THE METHOD OF MOMENTS SOLUTION

For many practical antennas and scatterers including linear antennas, the desired current distribution is obtained by nu­merically solving the integral equations. The Method of Mo­ments (MOM) is a technique to convert an integral equation to a matrix equation and hence solve the linear system by standard matrix inversion techniques. The MOM is very well documented in the literature (15), and only the basic steps will be briefly discussed below.

The magnetic field integral equation (MFIE) for the un­known current density can be rewritten as an inhomogeneous equation in operator form as follows:

L(Js) = 2n x H(r)

where m = 1,2,. . ., N; (■) is the inner product, the product of the two functions integrated over the do­main.

5. Express the set of algebraic equations given by Eq. (40) in the matrix form:

[Imn][an] — [gm]

where the matrix is given by

W L( Л) Wx, L( f2)

W2, L( f2) W2, L( f2)

J2*n<Wm, L( fn)) = {Wm, g)

[Imn] —

(43a)

where the right-hand side is a known quantity and L(Js) is an integrodifferential linear operator defined as

_WN, L( fN ) WN, L( fN ) ■■■ WN, L( fN )_

where aN and gN are the column vectors given by

(37)

a

1

L(JS) =Jsn x f Js(r’) x V’Gds’ 2п Jc

2

a

[aN] =

(44a)

XN.

W1, g)

<W2, g)

(38)

(44b)

where Js is the electric current on the wire, and G is the free space Green’s function.

Let us discuss the solution of inhomogeneous scalar equa­tion given by

L( f) = g

(39)

(40)

where f = f(x) is the unknown function to be determined, g(x) is a known function and L is a linear operator. The seven steps (16) in obtaining the solution of Eq. (38) is the same as the steps for the solution of Eq. (37).

The steps are as follows:

1. Expand f as

N

f=Y2 anfn

where the an’s are the unknown coefficients, and the fn’s are known functions of x known as expansion, or basis, functions.

2. Using Eq. (38) in Eq. (37), we get

N

У ‘ anL( fn ) = і

,<Wn, g).

Moment Method Solution for Radiation from Thin Wire

Two types of volume integral equations are used for the linear antenna and wire scatterer problem. These are the integral equation of Hallen type and the integral equation of Pockling – ton type. Hallen’s integral equation usually necessitates the postulation of a delta-gap voltage at the feed point and also requires the inversion of an (N + 1)-order matrix. The advan­tages of Pocklington’s integral equation is that it is possible to incorporate different source configurations and it requires inversion of a matrix of order N.

For a current-carrying perfectly conducting wire, the Hal – len’s integral equation obtained by solving the scalar wave equation is given by (1)

Iz (z’)6X^ _ ^ dz’ = cos (kz) +B2 sin(fe)] (45)

4п R

(41)

3. Define a suitable inner product f, g) defined in the range L of x:

< f, g) = f f (x)g(x) dx Jd

where

Iz(z’) = the current flowing through an elementary length of the wire

R = distance of the observation point from the elemen­tary length

I = the total length of the center-fed wire e, u, a = 0 are the primary constants of the medium in

which the antenna is radiating and k is the derived secondary constant, namely, the wave vector of the medium

^+k2)G{z, Z)

B1 and B2 are constants to be determined The Pocklington’s integral equation is given by

Iz (z’)

dz’ = —joeE[2]z (atp = a) (46)

exp(—jkR) 4jxR

where for thin wire the radius a < A, the free-space Green’s function G(z, z’) is given by

G(z, z’) = G(R) =

with R = /a[3] + (z — z’)2

Elz = the incident field (this is the source field for antennas and scattered field for scatterers)

(47)

able in Richmond’s work (17) and also in Ref. 8 are re­produced in Table 1.

2. Subdomain Expansion Functions. Subdomain expan­sions are attractive, convenient, and less expensive in terms of computer time. This stems from the fact that the current is matched on part of the integration path, whereas for the entire domain the integration is per­formed over the whole path and for all N terms of the expansions and coefficients determined.

Miller and Deadrick (8) provide a table containing the many basis and weighting functions which have been tested in com­puter runs. This table also compares the suitability of the use of different functions in different problems. The table is too big to reproduce here, and it is left to the reader to look up. It tabulates the method, integral equation type, basis function, current conditions for interior and end segments, the basis function in terms of unknown for unknown and end segments, weighting function for interior and end segments, number of unknowns, and specific comments on the applicability of the expansion functions.

THE METHOD OF MOMENTS SOLUTION

(48a)

(48b)

(48c)

The availability of high-speed computers, graphics, and soft­ware packages, along with the enormous development of mi­crocomputers, has made electromagnetic numerical computa­tion extremely viable. An attractive feature of numerical methods is their ability to handle arbitrarily shaped and elec­trically large bodies and bodies with nonuniformity and an­isotropy where exact solutions can only at best provide some physical insight. For large problems, it is possible to get a linear system with a minimum set of equations to achieve a certain accuracy.

An account of the area of numerical computation of thin wire problem is well-documented in the literature (1,3,6-13). As described in Ref. 8, there are many important computa­tional issues involved in thin wire problems. These are (a) segmentation of the structure and the correct number of seg­ments, (b) choice of right current expansion functions, (c) the thin wire approximation (radius a < A), (d) Roundoff errror due to matrix factorization, (e) near-field numerical anomaly, (f) treatment of the junctions of the segments, (g) wire-grid modeling, and (h) computer time required. Also, the errors (7) involved are of concern. There are two types of errors encoun­tered: (a) the physical modeling error, because in the absence of an exact solution for a variety of semicomplex and complex stuctures, it is the natural departure of the assumed struc­tural details from the actual structure, and (b) the numerical modeling error, since all numerical methods are approximate but sufficiently accurate for the application.

Formulation

Before we discuss the formulation of the thin-wire integral equation, comments on the expansion functions used in this study are in order. There are broadly two types of expansion functions:

The Electric Field Integral Equation and Its Matrix Representation

Figure 5 gives the geometry of the arbitrarily oriented straight wire. The wire is broken into segments, or subsec­tions. The mini numerical electromagnetic code (MININEC) relates the current distribution on the wire due to the inci­dent field. The integral equation relating the incident field Ei, magnetic vector potential A, and electric scalar potential ф are given by

—Ei • t = —joA • t — t • Уф

where

A = — f I(s)S(s)k(r)ds 4n J

Ф = ^ j q(s)k(r)ds

Table 1. Entire Domain Current Expansions Using Different Polynomials

A. The Polynomials

Fourier: I(z) = Ij cos(^x/2) + I2 cos(3nx/2) + I3 cos(5nx/2) MacLaurin: I(z) = Ij + I2x2 + I3x[4] + •••

Chebyshev: I(z) = I1T0(x) + I2T2(x) + I3T4(x) + ••• Hermite: I(z) = IjH0(x) + I2H2(x) + I3H4(x) + ••• Legendre: I(z) = I1 P„(x) + I2P2(x) + I3P3M + ••• where -1/2 < x = 2z/L < 1/2

B. Typical Results for Functions

L = 0.5A; a

= 0.005A; ft =

9

О

0

In

Fourier

MacLaurin

Chebyshev

Hermite

Legendre

1

3.476

3.374

1.7589

8.2929

2.2763

2

0.170

4.037

1.5581

14.3644

2.1005

3

0.085

3.128

0.0319

4.4135

0.0655

4

0.055

4.101

0.0112

0.3453

0.0421

5

0.040

1.871

0.0146

0.0073

0.0322

When the pulse functions of Eq. (52) are inserted in parenthe­ses, we obtain

0

1

■■■■■■■■■ n

-1

n

n+1 ■■■■

■■■■ N

-1

N

(a)

pi………. pn-1

pn+1 pN-1

n—1

n+1 ■

■ N-1

E {Sm ) ■

N

n

0

s™ — s

— s

5m+1

m1

m

m

Sm-1/2 +

m+1/2

2

2

ss

s

m1

m+1

m

m

= joiA(sm ) ■

Sm-1/2 +

S

m+1/2

(55)

2

2

The exact kernel treatment developed above is for observation points on source segments. For observation points near but not on the source, a segment has been developed by Wilton and MININEC has incorporated it (16).

Expansion of Currents

The currents are expanded in terms of pulse functions as shown in Fig. 5, excluding the end points where the currents are chosen as zeros to satisfy boundary conditions at the ends. The current expansion is given by

n

(b)

Figure 5. Wire segmentation with pulses for current and charges. (a) Unweighted current pulses. (b) Unweighted charge representa­tion. The whole length is broken into several segments. Each segment is assigned a pulse, and the pulses represent the assumed current distribution.

t is a unit vector tangential to the wire at any distance along the path of integration which is the length of the wire and k(R) is given by

I(s) = Y InPn (s)

(56)

2п exp(-jkR)

(49)

Using this current expansion in Eq. (48b) and after mathe­matical manipulations which are available in Ref. 1 and are not detailed here, we get the linear system matrix equation in N unknowns:

R

ф =0

The continuity equation given below determines the relation­ship between the charge q(s) and the rate of change of current with distance:

[Vm ] = [Zmn][In ]

(57)

1 dl

<?(s) = —–Г-

jo ds

(50)

where m, n = 1, 2, . . ., N, [Zmn] is the square impedance matrix, and [Vm] and [In] are applied voltage and current col­umn vectors:

The MININEC solves the integral equation using the follow­ing steps:

1. The wires are divided into small equal segments such that the length of the segment is still large compared to the radius of the wire (a < A, typically 1/100th of a wavelength). The radius vectors m, n = 0, 1, 2, . . . are defined with reference to a global origin.

2. The unit vectors are defined as

1

k (rm+1/2 rm-1/2 ) ‘ (sn + 1/2tm, n,n + 1/2

Zm — —

4п jrne

m+1 /2,n, n + l Sn + 1 – Sn

m+1/2 ,n-1,n

n-1/2tm, n-1/2,n

) –

(58)

+ sn

+ ■

sn s

n-1

tm-1/2,n, n-1 tm-1/2,n-1,n

+

sn+1 – sn

ss

n1

n

(51)

n+1 /2 ‘

‘n+1 ~’n

K+l ~Гп

The testing and expansion functions are pulse functions which are defined by

This matrix has elliptical integrals involved in it. These ellip­tical integrals can be evaluated numerically.

The above equations are valid for any radius other than small, for which the expression for ф breaks down and Har­rington (18) provided an approximate formula for ф. This is given by

(59a)

(59b)

(60)

Pn ( s ) =

(52)

2n As L a.

exp( jkrm )

n ° n – 1

2

rn 1 + rn

(54)

r

r

n + 1/2 ■

2

where the points sn+1/2 and sn+1/2 are the segment midpoints and are given by

Sn+1 + Sn Sn +1/2 — 2 ’ Sn-l/2

In terms of global coordinates,

1 for sn-1/2 < s < sn+1/2

0 otherwise

_ Гп +Г„_! гг —1/2 2

ss

(53)

Inclusion of Nonradiating Structures

The Ground Plane. When the wire structure near the ground plane is assumed to be perfectly conducting, an image

4п rm

The integral is given by

tm, u,v = / k0 (sm – s’) ds’

* = ^_іпГд£1_Д

nAs L aJ 4п

for m = n

for m

is created. The structure and the ground plane is equivalent to the structure and the image.

The voltage and current relationship is given by

THE METHOD OF MOMENTS SOLUTION

(61)

Figure 7. Geometry of the Tee antenna. Typical dimensions are shown.

+ z,

m,2N-n+1

Wire Attached to Ground. When a wire is attached to the ground on one or both sides, there will be a residual compo­nent of current at one or both ends. In this case, a current pulse is automatically added to the end point in the formu­lation.

7′ — 7

^mn — ^ mn

where

[Vm ] = [Z’mnJIn

Lumped Parameter Loading. If an additional complex load is added to the perfectly conducting wire (Fig. 6), there will be an additional voltage drop created at that point if the location of the load (Zl = Ri + jXl) is at a point of nonzero pulse func­tion. The impedance matrix is modified to to meet the following requirements: (a) the segmentation den­sity, (b) thin-wire criteria, (c) small radius calculation, (d) step changes in wire radius, (e) spacing of wires, (f) loop an­tenna, and (g) monopoles and antennas above ground.

Zmn +Zlt Zmn

Z’ =

mn

[Vm ] — [ZLJUn ]

where Z’mn is the modified impedance matrix and is given by

for m — n for m — n

Validation of the MININEC Code

Extensive work has been reported on the validation of the MININEC. Numerous validation runs have been carried out

Operation of Currents-LU Decomposition

The operation is oriented around the Menu shown below. Here we describe the DOS version (19,20), but the Windows version is also available (21-23).

MENU

1 – COMPUTE/DISPLAY CURRENTS

2 – CHANGE EXCITATION

3 – CHANGE FREQUENCY

4 – CHANGE LOADING

5 – LOAD GEOMETRY

6 – SELECT OUTPUT DEVICE

7 – RETURN TO SYSTEM SUPERVISOR 0 – EXIT TO DOS

SELECTION (1-7 OR 0)?

(a)

Some Examples Using MININEC

Tee Antenna. Figure 7 shows the geometry of the Tee an­tenna fed from the base by a coaxial line. The impedance cal­culations of the Tee antenna using different computer pro­grams including CURLU in MININEC and have been compared with measurements (25).

Near and Far Fields. The near – and far-field programs (FIELDS) calculate near and far fields using the current dis­tribution on the structure obtained by integral equation for­mulations. The current distribution can be computed using three programs: CURLU, CURTE, and CURRO. The current distribution can be computed using perfect and imperfect grounds, although the real ground corrections are applied to the far field only. The real ground correction is included in the form of reflection coefficients for parallel and perpendicular polarizations. For details, the reader is referred to Chap. 8 of Ref. 7. The menu is given below. User input(UI) means the user is expected to respond at that point.

Overlapping pulse

Wire 2

THE METHOD OF MOMENTS SOLUTION

Wire 1′

(b)

THE METHOD OF MOMENTS SOLUTION

Figure 6. Overlap scheme used at a multiple junction of wires. (a) Wire 1 with no end connections. (b) Wire 2 overlaps onto wire 1. (c) Wire 3 overlaps onto wire 1.

100 200 300 400

Radial distance (m)

Figure 9. Monopole near fields: Electric fields Ez and Er versus ra­dial distance.

0

THE METHOD OF MOMENTS SOLUTION

THE METHOD OF MOMENTS SOLUTION

103

102

4 8 12 16

Horizontal distance (m)

Figure 8. Monopole near fields: Ez versus horizontal distance.

0

£

>

33

ф

10[5]

20

500

10"

10"

10"

0

MENU

1 – COMPUTE NEAR FIELDS

2 – COMPUTE FAR FIELDS

3 – SELECT/CHANGE ENVIRONMENT

4 – SELECT/CHANGE CURRENTS FILE

5 – SELECT OUTPUT DEVICE

6 – RETURN TO SYSTEM SUPERVISOR 0 – EXIT TO DOS

SELECTION (1-6 OR 0)? User Input

NAME OF INPUT CURRENT FILE? User Input (UI) ELECTRIC OR MAGNETIC NEAR FIELDS (E/M)? User Input

X-COORDINATE Y-COORDINATE

INITIAL VALUE? UI INITIAL VALUE?

INCREMENT? UI INCREMENT?

NO. OF PTS? UI NO. OF PTS?

Z-COORDINATE UI INITIAL VALUE? UI UI INCREMENT? UI

UI NO. OF PTS? UI

PRESENT POWER LEVEL IS : CURRENT VALUE CHANGE POWER LEVEL (Y/N)? UI NEW POWER LEVEL (WATTS)? UI

Once the parameters are specified, the near – and far-field results are printed out in words. Figures 8 through 15 show the near-field characteristics of the monopole.

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