx' dy '
1
  4
0
( x  x' ) 2  ( y  y ' ) 2  ( z  z ' ) 2
(8.15)
If  ( x, y ) is known (which is often assumed in solving simple electrostatic problems, i.
e. charged density is specified), the potential function V can be computed directly. But in
practical problems, often the charge distribution  ( x, y ) is not known. The equation
(8.15) is an example of integral equation as the unknown  ( x, y ) appears under the
integral. To solve the unknown charge distribution we apply the method of moments. The
procedure is explained below:
We subdivide the plate into N squares of side 2b as shown in figure 8.5. The
n th square is denoted by S n and 2b 
2a
N
.
We approximate the charge distribution as:
N
 ( x' , y ' )    n f n ( x' , y ' )
(8.16)
n 1
on S n
1
where f n ( x' , y' )  
0 on S m m  n
Here, the functions
f n ( x' , y' ) are called the expansion functions or basis
functions and  n are the coefficients. Basis functions can be of different types; here we
have considered pulse basis functions, which have unit magnitude over some domain and
are zero everywhere else. With N sufficiently large, equation (8.16) closely approximates
the actual  ( x ' , y ' ) . To solve for the charge distribution  ( x ' , y ' ) approximately, the
unknown coefficients  n are to be determined. From the given boundary condition we
note that on the surface of the plate V ( x, y,0)  V0 and this condition can be used to
determine the unknown coefficients  n .
Using the approximate charge distribution given by equation (8.16), let us now
evaluate the equation (8.15) at the mid point ( xm , y m ,0) of each of the S m . The potential
Vm at the midpoint of S m is given by:
N
a a
Vm 

1
  4
a a
n 1
n
f n ( x' , y ' ) dx' dy '
(8.17)
( x m  x' ) 2  ( y m  y ' ) 2
0
The above equation can be written as
N
Vm    nVmn
(8.18)
n 1
where Vmn is the potential at the center of S m due to unit charge placed on S n .
For m  n ,
b b
Vmn 
1
  4
b b
1
0
x' 2  y ' 2
dx' dy ' 
2b
0
ln( 1  2 )
(8.19)
For m  n , treating the unit charge on S n as a point charge located at the mid point
( xn , y n ) of S n ,
Vmn 
b2
0
xm  xn 
2
  ym  yn 
2
(8.20)
N
As Vm    nVmn  V0 , considering the potentials at all the N sub sections we can write
n 1
 V11 V12
V
 21 V22
 .
.

.
 .
VN 1 V N 2
. . V1N    1  V0 
. . V2 N    2  V0 
. .
.  .    . 
   
. .
.  .   . 
. . V NN   N  V0 
Or,
V 
 V0
(8.21)
The unknown coefficients  n s can be computed as
  V 1V0
(8.22)
Figure 8.6 shows the charge distribution obtained using the MoM technique for V0  1V ,
2a  1 m.
Fig. 8.6: Approximate charge density along the side of the plate
Another parameter of interest is the capacitance of the plate.
Q 1
C 
V0 V0
a a
   ( x, y)dxdy
(8.23)
a a
With  n known, the capacitance of the plate can be approximated as:
C
1 N
 n sn
V0 n 1
(8.24)
Based on the discussions we had so far, we summarize the steps involved in MoM
solution.
We consider an inhomogeneous equation
L( f )  g
(8.25)
where L is a linear operator, g is a known function (excitation) and f is the unknown
function to be determined.
If we consider our previous example,
f ( x, y )   ( x, y )
g ( x, y)  V0 , x  a, y  a
a a
L( f ) 
  4
a a
f ( x' , y ' )
0
( x  x' ) 2  ( y  y ' ) 2
dx' dy '
We expand f in a series of functions
f   n f n
(8.26)
n
Where  n are constant. The set f n is called the expansion function or basis function. For
exact solution, n   , but in practice truncated to a finite value.

n
L( f n )  g
(8.27)
n
We define a set of weighting functions or testing functions w1 , w2 .......wN in the range of
L and take the inner product of equation (8.25) with each of the wm

n
wm , f n  wm , g
(8.28)
n
A scalar product w, g is defined to be a scalar satisfying
w, g  g , w
(8.29(a))
bf  cg, w  b f , w  c g, w
(8.29(b))
g*, g  0
if g  0
(8.29(c))
g*, g  0
if g  0
(8.29(d))
Here b and c are scalars and * indicates complex conjugation.
The inner product corresponding to our previous example is of the form
a a
w, g 
  w( x, y) g ( x, y)dxdy
(8.30)
a a
Similarly, the testing function for our previous example is
wm   ( x  xm ) ( y  y m )
(8.31)
i.e. the testing functions are Dirac delta functions. Such choice of testing function is
called point matching.
The equation (8.26) can be reduced to
 Amn n  gm
(8.32)
where
 w1, Lf1

w Lf
 Amn    2, 1
 .
 .
w1, Lf 2
w2, Lf 2
.
.
. .

. .

. .
. .
 1 
 
n   2 
 . 
 
 . 
 w1, g 


 w2, g 
gn  

 . 
 . 


If
 Amn  is non-singular we can write
 n   Amn  g m
1
(8.33)
and f can be calculated by using equation (8.24).
In our example we have used pulse type basis function and point matching, that is,
Dirac delta function as testing or weighting function. In general the basis functions may
be sub-domain basis functions (where the structure is subdivided into N nonoverlapping segments. The basis functions are defined in conjunction with limits of one
or more segments) or entire domain basis functions (which exist over the full domain).
In particular, when f n  wn , this case is known as Galerkin’s method.