NATL INST OF STAND & TECH

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NBSIR 82-1659

NBS

PUBLICATIONS

A STUDY OF THE ELECTROMAGNETIC

FIELDS DISTRIBUTION INSIDE

BUILDINGS WITH APERTURES EXCITED

BY AN EXTERNAL SOURCE

Mark T. Ma

M. G. Arthur

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

February 1982

*QC

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MAY 17 1982

hob OCXA STUDY OF THE ELECTROMAGNETIC

FIELDS DISTRIBUTION INSIDE

BUILDINGS WITH APERTURES EXCITED

BY AN EXTERNAL SOURCE

Mark T. Ma

M. G. Arthur

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

February 1982

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige. Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler Director

CONTENTS

Page

1. INTRODUCTION 1 2. A THREE-DIMENSIONAL CONDUCTING

RECTANGULAR BOX 3

2.1 Magnetic Field Integral Equation 4

2.2 Generalized Impedance Approach 7

3. LOSSY BUILDINGS WITH SMALL APERTURES 9

4. LARGE BUILDINGS AND APERTURES WITH LOSSY

GROUNDS— A HYBRID APPROACH 20

5. MEASUREMENT REQUIREMENTS AND PROCEDURES 31

6. COMPUTING REQUIREMENTS AND PROCEDURES 34

6.1 CURRENT Program (FORTRAN) 35

6.2 FIELD Program (FORTRAN).. 42

6.3 CURRENT Program (HPL) 43

6.4 FIELD Program (HPL)..... 44

6.5 HPL File Structure on Magnetic Tape 45

6.6 Print Options for HPL Programs 45

7. NUMERICAL RESULTS FOR A PRACTICAL BUILDING 47

8. CONCLUDING REMARKS 66

9. ACKNOWLEDGMENTS 66

10. REFERENCES 67

APPENDIX A. FORTRAN PROGRAMS 68

A. 1 CURRENT Program 63

A. 2 FIELD Program "9

APPENDIX B. HPL PROGRAMS ^6

B. l CURRENT and FIELD Programs 36

B. 2 Variable Assignments 1 "3

B .3 Instructions for Running HPL Programs 133

FIGURES 112

A STUDY OF THE ELECTROMAGNETIC FIELDS DISTRIBUTION

INSIDE BUILDINGS WITH APERTURES EXCITED BY AN EXTERNAL SOURCE

M. T. Ma and M. G. Arthur

Two special cases of the penetration of electromagnetic fields

into a cavity, building or box are formulated and analyzed. One

is to consider the case of a lossy cavity with small apertures in

free space, based on an application of the equivalence principle

and the use of a generalized network formulation. It is found

that the field strength at the aperture center is approximately

inversely proportional to the square-root of the conductivity of

the cavity walls and that high field levels can exist inside the

cavity under certain physical conditions. The second case is to

treat the problem of large buildings with large apertures on a practical lossy ground by a combination of theoretical approach

and limited measurement data. Field levels inside the building

for this latter case depend on the transmitter power, the

transmitter-to-building distance, the ground conductivity, and the

measurement antenna height relative to that of the transmitter.

Keywords: aperture; cavity; equivalence principle; field

distribution; slot.

1. INTRODUCTION

The capability of electromagnetic waves to penetrate into a building or a box which houses important, sensitive devices and equipment has always been of concern to industries and military installations. Generally speaking, it is

essential to know the level of the field strength and its component amplitudes

and phases inside the building before an assessment of hazardous effects on equipment performance and human health can be made. Genuine efforts have been

made to make on-site spot measurements inside an empty building for a known

transmitter at a specified distance away from the building. It is still

impossible from these spot measurements to make any meaningful predictions of

the highest field level and its polarization inside the building, which are

the ultimate goals of this study. Thus, to achieve these goals, systematic

measurements of component field amplitudes and phases along three orthogonal

directions at an interval distance of no more than one quarter wavelength

would be required. These systematic measurements are, of course, very timeconsuming. For a transmitter operated at 150 MHz (A -

1 , which is typical or

a land mobile radio, and a practical storage building ot the si_e of \ \

X iox , a huge number of measurement data in the order of one third of a

million data points have to be taken and analyzed. Furthermore, these

1
measurement data are only applicable to the specific building and depend on construction material, building size and shape, the aperture (doors or windows)

size and shape, the ground and surrounding conditions, the operating frequency of

the transmitter, and the transmi tter-to-bui 1 di ng distance under study.

To circumvent this measurement impracti cal i ty , a theoretical study of the

generalized three-dimensional problem was deemed desirable. However, in view

of the complexity of the problem involved and the enormous demand on the

computing facility for a practical size of building, only special cases for a CW excitation subject to certain physical constraints are considered in this

report.

In Section 2, the problem of a three-dimensional closed conducting

rectangular box in free space illuminated by a linearly polarized, normally

incident, plane wave is considered with two different approaches. One is by

the magnetic field integral equation with appropriate boundary conditions [1-

4], and the other is through the concept of a generalized impedance matrix.

Since there is no opening on the box surface in this case, no field can penetrate into the box. Numerical results for the induced surface current

densities, on a limited basis are given therein, based on which the field

scattered by the box may be computed.

The problem is then extended to the case of lossy rectangular buildings or boxes with small slot apertures in free space, which is presented in Section

3. Field strengths inside the box are shown to be dependent on the aperture

and box sizes relative to the operating wavelength and the material of the box

wal 1 s

It is found that the field strength at the aperture center is

approximately inversely proportional to the square-root of the conductivity of

the wall material and that high levels of the field strength can exist inside

the box at natural resonances and under a set of special physical conditions.

In Section 4, large buildings and large apertures located on a practical

lossy ground are analyzed with a combination of theoretical consideration and

limited set of measurement data. Requirements on the measurement equipment and

computing facilities for this case are outlined in Section 5 with the

measurement and computing procedures presented in Section 6. Numerical results

of the approximate induced current densities and the approximate field

distribution inside the building for a known transmitter frequency, power, and

the transmi tter-to-bui 1 di nq distance are given in Section 7. Physical

2
interpretations of these numerical results and the limiting conditions under

which the results have meanings are discussed in the concluding section. A relevant FORTRAN program used to obtain the numerical results is listed in

Appendix A. A simplified version using the HPL* suitable to Model HP 9825*

capable of making a quick on-site check with a much reduced matrix size is

presented in Appendix B.

2. A THREE-DIMENSIONAL CONDUCTING RECTANGULAR BOX

The geometry depicted in figure 1 may be used to represent a theoretical

model for the problem of the penetration of electromagnetic waves into a rectangular cavity (building or box). The source located outside the cavity

may be a CW or pulsed signal of arbitrary polarization and frequency, incident

on the cavity with an arbitrary angle. The cavity walls are generally made of

non-perfectly conducting material, and the cavity itself is situated on a homogeneous flat lossy earth represented by the ground constants of

permeability (y), permittivity (e) and conductivity (a).

The objective of formulating and analyzing this problem is to determine

the spatial field distribution and locations of the highest levels of field

strength inside the cavity as a function of many parameters such as the sizes

of the cavity and the aperture, the ground constants, etc. Obviously, this

type of problem is too complicated to be treated rigorously using Maxwell's

equations with appropriate boundary conditions. To simplify the problem, a three-dimensional perfectly conducting rectangular box without any apertures

located in free space, illuminated by an external source, as shown in figure 2 may be considered as a first approximate model to determine the induced surface

current densities, from which the scattered field outside the box may then be

calculated. The numerical results so obtained can be compared with the

existing measurement data on radar cross-secti ons to assess the validity of the

formulation and the accuracy of the numerical computations.

"Certain commercial equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by tie ie’v.

Bureau of Standards, nor does it imply that the materials or equipment

identified are necessarily the best available for the purpose.

3
2.1 Magnetic Field Integral Equation

Two approaches have been available for deriving the necessary integral

equations to determine the current densities induced on the surfaces of the

box in figure 2. The first is to formulate the electromagnetic boundary-val ue problem in terms of the magnetic field integral equation, which takes the

following form [5]:

j

J (r) = n X H (r)+T^nX} j (?') X V'G(r.r') ds' (1)

L 1 4tT S

where H. (r) is the known incident magnetic field, J (r) is the unknown

tangential current density induced on the six box surfaces,

G (r~r') = exp ( - j kR ) /R (2)

R = \r - r'\ = [(x - x')

2 + (y - y'f + (z - z')

2

]

1/2

(3)

r = xx+yy + zz (4)

r' = x x' + y

y' + z z' (5)

x, x-', y, y'

,

z, z' = coordinates on the box surfaces

x, y and z = unit vectors in x, y, and z directions

V" ( 6 )

n = a unit vector in the outward normal direction to each box

surface

S = the area of each of the six surfaces,

and the horizontal bar across the integral sign in (1) denotes that only the

principal value will be taken.

4
Since there are two components of J (r) on each surface of the box,

equation (1) results in 12 coupled scalar integral equations for the component

current densities. As an example, in reference to figure 2, we have

H. (r) = x H. (r) + z H. (r) , (7)

1 IX 1Z

for the wave propagating in the y -direction, normal to Face I (right). Tnus,

for faces I and II (right and left),

3 (r" ) = x J x

( r' ) + z J

z

( r" ) , (8)

for faces III and IV (top and bottom),

3 {?') = x J (?' ) + y J (r-), (9)

^ y

for faces V and VI, (front and back)

j {?'

)

= y Jy

(?') + z J

z (^)» ( 10 )

and

where

and

V" G (r, r') = (x

3x"

+ y

9y^

+ z 3Z :) exp ( - j kR)/R

x it + y it + Z\i) y

x y z

( 11 )

ip = ( jk + i)(x - x^) exp (- jkR)/R

2

, (12)

X

\py = (jk +{)(y-yO exp ( - j kR ) / R

2

, (13)

ip z = (jk +-^)(z-z") exp (-jkR)/R

2

. ^14^

5
The integrand of (1) then becomes

for Faces I and II,

x (-J

z fy)

+ y Ux

J z

" J x ^ + z ( J x ^y^’ (15)

for Faces III and IV,

X ( Jy

Xp z

) + y (“J

x + Z (J

x Vy “ Jy)> (16)

and for Faces V and VI

x ( Jy^z

" J

z ^y) + y ( J

z ^x

) + z ( _ ^x

Jy)‘ (17)

After taking the principal value of (1), we finally have,

for Face I where n = -y.

7

J (r)

l V

and

J

x

( r ')yls'+ 4!/ J (r')lMis' (18)

II, III, IV y III, IV, V, VI y

T J (r) = H. (r) - — /

J (r') ij; ds" + — J

J ( r ") ^ ds " (19)

2 z lx 4tt

J z y 4tt

j y z

1 1 , V , VI

y III, IV, V, VI

y

Equations similar to (18) and (19) can be obtained for Faces II through VI,

resulting in exactly 12 coupled scalar integral equations. If the method of

moments [6] is used to solve for this system of equations by subdividing each

O O surface into n x n patches, we will end up 1 2n ^ x 12n simultaneous linear

algebraic equations. When the value of n is sufficiently large to insure

meaningful results, the inversion of a matrix size involved may be prohibitive

even with the availability of a large computer capacity. The results of |J | and |

J

z | on Face I for a (0.5x)^ cube illuminated by a broadside incidence

plane wave H^

z with unity strength are presented in figure 3, which were obtained with 8x8 patches [5]. Similar results for other box surfaces are also available [5]. Based on the results of surface current densities, the

scattered fields outside the cube may then be computed. For larger sizes of

cube or arbitrary incidence, the numerical procedures are more involved.

6
2.2 Generalized Impedance Approach

An alternative approach to solving the problem depicted in figure 2 is by

using the concept of the generalized impedance matrix. Basically, the electric

field scattered by the box shown in figure 2 can be determined by

E = - jwA - V<}>

, (20)

where A and <j> are, respectively, the magnetic vector potential and the electric

scalar potential, satisfying the following equations:

A = /

s

3 (r) G (r, r ) ds' (21)

and

4> =

- 3^4^ J s v '« j C?') G(r, r'

)

ds' (22)

where J and G have been explained or defined in (1) and (2).

On the surface of a perfectly conducting box, the tangential component

of the total electric field vanishes, or

i ,tan

- - E

's ,tan

(23)

where E ; is the tangential component of the electric field at the box

I 5 L a H — urface due to the incident signal, and E

s tan is the tangential component of

lat in (20).

Combining (20) and (23), we obtain

E.

i

,

tan [

^

f

s

j (f') G

(’r - "

r") ds

'

+

l

4 tt jwe

J (r) . V' G (r. (24)

= L [J (r')]

where the linear operator L represents the necessary double integrals.

7
Using the standard method of moments [6], we subdivide the surface S into

patches and expand the components of the unknown surface current density

function J (r') as follows:

n J. (r") = j-th component of J lr') = V If (25)

j L, n n n=l

where f is a set of chosen basis functions, and I ' s are complex coefficients

to be determined.

Substitution of (25) into (24) yields

where

(26)

(27)

' < w

i’

Lf

i>

<W

r

Lf

2

> ... <w

x

. L V'

rz 1 =

L mn J <w

2

, Lf

l> <w

2

. Lf

2

> • •

• ^

j Lf

N

> (28)

, <W

N»

Lf

l>

• • • <W

NLf

2

> • • • » Lf

N >.

E. .> '

1 ,J

<W M’ •> N 1 ,J

(29)

E1 = the j-th component of E. in (24),

l ,tan

8
W 2’ *•* const itute a set of testing functions which are tangential to

S, and <a,b> represents an inner product defined as:

<a,b> = / ab ds' (30)

The elements of [Z

mr) ] in (28) may be called generalized impedances; those of

[I

p

] in (27) generalized currents; and those of [V ] in (29) generalized

voltages representing the sources. When [Z ] is nonsingular which is true in

general, its inverse [Zmn ]

-

^ exists. Then, we have

[IJ = [Z ]

_1

[V ] (31)

n

J L

rnn J L m

J v ‘

which is the set of coefficients required in (25) for obtaining an approximate

component of j(r').

Specific numerical results of | J /H^

z | and |J Z /H^Z | on Face I of figure 2 for (O.U) 3 cube with the same normal incidence are given in figures 4 and

5. From figure 4, it is clear that I

J

x / H -j z I 1S again the dominant response

for the particular plane-wave incidence being considered.*

3. LOSSY BUILDINGS WITH SMALL APERTURES

For electromagnetic sources at VHF impinging on a practical building

size, the box represented in figure 2 will be electrically large, and thus,

the methods outlined in Section 2 will be very time consuming, if not

impossible. In this section, a different approach to solving the problem of

large cavities is considered. In addition, apertures representing doors or windows, and the case of non-perfectly conducting building walls are also

included. For the sake of simplicity, only the small aperture in the form of

a horizontal narrow slot in an infinite ground plane as shown in figure 6 is

studied first.

The approach is based on an application of the equivalence principle L

: ] and the use of a generalized network formulation [8]. By this approach the

small aperture can be replaced by an equivalent magnetic current defined by:

M = y x E , (32)

d

*Private communication, D.K. Cheng, Syracuse University, Syracuse, New York.

9
where y is the unit vector in the direction of propagation of the incident

The field in region a of figure 6 is that produced by the incident wave

and that due to M over the aperture with the aperture covered by a perfect

conductor. The field in region b is that produced by -M over the aperture

with the aperture covered by a perfect conductor. The basic problem here is

component of the magnetic field is continuous across the aperture. This

eventually leads to another integral equation involving the unknown M. The

technique similar to that presented in Section 2.2 namely,

approxiinati ng M by with unknown coefficients V

n

, may then be utilized.

For narrow slots of a length not greater than three-quarters of a wavelength, there is evidence that M in the slot can be closely approximated

by an equiphasal half-cosine wave [9]. This suggests that we may simplify the

problem at hand considerably by using a one-term expansion for M and write

where L Y is the length of the slot under the condition L v < 3/4 a. The matrix A X equation is then reduced to a single scalar equation representing the

generalized Ohm's law:

the determination of M, with the boundary condition that the tangential

M = V M

o

= X V cos (ttX/L ) X

(33)

(

y

a + Yb

) v = I (34)

where V is a generalized voltage to be determined, Yd and Y

b are generalized

admittances (in the si ot-aperture) of the half-space and the cavity

respectively, and I is the generalized current source given by:

1 = 2 f M . H . ds

,

J 0 1 apert.

(35)

with H representing the incident magnetic field over the si ot-aperture in

free space.

10
In (34) the generalized half-space admittance in region a is [8].

Y

3 = -2 / M

q

. H (M

q

) ds (36)

apert

where H (M ) is the magnetic field produced and radiated to free space by

In region b, the admittance may be expressed in terms of the mode

characteri sti cs in the cavity [9],

Y b = -j z V Y. cot k.d (37)

where and k- denote, respectively, the characteri stic admittance and the

wavenumber of the ith mode, and the coefficient A.j is given by:

A. = f M . y x e . ds , (38)

apert

with e. being the normalized ith-mode transverse electric-field vector.

If there exists only one transmission mode in the waveguide in region b;

that is, if all modes except the i=l mode are cut-off, then all and k- are imaginary except Y^ and k-p We write [9a]

Y

i = j6

i (i * 1) (39)

k

i = - j “i

(i * 1) . (40)

For i = 1 , Y-j is the (real) characteri sti c admittance of the dominant mode, and

k

l = 3i

- J a

i

, l 4 l)

where ^ is the phase constant which is related to the guide wavelength,

of the dominant mode

3l

= 27r/A

gl

and oj is the attenuation constant due to power loss in cavity walls.

( 42 )
Based on the above, we can construct a generalized equivalent network for

the cavity-backed si ot-aperture shown in figure 6 . The network consists of a current source I in parallel with Yd

, together with

j B = j z B

i coth a .d (43)

i*l

and a terminated transmission line coupled through an ideal transformer with a turns-ratio Ap This is given in figure 7.

In order to study cavity fields, it is essential to consider that the

equivalent transmission line is terminated by a nonzero surface impedance

z

s

- R

S + * d + j) T

= (1 + j)

TT f y a

(44)

where a is the conductivity of the cavity material, and for the dominant

TE^q mode in an a x b rectangular waveguide is given by

a i nb

/ 1 - x/2a)

[1 (45)

n being the intrinsic impedance of the medium inside the cavity. For air, n is approximately 377 n. For cavity walls made of a good conductor the reactive part of Z

s

, X

$

, has an insignificant effect on the resonant frequency of the cavity and may be

neglected. The real part, R

$

, however influences the Q and field calculations

in a major way; it must be retained, as was done in figure 7.

The input admittance of a lossy transmission line of characteri Stic

admittance Yp wavenumber kj = - jap length d, and terminated in R

s is

Y.in

r 1 + tan (

3

1 d - j ot x d

)

\ Y

1 R

S + j tan ( 3 ^

- ja^) . (46)

12
At natural resonance.

0 id - n-n, n = 1 , 2 , . .

(47)

Equation (46) becomes

< 1 + Y.R tanh ai d \ Y = Y

s l in 1 { Y

1 ks + j tan a] d (48)

The quantity ot^d is usually very small for highly conducting walls, and it is

safe to approximate tanh cqd by cqd and to neglect the term Y^R

S tanh a^d in

comparison with 1. We have, from (48)

Y.

•i n (49)

where

r d

- aid (50)

is a dimensionless quantity. Thus at natural resonance, Y^

n is purely real.

Tne input impedance of the low-loss transmission line terminated in R

$

in

the neighborhood of natural resonance is (^d not exactly equal to pn)

where

Z.in Y^—

= R + jX

' i n

( 51 )

R = r d/Y

l ‘ r d

Z

l (52)

and

X = Zjtan B^d. (53)

The Q of the cavity can be obtained from the definition

dX

duj

(54)

13
Using (52) and (53) for the dominant mode, we have

Q = 7^— Hr)

2 secVd (55)

r d

A

gl A 1

where the relation

»gl

* * 1 ' ( 56 > has been used.

At natural resonance.

d n=l ,2

,

(57)

Equation (55) becomes

Q

=

TT d Vgl

A (- hTT 2r

d

<r,

2 (58)

In order to determine the fields in the si ot-aperture and inside the

cavity, we must find V from (34). In other words we need to find I, Y a and

Y b

. Assuming an incident plane wave with

H.

l xH e

o

-jky -jky

(59)

where k = 2tt/a, I can be evaluated easily for a narrow L

x x Ly slot by using

(33) and (59) in (35).

I _4

tt n

L L E x y o (60)

The equivalent half-space admittance Y a can be computed from (36) in

terms of an electric vector potential F and a magnetic scalar potential <f> m due

to the magnetic current M

Q

. That is,

Y 3 = 2 / M . (jwF + v$ ) ds (61)

o m apert

14
where

F = — f m ex P(-j k ^ - r

4tt ^ o - - apert. r - r'

ds- (62)

and

J

4tt p go o apert.

/ (V- • M ) exp(-jk|r - r

ds (63)

These integrals have to be evaluated numerically.

The cavity admittance is obtained by combining (37), (43), and (49)

b 2

Y = A, Y. + jB. 1 in J (64)

It is also necessary to find the turns-ratio of the ideal transformer shown

in figure 7 for the dominant mode as well as the A- (i = 2,3...) for the

higher-order modes in (43) from (38). For an a x b rectangular waveguide, the

normalized ith mode function for TE mn modes (m+n/O) is [10],

where

e . =

i

e ome on

ab [(f) + 0

2

r

l/2

/- nir \ /muxf . s

. [x (^-) cos (-

) sin (—g-)

~ /mii> • ,mirX'' x <nTiZ' n + y (—) sin (-g—) cos (-g—)],

(65)

e om

/ 1, m = 0

2, in = 1 ,2, . .

(66)

15
The normalized ith mode function for TMnm modes (m/0 and n/0) is [10]

e

i J ab

t(^) + 0

2 rl/1

r /II ITT \ / 1 .[ X (—) cos (-

\

• t iittz \ -) si n (-—=-) ( 67 )

-z (t-~) sin (——) cos (—K—)]

In (65) and (67), x' = x + a/2 and z' = z + b/2. Substitution of (33), (65)

and (67) in (38) and integration yield the values of for all TE and TM

modes.

The wavenumber, k.j , of the ith mode can be expressed in terms of its cutoff wavelength X-j c and the correspondi ng cut-off wavenumber k^,

where

k /l - < kA ic>

2

• * < *ic

-jk.Jl - (x.

c /x)\ \>xu

( 68 )

(69)

(70)

The characteristic admittance for the ith mode is

k.j /oou 5 TE Modes (71)

we/k

9 1M Modes. (72)

Having found the values of A

n

- , k^ and for the dominant and higher-order TE

and TM modes, we are then in a position to determine Y b from (64) in

conjunction with (43) and (49). With Y

d and Y b known, the generalized voltage

V can be found from (34) using the value of I in (60).

16
Tne transverse electric field inside the cavity for the dominant TE 1Q

mode is

where

E = zVA C cos (—

)

It 1 a

sin k (d-z)

ttX , 1' sin k ^d

(73)

C (74)

The field distribution for the higher-order modes has the following

expressions

E.

it

E VA.e.

i ^ 1

l l

sin k . (d-z)

i sin k d

(75)

These field distributions can be computed for a rectangular cavity of given

dimensions, made of a material with known conductivity and having a slotaperture of specified width and length.

We now present some numerical results for the following aspects of a cavity-backed si ot-aperture problem at natural resonance: cavity Q, field

amplitude in the slot-aperture, and field distribution inside the cavity. The

effects of the cavity material on these results are considered. For

computational purposes we choose the following parameters:

Frequency

Wavel ength

Cavity dimensions:

Aperture dimensions:

f = 100 MHz

A = 3 m

a = 0.762 a

b = 0.339 a

d = A

g

/ 2 = 0.663a

L

x = 0.25a

Ly= 0.02 a

17
The cavity quality factor for the dominant TE-^q mode, Q, is calculated for

three different wall materials; namely copper, aluminum and iron, using (58)

with n = 1, which are given in Table 1.

Table 1

Cavity quality factor, Q, for dominant mode at

natural resonance, (x = 3 m )

Cavity material

Copper

A1 umi num

I ron

Conductivity, g

5.80x1

0

7 S/m

3.54xl0 7 S/m

0.95x1

0

7 S/m

Quality factor, Q

9. 19xl0 4

7 . 1 8x1

0

4

3. 72xl0 4

The electric-field distribution in the si ot-aperture at z = 0 is

approximately a half-cosine whose normalized amplitudes, |E |/E , are calculated in accordance with (73) for the three wall materials and tabulated

in Table 2.

Tabl

Normalized amplitude of

si ot-aperture (x=y=z=0)

Cavity Material

Copper

A1 umi num

Iron

_2

electric field in the

at natural resonance.

Normalized electric-field

ampl i tude ,|

$

| /E

Q

8.1 6xl0“ 5

10.45xl0“ 5

20. 17x10“'*

The amplitude of the electric field in the aperture is seen to be several

orders of magnitude smaller than that of the incident wave. It is most

interesting to note that the field there is almost exactly inversely

proportional to the square-root of the conductivity of wall material.

18
The amplitude of the electric field inside the cavity at y

= d/2 has also

been calculated for different wall materials. For the present example, it is

several orders of magnitude larger than that at the si ot-aperture ; it is more

than three times that of the incident wave; and it is practically independent

of the conductivity of the wall material. Table 3 is included to emphasize

these points.

Table 3

Normalized amplitude of electric field at the cavity center,

x=z=0, y= d/2 at natural resonance

Normalized electric-field

Cavity Material amplitude, |E |/E

Copper

Aluminum 3.29

I ron

Figure 8 shows a plot of |E lt |/Eg along the center axis x = z = 0, indicating

that it is at least possible to produce a high field strength inside the

cavity. Note, however, that the specific numerical results given thus far are

good only at the first natural resonance when the dominant TE^g mode exists

and the slot is not too long. For larger slots, more terms should be added in

(33) and the computations will be much more complicated.

19
4. LARGE BUILDINGS AND APERTURES WITH LOSSY GROUNDS— A HYBRID APPROACH

The analysis given in the preceding section is valid only for small

apertures and for the building (cavity) in free space. With buildings of

practical sizes for the purpose of storage, the buildings and apertures (doors

or windows) are usually large at VIH. Furthermore, the buildings and the

exterior transmitters are above an imperfect ground and the building walls may

be made of some reinforced material which can hardly be considered as perfectly conducting. In view of the outlines presented in Section 2, a rigorous analysis of a practical three-dimensional model based on the

Maxwell's equations and appropriate boundary conditions is certainly not a simple task. For this reason, a hybrid approach —a combination of equations

and measured data —together with some necessary simplifications and practical

restrictions is proposed in this section.

The field distribution inside an empty building is considered to be

produced by a direct component due to the source being operated outside the

building and a scattered component re-radiated by an equivalent set of surface

current densities induced on the building walls (including the floor and

ceiling) which would exist if the wall surfaces were infinitely thin and

perfectly conducting.

The geometry and coordinates used here are shown in figure 9, where the

building is hollow, situated on an imperfectly conducting ground, with only

one door (aperture) appearing on Plane m^m^m^m^ (xz-plane) or Plane 3. The

floor plane (m^m^nigm,-) is designated as Plane 1, the front (m^m^nigm,-) as Plane

2, the ceiling (m^m^m^mg) as Plane 4, the back (m^m^m^mg) as Plane 5, and the

left (rTigirigm^mg) as Plane 6. There is a vertically polarized signal from a transmitter located at x = D

c

, y

= -y

t

, and z = h^ , where D

c is the door

center at floor level as indicated in figure 9.

20
Tne basic equation needed here for calculating the scattered field is the

combination of (20), (21), and (22):

E

s (0 = - / J (r 1 ) G (r, r' ) ds

+ jdbl v / v

' • J (^') G (r» r') ds*

= . j

j ( r - ) G (f } f- ) ds '

- —^ V /

J (r 1 )

. v‘ G (r, r') ds 1 . (76)

After approximating the double integral in (76) with a double summation by

subdividing each of the six planes into m x n rectangular patches, assuming a constant unknown value for the surface current density on each patch and

expressing the final result into components, we obtain

j4„E (x,y,z) = mill 4mn G

mnl AXAy

m n

in n

( 3) xmn ~rnn 3 + coy l l J' ' G „ AZAX

- — III

I

(1)

1+JkRmnl

. xmn n

2 m n ’ R mnl

+ r(x-x’) LV + (y-y')<f *da ,Ig .&x&y

nr xmn ' n

y ymn inn 1 j mnl ^

_ J_ y y r(y-y') j(

2

) + (z-z 1 ) ] a 0 G 0 AyAz

we L L uy mn

v n

; zmn

J mn2 mn

m n

(j0 £ m n

nr ymn

1+jkR

-Z7ll\ 4mn

:^

!l + C( z -^J zTMn

+ ( x -^) J xTM>inn3

R mn 3

Gmn3 AZAX>

(77)

21
J 4TTE y

(x ,y

and

j 4tt E

z (x,

Z ) = “P I I ji

y2 Gmnl

m n J

* mil J y2 Gmn2 AyAZ

m n

- — l l \ J (1)

1 + [(x-xO J (1) +(y-y^)J (1) ]b Ig -I AxAy we L L

| ymn v nr xmn

J u' ymn J mnlj inn 1 J m n K mnl

- h {

J)mn ^2

mn2

+ C( y

- ym>

J imn

+ < z

' z

n

> J imn 3 b mn 2 \

Gmn2AyAZ

m n mn2

- ~ l l T ( z_z ') + ( X_XJ ) b 7]aza x , (78)

we m n m zmn V n xmn mn 3

j

v '

' z > - mil 4

2

„> Gmn2 AyAz

m n

( 3 ) + mil G

rnn3 AZAX

m n

- ~ y y (( x-x '') ^ + iy-y') c , G n AXAy

008 m n ^ xmn J

n ^mn inn l mn ^ ^

l + r (y _ r)J ( 2

) + (z _ z0J ( 2) we t £ 1 zmn ^2

J ymn v n' m n

(2)

UJkR

mn2

zmn zmn mn2] c mn2jGmn2AyAZ

— I l jj we

m n I

(3)

1+jkR mn3

zmn

p 2

K mn3

|(z-z^)J^

3

) + (x-x") J

^ 3 c 0 1 g q aza x , O nr zmn v n' xmn/1 mn3j mn3

( 79 )

22
where

a mnl = (x - X')

ITT s

i

3

^mnl = (y - y

n> s

i

3

c mnl

= zS

x

5

a mn2

= xS^ 3

^mn2 = (y - S 2

3

C mn 2 = (z - Z

S> S

2

3

d mn3 = (X - S

3

3

^mn3

ys

3

3

c mn3

= (Z - z m>

S

3

3

2 2

k R . mm

- 3 - j 3kR . mm

R . mm

( 80 )

(81)

k

fMHz

2n 2tt f

" X '300 MHz’

= operating frequency in MHz ( 82 )

23
and

x' = (m - 1) a x + x m

v ' o

x; = (n - 1) A x + x

Q

y‘ = (m - J 1) a y + y

m

v ' o

yn = (n - 1) A y + y

0

m,n = 1,2,3 (83)

x

Q

, y

Q

, and z q being the possible distances from which the patches are counted. Note that, for simplicity, we have dropped the subscript s in (77)

through (79).

A detailed explanation for (77) is now in order. The quantity E x on the

left-hand side represents the x-component of the scattered complex field

strength inside the building at a specified location (x, y, z) contributed by

the induced surface currents. In reality, this is supposed to be known as it can be obtained by measurements. Let the amplitude and phase be denoted by

V , and ^ for the measured voltage due to the x-component of the total field

at (x, y, z). The amplitude of the x-component of the total field itself is

then |E. | = V /i , where the effective length of the dipole, % , is

approximately equal to (0.6-0. 9) of the actual dipole length, depending on the

frequency. The phase information can be secured by using a reference

antenna and a vector voltmeter. This implies that the x-component of the

V total If E. represents the incident field from the

source, and E scattered field in (76), the total field becomes

s

v '

E t

E.is+ E , or E = E - E.

s t 1 (84)

24
The x-component of (84) is, therefore.

E = x ( 85 )

where E^ x has been neglected in view of the fact that the incident field is

assumed z-polarized.

Separating (85) into the real and imaginary parts, we have

V Re E x

= cos

*x » (36)

X and

V Im E = — si n . X t

x (87)

The first term on the right-hand side in (77) is the x-coinponent field

contributed by the x-component of J (r') on the floor (xy-plane where z

1 = 0,

or Plane 1) with representing the x-component of the source concentrated

t h at the center of the mn

Ln

patch in Plane 1, Gmnl = exp(-jkRmnl )/Rmnl , 2 2 2 1 /2 Rmni

= [(* - x') + (y - y„) + z ] and (xj, y'. 0) representing the

location of the mn

L patch. The second term on the right-hand side is the

m

contribution due to the x-component of the surface current, J^, on Plane 3

(xz-plane) where Gmn3 = exp(-jkRmn3 )/Rmn3 , R

fnn3 = [(x - x^)

2 + y

2 +

2 1/2

(z - z') 1

7

. The first two terms on the right-hand side of (77) are,

' nr therefore, directly from the first term of (76).

The remaining three terms in (77), derived from the second term of (76)

after performing the necessary mathematical operations of divergence ana

gradient, represent contributions from various components of the surface

currents on Planes 1, 2 (yz-plane), and 3 (zx-plane). Similar explanations

for the various terms in (78) and (79) can also be made.

In (78), we note, for the left-hand side, that

E y

9 ( 88 )

25
yielding

V Re E =

_ —y COS tf, *y y

and

Im E —— sin \ii ix ,

( 89 )

where and ^

are respectively the measured amplitude in volts and phase in

degrees due to the y-component of the total field, and % = % . Once again,

y x

E. has been omitted.

iy

In (79), we note, for the left-hand side, that

V

E = E. - E. =

Z tz 1 z

i, - E. r

z 1Z (90)

where V

z and ^ are the respective measured voltage amplitude in volts and

'z ~x

the z-component of the incident field above a lossy ground is approximately

phase in degrees due to the total z-component of the total field, and

the z-comi

given by:

730 PG

E. =•! 1 + R exp

iz 1 v

K [-jMr r

r)]]' exp (j k r) exp (jkr

0

) (91)

where

r

1

h

input power to the transmitter, in watts,

transmitting antenna gain (numeric)

distance between the transmitter at (D , -y^, h

fc ) and the

measurement point at (x, y, h ) 2 2 m 2 1/2

[(x-D

c

) + (y + y

t

) + (h m

- h

t ) J

7

, in meters,

distance between the transmitter image and the measurement point

2

i 1 /

2

(<

r =

R,, =

x-E> c

) + (y + y

t

) + (h m

+ h t ) J

7

, in meters,

feed point of the measuring antenna above ground, in meters,

distance between the transmitter and the reference antenna, in meters,

Vertically polarized ground reflection coefficient

(cose - n ' )/ (cose + n'

)

, (92)

and

90° - tan' h

\ m

h t )/ /(x - D

c

)

2 + (y + y t

)

2J

. (93)

26
In (92), n is a complex factor involving the ground constants, which may be

determined empirically by measuring the z-component field strength called E' at a convenient point. This convenient point may be at the door aperture (D , 0, h ) when

the door is electrically large. Thus,

E' = | E

1

1 - z

1 z no

Az = E1 + R

v e xp(-jkrJ + jkr')] 7 ^—^ exp(-jkr')exp(jkr

o ), (94)

where

/

2 2 'y

t + (h m

- h

t ) ,

r' = Jy i h+ < h m

f h

t

>

2 (95)

From (94) an approximate value for R

v (here called Rj) at this particular convenient

point may be determined. The angle e at this particular point becomes

S' = 90» . tan1 ((hm

- h

t )/yj (96)

Substituting (96) and R^ obtained from (94) into (92), we can compute

n' = cose' (1 R')/(l + R 0 • V

7 v V7 (97)

With n

1 so computed, the ground reflection coefficient R

v for a general measurement

location can be determined from (92) and (93).

Note that in (77) to (79), the correspond!

- ng contributions from Planes 4,

5, and 6 have been ignored in order to reduce the number of equations

involved. This can be justified as long as the measurement data to be used

for E , E , and E are taken from the region near the origin m

1 (see figure

x y z i 9). Of course, when the measurement data taken from the region near the corner m? are to be used. Plane 5 (back) will be more important than Plane 2

L (2)

(front). Under this ci rcumstance, we replace in (77) to (79) by

,( 5

) i( 2

) zmn

b* c mn5 = < z

Jymn’ J zmn

by J zmn’

a mn2 by a TMn5

“ W ' S 5’ b mn2 by b mn5 ' y

.n

S

5

c

-nn2

2 2

z n) S

5

5 S

2 by S

5 R mn 5

3 " j 3kR

mn5^

/R mn5 ’ and R nn 0 by

R mn5 = ((X - W)

2 + (y - y') + (Z - Z')J1/2 where W is the width of the

building (see figure 9).

27
When the measurement data are taken from spaces where contributions from

(4 \ all the planes are considered equally important , the contributions by J^'and

from Plane 4 (ceiling), and from Plane 5 (back), and

ymn

x ymn zrnn

v '

and from Plane 6 (left) can be easily added to (77) through (79),

with similar definitions for amni , bmni , cmni , Rmni , and Smni ,

i = 4, 5, and

6. Of course, the number of unknowns will then be doubled.

When the assumption made in (77) through (79) is justified, there will be

six different complex surface currents with two from each plane, yielding 1 2mn

real unknowns. The left-hand side of (77) through (79) gives six data values

(ReE.j and ImE^, i = x, y, and z) at each measurement point. Thus, if p is

designated as the number of locations at which measurements of three

orthogonal components (both amplitude and phase) are made, there will be 6p

equations. Then, we require

6p > 12 inn or p > 2 mn , (98)

in order to be able to solve for the system of equations by matrix inversion.

In (98), we have assumed that the same number of patches (mn) are considered

for each of the three planes. On the other hand, should a different number of

patches be taken into account for different planes, then we have,

6p > 4 jm (1)

n (1)+ m

(2)

n (2)

+ m

(3)

n (3)

]

,

or

p >| +m< 2 >n(

2

> +m( 3 >n<

3 >] , (99)

where m^and n^) are the number of patches considered from Plane i,

i= 1, 2, and 3.

28
It should be emphasized that although (77) through (79) are important for

computing piecewisely the unknown surface currents with a set of measurement

data taken at some points near the planes of interest, the same equations are equally useful for computing the field strengths at points other than those

data points as soon as the unknown surface currents are determined. In this

latter process, the computation is strai ghtforward and does not involve matrix

inversions. For this reason, this latter computation is designated as the

forward computation.

Naturally, the larger the number of patches to be considered at a time,

the more accurate are the numerical results. The actual number of patches

used is ultimately determined by compromising the required accuracy and the

computing capacity and cost. Since there is no a priori knowledge about the

accuracy of the results, we recommend to begin with a small number of in and n, and then proceed to obtain a solution for the surface currents. These

currents can, in turn, be used to perform a forward computation, also using

(77) through (79), for the field at different locations where the measured

data are also available. A comparison of the computed and measured results on fields should reveal the degree of accuracy by the procedure so outlined. If

the accuracy is at an acceptable level, the originally assumed small number

for m and n is considered satisfactory. Otherwise, a large number for m and n has to be used to improve accuracy.

This process may be verified by on-site measurements if a small computer

with reasonable capacity is available.

In principle, whenever the equal sign holds in (98) or (99), a unique

solution is always obtainable. In practice, however, we do prefer to have p larger than the minimum required such that the number of equations is larger

than the number of unknowns. Under this condition, the solution obtained in

the sense of least square fit will minimize large deviations which may result

because of the potential errors contained in the measured data.

Z9
With an accurate and approximate set of piecewise solutions for the

surface currents obtained from (77) through (79), we then use them to compute

the field strength everywhere else inside the building so that the

el ectromagnetic environment of the entire building may be assessed. In general, the space near the door (when it is open) for a large building and

door should experience relatively high field strengths. Therefore, careful

and regular measurements should be made for the space close to Planes 2, 3,

and 5 in order to obtain more accurate results for the surface currents on Planes 1, 2, 3, and 5 near the door, which are important to compute the field

strengths for the front space near the door. With such a hybrid analytical

procedure one can avoid a large scale of tedious and systematic measurements,

which would normally be required. Sample numerical results using this hybrid

procedure are given in Section 7.

30
5. MEASUREMENT REQUIREMENTS AND PROCEDURES

The equipment required for measuring the field strengths consists of

three orthogonal dipoles, a pole for supporting the antennas, an antenna

selector switch box, a vector voltmeter, a three-channel digital

voltmeter/printer, and a reference antenna (monopole), connected as given in

figure 10. A movable platform on which the antennas and the selector switch

box are mounted is also required. The digital vol tmeter/pri nter prints out

three amplitudes in volts and three phases in degrees.

The measurement procedures may be outlined as follows:

(1) Select an origin for the building as shown in figure 9. Identify

x, y, and z directions.

(2) Measure the building width W, depth D, and height H

} all in

meters. These may be obtained from the base where the building is

being investigated. When H is larger than 5 meters, it may be

ignored since the contribution from Plane 4 (ceiling) may be

del eted.

(3) Measure the width and height in meters of the door. Determine D

c (see figure 9) in meters.

(4) Divide the floor (Plane 1) into regular grids by drawing straight

lines parallel to the x- and y- axes such that ax and Ay are x/4 or

less, where x is the operating wavelength. Let ax = Ay although

they are not required to be equal in principle. Record ax and Ay

in meters. Observe the intersecting points. Also choose az.

(5) Connect the measurement equipment as shown in figure 10, with one antenna in x-direction, another in y-direction, and a third in zdi recti on (vertical).

(6) Record the feed height h of the measurement antennas above the

floor, in meters.

(7) Set the transmitter outside the building at x = D

c

, y

= -y

t

, and z

= h t

. Record y t and h

t in meters.

31
( 8 ) Set the reference antenna at a convenient location outside the

building. Record the distance r Q

in meters between the transmitter

and the reference antenna.

(9) Record the frequency f in MHz of the transmitting antenna which is

assumed to be vertically polarized (E^

z only), the input power P in

watts, and the antenna gain G (numeric).

(10) Determine x

Q

, yQ

, and z Q

in meters required in (83). This is

necessary because of the size of the moving platform on which the

measurement antenna is mounted.

(11) Mark the first intersecting point [obtained in (4)] near the origin

where measurements are possible as position (1,1). Number the next

intersecting points toward the +x-direction as (2,1,), (3,1) etc.

to form the first row. Similarly, number the next intersecting

points toward the +y-direction as (1,2), (1,3), etc. to form the

first column.

(12) Make as many columns as possible between Planes 2 and 5. Make as many rows as possible up to y

= D/2.

(13) One person is assigned to the transmitter to turn on and off of the

transmitting signal as instructed.

(14) Another person is assigned to record the signal from the printer.

(15) A third person is assigned to align the feed point of the

orthogonal dipoles to position (1,1). Once done, he asks the

transmitter operator outside the building to key on the signal,

walks away from the measuring antennas, and asks the printer

operator to record the amplitudes and phases of the three component

voltages (V , V , V , ^ , 4/, 41 ) in volts and degrees

a y z x y z respectively, together with the measurement position.

32
(16) Once the signals are recorded, the person assigned to move the

measuring antennas asks the transmitter operator to turn off the

signal. He then moves the measuring antenna to position (2,1),

signals the transmitter to be turned on, and repeats the process.

(17) Take and record at least 10 rows and 5 columns each along Planes 2 and 5.

(18) Take and record a few points around the center of the building (x

W/2, y

= 0/2).

(19) Take and record a few points at the door aperture (y = 0).

(20) Raise the antenna feed point by x/4, and repeat the same process

starting position (1,1).

33
6. COMPUTING REQUIREMENTS AND PROCEDURES

This section contains a discussion of four computer programs used to

solve (77), (78), and (79) given in section 4. The main part of the

discussion deals with two FORTRAN IV programs, one that computes the currents,

J , from the known measured field strengths, and another that computes field

strengths, E, from given currents. A second pair of programs, using HPL

language, is based on the FORTRAN programs but is designed for a smaller

desktop computer.

When computing currents, a 6p-by-12mn matrix is formed, which can become

large if p, m, and n are large (p, m, and n are defined in section 4). Thus

the computer's central memory storage capacity requirement will be large. For

example, let m and n be equal to 2. This defines a 2 x 2 x 2 -point

measurement cell which we consider to be the smallest data set that will

produce meaningful results. The matrix size is 48 x 48 elements. To compute

the currents in Planes 1, 2, and 3 requires a central memory containing

approximately 25,000 storage locations. If m and n are equal to 6 (i.e.,

a 6 x 6 x 2 -point data set), the matrix size is 216 x 120 elements and

approximately 250,000 storage locations are required. For a computer that is

too small to execute this program even for the smallest (48 x 48) matrix, the

program is divided into two parts as described in section 6.3 below.

When computing field strength, no matrix operations are required, and the

required amount of storage does not strongly depend upon the data set size.

Approximately 20,000 storage locations are adequate for a 6 x 6 x 2 -point

data set.

The actual computer size required will depend upon the individual

computer system's archi tectural details such as compiler design, word length

capacity, exponent range, on-line storage capability, etc. The large computer

used to execute these programs at NBS has the following characteri sti cs

e.

a.

d.

c

b.

NOS operating system

400K words (octal) central memory capacity

60-bit word size

±322 exponent range

~10 7 instructions per second

34
6.1. CURRENT Program (FORTRAN)

The FORTRAN program used to solve (77), (78), and (79) for the induced

currents in Planes 1, 2, and 3 is given in Appendix A, section A.l. In its

present form, this program contains a large amount of redundancy as a result

of the decision to mimic the original algebraic equations of the problem.

This allowed the program to be proof-read for accuracy by project personnel

who have only a casual acquaintance with FORTRAN language. For a large

program such as this one, much debugging cost can be saved by this approach.

A more efficient program could be written, but little overall machine savings

would result because the vast majority of computer memory is consumed by the

6p-by-12mn matrix. The program as shown is considered adequate for the

present purposes.

The following background discussion will be helpful for understanding

this program.

Equations (77), (78), and (79) can be written in matrix notation as

fol 1 ows

[E] = [A] [J] , ( 100 )

where [E] is a u-vector, [ 3 ] is a v-vector, and [A] is a u-by-v matrix. In our case, [E] is a 6p - element column matrix (u = 6p),

[E]

P

E (x, y, z)

X

E (x, y, z)

y

E

z (x, y, z)

[ 101 )

and [ 3 ] is a 12mn - element column matrix (v = 12mn), The elements in matrix

[A] are the coefficients of the currents J as found in (77), (78) and ('9'.

The solution of (100) for the currents, [ 3 ], makes use of the "OR

factorization" [11] of matrix [A], thus converting (100) into a for 1 that c a

be solved directly by back-substitution [

1 2 j . In the QK fac t. or i z a t. i o e t a d ,

an orthogonal matrix, [Q], and an upper triangular matrix, [R]. are e

such that

[Q] [R] = [A]. ( 102 )

35
An orthogonal matrix has the property.

[Q]

t [Q] = [I], ( 103 )

where [Q]

T

is the transpose of [Q], and [I] is the unit (or identity) matrix.

[I]

10 0 .. . 0 10 ...

0 0 .... ....00

...010

...001

(104)

An upper triangular matrix has the form.

[R]

r r r r r . o r r r r . o o r r r . ooo r r . etc.

J

(105)

in which elements r^j are equal to zero when i > j.

From (102) and (103)

[R] = [Q]

T [A], (106)

Multiplying (100) through by [Q]f and applying (106) gives

[Q]

T [E] = [Q]T [A] [J]

= [R] [J].

[107)

Because [R] is upper triangular, (107) is easily solved for the elements

of [J] by back-substitution, starting with the last (12nm-th) element. Thus,

J 12mn

(q E)

r r6p,

6p

12inn

(108)

36
where J is the last element in 12mn is the last element in [3], (q E) 6p

[Q] [E], and

^ 2mn

is the correspondi nq element in [R]. Having found

we next e i emeirt ^I2mn-1’ ^12mn-2 anci so on un1: ii a i 1 elements

in [3] have been found.

The solution thus obtained is a least square solution [13] of the system

of equations.

The program as shown in section A.l is based upon the following assumptions

a. The field strength measurement data are stored in a computer file in

the specific ordered sequence as shown in Tables 1 and 2, section 7. Each

data value occupies a specific position in the data grid that corresponds,

poi nt-for-poi nt , with the measurement grids discussed in section 5.

b. The intervals ax and Ay , between the data points, are constant in

both the X- and Y-di recti ons . They are designated DX and DY in the program,

and may be either equal or not equal to each other.

c. Measurement data are obtained at two levels above Plane 1 (floor).

The lower level height above Plane 1 is identified as ZSTART, and the upper

level is OZ above ZSTART.

d. Out of the complete set of measurement data, the number of data point

locations in the X-direction at the lower (first) level that are used in these

computations is 2. This is identified as NXTOTH(l) = 2, which freely

translated, reads, "the TOTa l J^umber of _X-di recti on data locations in the

first level Held for computation use is equal to two." Similarly,

NXT0TH(2) = 2 identifies that there are two X-direction data locations at the

upper (second) level used in the computations. NYTOTH(l) and NiT0TH(2^ r sfer

to the number of Y-direction data locations in the first and second levels,

respectively, used in the computation. Bear in mind that there are six data

numbers at each location; viz., two each (the real and imaginary partsl t e

three (X, Y and Z) orthogonal directions.

e. Only the currents in Planes 1, 2 and 3 are calculated. his is

keeping with the discussion in section 4.

37
Assumption a allows the data at any desired grid location to be used in

the computation. The computer selects the desired data according to program

statements beginning with the letters IXH and IYH. This is discussed further

below.

The consequence of assumption b is that the coordinates of the data used

in the computations are calculated by the computer for use in the program

rather than being provided to the computer as input data. The formulae for

coordinate computation has the form,

x = (m - 1 )ax + x(start) , (109)

where m is an integer-index (m > o) that specifies the desired data point.

Assumptions c and d are arbitrary and can easily be changed if desired.

We have executed this program for data sets as large as six (6) in each of the

X- and Y-directions (i.e., 6 data point locations in the X-direction, 6 in the

Y-direction, and 2 in the Z-di recti on ) . If measurement data had been taken at more than two levels, a simple modification of the program would allow it to

be used. However, the maximum size of the data set is limited by the

computer's central memory storage capacity as stated above.

The consequence of assumption e is that the program simply omits

statements that would be used to calculate currents in Planes 4, 5, and 6.

These statements can be added as desired, using the same basic code as used

for Planes 1, 2, and 3. We have, for example, a second program, like the one shown in section A.l, that calculates currents in Planes 1, 3, and 5. For discussion, the program may be divided into five parts as follows:

1. Lines 1 through 56 contain basic information such as variable

definitions, dimension statements, values of constants, and selectable

parameter values.

2. Lines 57 through 237 accomplish the computation of the elements of

coefficients matrix [A]. These computations, for the most part,

involve the geometry of the problem, i.e., the locations of the

currents relative to the locations of the points at which field

strength data were measured.

3. Lines 238 through 244 deal with subroutine CQRDCT by which the QR

factori zation of matrix [A] is accomplished.

38
4. Lines 245 through 309 construct matrix [E] by reading in the

measurement data from the data file and modifying it according to (90),

et seq., of section 4.

5. Lines 310 through 359 (the end of the program) accomplish the solution

of (100) for the currents by use of subroutine CQRSLT, which is the

back-substitution procedure for solving (107).

4 finely detailed discussion of this program will not be given because

familiarity with FORTRAN programming is assumed on the part of the reader.

However, certain features should be briefly described. These are grouped

according to the five parts of the program listed above.

Part 1 (lines 1 through 56). The statements that begin with the letters

IXH and IYH determine which data are used in the computation of currents.

They do this by defining an index that is used at various places throughout

the program to select the desired data values (from the data file) and to

compute their coordinate locations.

These statements have the form,

IXH(a,b) = c, (110)

where a, b and c are integers. The letter I means that this is an index. X means it pertains to the X-direction. H means that the index is held in

memory for use as needed. Integer a is an ordinal number signifying the

sequential order of this index. For example, if a = 1, then this is the " i rst

index of its particular kind that will be used in later do-loops. Integer b

refers to measurement grid level. If b = 1, the index refers to data in the

first (lower) measurement grid; b = 2 refers to the second (upper) grid.

Integer c is an ordinal number that refers to the row or column in the chosen

measurement grid where the desired data point is located. For example, if c =

1 in eq. (110), the index refers to the first data point in the X-direction.

39
Here are two examples. In words.

IXH (1,2) = 3

reads, "the first data point used by the do-loop from the second (upper)

measurement qrid is the third point in the X-direction in that measurement

grid." Similarly,

IYH (2,1) = 9

reads, "the second data point used by the do-loop from the first (lower)

measurement grid is the ninth point in the Y-direction in that measurement

grid."

The user may change the integer c to select any particular sets of data

that he desires to be used in the computation of currents. The computer will

calculate the coordinates of those data and use them whenever they are required in the program.

As mentioned above, the program printed in section A.l is written to call

two data points from each level in each of the X- and Y-di recti ons . If more

data points are desired, suitable IXH or IYH statements are added. For

example, if a third data point is to be added from the first level measurement

grid in the X-direction, this statement would be added:

IXH( 3, 1) = c, (111)

where c is defined above. In this case the NXTOTH (1) statement must be

changed to read

NXTOTH ( 1 ) = 3,

since now there are a total of three data points in the X-direction that will

be incorporated by the program. The number to the right of the equals sign in

the NXTOTH and NYTOTH statements must be the same as integer a in the

correspondi ng IXH and IYH statements.

40
— rt 2 (lines 5 7 through 237). Coordinates of field locations within the

building and of current locations in Planes 1, 2, and 3 are calculated in this

part of the program. The former are designated by the letters X, Y, and Z,

and the latter by XP, YP, and ZP.

A;->o calculated here are the constants a • , b • c • G • R anH mm 5 mm 5 Lmm » hnni’ K mni 5 dna

S

i# which are defined in section 4.

Using this information, the computer calculates current coefficients CJX,

,.JY, and CJZ. These coefficients are structured by the computer as the

elements of matrix [A]. In general , the coefficients are complex and are comprised of several terms. For example, the coefficient of the

current is, from (77),

CJX1

mn 4tt 4tTU)£

1+jkR

mn 1

R

1,11 1

, t „ \ o + (X-X" )a , 2

v m ' mnl

mnl

G

mnl

^ ( 112 )

However, each coefficient reduces to a single complex number pair.

For NXTOTH(l) = 2, NXT0TH(2) = 2, NYTOTH(l) = 2, and NYT0TH( 2) = 2,

and

p

- 2 + 2+ 2+ 2=8,

m = 2,

n = 2,

6p = 6 x 8 = 48,

12mn = 12 x 2 x 2 = 48.

Thus [A] is a 48-by-48 element matrix.

Although this part of the program appears to be lengthy, it actually

contains a basic code that is repeated three times, once each for Planes 1, 2,

and 3.

Part 3 (lines 238 through 244). Subroutine CQRDCT is shown in section

A. 2. This is a compact form of a public-domain program called CQRDC, which

stands for Complex QR Decomposition [14]. CQRDCT was written especially fo r -

this project by P. N. Swarztrauber [15]. Consult reference [14] for a description of this subroutine.

Notice that the symbol A is used for two different matrices, neitner of

which is the matrix [A] discussed above. First, it defines a doubly

subscripted complex array that contains matrix [A] of (100). Secono ,

: defines the upper triangular matrix [R] of (102) used in the solution o'

( 107 ).

41
Part 4 (lines 245 through 309). The measurement data is read from the data

file (line 288), converted to field strength by dividing the voltage readings

by £ , and corrected for the incident field per (90). The three components of the

scattered field are structured as the elements of matrix [E].

Part 5 (lines 310 through 359). Subroutine CQRSLT is shown in section A. 3.

This is a compact form of a public-domain program called CQRSL, which stands for

Complex QR Solver [14]. CQRSLT was written especially for this project by P. N. Swarztrauber [15]. Consult reference [14] for a description of this subroutine.

The solution to (107), and thus the solution to (100), is written out as the

12mn elements of matrix [3]. These are designated by the symbol XS.

6.2 FIELD Program (FORTRAN)

The FORTRAN program used to solve (77), (78), and (79) for the field

strengths E, given the induced currents, 3 , is given in Appendix A, section

A. 2. This program employs much that is used in the program listed in section A.l,

and like that program, it was written for greatest proof-readi ng ease rather than

for greatest efficiency.

The form of the program is based upon the same five assumptions listed in

Section 6.1 with the following exceptions:

a. Input data is the current matrix [ 3 ], which is stored in a computer file

in a specific ordered sequence that corresponds, poi nt-for-poi nt, wi th the

projections of the measurement data locations.

b. The locations of the computed field strengths can be at any arbitrary

points chosen by the user. The discussions of this program is divided into the following four parts:

1. Lines 1 through 54 contain the same basic information discussed in section

6 . 1 .

2. Lines 55 through 127 read the input data from the computer file that contain

matrix [ 3 ], and prints out the matrix elements for verification.

3. Lines 128 through 289 compute the coefficients matrix [A], and then solve

(100) for the elements of field strength matrix [E],

4. Lines 290 through 349 (the end of the program) add the incident field per

(90), and print out the computed fields strength at the selected field

point(s).

42
A brief discussion of this program follows:

Part 1 (lines 1 through 54). This part of the code is identical to its

counterpart in the preceding section, except that the coordinates of the point

at which the user desires to compute the field strength are defined as XFP,

YFP, AND ZFP. They may be changed at will.

Part 2 (lines 55 through 127). The input data are those values of 3

that are selected according to their coordinate locations. These in turn,

are determined by the indices IXH and IXY as discussed in section 6.1. The

elements of the current matrix [ 3 ] that is formed from the input data are defined by the symbol XS.

The coordinates of the current locations in Planes 1, 2, and 3 are calculated by the same code discussesd in section 6.1. They are designated by

the letters XP, YP, and ZP.

Part 3 (lines 128 through 289). Constants a mni , bmni , cmni , Gmni , Rmni , and S.j , and the current coefficients CJX, CJY, and CJZ are calculated by the same code discussed in section 6.1. A major difference in the two programs

occurs at this point; namely, this program computes the elements of matrix [A]

and solves for the elements of matrix [E] point-by-point without the need to

construct matrix [A] in toto. This greatly reduces the central memory size

requirements as compared with the program of section 6.1. At this point, the

incident field contribution has not yet been added in.

Part 4 (lines 290 through 349). The incident field strengths are calculated by the same code discussed in section 6.1, and used to correct the

computed fields per (90).

6.3 CURRENT Program (HPL)

The HPL program used to solve (77), (78), and (79) for the induced

currents in Planes 1, 2, and 3 is given in Appendix B, section B.l. It is

divided into two parts so that a small desktop computer can he used. Part 1 computes the elements of the coefficients matrix [A], and performs the OR

factorization given by (106) and (107). Part 2 computes the elements of the

current matrix [ 3 ].

This program is based on the FORTRAN program discussed in section 6.1.

It uses the same computational approach, and the same assump, ions non > ov-"

for assumption a (pertaining to the input data). In the HPL P'oora ,

coordinates of the data points that are to be used in the co 'nut a*, o a e

first manually entered in accordance witli prompting instructions o

1 o \ i *.o > . o \

43
the computer. Later, the field data values are manually entered, again

following the computer's prompting. Having obtained the data in this way, the

computer calculates the solutions to (77), (78), and (79) by the same methods

discussed in section 6.1. These current values and their coordinates are stored on the HPL program tape for subsequent use by the FIELD program

described in section 6.4, below.

A detailed discussion of the design of this program will not be given

because (a) familiarity with HPL programming is assumed on the part of the

reader, and (b) the general design of the program is closely similar to the

corresponding FORTRAN program. Refer to the discussion given in section 6.1

for information on these programs.

Variable assignments are given in section B.2. Instructions for

executing the HPL CURRENT program are given in section B.3.

6.4 FIELD Program (HPL)

The HPL program used to solve (77), (78), and (79) for the field strength

at any arbitrary location within the building, after the currents are obtained, is also given in Appendix B, section B.l. It is based upon the

FORTRAN program discussed in section 6.2. It uses the same computational

approach, and the same assumptions apply. In the HPL program, the coordinates

of the field points are manually entered in accordance with prompting

instructions provided by the computer. The current values and their

coordinates are obtained from the CURRENT program (see section 6.3, above).

The general design of this program is closely similar to the

correspond! ng FORTRAN program. Refer to the discussion given in section 6.2

for information on these programs.

Variable assignments are given in section B.2 (same as for HPL CURRENT

program). Instructions for executing the HPL FIELD program are given in

section B.3.

44
6.5 HPL File Structure on Magnetic Tape

Both the HPL CURRENT and the HPL FIELD programs are on a single magnetic

tape. They are structured such that (a) each program may be run independently

of the other, or (b) the two programs can be run in uni nterrupted sequence.

However , the CURRENT program must have been run before the FIELD program can

be run alone because the latter requires computed information from the

former . Once the CURRENT program has been run and its output stored on the

taoe, the FIELD program can be run any number of times to compute the field

strengths at any number of arbitrary field points. Each of these field

computations will pertain to the one set of currents computed by the CURRENT

program, of course.

HPL CURRENT program, Part 1, is stored in file #1 of the magnetic tape.

CURRENT program. Part 2, and the FIELD program stored in files #2 and #3,

respectively. Computed currents and their coordinates from the CURRENT

program are stored in file #4.

6.6 Print Options for HPL Programs

The two HPL programs are provided with a means to either permit or prevent the printing out of certain numerical information that is available

within the computer. This is accomplished by the use of flags. The user can determine whether or not to print this information by modifying the programs

before running them. Some of the printable information is useful primarily as

an aid in diagnosing the cause of problems and would not normally be printed

out in a routine run. Other information, such as the computed currents J, may

or may not be needed in hard-copy form. Printing out any information

lengthens the time required to run the programs.

The print option applies to the following information. The procedure for

setting the print option flags is described below.

1. Flag 1 controls the printing of the six constants.

a. OMEGA = 2*PI*FREQ, where FREQ = test frequency

b. MU = permeability of free space

c. EPSILON = permittivity of free space

d. KAY = OMEGA*|/(MU*EPSILON)

e. 0MS4P = 0MEGA*/MU/(4*PI)

f. S04PE = 1/(0MEGA*4*PI*EPSIL0N)

45
2 . Flag 2 controls the printing of the A coefficients.

3. Flag 3 controls the printing of the quantities u and v defined in

section 6.1. In the HPL programs, u and v are designated by MA and NA,

respecti vely

4. Flag 4 controls the printing of the diagonal of the matrix [R].

5. Flag 5 controls the printing of E

z

, defined by (90).

6. Flag 6 controls the printing of the residual of the least squares

approximation involved in the solution of (100).

7. Flag 7 controls the printing of the currents J.

Items 1, 2, 3, 4, and 6, above, are primarily useful for diagnostic

purposes, and are not normally printed out. E

z (item 5) may be of interest in

certain cases. The current J (item 7) may be printed out to obtain an idea of

the distribution of the induced currents in the walls of the rectangular

box. These currents are, of course, the output of the CURRENT program, and

they are input data to the FIELD program. Thus they may be printed out when

it is desired to verify that the same set of values appear in both programs.

In the CURRENT program, statements 6 through 12 of Part 1 pertain to

these flags. In the FIELD program, statements 6, 7, and 8 apply. To prevent

printing, the syntax following the colon is "sfg"; to permit printing, the

syntax is "cfg". The program listing given in section B.l show the flags set

to prevent printing of all the items listed above.

46
7. NUMERICAL RESULTS FOR A PRACTICAL BUILDING

The result for a building at a California site is hereby presented.

Appropriate input data are given below.

W

D

H

Door size

*t

P

G

f

X

AX

X

0

y0

= building width = 8.763 m.

= building depth = 24.232 m.

= building height > 5 m. (The ceiling plane may be neglected).

= 4.267 m by 3.658 m. (el ectri cal ly large).

= door symmetrical point = 3.048 m. (The door is not

symmetrically situated on Plane 3, however).

= horizontal distance between the transmitter and the

door = 9.906 m.

= height of the transmitter above ground = 1.219 m.

= input power to the transmitter = 45 W.

= transmitting antenna gain = 0.3, including the ground effect.

= operating frequency of the transmitter = 148.1 MHz

= correspond!

- ng wavelength = 2.026 m.

= height above ground of the feed point of the measuring antenna

1.4478 m. or 1.9050 m. (two antenna heights)

= Ay = size of the grid = 0.6096 m. az = 0.4572 m.

r distance of Position (1,1) from Plane 2 = 0.9144 m

I distance of Position (13,1) from Plane 5 = 0.5334 m.

= distance of Position (1,1) from Plane 3 = 1.0668 m.

47
z Q

= distance from the floor plane to the edge of the first grid on

Planes 2 and 5=0

r n = distance between the transmitter and reference antenna = 9.684 m.

There are 16 rows and 13 columns. The actual measured data are tabulated as

fol 1 ows

Table 4

+x

vy *y

V

Z *7

Position in volts in deg. in volts in deg. in volts in deg

(i.i) 0.040 -171.4 0.030 -46.3 0.030 107.3

(2,1) 0.034 -145.6 0.013 -25.0 0.090 116.6

(3.1) 0.009 -97.4 0.022 168.5 0.146 110.1

(4,1) 0.014 41.7 0.024 -104.6 0.132 101.2

(5,1) 0.009 68.5 0.041 -73.0 0.088 116.9

(9,1) 0.010 81. 0.014 -116.3 0.010 -90.4

(10,1) 0.009 95.2 0.009 -175.4 0.011 -157.5

(11,1) 0.009 56.3 0.010 135.8 0.015 18.9

(12,1) 0.006 39.7 0.005 42.4 0.011 3.0

(13,1) 0.003 -24.5 0.011 -14.1 0.012 -158.5

48
( 1 . 2 ) 0.025 101.8 0 .

( 2 . 2 ) 0.044 118.7 0 .

( 3 , 2 ) 0.026 87.7 0 .

( 4 , 2 ) 0.010 - 20.5 0 .

( 5 , 2 ) 0.006 - 131.9 0 .

( 1 , 2 ) 0.008 40.7 0 .

( 10 , 2 ) 0.009 54.7 0 .

( 11 , 2) 0.007 116.7 0 .

( 12 , 2 ) 0.006 139.8 0 .

( 13 , 2 ) 0.005 96.3 0 .

( 1 , 3 ) 0.011 - 23.7 0 .

( 2 , 3 ) 0.029 39.5 0 .

( 3 , 3 ) 0.018 10.2 0 .

( 4 , 3 ) 0.015 - 111.7 0 .

( 5 , 3 ) 0.015 - 158.4 0 .

( 9 , 3 ) 0.019 - 34.4 0 .

( 10 , 3 ) 0.020 - 27.0 0 .

( 11 , 3 ) 0.006 - 70.6 0 .

( 12 , 3 ) 0.008 - 153.5 0 .

( 13 , 3 ) 0.008 - 82.3 0 .

- 110.6 0.043 - 24.0

87.1 0.044 - 3.2

101.2 0.112 1.6

165.4 0.131 - 9.7

- 154.8 0.099 5.4

- 100.6 0.012 - 133.7

54.6 0.027 133.2

132.1 0.024 42.4

- 35.5 0.005 22.0

- 20.6 0.020 178.1

167.7 0.048 - 112.5

43.1 0.083 - 122.9

29.4 0.127 - 111.3

22.5 0.140 - 106.9

59.9 0.096 - 112.2

161.2 0.039 - 174.4

81.1 0.027 120.4

130.5 0.027 - 35.7

- 72.3 0.022 - 103.7

- 37.6 0.025 166.7

,030

,034

,056

,025

,023

010

007

009

002

009

034

039

047

007

Oil

003

013

Oil

002

016

49
( 1 , 4 ) 0.015 - 167.6 0 .

( 2 , 4 ) 0.031 - 74.7 0 .

( 3 , 4 ) 0.038 - 105.4 0 .

( 4 , 4 ) 0.026 - 175.5 0 .

( 5 , 4 ) 0.029 110.0 0 .

( 9 . 4 ) 0.005 -118.3 0 .

(in, 4) 0.010 -19.3 0 .

(11,4) 0.014 - 4.7 0 .

(12,4) 0.003 9.2 0 .

(13,4) 0.004 -122.6 0 .

( 1 , 5 ) 0.017 95.4 0 .

( 2 , 5 ) 0.016 - 133.1 0 .

( 3 , 5 ) 0.023 - 177.4 0 .

( 4 , 5 ) 0.018 115.1 0 .

( 5 , 5 ) 0.026 68.7 0 .

( 9 . 5 ) 0.027 134.6 0 .

( in , 5 ) 0.008 133.7 0 .

( 11 , 5 ) 0.007 16.4 0 .

( 12 , 6 ) 0.012 43.8 0 .

( 13 , 5 ) 0.006 62.8 0 .

119.0 0.046 130.2

- 15.8 0.098 127.6

- 75.7 0.120 147.0

- 128.6 0.114 161.7

69.9 0.094 148.9

117.6 0.034 105.2

81.7 0.023 35.1

- 147.8 0.040 - 73.2

- 129.7 0.019 - 119.0

- 50.9 0.030 104.5

73.1 0.042 23.9

179.5 0.098 32.2

180.0 0.114 40.3

150.6 0.104 41.5

113.6 0.099 37.5

00

• 0.044 41 .3

11.9 0.019 - 0.8

- 115.9 0.029 - 131.3

- 148.7 0.018 148.5

- 145.9 0.045 66.7

041

021

039

025

007

012

015

008

010

012

026

008

034

041

017

Oil

014

006

017

012

50
( 1 . 6 ) 0.022 - 14.8 0 .

( 2 , 6 ) 0.011 119.4 0 .

( 3 , 6 ) 0.021 102.1 0 .

( 4 , 6 ) 0.020 29.1 0 .

( 5 , 6 ) 0.027 - 38.0 0 .

( 9 , 6 ) 0.016 80.7 0 .

( 10 , 6 ) 0.008 - 125.3 0 .

( 11 , 6 ) 0.011 - 127.4 0 .

( 12 , 6 ) 0.004 59.7 0 .

( 13 , 6 ) 0.007 47.0 0 .

( 1 , 7 ) 0.019 - 131.2 0 .

( 2 , 7 ) 0.013 128.5 0 .

( 3 , 7 ) 0.012 30.1 0 .

( 4 , 7 ) 0.026 - 51.7 0 .

( 5 , 7 ) 0.028 - 89.0 0 .

( 9 , 7 ) 0.021 - 29.5 0 .

( 10 , 7 ) 0.012 - 115.4 0 .

( 11 , 7 ) 0.016 - 166.2 0 .

( 12 , 7 ) 0.006 170.5 0 .

( 13 , 7 ) 0.004 56.4 0 .

0.1 0.061 - 92.5

63.0 0.103 - 76.1

102.5 0.126 - 70.9

78.4 0.112 - 76.4

29.2 0.094 - 67.2

- 9.6 0.016 - 51.7

- 45.3 0.009 - 129.5

35.2 0.037 180.0

150.0 0.013 120.8

- 165.8 0.036 3.5

- 79.9 0.054 163.5

- 71.5 0.078 - 179.0

3.0 0.110 178.0

- 13.1 0.123 175.0

- 57.2 0.098 176.0

- 96.1 0.018 - 102.8

- 121.4 0.005 - 34.0

94.2 0.024 116.0

61.3 0.010 - 53.9

- 139.3 0.054 - 64.9

020

012

046

056

030

014

020

008

Oil

015

008

014

027

043

038

020

023

013

020

003

51
( 1 , 8) 0.025 130.4 0 .

( 2 , 8 ) 0.018 60.4 0 .

( 3 , 8 ) 0.020 - 21.5 0 .

( 4 , 8 ) 0.021 - 58.6 0 .

( 5 , 8 ) 0.012 140.3 0 .

( 8 , 8) 0.020 - 103.0 0 .

( 10 , 8 ) 0.017 160.9 0 .

( 11 , 8) 0.021 132.7 0 .

( 12 , 8 ) 0.004 152.9 0 .

( 13 , 8) 0.003 - 101.8 0 .

( 1 . 9 ) 0.032 31.2 0 .

( 2 , 9 ) 0.019 2.6 0 .

( 3 , 9 ) 0.010 - 66.9 0 .

( 4 , 9 ) 0.002 - 115.3 0 .

( 5 , 9 ) 0.020 62.6 0 .

( 9 , 9 ) 0.026 - 158.9 0 .

( 10 , 9 ) 0.025 108.5 0 .

( 11 , 9 ) 0.025 78.5 0 .

( 12 , 9 ) 0.013 - 48.0 0 .

( 13 , 9 ) 0.014 - 100.2 0 .

158.6 0.077 70.3

- 163.7 0.109 71.7

- 118.1 0.130 82.4

- 116.7 0.131 89.4

- 146.5 0.092 80.9

- 134.3 0.015 - 22.6

- 166.6 0.025 8.6

94.9 0.034 56 .

5

19.9 0.004 102.9

- 38.2 0.036 - 120.5

105.0 0.070 - 55.0

129.1 0.102 - 47.5

172.5 0.119 - 18.8

146.0 0.113 - 13.0

142.8 0.102 - 29.9

137.8 0.017 - 152.9

126.8 0.017 - 112.8

52.6 0.003 - 11.3

- 42.3 0.029 126.1

- 112.1 0.028 159.9

,021

,028

,032

,037

,043

020

028

015

025

018

025

019

020

038

059

018

024

013

022

019

52
(1,10) 0.027 -55.6 0.018 24.4 0.069 -136.3

(2,10) 0.017 -88.4 0.020 7.9 0.110 -134.8

(3,10) 0.004 145.0 0.018 31.5 0.117 -122.7

(4,10) 0.009 -158.2 0.040 59.1 0.105 -118.6

(5,10) 0.029 -60.9 0.051 63.4 0.097 -133.5

(9,10) 0.016 139.0 0.034 102.2 0.023 171.3

(10,10) 0.017 21.2 0.023 65.1 0.027 -138.4

(11,10) 0.019 -26.3 0.022 -53.7 0.024 -89.6

(12,10) 0.013 -96.2 0.018 -122.3 0.011 7.4

(13,10) 0.006 -145.3 0.021 -179.0 0.024 88.0

In addition, at x = D

c

= 3.048 m and y

= hm

= 1.448 m (door aperture),

V

z

= 0.119v and = - 40.2 deg.

A few important conclusions may be drawn from the above table.

(1) On the average, V

z is the dominant component as it is expected.

(2) The values of V

z are consi sitently larger at (i, j) with i = 3,4 and

5 for all j's, since these positions are in the lit region directly

illuminated by the source. The values of V

z are relatively weaker

at (i,j) for i = 9 through 13 near plane 5, since those positions

are in the shadow region.

(3) For i = 3,4, and 5 (lit region), values of V

z vary between a high of

0.146 volt and a low of 0.088 volt, indicating a standing-wave type

of field distribution inside the building.

(4) Phases of the component fields change rather irregularly.

53
In view of the subject presented in Section 3, one particular component

field strength or polarization with respect to the narrow slot which may

appear on an object to be placed inside the building may have more penetration

effect than the other components or polarizations. For this reason,

measurements of the component field strengths rather than the total field

amplitude (|E

X |^ + |E y

|^ + |E

Z | 2 )^ should prove more meaningful.

A second set of measured data with a new measuring antenna height

(h m

= 1.905 m) is tabulated as follows:

Table 5

Gri d

Position

V

x

in volts in deg.

vy

in volts in deg.

v

z

in volts

^z

in deg.

tun 0.055 179.0 0.055 -49.7 0.059 87.5

(2,1) 0.063 -165.9 0.037 176.3 0.113 115.6

(3,1) 0.015 171.6 0.074 176.3 0.157 111.9

(4,1) 0.028 63.1 0.034 -154.6 0.153 101.0

(5,1) 0.030 91.4 0.046 -61.2 0.119 121.4

(9,1) 0.025 146.3 0.008 -8.5 0.022 -54.0

(10,1) 0.020 143.8 0.012 -18.8 0.016 -152.3

(11,1) 0.014 139.6 0.017 -99.3 0.026 66.5

(12,1) 0.010 114.5 0.009 175.6 0.017 26.9

(13,1) 0.008 62.9 0.020 95.2 0.020 -137.1

54
( 1 . 2 ) 0.037 99.3

( 2 , 2 ) 0.063 118.6

( 3 , 2 ) 0.030 98.7

( 4 , 2 ) 0.017 - 9.2

( 5 , 2 ) 0.012 - 73.3

( 4 , 2 ) 0.022 43.1

( 10 , 2 ) 0.013 52.0

( 11 . 2) 0.009 155.6

( 12 , 2 ) 0.012 179.0

( 13 , 2 ) 0.006 75.1

( 1 , 3 ) 0.030 - 17.5

( 2 , 3 ) 0.037 22.1

( 3 , 3 ) 0.030 - 19.4

( 4 , 3 ) 0.045 - 79.7

( 5 , 3 ) 0.026 - 98.3

( 9 , 3 ) 0.038 - 18.3

( 10 , 3 ) 0.028 - 21.4

( 11 , 3 ) 0.011 - 58.8

( 12 , 3 ) 0.011 - 128.1

( 13 , 3 ) 0.008 - 79.1

042 - 117.1 0.060 - 22.5

034 124.1 0.083 - 8.2

064 114.5 0 . 14,0 - 3.6

024 156.8 0.152 - 5.3

019 - 138.2 0.124 2.2

003 -41 .5 0.027 - 137.8

007 - 18.7 0.035 143.3

010 - 114.1 0.037 54.7

003 132.4 0.011 5.7

017 61.7 0.031 169.0

044 170.5 0.068 - 129.6

029 60.7 0.096 - 124.6

046 21.5 0.140 - 106.9

025 - 8.3 0.154 - 100.2

010 46.0 0.122 - 114.5

006 179.8 0.048 - 142.7

008 162.6 0.030 125.4

015 - 173.0 0.041 - 19.7

008 80.1 0.031 - 79.1

024 34.7 0.032 177.2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

55
( 1 , 4) 0.020 - 132.7 0 .

( 2 , 4 ) 0.035 - 64.5 0 .

( 3 , 4 ) 0.038 - 94.9 0 .

( 4 , 4 ) 0.035 - 164.3 0 .

( 5 , 4 ) 0.032 154.4 0 .

( 4 , 4 ) 0.011 - 92.3 0 .

( 10 , 4 ) 0.008 - 13.9 0 .

( 11 , 4 ) 0.008 53.2 0 .

( 12 , 4 ) 0.007 86.4 0 .

( 13 , 4 ) 0.003 - 79.4 0 .

( 1 , 5 ) 0.021 120.2 0 .

( 2 , 5 ) 0.011 - 135.5 0 .

( 3 , 5 ) 0.025 - 178.3 0 .

( 4 , 5 ) 0.046 128.8 0 .

( 5 , 5 ) 0.040 107.6 0 .

( 4 , 5 ) 0.033 162.9 0 .

( 10 , 5 ) 0.016 148.9 0 .

( 11 , 5 ) 0.012 100.3 0 .

( 12 , 5 ) 0.020 86.1 0 .

( 13 , 5 ) 0.009 80.4 0 .

133.0 0.071 134.4

- 43.3 0.123 138.7

- 67.0 0.140 155.3

- 89.0 0.132 166.5

- 6.0 0.123 159.1

128.9 0.039 116.9

157.3 0.028 39.5

- 152.7 0.050 - 52.9

- 90.1 0.024 - 103.5

15.8 0.036 128.5

96.5 0.064 24.2

179.5 0.110 35.5

- 162.0 0.127 49.4

171.1 0.123 50.5

134.2 0.123 50.4

72.0 0.050 56.5

56.0 0.020 11.6

- 131.6 0.035 - 118.3

- 101.9 0.025 164.1

- 31.2 0.054 87.1

044

012

052

034

008

012

014

022

005

023

027

015

040

038

019

022

010

018

017

Oil

56
( 1 . 6 ) 0.017 13.1

( 2 , 6 ) 0.010 95.8

( 3 , 6 ) 0.029 82.4

( 4 , 6 ) 0.029 42.0

( 5 , 6 ) 0.031 - 11.2

( 9 , 6 ) 0.016 110.6

( 10 , 6 ) 0.013 - 113.6

( 11 , 6 ) 0.011 - 142.8

( 12 , 6 ) 0.014 100.4

( 13 , 6 ) 0.008 91.7

( 1 , 7 ) 0.020 - 109.8

( 2 , 7 ) 0.011 173.3

( 3 , 7 ) 0.010 42.7

( 4 , 7 ) 0.027 - 39.8

( 5 , 7 ) 0.033 - 75.1

( 9 , 7 ) 0.020 - 18.6

( 10 , 7 ) 0.021 - 100.2

( 11 , 7 ) 0.019 - 127.4

( 12 , 7 ) 0.002 163.4

( 13 , 7 ) 0.008 46.5

0.022 37.2 0.087 - 76.1

0.020 103.5 0.118 - 70.6

0.047 116.6 0.130 - 61.4

0.048 98.6 0.120 - 63.1

0.028 50.5 0.107 - 59.5

0.024 29.4 0.020 - 53.9

0.012 3.5 0.015 - 127.9

0.009 152.4 0.043 - 170.3

0.007 - 142.9 0.015 130.1

0.015 - 89.4 0.046 17.8

0.013 - 44.7 0.078 178.1

0.014 - 14.9 0.098 176.1

0.034 32.5 0.122 - 168.5

0.041 3.9 0.130 - 167.9

0.042 - 47.9 0.111 - 171.4

0.024 - 70.5 0.019 - 91.0

0.013 - 128.2 0.003 2.1

0.023 120.3 0.027 118.0

0.015 90.9 0.018 - 29.4

0.011 - 71.0 0.055 - 53.0
( 1 , 8 ) 0.030 152.6 0 .

( 2 , 8) 0.015 146.7 0 .

( 3 , 8 ) 0.014 - 84.4 0 .

( 4 , 8) 0.014 - 136.6 0 .

( 5 , 8 ) 0.024 154.0 0 .

( 9 , 8) 0.019 - 101.6 0 .

( 10 , 8 ) 0.013 172.6 0 .

( 11 , 8) 0.013 134.2 0 .

( 12 , 8 ) 0.008 - 10.5 0 .

( 13 , 8 ) 0.007 - 32.5 0 .

( 1 , 9 ) 0.024 55.3 0 .

( 2 , 9 ) 0.016 - 2.6 0 .

( 3 , 9 ) 0.009 - 58.2 0 .

( 4 , 9 ) 0.005 141.9 0 .

( 5 , 9 ) 0.028 82.7 0 .

( 9 , 9 ) 0.019 - 153.1 0 .

( 10 , 9 ) 0.028 98.8 0 .

( 11 , 9 ) 0.017 64.6 0 .

( 12 , 9 ) 0.017 - 77.2 0 .

( 13 , 9 ) 0.015 - 111.1 0 .

- 149.6 0.083 86.0

- 127.2 0.109 84.7

- 79.8 0.127 97.1

- 86.7 0.121 101.9

- 115.5 0.093 92.1

- 109.3 0.018 - 1.2

- 158.4 0.027 19.5

93.4 0.032 63.9

45.8 0.003 - 178.8

- 61.7 0.038 - 111.6

162.2 0.083 - 33.5

- 177.7 0.112 - 31.3

- 170.5 0.118 - 15.3

172.2 0.109 - 5.9

161.8 0.100 - 23.4

- 178.4 0.016 - 135.9

125.2 0.018 - 106.3

36.5 0.002 1.4

- 8.1 0.028 144.8

- 114.4 0.036 164.0

018

021

028

038

049

034

022

028

025

019

014

017

023

032

041

017

013

021

018

013

58
-126.4 (1,10) 0.025

(2,10) 0.015

(3,10) 0.004

(4,10) 0.006

(5,10) 0.029

(9,10) 0.011

(10,10) 0.021

(11,10) 0.012

(12,10) 0.020

(13,10) 0.011

-55.6 o.mo

-67.2 0.007

41.8 0.012

-16.5 0.033

-30.1 0.041

107.7 0.035

45.5 0.015

-15.8 0.022

-120.5 0.009

-164.6 0.021

23.6 0.077

3.8 0.111

108.5 0.107

103.5 0.079

86.0 0.087

118.0 0.030

71.5 0.028

-44.6 0.025

-102.3 0.011

163.7 0.028

-127.5

-117.1

-114.6

-121.9

178.3

-129.1

-69.1

37.6

102.3

Note that the general conclusions made for the data presented in Table 4

with h n

= 1.448 m. hold good also for the data presented in Table 5 with h^ =

1.905 m. The highest value for V

z is now 0.157 volt (against 0.146 volt for

Table 4) at position (3,1). This result is expected from the behavior of the

ground reflection coefficient R

y given in (92). Thus, if possible,

measurements should be made with the feed point of the measuring antennas

farther away from the floor. Of course, this is limited by the supporting

pole and the convenience of making measurements.

59
Based on the measured data presented in Tables 4 and 5, and the hybrid

approach represented by (77) to (79), we obtain the following results.

(1) When the number of patches to be used is eight on Planes 1, 2, and 3

represented by m^) = n^ = m^) = n^ = m^) = n(3) = 2, the minimum number

of locations, p, where the measured data are needed is also eight in

accordance with (98) or (99). Thus, we choose the measurement positions

(1.1). (2.D , (1.2) and (2,2) with two measuri nc1 antenna heights so that the

equal sign in (98) or (99) holds. The speci fic results are:

for Plane 1 (floor) where z' = 0,

position (i.i) J

x = 0.5750 + j0.3598 Jy

= -0.4200 - jO.4349

position (2,1) J

x

= 0.4628 + j 1.0371 Jy

= -0.3880 + j 0.3937

position (1,2) J

x

= 0.2582 - jO.3361 Jy

= -0.2460 - j 0.586

5

position (2,2) J

x

= -0.9534 - j 0.5706 Jy

= 0.7870 + j 0.2328, (10

for Plane 2 (front) where x' = 0,

position (1,1) Jy

= 1.3713 - j 0.4874 J z

= -0.9528 + ,10.2119

position (2,1) Jy

= -0.1948 + j 0.1866 J

z = -0.6499 - j 0.0266

posit i on (1,2) J

y = -0.9331 + jO.1377 J

z = 0.1453 + j 0.8761

position (2,2) J y

= 0.1786 + j 0.0755 J z

= -0.2295 - j 0.87 28, (1

and for Plane 3 (right) where y' = 0,

position (1,1) J

x = -2.1408 + j 0.0044 J z

= -1.2517 + jO. 3303

position (2,1) J

x = -0.1024 -

j 0. 1476 J

z = -0.9308 - j 0.3842

position (1,2) J

x = 1.5798 + j0.2727 J

z = 0.0751 -

j 1.2033

posi ti on (2,2) J

x = 0.0568 - ,10.1989 J

x = 0.0481 + j 1.2338 (10:

(2) When = n(^) = m^ 2 ) = m^) = 3 and n^ = n(3) . 2, we require p > 14.

Actually we ichoose the measurement positions (1, 1), (2,1) , (3,1), (1 ,2), (2,2),

60
(3,2), (1,3), (2,3), and (3,3), with two measuring

positions. Thus, the greater sign in (99) holds,

for Plane 1 where z‘ = 0,

antenna heights, a total of 18

The 1 east-square-f it results are:

position (i,i) J

x

= 0.0496 + jO.0705 Jy

= 0.0705 - j 0.0432

position (2,1) J x

= -0.0494 - j 0.1247 Jy

= 0.1369 + jO.0110

position (3,1) J

x = 0.0274 + j 0.0885 Jy

= -0.0898 - j 0.0077

position 0,2) J

x = -0.1276 - j 0.0836 Jy = -0.0908 + j 0.0498

position (2,2) J

x = 0.1340 + j 0.1464 Jy

= -0.1433 - j 0 . 0202

position (3,2) J

x = -0.0712 - j 0.1124 Jy

= 0.0780 + j 0 . 0154

position (1,3) J

x

= 0.0627 + j 0.0519 Jy

= -0.0725 - j 0.06 10

position (2,3) J

x

= -0.0536 - j 0.0808 Jy

= 0.0972 + j 0.0545

position (3,3) J

x

= 0.0201 + j 0.0589 Jy

= -0.0353 -

j 0 . 033 1 ,

Plane 2 where x

1

:= 0,

position (1,1) Jy

= 0.0350 - j 0.0439 J

z = 0.0174 -

j 0 . 0146

position (2,1) Jy

= -0.0316 + j 0.0370 J

z = 0.0364 - j 0.0289

position (3,1) Jy

= 0.0447 - J0.0075 J

z = 0.0275 - jO.0140

position (1,2) Jy

= -0.0170 + j 0.0 185 J

z

= 0.0033 - j 0.0340

position (2,2) Jy

= 0.0137 -

j 0 . 0129 J

z

= 0.0339 + j 0 . 0263

position (3,2) Jy

= -0.0270 + j 0.0049 J

z = -0.0204 - jO.0117,

for Plane 3 where y' - 0,

position (1,1) J

x

= 0.0505 - j 0.005

J

z = 0.0254 + j 0.006

position (2,1) J

x

= -0.0710 + j 0.0250 J

z = 0.0001 - j 0.0 125

position (3,1) J

x

= 0.0321 - jO.0213 J

z = -0.0046 4- j 0 . 0012

posi ti on 0,2) J

x

= -0.0446 - j 0.0084 J

z = -0.0581 + j 0 . 0352

position (2,2) J

x

= 0.0638 - j 0.0048 J

z = 0.0138 - j 0.0600

position (3,2) J

x

= -0.0298 + j 0.0080 J

z

= -0.0143 + j 0 . 03 15.

(103)

(104)

(105)

61
(3) When we use m^) = = m^) = m^) = 4 and n^ = n^) = 2, we require

p > 16. Actually we choose the measurements positions, (1,1), (2,1), (3,1),

(4,1), (1,2), (2,2), (3,2), (4,2), (1,3), (2,3), (3,3), (4,3), (1,4), (2,4),

(3,4), and (4,4) with two measuring antenna heights, a total of 32

positions. Thus, the greater sign in (99) also holds here. The specific

results are: (also in the sense of least square-fit)

for Plane 1 where z' = 0,

posi ti on ( 1 . 1 ) = -0.0642 - JO .0120 Jy

= 0.0770 + jO.0337

position ( 2 . 1 ) J

x

= 0.1257 - j 0.0 152 J v= -0.1334 + j 0.0348

position (3,1) J

x = -0.0948 + jO.0174 Jy

= 0.1441 - jO.0577

position (4,1) J

x 0.0456 + j 0.0060 J v

= -0.0852 + j 0.0102

position ( 1 , 2 ) J

x

= 0.1129 -

j 0.0859 J y

= -0.0985 -

j 0.0197

position ( 2 , 2 ) J

x = -0.1797 + j 0.1288 Jy

0.1543 -

j 0 . 0446

position (3,2) J

x = 0.1447 - j 0.0659 Jy

= -0.1803 + jO.0552

position (4,2) J

x = -0.0770 - jO. 0015 J y

0.1098 + j 0.0063

posi ti on (1,3) J

x

= -0.1266 + j 0.1000 °y = 0.0750 - jO.0255

position (2,3) J

x

= 0.1949 - j 0.1147 Jy

= -0.1379 + j 0.0678

position (3,3) J x

= -0.1585 + j 0.0349 Jy

= 0.1776 - j 0.0469

posi ti on (4,3) J

x = 0.0717 + j 0.0195 J y

= -0.1044 - j 0.0243

posi ti on (1,4) J

x

= 0.0913 -

j 0 . 03 14 Jy

= -0.0681 + jO.0279

position (2,4) J

x = -0.1295 + j 0.0254 Jy

= 0.1222 - jO. 0554

position (3,4) J

x

= 0.0932 + j 0.0184 Jy

= -0.1312 + j 0.0240

position (4,4) J

x

= -0.0277 - j 0.0291 Jy

= 0.0611 + j 0.0242,

62
for Plane 2 where x' = 0

position (i.i) Jy

= 0.0143 -

j 0 . 0057 J

z = -0.0125 - j 0.0083

position (M) Jy

= -0.0419 - j 0.0099 Jz = -0.0105 - j 0.0079

posit i on (3,1) Jy = 0.0522 + j 0 . 0234 Jz = 0.0185 + j 0 . 01 49

position (4,1) Jy

= -0.0159 - j 0.0256 J z -0.0123 - j 0.0180

position (1,2) Jy

= -0.0009 -

j 0 . 0027 J

z = 0.0548 + j 0.0570

position (2,2) Jy

= 0.0091 + j 0.0 144 J

z

= -0.0625 - j 0.0301

position (3,2) Jy

= -0.0182 - jO.0143 J

z = 0.0282 + j 0 . 0 1 7 9

position (4,2) Jy

= 0.0062 + jO.0136 J

z -0.0004 - j 0.0042,

and for Plane1 3 where y

1 0

position (i.i) J

x = 0.0110 -

j 0 . 0056 J

z

= 0.0650 - j 0.0023

position (2,1) J

x

= 0.0116 + j 0.0274 J

z = -0.0477 + j 0.0469

position (3,1) J

x

= -0.0410 - j 0.0080 J

z = 0.0291 - j 0.0498

position (4,1) J

x = 0.0218 - j 0.0 133 J z

= 0.0016 + j 0.0209

posit i on (1,2) J

x = -0.0141 - jO.0102 J

z = -0.0973 - j 0.07 57

position (2,2) J

x

= 0.0085 - jO.0141 J

z = 0.1167 + j 0.0304

posi ti on (3,2) J

x = 0.0148 + jO.OllO J z

= -0.0770 -

j 0 . 0 2 2 6

position (4,2) J

x

= -0.0083 + j 0 . 0057 J

z

= 0.0036 + jO . 0169.

Results for other cases with m(-*-) = n(D = m( 2 ) = m

1(3) = 5, n

(: 2 ) =n(3) =

(107)

(108)

and

m (D = „(D . m

(2) , m O) . 6> n (2) . n

( 3) . 2 are also available but not listed

here.

Comparing the results in (100) with those corresponding ones in (103) and

(106), we find that J Y and J in Plane 1 vary substantially without clear

indications of convergence to certain values when the total number of patches used

is increased up to 4 x 4 x 2. This is not unexpected because the incremental

63
interval of ax = AY = 0.6096 m (approximately 0.3x) is too large. Should a smaller

increment in the order of ax = AY = 0.2 m (or O.lx) be used, a tendency of

convergence for J Y and J may be established. Same comments also apply to and J x y y l

in Plane 2 and to J

x and Jz in Plane 3.

When the equivalent surface current densities obtained in (100) through (108),

however imperfect as they are, are used in (77) to (79) for computing the field

strengths at spaces inside the building, the results are still quite interesting as

seen from the summary presented below.

With the current densities in (100) to (102), based on eight patches (2 x 2 x

2), substituted in (77) to (79), we obtain the field strength components at x =

0.9144 m (x Q

), y

= 1.0668 m (y Q

), and z = 1.4478 m (h m ) as follows:

E

x = 0.0714 7-171.4° , E y

= 0.0536 /-46.3° , E

z = 0.0536 /1Q7.20 V/m.

Since the effective length of the dipoles being used in the measurement is

approximately a = i = % = 0.5527 m, the received voltage components becomes

x y z

v

x

= 0.0395 7-171.4° , V

y

= 0.0296 7-46.3° , V

z = 0.0296 /1Q7.2° v,

which are, of course, the same as those appearing at position (1,1) in Table 1

because position (1,1) is one of the data points. Similarly, the same results at

positions (1,2), (2,1) and (2,2) in Table 1 and those in Table 2 with x = 0.9144 m,

y

= 1.0668 m and z = 1.9050 m (a second measurement height) are also recovered.

The voltage components computed at x = 1.2192 m, y

= 1.3716 m and z = 1.6764 m,

which are not one of the original data points, with different number of patches are:

(1) With 2 x 2 x 2 or the results in (100) through (102),

V

x = 1.5958 /160.0° , Vy

= 1.0780 7-14.8° , V

z = 0.0584 /129.7° v;

(2) With 3 x 3 x 2 or the results in (103) through (105),

v

x

= °* 0663 /162,7° , Vy

= 0.0037 /27.5° , V

z = 0.0781 /86.2° v;

64
(3) With 4 x 4 x 2 or the results in (106) through (108),

V

x = 0.0531 7-175.1° , Vy

= 0.0246 /78.9° , V

z = 0.0776 /83.7° v.

For other different numbers of patches such as 5x5x2, 6x6x2 and 7 x 6 x 2 of

which the current densities are also available though not explicitly listed, the

results of voltage components are also presented below for the analysis purpose.

Number of

Patches

Vx

in volts

il; T

x

in deg.

vy

in volts

*y

in deg.

v

z

in volts

*z

in deg.

5x5x2 0.0433 155.8 0.0245 125.6 0.0811 81.1

6x6x2 0.0539 173.2 0.0124 -66.7 0.0828 90.3

7x6x2 0.0407 165.1 0.0017 31.2 0.0807 84.1

Clearly, the voltage components or their field strengths counter parts do converge

to reasonable levels when the number of patches used in the computation is

increased. Furthermore, the z component is still the dominant one as it should be

even with the number of patches as small as 3 x 3 x 2. The y-component is the

weakest as it should be, since the direction of propagation is along the y-axis.

This simple analysis indicates the method proposed in this report for computing the

field strength inside a building based on the limited number of measurement data may

be satisfactory to a certain degree. With large computing facilities available to

accommodate a larger number of patches, the accuracy in computation and the ability

to predict the field strength distribution inside a building similar to the geometry

shown in figure 9 can be improved.

65
8. CONCLUDING REMARKS

The problem of determining the field distribution inside an empty building or cavity with apertures due to an external source has been formulated for two

different configurations. One of them requires that the aperture take the

form of a horizontal narrow slot contained in an infinite conducting plane,

that the cavity wall is lossy, that the cavity is in free space, and that the wave from the external source is normally incident on the slot. For this

case, the problem is solved by an application of the equivalence principle and

the generalized network presentation. It is found that at natural resonances

the field amplitude in the slot aperture is inversely proportional to the

square-root of the wall conductivity and that a field of high strength can exist inside the cavity.

The other configuration is large buildings and apertures situated on a lossy but homogeneous ground. For this latter case, the field distribution

inside the building is determined by an analytical means together with a limited number of measurement data. This approach enables an engineer to make

a quick on-site check with a moderate computing facility available, even though the accuracy is not very good. To improve the accuracy in the final

computed results, a larger computing facility will be required.

9. ACKNOWLEDGMENTS

The authors are thankful to M. Kanda and G. Reeve for their suggestions

and comments during the course of this

programming from P. N. Swarztrauber is

leading to the results reported herein

Communi cati ons-El ectroni cs Engi neeri ng

study. Help received in computer

gratefully acknowledged. The work

was partly supported by U.S. Army

Installation Agency.

66
10. REFERENCES

[1] Mittra, R.; Ral mat-Sami i , Y.; Jamnejad, D.V.; Davis, W.A. A new look at the

thin plate scattering problem. Radio Sci . 8 (10): 869-875; 1973 October.

[2] Poggio, A.J.; Miller, E.K. Integral equation solutions for three-dimensional

scattering problems, chapter 4 of Computer techniques for electromagnetics,

R. Mittra, Ed. New York, NY: Pergamon Press; 1973.

[3] Mentzer, J.R. Scattering and diffraction of radio waves. New York, NY:

Pergamon Press; 1955.

[4] Richmond, J.H. A wire-grid model for scattering by conducting bodies. IEEE

Trans, on Antennas and Prop. AP- 14(6) : 782-786; 1966 November.

[5] Tsai, L.L.; Dudley, D.G.; Wilton, D.R. Electromagnetic scattering by a three-dimensional conducting rectangular box, J. of Applied Physics.

45(10): 4393-4400; 1974 October.

[6] Harrington, R.F. Field computation by moment methods. New York, NY:

McMillan; 1968.

[7] Schelkunoff, S.A. Some equivalence theorems of electromagnetics and their

application to radiation problems. Bell System Tech. J. 15: 92-112; 1936.

[8] Harrington, R.F.; Mautz, J.R. A generalized network formulation for aperture

problems. IEEE Trans, on Antennas and Prop. AP-24 (6): 870-872;

1976 November.

[9] Harrington, R.F.; Mautz, J.R. Electromagnetic transmission through an aperture in a conducting plane. AEU, 31: 81-87; 1977.

[9a] Cheng, D.K.; Liang, C.H. Electromagnetic fields in cavities with a slotaperture at natural resonance. IEEE International Symposium on Electromagnetic Compatibility: 489-494; 1981.

[10] Collin, R.E. Field theory of guided waves. New York, NY: Mc-Graw-Hi 1

1

1960.

[11] Stewart, G.W. Introduction to matrix computations, chapter 5, esp. p. 214-

and 217 et seq. New York, NY: Academic Press; 1973.

[12] ibid., p. 108.

[13] ibid.k p. 217 et seq.

[14] Dongarra, J.J.; Moler, C.B.; Bunch, J.R.; Steward, G.W. UNPACK user's

guide. Philadelphia: SIAM Publications; 1979. pp 9.1.

[15] CQRDCT and CQRSLT were modified from CQRDC and CQRLS by Dr. Paul N. Swarztrauber, Scientific Computing Division, National Center for Atmospheric

Research, Boulder, Colorado while on loan to the Scientific Computing

Division, Center for Applied Mathematics, Nat. Bur. of Stand. { .S. . Boulder, Colorado.

67
APPENDIX A. FORTRAN PROGRAMS

A.l CURRENT Program

PROGRAM CURRENT ( IN PUT * OUTPUT* TAPE5- INPUT, T APE6-0UTPUT* TAPE7

)

C THIS PROGRAM READS FIELD DATA IN VOLTS

C AND COMPUTES CURRENTS IN PLANES 1* 2, AND 3

COMPLEX 0MS4P*S0*PE*GMN1*GNL2* 6ML3* S1*S2*S3*AMN1*ANL2* AML 3

1 *BMN1*BNL2*BML3*CMN1*CNL2*CML3*CJX1,CJX3,CJY1,CJY2

2 *E(24 >*CJZ2*C JZ3, A,DIAG(2<»)*C1*RSUBV* ESUBZ,XS(2<»)

INTEGER JPVT(24)*NXT0TH(2)*NYT0TH(2)*IXH(13*2)*IYH(16*2)

COMMON A(24*24)

REAL MU* K A Y* L SUBZ

NPRNT1-0

NPRNT2 1 NPRNT3 1 NPRNT4’

NPRNT5*

C NPRNT1

C NPRNT2

C NPRNT3

C NPRNT^

C NPRNT5

CONTROLS

CONTROLS

CONTROLS

CONTROLS

CONTROLS

MDIMA-24

NXTOTHUi -2

NXTOTH(2)-2

N YTOTH ( 1 ) *2

NYTOTH<2)-2

I XH ( 1 * 1 ) *1

I XH ( 1 * 2 ) -1

I X H ( 2 * 1 ) *2

IXH<2*2»«2

IYHC1* 1)*1

I YH ( 1 » 2 ) * 1

I YH ( 2 > 1 ) 2

PRINTING OF 0MEGA*MU*EPS*KAY*0MS4P*S04PE

PRINTINT OF XP,YP,ZP AND COEFFICIENTS

PRINTING OF M A » N A PRINTING OF DIAGONALS

PRINTING OF RESIDUALS

IYH(2*2 )»2

DX- • 6096

DY-.6096

DZ *572

DXDY-DX+OY

DXDZ»DX*DZ

D YDZ-DY*DZ

XST ART 91**

YSTART *1 #0668

ZSTART»1.**78

FREQ-1 81

8 P I-*. * AT AN ( 1 •

)

OMEGA-2. *PI*FREQ

68
45

50

55

60

65

70

75

80

05

90

95

i

Cc

cc

cc

MU*4.E-7*PI

EPS-1. E-9/(36.*PI)

LS-1

LF-2

KAY-OMEGA*SQRT(MU*EPS

)

0MS4P»CMPLX(0.,0MEGA*MUm.*PI )

)

S04PE —CMP LX <0., .25/ (0MEGA*PI*EPS ) ) IF (NPRNT1.GT.0) WRITE (6,15) OME G A , M U, E P S , K A Y , OMS 4P» S 04 P E 15 F ORMAT ( *0 THE VARIABLES OMEGA, MU, EPS# KAY, 0MS4P , S04PE ARE

1LISTED BELOW* / ( 8E15 • 6 )

)

IF—

DO 100 IZ-LS,LF

Z-(FL0AT(IZ)-1.)*DZ+ZSTART

NXTQT-NXTQTH ( I Z

)

NYTOT-NYTOTH(IZ)

DO 100 IR -1 , NYTOT

I Y= I YH { I R , I Z ) Y-(FLOAT( IY)-1. )*DY+YSTART

DO 100 IC-1,NXT0T

IX-IXH( IC, IZ ) X«{ FLOAT ( IX)-1.)*DX*XSTART

IF-IF+3

IF(NPRNT2.GT.0) WRITE (6,1) X,Y,Z _ TCI

F OR MAT ( /*1THE FOLLOWING COEFFICIENTS CORRESPOND TO THE FIEL

1P0INT */* X-*F8.4*, Y-*F8.4*, Z**F8.4/)

COMPUTE COEFFICIENTS IN THE FLOOR OR Z-0 PLANE

(SUBSCRIPT 1, BOTTOM XY-PLANE, FLOOR)

JU —1 NYT-NYT0TH(1

)

DO 101 IN-1, NYT

N- I YH ( IN, 1

)

YP- ( FLOAT ( N )-l. )*DY*YSTART

NXT-NXTOTH(l)

DO 101 I M - 1 , N XT

M-IXH( I M, 1

)

XP-(FL0AT(M)-1. )*DX+XSTART

JU-JU+2

R MN1 -SORT ( (X-XP)**2+(Y-YP)**2+Z**2

)

GMN1-CMPLX(C0S(KAY*RMN1),-SIN(KAY*RMN1) ) /RMN1

Sl»CMPLX(KAY*KAY*RMNl*RMNl-3.,-3.*KAY*RMNl ) /RMN1**4

AMN 1 - ( X— X P ) *S 1

BMN1- ( Y—YP )*S1

CMN1-Z*S1

IFCNPRNT2.GT.0) WRITE (6,31) RMN 1 ,GMN1 , SI

,

1AMN1,BMN1,CMN1 .. .. ., c1n

FORMAT ( *0 RMN1-*F10.6*, GMNl -*2 F

0

6* , S1»*2F10.6/

1* AMN1-*2F 10.6* , BMN1«*2F10.6*, C MN 1 * *2 F 1 0 . 6 )

69
100

105

110

115

120

125

130

135

140

145

ooor>

ooo

o

o

o

or>o COMPUTE COEFFICIENTS OF JX AND JY CORRESPONDING TO E SUB X CJX1«-0MS4P*GMN1*DXDY-S04PE*(CMPLX( 1 .,KAY*RMN1 ) /RMN1**2

1 4-(X-XP)*AMNl)*GMNl*DXDY

CJY1— SQ4PE*(Y-YP)*AMN1*GMN1*DXDY

A ( IF, JU) *C JX1

A( IF, JU+1)»C JY1

IF(NPRNT2.GT.Q)WRITE(6,2) XP , YP , C J X 1 , C J Y1

2 FORM AT ( *0 COEFFICIENTS OF JX AND JY CORRESPONDING TO

IE SUB X AT XP»*F8.4*, YP-+F8.4* AND ZP-O. AR E * / ( 4E 1 5 . 6 ) ) COMPUTE COEFFICIENTS OF JX AND JY CORRESPONDING TO E SUB Y CJX1--S04PE*(X-XP)*BMN1*GMN1*DXDY

CJY1— 0MS4P*GMN1*DXDY-S04PE*(CMPLX( 1 KAY+RMN1) /RMN1**2

1 (Y-YP)*BMN1)*GMN1*DXBY

A(IF+1,JU) -CJX1

A(IF+1, JU*1)«CJY1

IF(NPRNT2.GT.0) WRITE (6,3) XP,YP,CJX1,CJY1

3 FOR MAT ( *0 COEFFICIENTS OF JX AND JY CORRESPONDING TO

IE SUB Y AT XP-*F8.4*, YP«*F8.4* AND ZP«0. AR E* / ( 4E1 5 . 6 )

)

COMPUTE COEFFICIENTS OF JX AND JY CORRESPONDING TO E SUB Z CJX1»-S04PE*( X-XP )*CMN1*GMN1*DXDY

C JY1»-S04PE*( Y-YP) *CMN1*GMN1*DXDY

A( IF*2,JU> -C JX1

A(IF+2»JU*1)-CJY1

IF(NPRNT2.GT.O) WRITE (6,4) XP,YP,CJX1,CJY1

4 FORMAT ( *0 COEFFICIENTS OF JX AND JY CORRESPONDING TO

IE SUB Z AT XP-*F8. 4* , YP«*F8.4* AND ZP-O. ARE* / ( 4E 15. 6 ) /

)

101 CONTINUE

COMPUTE COEFFICIENTS IN THE SECOND OR X-0 PLANE

(SUBSCRIPT 2, NEAR YZ-PL ANE > FRONT WALL)

DO 102 L*L S, LF

ZP- ( FLOAT ( L )-l. )*DZ*Z START

NYT-NYTOTHCL

)

DO 102 IN-1, NYT

N * I YH ( I N, L ) YP- (FLOAT (N)-l. )*DY*YSTART

JU-JU+2

RNL2-SQRT( X**2M Y-YP )**2+( Z-ZP )**2

)

GNL2«CMPLX(C0S(KAY*RNL2),-SIN( KAY*RNL2) ) /RNL2

S2-CMPLX(KAY*KAY*RNL2*RNL2-3.,-3.*KAY*RNL2) /RNL2**4

ANL2-X*S2

BNL2-( Y-YP )*S2

CNL2* ( Z-ZP ) *S2

IF (NPRNT2.GT.0) WRITE (6,32) RNL 2 , GNL 2 , S2

,

1ANL2,BNL2,CNL2

70
150

155

160

165

170

175

180

185

190

195

oooo

ooo

ooo

ooo

32 FORMAT (*0 RNL2«*F10.6*, GNL2 **2F10 • 6* , S2«*2F10.6/

1* ANL2»*2F10.6*, BNL2»*2F10.6*, C NL 2 * *2 F 1 0 . 6 ) COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB X

C JY2»-S04PE* ( Y-YP ) *ANL2*GNL2*DYDZ

C JZ2«-S04PE* ( Z-ZP) *ANL2*GNL2*DYDZ

A{ IF,JU)*CJY2

A(IF, JU+1 1-CJZ2

IF (NPRNT2.GT.0) WRITE (6,5) YP, ZP,C JY2, C JZ2

5 FORMAT ( *0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SUB X AT XP*0., YP«*F8.4* AND ZP«*F8.4* ARE* / ( 4E 15. 6 )

)

COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB Y

C JY2«-0MS4P*GNL2*DYDZ-S04PE*(CMPLX( 1 .,KAY*RNL2)/RNL2**2

1 ( Y-YP )*BNL2 )*GNL2*DYDZ

CJZ2«-S04PE*(Z-ZP)*BNL2*GNL2*DYDZ

A ( I F *1 , J U ) *C J Y2

A ( IF*1,JU+1)-CJZ2

IF (NPRNT2.GT.0) WRITE C6,6) YP,ZP,CJY2,CJZ2

6 F ORMAT { *0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SUB Y AT XP*0., YP-+F8.4* AND ZP»*F8.4* ARE*/(4E15.6)

)

COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB Z

C JY2 —S04PE* ( Y-YP )*CNL2*GNL 2*0 YDZ

CJZ2*-0MS4P*GNL2*DYDZ-S04PE*(CMPLX( 1 .,KAY*RNL2 ) /RNL2**2

1 CZ-ZP)*CNL2S*GNL2*DYDZ

A(IF*2, JU) -C JY2

A(IF*2>JU*1)*CJZ2

IF(NPRNT2.GT.O) WRITE (6,7) YP,ZP,CJY2,CJZ2

7 F OR MA T ( *0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SUB Z AT XP-O., YP *F8 • 4* AND ZP»*F8.4* ARE* / < 4E 15. 6 ) / ) 102 CONTINUE

COMPUTE COEFFICIENTS FOR THE THIRD OR Y«0 PLANE

(SUBSCRIPT 3, NEAR XZ-PLANE, RIGHT WALL)

DO 103 L*LS, LF

ZP« (FLOAT CL )-l. )*DZ*ZSTART

NXT*NXTOTH ( L

)

DO 103 IM*1,NXT

M*I XH( IM, L ) XP« (FLOAT (M)-1.)*DX4-XSTART

JU-JU+2

RML3*SQRT ( (X-XP)**2*Y**2*( Z-ZP )**2

)

GML3«CMPLX(C0S(KAY*RML3),-SIN(KAY*RML3) ) /RML3

S3«CMPLX(KAY*KAY*RML3*RML3-3.,-3#*KAY*RML3) /R ML 3**4

AM L 3 * ( X— X P ) * S 3

BML3-Y*S3

CML3«( Z-ZP )*S3

71
200

205

210

215

220

225

230

235

240

245

250

ooo

ooo

ooo

IFCNPRNT2.GT.0) WRITE (6,33) R ML 3, GML 3 , S 3

,

1 AML3, BML3 , CML3

33 FORMAT ( *0 R Ml 3- *F 10 • 6* » GML 3-*2F 10. 6*> S3«*2F10.6/

1* AML3«*2F10.6*, BML3-+2F10 .6*, C ML 3 «*2F 1 0 . 6

)

COMPUTE COEFFICIENTS OF JX AN0 JZ CORRESPONDING TO E SUB X

C JX3—0MS4P*GML3*DXDZ-S04PE*(CMPLX( 1. ,KAY*RML3) /RML3**2

1 +( X-XP)*AML3)*GML3*DXDZ

CJZ3--S04PE*(Z-ZP)*AM13*GML3*DXDZ

ACIF, JU)«CJX3

A ( I F , J U+l ) *C J Z3

IF(NPRNT2.GT.O) WRITE (6,8) XP,ZP,CJX3,CJZ3

8 FORMAT ( *0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB X AT XP-4F8.4*, YP-0. AND ZP-+F8.4* AR E * / ( 4E 1 5 . 6 )

)

COMPUTE COEFFICIENTS OF JX AND JZ CORRESPONDING TO E SUB Y CJX3— S04PE*( X-XP )*BML3*GML3*DXDZ

C JZ3 —S04PE*( Z-ZP)*BML3*GML3*DXDZ

A(IF*1,JU)-CJX3

AdF+l, JU*1)-CJZ3

IFCNPRNT2.GT.0) WRITE (6,9) XP,ZP,CJX3,CJZ3

9 FORMAT ( *0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB Y AT XP-4F8.4*, YP-O. AND ZP-*F8.4* AR E * / ( 4E 15 . 6 )

)

COMPUTE COEFFICIENTS OF JX AND JZ CORRESPONDING TO E SUB Z CJX3*-SQ4PE*(X-XP)*CM13*GML3*DXDZ

C JZ3»-0MS4P*GML3*DXDZ-S04PE*(CMPLX( 1 ,»KAY*RML3)/RML3**2

1 ( Z-ZP)*CML3)*GML3*DXDZ

A( IF + 2, JU) -CJX3

A ( IF *2 » JU + 1 ) *C JZ

3

IF(NPRNT2.GT.O) WRITE (6 >10) XP,ZP,CJX3,CJZ3

10 FORM AT ( *0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB Z AT XP-+F8.4*, YP-O. AND ZP-+F8.4* AR E * / ( 4E 1 5 . 6 ) /

)

103 CONTINUE

100 CONTINUE

NA-JU+1

MA-IF+2

IF(NPRNT3.GT.0)WRITE<6,13) MA,NA

13 FORMAT ( *0 MA-*I5* NA-*I5/)

CALI CQRDCT(MDIMA,MA,NA,A,JPVT,DIAG)

IF (NPRNT4.GT.0) WRI TE (6,12 ) (DIAG(I),I*1,NA)

12 F0RMAT(*0DIAG0NAl*//(8E15«6) ) Cl -CMP LX ( .376681,-. 004938)

DT0R-PI/180.

FTOM-. 304801

KAY-177. 72*DT0R

X PP-10. F TOM

YPP--32 • 54 FTOM

WR I TE ( 6, 17

)

72
255

260

265

270

275

280

285

290

295

300

o

o

o

17 FORMAT!/* INPUT DATA AND E SUB Z FROM THE ANTENNA AT THE

IF IE LD POINTS*/ /3(3X*AMP*4X*PHASE*1X) > 1 1 X* E SUB Z*/)

C READ THE OBSERVED FIELD DATA

C KX-0

K Y* 1 K Z * 1 K YF *16

IF —

DO 200 IZ*LS,LF

Z*(FLOAT( I Z

) -1 • )*DZ+ZSTART

IF (IZ.EQ.2 ) GO TO 201

ZD-.75*FT0M

TC*80.*DT0R

GO TO 202

201 ZD«2.25*FT0M

TC-99.+DT0R

202 NYTOT-NYTOTHUZS

DO 200 I N* 1 > N YTOT

I Y*IYH(IN>IZ)

Y»( FLOAT ( I Y

) — 1 • )*BY+Y$TART

NXTOT-NXTOTH ( IZ J DO 200 IM-1, NXTCT

I X-I XH( IM> I Z ) X*(FLOAT( I X ) - 1 • )*DX+XSTART

I F « I F 3

217 KX«KX*1

IF(KX.LE.IB) GO TO 218

KX-1

K Y-KY+1

IF (KY.LE.KYF ) GO TO 218

KY-1

K Z -KZ*

K YF *1

IF(KZ.GT.2) GO TO 219

218 READ ( 5» 14 ) AMPX, PHSX, AMPY, PHSY, AMP Z , PHS

14 FORMAT ( 6E10.3 ) IF((IX.NE.KX).QR.<XY.NE.KY).OR.(IZ.NE.KZ) ) GO TO 217

REMOVE DIRECT ANTENNA CONTRIBUTION FROM THE OBSERVED FIELD DATA

L SUB Z* • 56

AMPX*AMPX /LSUBZ

AMPY*AMPY/LSUBZ

AMPZ*AMPZ /LSUBZ

R SUB T* SORT ( ( XPP-X ) **2* ( YPP-Y ) **2 ) THETA-90. *DTOR-AT AN (ZD/RSUBT)

RSUBV* ( COS ( THETA) -Cl )/ (COS ( THETA )*C1

)

R* SORT (R SUB T**2*ZD**2)

ARG*KA Y*R *TC

ESUBZ-(1.*RSUBV)*18.515939/R*CMPLX(C0S(ARG) »- S IN ( ARG )

)

WRITE (6* 16 ) AMPX,PHSX,AMPY,PHSY,AMPZ,PHSZ,ESUBZ

73
305

310

315

320

325

330

335

340

3 4 5

350

355

16 F0RMAT(F8.6,F8.2»F8.6,F8.2,F8.6,F8.2,2E15.6)

E ( I F ) -AMPX*CMPLX(C0S (DT0R*PHSX), SIN ( DT0R*PHSX ) ) E( IF + 1) -AMPY*CMPLX(COS(DTOR*PHSY),SIN(DTOR*PHSY)

)

E ( IF +2) «AMPZ*CMPLX ( C OS ( DTOR * PHSZ ) , S I N ( DTOR * P HS Z ) )-ESUBZ

200 CONTINUE

219 CALL CQRSLT(MDIMA,MA,NA,A,JPVT,DIAG, E, XS,RESID)

RESID-RESID/FLOAT(MA)

IF(NPRNT5.GT.Q) WRITE<6,21) RESIO

21 FORMAT ( / RESIDUAL-+E15.6)

WRITE (6»18)

18 FORMAT ( / CURRENT IN THE ZP«0 PLANE*/4X*XP*6X*YP*15X

1*J SUB X*24X+J SUB Y*/)

JU--1

NYTOT -NYTOTH ( 1

)

NXTOT-NXTOTHU ) DO 301 IN-1, NYTOT

N- 1 YH ( IN , 1

)

YP-CFLOAT (N)-l* )*DY+YSTART

DO 301 IM»1,NXT0T

M« I XH ( I M, 1

)

XP-(FL0AT(M)-1. )*DX+XSTART

JU-JU+2

WRI TE ( 6, 22 ) XP,YP,XS(JU),XS ( JU+1

)

WRXTE(7,23) X S ( J U ) , X S ( JU + 1

)

22 F0RMAT«2F8.4,4E15.6)

23 FQRMATC 4E15.6)

301 CONTINUE

WRXTEl 6,19)

19 F OR MAT ( /* CURRENT IN THE XP«0 PL ANE* /4 X* YP*6 X* ZP*15 X 1*J SUB Y*24X* J SUB Z*/)

DO 302 L »L S , L F ZP«(FL0AT(L)-1.)*DZ+Z START

NYTOT-N YTOTH ( L

)

DO 302 IN-1, NYTOT

N-I YH( IN, L

)

YP*( FLOAT (N)-l. )*DY+YS TART

JU-JU+2

WRITE (6, 22) YP,ZP,XS(JU),XS< JU+1)

WR I TE ( 7 , 2 3 ) XS(JU),XS( JU+1)

302 CONTINUE

WRITE (6,20 ) 20 FORMAT ( /* CURRENT IN THE YP-0 PLANE*/4X*XP*6X*ZP*15X

1*J SUB X*2 4 X* J SUB Z*/)

DO 303 L *L S , L F ZP- ( FLOAT (L)-l. )*DZ+ZSTART

NXTOT-NXTOTH ( L

)

DO 303 IM«1,NXT0T

M - 1 XH ( IM , L ) XP- ( FLOAT <M)-1. ) DX + XST ART

JU-JU+2

WRITE (6, 22) XP,ZP,XS(JU),XS(JU+1)

WRITE(7,23) XS ( JU), X S ( JU+1)

303 CONTINUE

STOP

END

74
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

SUBROUTINE C QR DC T ( M D IM A, M , N , A , J P VT , D I AG

)

SUBROUTINE C QR DC T COMPUTES THE QR FACTORIZATION OF THE MATRIX A WHICH CAN BE INPUT TO SUBROUTINE CQRSLT IN ORDER TO COMPUTE A LEAST SQUARES SOLUTION TO THE EQUATIONS AX-B.

INPUT PARAMETERS

MDIMA THE FIRST OP ROW DIMENSION OF A

M NUMBER OF ROWS IN A

N NUMBER OF COLUMNS IN A

A A DOUBLY SUBSCRIPTED COMPLEX ARRAY WITH DIMENSIONS (MDIMA, N)

THAT CONTAINS THE MATRIX WHOSE OR DECOMPOSITION IS

TO BE COMPUTED.

OUTPUT PARAMETERS

A CONTAINS THE UPPER TRIANGULAR MATRIX R AND INFORMATION

THAT CAN BE USED TO EFFICIENTLY DETERMINE THE MATRIX

Q. THIS ARRAY IS INPUT TO SUBROUTINE CQRSLT WHEN A LEAST SQUARES SOLUTION IS COMPUTED.

J P VT A SINGLY SUBSCRIPTED INTEGER ARRAY OF LENGTH N THAT

CONTAINS INTERMEDIATE RESULTS THAT ARE INPUT TO SUBROUTINE

CQRSLT IF A LEAST SQUARES SOLUTION IS COMPUTED.

D I AG A SINGLY SUBSCRIPTED COMPLEX ARRAY OF LENGTH N THAT

CONTAINS INTERMEDIATE RESULTS THAT ARE INPUT TO SUBROUTINE

CQRSLT IF A LEAST SQUARES SOLUTION IS TO BE COMPUTED.

INTEGER JPVT(N)

COMPLEX A(MDIMA,N),DIAGCN), AHOLD

DO 8 1*1,

D I AG ( I

) *0.

J P VT ( I

) * I 8 CONTINUE

DO 1 J * 1 * N

C WRITE ( 6,753 ) J 753 FORMATt* J-+I5)

MMJ«M-J*1

EMA XN-O.

DO 2 K * J » N

ENORM*VNORM(MMJ, A( J,K)

)

I F < ENORM.LE.EMAXN) GO TO 2

JPVTH-K

EMAXN*ENORM

2 CONTINUE

I F ( EMAXN. EQ.O. ) RETURN

JPVT(J)*JPVTH

DO 3 I *1 » M

75
OOOOOOOOOOOOOOOOOOOOOOOOO

AHOLD* A ( I » J ) A(I»J)*A( I, JPVTH)

A ( I > J P VTH ) *AHOLD

3 CONTINUE

C COMPUTE THE Q VECTOR

VNA*VN0RM(1, A( J, J )

)

IF ( VNA.NE .0. ) GO TO 5

D I AG ( J

) “-EMA XN

GO TO 6

5 DXAG(J) «-EMAXN*A(J#J ) / VN A 6 A(J,J)*A(J,J )-DI AG ( J

)

QNORM*VNORM(MMJ, A ( J, J )

DO 7 I J * M A(I# J

) *A( I, J ) /ONORM

7 CONTINUE

IF(J.EQ.N) RETURN

JP1-J+1

DO 4 K * J P 1 » N

CALL QTX(MMJ#A(J»J)*A(J>K)*A(J*K))

4 CONTINUE

1 CONTINUE

RETURN

END

SUBROUTINE CQRSLT(MDIMA,M,N,A,JPVT,DIAG>B,X,RESID)

SUBROUTINE CORSLT COMPUTES THE LEAST SQUARES SOLUTION OF

AX-B USING THE OR DECOMPOSITION THAT IS COMPUTED BY

SUBROUTINE CQRDCT.

INPUT PARAMETERS

MDIMA THE FIRST OR ROW DIMENSION OF A

M THE NUMBER OF COLUMNS OF A.

N THE NUMBER OF COLUMNS OF A.

A A TWO DIMENSIONAL COMPLEX ARRAY THAT CONTAINS THE

INFORMATION FOR THE OR DECOMPOSITION OF A THAT WAS

COMPUTED BY SUBROUTINE CQRDCT.

J PVT A SINGLY SUBSCRIPTED INTEGER ARRAY OF LENGTH N THAT

CONTAINS RESULTS THAT WERE COMPUTED BY SUBROUTINE CQRDCT.

D I A G A SINGLY SUBSCRIPTED COMPLEX ARRAY OF LENGTH N THAT

CONTAINS RESULTS THAT WERE COMPUTED BY SUBROUTINE CQRDCT.

B A SINGLY SUBSCRIPTED COMPLEX ARRAY OF LENGTH M THAT

CONTAINS THE RIGHT HAND SIDE OF THE EQUATIONS AX*B.

76
oooooooo

OUTPUT PARAMETERS

X A SINGLY SUBSCRIPTED COMPLEX ARRAY OF LENGTH N THAT

CONTAINS THE LEAST SQUARES SOLUTION OF THE EQUATIONS AX-B.

RESID THE L 2 NORM OF THE RESIDUAL VECTOR R-B-AX.

COMPLEX A(MDIMA,N),DIAG(N)*B(M) ,X<N)#SUM, XHOLD

INTEGER JPVT(N)

DO 1 I *1 * N

X C I

) -0.

IF(VN0RM(1#DIAG(I ) ) • EQ • 0 • I GO TO 1 NM-I

1 CONTINUE

DO 2 J *1 > NM

MMJ *M-J+1

CALL QTX(MMJ>A(J>J)>B(J)>B(J )

)

2 CONTINUE

LB-M-NM

RESID*0.

IF(LB.GT.O) RESID* VN0RM(LB»B(NM+1)

)

C BACKSUBSTITUTION

X(NM)*B(NM)/DIAG(NM)

NMM1 *NM-1

DO 5 J*1»NMM1

K * N M —

J

KP1*K*1

SUM *0

DO 6 L*KP1>NM

SUM* SUM + A ( K# L )*X(L)

6 CONTINUE

X(K)*( B(K)-SUM) /DIAG(K)

5 CONTINUE

C UNSCRAMBLE THE SOLUTION

DO 9 1*1* NHM1

J *NM—

J P VTH* J P VT ( J ) XHOLD *X { J

)

X ( J )*X( JPVTH)

X( JPVTH)*XHOLD

9 CONTINUE

RETURN

END

77
r>

o

o FUNCTION VNOR M { M» X

)

FUNCTION VNORM COMPUTES THE NORM OF THE M-VECTOR X.

COMPLEX X ( M

)

XMAX-O.

DO 1 I * 1 * M XMAX*AMAX1(XMAX>ABS{ R E A L ( X{ I ) ) )>ABS(AIMAG( X ( I)

)

) ) 1 CONTINUE

VNORM-O.

IFtXMAX.EQ.O. ) RETURN

SUM«0.

DO 2 I *1 > M SUM-SUM+CREAL (X(I) ) /XMAX)**2+( AIMAG( X C I ) ) /XMAX)+*2

2 CONTINUE

VNORM*XMAX*SQRT ( SUM

)

RETURN

END

SUBROUTINE QTX ( M# 0, X , OX)

C COMPUTE 0 TIMES X WHERE 0 IS AN ELEMENTARY REFLECTOR AND

C X IS AN ARBITRARY M-VECTOR.

COMPLEX Q(M)

X ( M ) >QX(M)»QHX

OHX-O.

DO 1 I *1 * M

OHX«QHX*CONJG(Q(I ) )*X(I)

1 CONTINUE

QHX *QHX + OHX

DO 2 I-1>M

QX(I)«XCI)-OHX*Q(I)

2 CONTINUE

RETURN

END

78
1

5

10

15

20

25

30

35

*0

A. 2 FIELD Program

PROGRAM FIELD ( I N PUT , GUTP UT > TA p E 5 * I N P UT » T A P E 6 » OUTP 'JT ) C THIS PROGRAM READS COMPUTED CURRENTS AS DATA FROM FILE 123CXXX

C AND COMPUTES FIELD STRENGTHS AT A SELECTED FIELD POINT

COMPLEX 0MSAP,S0APE»GMN1,GNL2»GML3»S1>S2,S3,AMN1,ANL2,AML3

1 »BMN1,BNL2*BML3,CMN1,CNL2>CML3»CJX1,CJX3>CJY1,CJY2

2 >E(3)»CJZ2,CJZ3,C1»RSUBV>ESUBZ,XS(2A)

INTEGER NXT0TH(2)>NYT0TH(2),IXH(13,2),IYH(16,2)

REAL M U> K A Y NPRNT1*0

NPRNT2*0

NPRNT3=0

C NPRNT1 CONTROLS PRINTING OF OM E G A , MU > E P S > K A Y, OM S A P » S OA p

C NPRNT2 CONTROLS PRINTINT OF XP,YP,ZP AND COEFFICIENTS

C NPRNT3 CONTROLS PRINTING OF M A » N A N XTOTH ( 1 ) =2

N X TO TH ( 2 ) *2

NYTOTH(l) -2

N YTOTH ( 2

) * 2 IXH(1,1)=1

I XH(1,2) -1

I XH ( 2 * 1

) *2

I XH ( 2 » 2

)

s 2 IYH( 1» 1) *1

I YH ( 1 f 2 ) * 1

I YH ( 2 » 1 ) 2

I YH ( 2 > 2 ) * 2 X=0.9144

Y*1 .0668

Z«1.^78

DX* • 6096

DY= .6096

DZ* .4572

DXDY«DX*DY

DXDZ=DX*DZ

DYDZ = D Y*D Z XSTART*.91A4

YSTART*1. 0668

ZSTART*l.<fA78

FREQ= 1 .481 E8

PI = A,*ATAN(1. ) TPI -PI + PI

OMEGA*TP I*FREQ

MU=A.E-7*PI

E P S = 1 • E~9 / ( 36 • * P I

)

79
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50

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60

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70

75

80

85

90

95

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l S * 1 l F * 2 KAY®0MEGA*SQRT(MU*EPS)

0MS4P«CMPLX(0.,0MEGA*MU/(4.*PI )

)

S04PE=-CMPLX(0.,.25/(0MEGA*PI*EPS)

)

IF(NPRNT1»GT.0) WRITE (6,15) OMEGA, MU, EPS, KAY, 0MS4P,S04PE

15 FORMAT ( *0 THE VARIABLES 0M E G A , MU, E P S , K A Y, 0M S4 P , S 04 P E ARE

1LISTED BEL0W*/(8E15.6 ) ) READ CURRENTS IN THE ZP=0 PLANE, J SUB X AND J SUB Y

JU = -1

NYTDT^NYTOTH (1 ) N X T0T = N X TO TH ( 1

)

DO 301 IN*1, NYT0T

DO 301 IM=1»NXT0T

J U = J U + 2

READ(5,22) XS(JU) ,XS( JU+1)

22 FORMAT (4E15.6)

301 CONTINUE

READ CURRENTS IN THE XP*0 PLANE, J SUB Y AND J SUB Z

DO 302 L * L S , L F NYTOT s NYT0TH { L

)

DO 302 IN*1, NYTOT

J U® J U+ 2 READ(5,2 2) X S ( J U ) , X S ( J U+ 1

)

302 CONTINUE

READ CURRENTS IN THE YP®0 PLANE, J SUB X AND J SUB Z

DO 303 L s L S , L F NXT0T«NXT0TH(L I DO 303 IM»1,NXT0T

JU» JU*-2

READ(5,22) XS ( JU) , XS ( JU+1

)

303 CONTINUE

WRITE CURRENTS AND COORDINATES FOR CHECK

WRI TE (6,1 8)

18 FORM AT ( /* CURRENT IN THE ZP-0 PLANE*/4X*XP*6X*YP*15X

1*J SUB X*2 4 X* J SUB Y*/)

J U= -1

NYTOT *NYTOTH ( 1

)

NXTOT-NXTOTHd ) DO 321 IN*1, NYTOT

N = I YH ( IN, 1)

YP* (FLOAT (N )-l • )*DY+YSTART

DO 321 IM«1,NXT0T

M « I X H ( IM, 1 ) XP«(FL0AT(M)-1.)*DX+XSTART

80
100

105

no

115

no

125

130

135

no

H5

JU=JU+2

WRITE (6*322) XP,YP*XS(JU)*XS(JU+1)

322 FORMAT (2F8,A*AE15*6)

321 CONTINUE

WRI TE (6*19)

19 FORMAT(/* CURRENT IN THE X P *0 PLANE* / A X* YP *6 X* Z P*1 5

1*J SUB Y* 2 A X * J SUB Z*/)

DO 323 L* L S * L F ZP*(FLOAT(L ) -1 • )*DZ+ZSTART

N YTOT *NYTOTH ( l

)

DO 323 IN-1 * NYTOT

N*IYH( IN* L

)

YP « (FLOAT ( N

) -1 . )*DY-fYSTART

JU-JU+2

WRITE (6*322) YP,ZP*XS(JU)*XS(JU+1)

323 CONTINUE

WRI TE (6,20)

20 FORMAT ( / * CURRENT IN THE YP*0 P L A N E * / A X * X P * 6 X * Z P * 1 5

1*J SUB X* 2 A X * J SUB Z*/)

DO 3 2 A L*LS,LF

ZP«(FLOAT (L ) -1 • )*DZ+ZSTART

NXTOT -NXTOTH ( L

)

DO 32A I M = 1 , NXTOT

M * I XH ( IM, L ) XP*(FL0AT(M)-1« )*DX+XSTART

JU-JU+2

WRITE(6,322) XP,ZP*XSIJU),XS(JU+1)

3 2 A CONTINUE

I F = 1

E ( IF ) *0.

E ( IF + 1) =0.

E ( I F +2 ) *0 . IF(NPRNT2.GT.O) WRITE (6*1) X,Y,Z

1 FORMAT ( /+1THE FOLLOWING COEFFICIENTS CORRESPOND TO THE

1 F I F L D POINT */* X = *F8.A*» Y=*F9.A*, Z = *F8.A/)

CC COMPUTE COEFFICIENTS IN THE FLOOR OR Z-0 PLANE

C (SUBSCRIPT 1* BOTTOM XY-PLANE* FLOOR)

C JU = -1

N YT -NYTOTH ( 1

)

DO 101 IN*1*NYT

N = I YH ( IN, 1 ) YP=(FL0AT(N)-1.)*DY+YSTART

NXT»NXT0TH(1 ) DO 101 IM=1,NXT

M = I XH ( IM, 1 ) XP«(FL0AT(M)-1.)*DX+XSTART

JU-JU+2

R MN1 s SOP T ( ( X-XP ) **2+ ( Y-YP ) **2+Z**2

)

GMN1=CMPLX (C0S(KAY*RMN1) , — S I N ( KAY*RMN1 ) ) / RMN1

Sl-CMPL X( KA Y*K AY*RMNl*RMNl-3 .,-3.*KAY*RMN1)/RMN1**-,

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180

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AMN1«( X- X P ) * S 1 BMN1=(Y-YP)*S1

CMN1=Z*S1

COMPUTE COEFFICIENTS OF JX AND JY C 0 P R E S PON D I N G TO E SUB X CJX1=-0MSAP*GMN1*DXDY-S04PE*(CMPLX(1.,KAY*RMN1)/RMN1**2

1 +( X-XP ) *AMN1 )*GMN1*DXDY

CJY1=-S04PE*(Y-YP)*AMN1*GMN1*DXDY

E(IF)=E(IF)+CJX1*XS(JU)+CJY1*XS(JU+1)

IF(NPRNT2.GT.0)WRITE(6,2) XP,YP»CJX1,CJY1

? F0RMAT(*0 COEFFICIENTS OFF JX AND JY CORRESPONDING TO

IE SUB X AT XP»*F8.4*, YP = *F8.4* AND ZP-0. AR E * / ( A El 5 . 6 )

)

COMPUTE COEFFICIENTS OF JX AND JY CORRESPONDING TO E SUB Y CJX1*-S0APE*(X-XP)*BMN1*GMN1*DXDY

CJY1=-0MS4P*GMN1*DXDY-S04PE*(CMPIX(1.,KAY*RMN1)/RMN1**2

1 ( Y-Y 5 ) ARMN1 ) *GMN1*DXD

E(IF+1)«E(IF+1)+CJX1*XS(JU)+CJY1*XS(JU+1)

IF ( NPR NT2.GT.0) WRITE (6,3 ) XP,YP,CJX1,CJY1

3 FORMAT ( $ 0 COEFFICIENTS OF JX AND JY CORRESPONDING TO

IE SUB Y AT XP-+F8.4*, YP=*F8.4* AND ZP = 0. AR E* / ( 4E15 .6 ) ) COMPUTE COEFFICIENTS OF JX AND JY CORRESPONDING TO E SUB Z CJX1»-S0^PE*(X-XPJ*CMN1*GMN1*DXDY

C JY1--S04RE* ( Y-YP)*CMN1*GMN1*DXDY

E ( I F 2 ) * E ( IF + 2)+CJXl*XS(JU)+CJYl*XS( J U 1 ) IF(NPRNT2.GT.0)WRITE(6,^) XP,YP,CJX1,CJY1

4 F OR M AT ( *0 COEFFICIENTS OF JX AND JY CORRESPONDING TO

IE SUB Z AT XP-+F8.**, YP-*F8.A* AND ZP-O. ARE * /(4E 15. 6 ) / > 101 CONTINUE

COMPUTE COEFFICIENTS IN THE SECOND OR X«0 PLANE

(SUBSCRIPT 2 , NEAR Y Z-PL ANE , FRONT WALL)

DO 102 L = L S , L F ZP»(FL0AT(L)-1.)*DZ+Z START

NYT«NYTOTH(L ) DO 102 IN-1, NYT

N - I YH ( IN, L ) YP= ( FLOAT (N )-l . )*DY4-YSTART

JU-JU+2

RNL2-S0RT( X * *2+( Y-YP )**2+ ( Z-ZP )**2

)

GNL?«CMPLX(C0S(KAY*RNL2),-SIN(KAY*RNL2))/RNL2

S2=CMPLX( KAY*KAY*RNL2*RNL2-3.,-3.*KAY*RNL2) /RNL2**^

A NL 2 * X* S2

BN L 2 * ( Y- Y P ) * S 2 CNL2- ( Z-ZP )*S2

COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB X

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220

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230

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240

245

250

CJY2«-S04PE*(Y-YP)*ANL2*GNL2*DYDZ

CJZ2=-S04PE*(Z“ZP)*ANL2*GNL2*DYDZ

E(IF)»E(IF)+CJY2*XS(JU)+CJZ2*XS(JU+1)

IF(NPRNT2.GT.0)WRITE(6,5) YP,ZP>CJY2>CJZ2

5 FORMAT ( *0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SUB X AT XP*Q., YP=*F8.4* AND ZP = *F8.4* AR E * /

(

4 El 5 . 6 ) ) CC COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB Y

C CJY2=-0MS4P*GNL2*DYDZ-S04PE*(CMPLX(1.,KAY*RNL2)/RNL?**2

1 +(Y-YP)*BN12 )*GNL2*DYDZ

CJZ2«-S04PE*(Z-ZP)*BNL2*GNL2*DYDZ

E ( IF + 1) =E ( IF 41 )+C JY2*XS( JU) +CJZ2*XS ( JU + 1)

IF(NPRNT2.GT.0)WRITEC6»6) YP,ZP,CJY2,CJZ2

6 FORMAT (*0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SUB Y AT XP*0«, YP S *F8.4* AND ZP«*F8.4* A P E * / ( 4E 1 5 . 6 ) ) C

C COMPUTE COEFFICIENTS OF JY AND JZ CORRESPONDING TO E SUB Z C CJY2»-S04PEM Y-YP ) *CNL2*GNL2*DYDZ

C JZ2=-0MS4P*GNL2*DYDZ-S04PE* (CMPLX( l.,KAY*RNL2)/RNL2**2

1 ( Z-ZP)*CN12 )*GNL2*DYDZ

E(IF+2)»E(IF+2)+CJY2*XS(JU)+CJZ2*XS(JU+l)

IF(NPRNT2.GT.0)WRXTEC6*7) YP,ZP,CJY2,CJZ2

7 FORMAT ( *0 COEFFICIENTS OF JY AND JZ CORRESPONDING TO

IE SU8 Z AT X P ®Q » f YP»*F8.4* AND ZP=*F8.4* ARE* / ( 4E 1 5 . 6 ) /

)

102 CONTINUE

C

C COMPUTE COEFFICIENTS FOR THE THIRD OR Y*0 PLANE

C (SUBSCRIPT 3 » NEAR XZ-PIANE, RIGHT WALL)

C DO 103 L«LS»LF

ZP=(FL0AT(L)-1.)*DZ+ZSTART

N XT *NXTOTH ( L

)

DO 103 IM S 1»NXT

M*IXH( IM,L

)

XP S (FLOAT ( M

) -1 • )*DX+XSTART

JU= JU+2

RML3*S0RT( (X~XP!**2+Y**2+(Z-ZP)**2 ) GML3«CMPLX(C0S(KAY*RML3)»-SIN(KAY*RML3))/RML3

S3«CMPLX(KAY*KAY*RMl3*RML3-3.»-3.*KAY*RML3)/RHL3**4

A M L 3 = ( X-X P ) * S3

B ML 3* Y+ S3

CML3«( Z~ZP )*S3

C COMPUTE COEFFICIENTS OF JX AND JZ CORRESPONDING Tr E SUB x

CJX3*-0MS4P*GML3*DXDZ-S04PE*(CMPLX(1 .#kay*r w l?)

l 4(X-XP)*AML3)*GML3*DXDZ

CJZ3*-S04PE* ( Z—ZP)*AML3*GML3*DXDZ

E(IF)«E(IF)+CJX3*XS(JU)+CJZ3*XS(JU+1)

IF(NPRNT2.GT.0)WRITE (6>8) XP >ZP»CJX?»CJ -3

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270

275

280

285

290

295

300

305

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8 F OR MAT ( *0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB X AT XP«*F8.4*, YP*0. AND ZP»*F8.4* A R E * / ( 4E 1 5 . 6 )

)

COMPUTE COEFFICIENTS OF JX AND JZ CORRESPONDING TO E SUB Y

CJX3=-S04PE*(X-XP)*BML3*GML3*DXDZ

CJZ3=-S04PE*(Z-ZP)*BMt3*GMt3*DXDZ

E(IF+1)*E(IF+1)+CJX3*XS(JU)+CJZ3*XS(JU+1)

IF(NPRNT2.GT.0)WPITE(6»9) XP,ZP»CJX3,CJZ3

9 FORMAT (+0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB Y AT XP«*F8.4*, YP-0. AND ZP«*F8.4* AR E* / ( 4E 1 5 . 6 ) ) COMPUTE COEFFICIENTS OF JX AND JZ CORRESPONDING TO E SUB Z CJX3»-S04PE*(X-XP)*CML3*GML3*DXDZ

C JZ3=~0MS4P*GML3*DXDZ-S04PE* (CMPLX(1.,KAY*RML3)/RML3**2

1 ( Z-ZP)*CML3 )*GML3*DXDZ

E(IF+2)=E(IF+2)+CJX3*XS(JU)+CJZ3*XS(JU+l)

IF(NPRNT2.GT.O)WRITE(6,10) XP,ZP,CJX3,CJZ3

10 F0RMAT(*0 COEFFICIENTS OF JX AND JZ CORRESPONDING TO

IE SUB Z AT XP«*F8.4*, YP«0. AND ZP«*F8.4* AR E * / ( 4E 1 5 . 6 ) /

)

103 CONTINUE

100 CONTINUE

N A * J U+ 1 M A= I F + 2 IF(NPPNT3.GT.0)WRITE(6,13) MA,NA

13 FORMAT ( *0 MA«*I5* NA-+I5/)

ADD DIRECT ANTENNA CONTRIBUTION TO E SUB Z

j: Cl*CMPLX(.376681»-.004938)

DTOR=PI /180.

F TOM* .304 801

K AY*177. 72*DT0R

X P P = 1 0 . * F T 0 M YPP=-32. 5*FT0M

ZD=7-4.*FT0M

TC»(41.5573*Z+19.8333)*DT0R

RSUBT»SORT ( ( XPP-X)**2+(YPP-Y)**2

)

THETA=90.*DT0R-ATAN(ZD/RSUBT)

R SUBV= (COS (THETA) -Cl) / (COS (THETA )+Cl

)

R S QR T ( RSUBT**2 + ZD**2 ) arg=kay*r+tc

ESUBZ=(1»+RSUBV)*18«515939/R*CMPLX(C0S(ARG)>— SIN(ARG))

E(IF+2)«E(IF+2)+ESUBZ

WRITE E SUB X, E SUB Y, AND E SUB Z AT THE FIELD POINT

WR I TE ( 6 » 1 7 ) 17 FORMAT

(

f

E SUB X, E SUB Y, AND E SUB Z AT THE FIELD

1P0INT*//*X*7X*Y*7X*Z*3X,3(5X*AMP*4X*PHASE*1X)/)

AMPX = C ABS ( E ( IF ) ) AMPY = CABS( E ( I F 1 ) )

84
AMPZ=C ABS ( E ( I F + 2 ) ) PHSX=ATAN2 ( AIMAG< E( IF ) ),REAL (E (IF) ) ) /DTOR

PHSY=ATAN2(AIMAG(E(IF+1)),REAL(E(IF+1)))/DT0R

PMSZ=ATAN2(AIMAG(E(IF+2)),REAL(E(IF+2)))/DT0R

WRITE ( 6, 16 ) X,Y,Z,AMPX,PHSX,AMPY,PHSY>AMPZ,PHSZ

16 F0PMAT(3F8.4,F10.6»F8.2,F10.6,F8.2,F10.6.F8.2)

STOP

END
APPENDIX B. HPL PROGRAMS

B.l CURRENT and FIELD Programs

U

1 :

4 :

10

1 1

1 2

1 3

14

15

1 6

17

18

19

20

21

43

44

45

46

47

48

49

50

E u i 1 d i n g C h ar ac t e r i z at, i o n F' r o g r am a

" : E M T h e o r y d e u e 1 o p e d b y D r . M . ’ T . M a " : H F' L P i- o g r a rn a- a- u p e r i a- e d b y M . G . Fl r t h u r

1 H at i o n a 1 E u r e- -au o f S t an d an da-" : Boulder, CO 80303":

CURRENT

'FLAG 1 ( 'FLAG 2 C FLAG 3 i: 'FLAG 4 "FLAG 5

"FLAG 6

"FLAG 7

rad

24 ; Ml n

d i m A [ M

0 N T R 0 L 3 P R I N T N G 0 F C 0 N 3 T A N T S " : a f g ONTROLS PRINTING OF COEFFICIENTS":

0 N T R 0 L S P R INTI H G 0 F M A A H D H A " : s f

g

PR INTI NG OF D I AGONALS " : sf

q

1sf g

CONTROLS

CONTROLS

ONTROLS

ONTROLS

PRINTING OF E SUE Z":sfg 5

PR I NT I NG OF RES I DUAL " : sf g 6 P R INTI N G Ci F C U R R E N T S " : a f q 7

d i m spc

prt

f xd

N , 2 ] , E [ M , 2 ] , D C N , 2 ] ,

.T [ N ] , I [ 30 ] W i C 1 ] C C 5 0 , 2 ] , R [ 6 0 ] , XCN, 21

"CURRENT

4 PROGRAM" ; spc 2

24 > I C 1 3 ] “ENTER" : prt " C 0 0 R D I H A T E S 0 F FIELD POINT 3 : "

24

- P b e e p ; ent "ENTER XI. XI = ? "

, R [ 31 ]

“i cr

ii. J b e e p> ; e n t "ENTER X2. X 2 = '? "

, R [ 38 3 2 6 b e e pi : e n t. " E H T E R V 1 . V 1 = ? ")R[ 34]

2 7 b e €' p ; e n t "ENTER V 2Y2 = ? "

, R [ 39]

cL O b e e p ; ent "ENTER Zl. Z 1 = ? "

, R[ 37]

2 9 beep ; ent "ENTER Z2Z k‘ — ’? "

, R [ 4 0 ] 30 "PRINT ENTRIES":

31 prt ll ^ — 11 , RC31 ] , " X2=

, F: [ 3 8 ] , " Y 1 = "

, R [ 3 4 ] 32 prt II 17 ^ = ii , F [371 , " Z2=

, R [ 4 0 ]

34

-• P dap "ARE THESE NUMBER S C 0 F: R E C T '? ( Y N )

" 35 ent "

"

, W*

3 6 if U*[l]= "Y" or M T [ 1

]

= " y " ; g t o + 8

O ~7 b e e p ; €• n t. "RE-ENTER XI XI = ?

, R C 3 1 ] 3 8 beep ; e n t "RE-ENTER X2 . X 2 = ?

, R [ 3 8 ] 3 9 beep ; e n t "RE-ENTER VI Y 1 = ?

, R [ 3 4 ] 4 0 b e e p ; e n t "RE-ENTER Y2 . Y 2

1

, R [ 3 9 ] 41 beep : e n t "RE-ENTER Zl Zl _ r-i 1

, R [ 3 7 ] 42 hi e e pi ; a n t "RE-ENTER Z2 . Z2 = ?

, R [ 4 0 ] gto "PRINT ENTRIES"

"COEFFICIENTS" : RC38I-RC31 ] > R [ 6

]

R [ 3 9 ] - R C 3 4 ] > R [ 7 I R [ 4 0 ] - R C 3 7 ] } R [ 8 3 R [ 6 ] * R [ 7 ] } R [ 9 ] R [ 6 ] *R [ 8 ] }RC 10]

R C 7 ] *R [ 8 ] > R C 1 1 ]

86
51: 1 . 481e8>R[ 14]

52: 4 * at n < 1 ) } P 53s 2*F*RC14]>RC1S]

54: 4e -7*P } R [ 173

55: 1 e - 9 C36*P)>R[12]

56: 1 > I C 15] > I [ IS 3 > I [ 1 1

]

57: 2> I [ 14] > I C 17] > I [ 10]

5S : R C 1 8 ] *\ < R C 1 7 ] *R [ 1 2 ] > > R [ l e

]

59 : 0 > C C22, 1 ] ; R [ 1 8 ] *R [ 1 7 ]

" <4*P > > C [ 22 , 2 ] 60 : 0 > C [ 24 , 1 ] ; - . 25,- C R [ 1 8 ] *P*R [ l 2 ] > } C C 24 , 2 ] 61: if f 1 g 1 ; g t o +6

62 : prt “THE VRR I RULES OMEGA , MU , EPS I LONKRY , GMS4P , S04PE

63: prt " RRE LISTED EEL0W";flt 6

64: s p c 65 : prt R C 1 8 ] , R C 1 7 ] , R [ 1 2 ]

, R C 1 6 ]

, C [ 22 , 1 ]

, C [ 22 , 2 ]

, C C 24 , 1 ]

, C [ 24 , 2

]

66: spc 2

67: dsp "THIS CALCULATION TAKES S MINUTES"

68: -2 >

69: for K = I C 1 1 ] to I C 1 0

]

70: < K — 1 !' * R [ 8 ] + R [ 3 7 ] } Z 71: for J = I [ 1 8 ] to I [ 1 0

]

72: < J-l >*RC7]+RC34] >Y

73: for I = I C 1 5 ] to I [14]

74: < I - 1 > R [ 6 ] + R [ 3 1 ] > X 75: F + 3 > F

.76: if f 1 g 2 ; g t- o + 7

78: prt “THE FOLLOWING COEFFICIENTS CORRESPOND TO THE FIELD POINT

79: spc

80: fxd 4

8 1 : prt " X

=

"

, X

, " Y=

"

, Y

, " Z=

"

,

82: spc 2

83: -1 > LI 84: for N = I C 1 8 ] to I Cl —>

\ ] 85: < N- 1 >*RC7]+RC34] >R [ 32]

86f or- M = I C 1 5 ] to IC1 4 ] 87: al

1 >*RC6]+RC 3 1 ] >R [ 29]

88: U + 2 >U

89: N. ( ( X — R C 2 9 ] > 2 + < Y - R [ 32] >

'x

ii! + Z

"

izL ) } R [ 4 ] 9 0

C Q :E <RC16]*RC24] ') s R [ 24 ] > C [ 20, 1 ] ; -s i n ( P [ i b ] * R [ 2 4 ] R [ 2 4 ] > C [20,2]

91 : ; < f C 1 6 ] * R [ 2 4 ] ) A 2 - 3 j R [ 2 4] 4 } c [25, 1 ] , - J * R [ 1 y ] ir R [ 2 4] R [ 2 4] 4 > C [ 2 5 , 2 ] 92

( X

— R [ 2 9 ] j * C C 2 5 , 1 ] > c [2, 1 ^

< X

- R [29] j * c [ 'cL 5 9 2 ] > c [ 9 2 ] 93

C Y

— R [32] > * C [25, 1 ]

c [5, 1 ]; < Y

- R [32] > c [ cL 5 j 2 ] > L [

c9 2 ] 94: z+c [25, 1 ] >C[ 14, 1 ]

7 *CC 2 5, 2] > c [14, cL ] 95 : c 1 1 C M U L

x

( - C [ 2 2 , 1 3 , -c [ lL izL , 2] , C [ 2 0 9 1 ] 9 c [

2' 0 5 2 ] 9 CL i** , 1 ] , c [ ? 2] > 9 6

C [ 7 , 1

]

*R [ 9 ] > C [ 7 , 1

]

5 C [ 7 J 2] *R[ 9 ] > C [

"7

!» 2 1 97: 1

” R [ 2 4 ]

-'• 2 + C X - F: [ 2 9 ] > *C [ 2 , 1 ] > C [ 28, 1 3 98 : R [ 1 6 ] * F‘ C 2 4 ] s R [ 2 4 ] 2 + ( X

- R [ 29 ] > * C [ 2 9 2 ] > c [ 2 y 9 2 ] 99 : c 1 1 'CMUL' <C[24, 1

]

9 C [ 2 4 J

*=- 3 , c [ 28,1 3 * c [ cl' o 2 ] j c [ 2 8 j 1 ] , c [ 2 2] >

1 0 0 c 1 1 'CMUL'<C[28,1 3 , C[ 9 2], c [20, 1 ]

9 c [ 2 0 2 ] , c [ 2 8 , 1 ] , c [ 2 y « 2 ^ 1 101 CC 28, 1 ] * R [ 9 ] > C [ 2

8

9 1 ]

c [ 2 8 , 2 ] * R [ 9 3 > c [ 2 8 9 2 ] 102 C[ 7, 1 ]-C[23, 1 ] >C[ { , 1 3 9 CL i'" » L ] - C [ 2 y 9 2 ]

«. c [ r’ •I ] 103 —

c

[24,1 ] *

(

Y - R [ 3 2 ]

\ > C [ Q

, 1 ] ; - c [24, cl 3 * < v' - R [ 3 2 ] > > C [ 9 2 ]

1 04 c 1 1 ' OMUL ' < C [

9

, 1

]

9 C [ 9 9 2] , C [ 2 ,n, c [ 2 , 2 ] j c [ 9 J 1 ] , C C9 J 2 ] }

105 c I 1 'OMUL" <C[9, 1

]

n C [ 9 9 2 ] , C[ 2 0,1] 9 C [ 2 0 , 2 ] c [ 9 , 1 ] , c [ 9 , 2 ] > 106 CC 9 , 1 ] * R [ 9 ] / L [ 9 , 1 ] ; C [ 9 9 ]*R [ 9 ] > C c 9 , 2 3 107 CC 7 , 1 ] > A [ F , IJ

, 1 ] ; C [

“7

‘ !• *— ] > A [F, IJ ,21

108 CC 9, 1 ] >R[F, IJ+1 , 1 ] * C [ 9 9 2] > AC F , U+

9 2 ]

87
109

110

1 1 1

1 12

1 1 3

114

1 15

116

1 17

1 1 8

1 19

120

121

123

124

125

126

127

128

129

1 30

131

133

134

135

136

137

138

i f

Pr

prpr

sp

pr

®P

-

f 1 g 2 j g t o + 8

ct "COEFFICIENTS OF JX FIND JV"

t " C 0 R R E S P 0 N D I N G T 0 E S U E X A T "

; f x d 4

t "XF=" , RC293 , "YP=" , RE 32 3

, "AND 2P=0 ARE" ; f 1

t

c

t C E 7 , 1 ] , C [ 7 , 2 ] , C [ 9 , 1 ] , C C 9 , 2

r

C 1 1 C 1 1

4 , 1 ] * ( x

- R [ 2 Q 3 ) } c [ r

J 1 3 c [2 4 9 2 3 ( y - R [ 9 3 ) } C[ f 23

CMC l <: c [7, l 3 c [

“7

9 2 3 n C[ 5 , 1 3 •i c [ 5 9 4 3 c [ \ 9 1 3

, c [7 ,23 j CNL L

•" < C [7, l 3 •I C [

"7

j 2 3 9 C[ 4 0, 1 3 , c [ 2 0 , 2 3 9 c L *'

|l 1 3 , c [7, 2 3 >

1 I * R [ 9 3 > C [ 7 9 1 3 % c [ r

j 2 3 * F: [ 9 3 > c [

r>

9 2 3 CNL L

•- < - C [ 2 2 , 1 ] , - c [ 2 2 9 3 , c [ 2 0 9 1 3 , c [ 2 0 9 2 3 , C[ Q-• , 1 3 , C [ 9 23

1 3*R [ 9 3 > C [ 9 9 1 3 5 c [ 9 , Ci 3 *R [ 93 > c [ 9 9 2 3 2 4 ]

•- 2 + < Y -R [

4' 3

\\ * c [

Cj

9 1 3 > C [2 8 5 1 3

] *R[ 24 3/ R [ 2 4 3 2 + f V - R [ 3 2 3

|i ir c [

cr

9 2 3 > c [ 2

C;

9 2 3 CNL l

•" «: c [24 ? 1 3 n c [ 2 4 , 2 3 , c [2 y l 3 !» c [ 4 O ,, 2 3 j C[ 2 y , 1 3 , C [28 23

CNL L

- 1

C

[28 9 1 3 ? c [ 2 8 9 2 3 , c [2 0 n l 3 9 c [ 2 0 , 2 3 ? C [ 2 y , 1 3 , C [28 23

9 13* R [ 9 3 > C [ 2 C* 9 1 3 5 c [ 2 o , 2 3 * R [ 9 3 / c [ 2 8 9 2 3

1 3-C [28, 1 3 > c [ 9 , 1 3 !» c [ 9 9 3 -c [ 2 y j 2 3 > c L 9 9 2 3 C [ 7 , 1 3 } A [ F + 1 , U , 1 3 ; C [ 7 , 2 3 } A [ F + 1 , U , 2 3 C [ 9 , 1 ] > fi [ F+ 1 , IJ+ 1 , 1 3 ; C [ 9 , 2 3 > A [ F+ 1

, U+ 1, 2 3

l f sp

pr

pr

Prsp

pr

-P

f 1 g 2 ; g t o + 8

t '"COEFFICIENTS OF JX AND JV"

t " C 0 R R E S P 0 N D I N G T 0 E S U E Y AT"; f x d 4

t " X P = "

, R [ 2 9 3 , " V P

=

"

, R [3 2 3, "AH D ZP = 0 ARE"; fit 6

c

t C C 7 , 1 ] , C [ 7 , 2 ] , C [ 9 , 1 ] , C C 9 , 2

139 -C[ 2 4, 1 3*<X-R [29 3 j C [ r' , 1 3 ; - C [ 24,3 * -R [ cL 9 3

;i } c [7 9 3 140 c 1 1 •"CNUL-' <C[ “7

r q 1 3 , c [7 n 23 , C[ 14 ,13, c [ 14 9 3 9 c [ J7 , 1 3 , C [ r

j c- 3 > 141 c 1 1 C M U L <C[ II 1 3 , c [7 !i 2 3 , C [ 2 0 ,13, c [ 2 0 9 ^ 3 9 c [ i

-

' , 1 3 , C[ J7

9 c- 3 > 142 C [ 7 9 1 3 * F.: [ 9 3 >C [7, 1 3 ; C[ r

9 3*R [ 9 3 > C [ r 23

143 -C [ 2 4, 1 3*<Y-R [:] 2 3 CL 9 , 1 3 ; - C[ 24,2 3 * C V -R [ y 2 3 ) > c [ 9 9 ^ 3 144 c 1 1 -CNUL'" <C[ 9, 1 3 , c [9 j 23 , C[ 14 ,13, c [ 14 9

l=- 3 c [ 9 , 1 3 , C[ 9 9>=- 3 > 145 c 1 1 -'CNUL-' <C[ 9, 1 3 , c [9 9 23 , CL 20 ,13, c [ 20 9 ^ 3 9 c [ 9 , 1 3 , C[ 9 9 c- 3 > 146 C [ 9 9 1 3*R[93 >C [:s 1 C[ 9 , 4 3*R [9 3 > C [ 9 J 2 3 147 C [ 7 9 1 3 > A [ F + 2 , u, 1 3 ; c [7 1 23 > A [ F + 2 , U , 2 3 148 C [ 9 9 1 3 > A [ F + 2 , LH-1

j 1; c [ 9, 2 3 > A [ F + 2, u + 1, 2 3 149 i f f 1 g 2 ; g t o + r! 150 spc

151 r t "COEFFICI ENT C; 0 F j y AH D J Y

152 prt "CORRES F'O HIn H G TO E y IJE Z AT" ; f d 4 153 p r t

" X P

=

, R[2 93 ,

II YP II

J R [ 32 3

II 9 AND z P = 0 A R EII n f 1

6

154 3 p c 155 prt C: [ 7 , 1 3 , C [

1'' 2 3 , c [9 J 1 3 ,C[ 9, 23

156 spc

157 n e x t- M 158 n e x t- N 159 t" o r L = I [ 1 1 3 t l“l 1 [ 10 3

1 6 0 < L- 1

'.> * R [ 8 3 + R [ 37 3 / R [ 3 6 3 161 f o r N = I [ 18 3 t O 1 [ 17 3

1 62 < N

- 1 > * R [ 7 3 + R [ 34 3 > R [ y 2 3 163 U + 2 } IJ 164 \ (. X 4 + ( Y - E‘. [32 3 : 2 + < Z

“ R [3 6 3 > 2 > > R [ 4

cr 3 165 c o s R [ 1 6 3 * R [ 2 5 3 > R [ 2 5 3 > C [21 , 1 3 ; -s 1 n < R [ 1 6 3 ir R [ 25 3 > /R [

•~i cr 3 > 166 < < R [ 1 6 3 * R [ 2 5 3 - y .!• / R [ 25 3 4 } c [26, 1 3 n 3* R [ 1 y ] * R [ 2 5 3 R: [

“1 cr [21

a 4 }

, 21

C [26, 2 3

88
167

169

169

170

171

172

1 7 J 174

175

176

177

178

179

180

181

1 82

1 8 o 184

185

1 86

187

188

189

190

191

192

1 93

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

2 1 2 213

214

215

216

217

218

219

220

221

222

223

224

X fc c [ 26, 1 3 >CI 3 * l 3 .1 X *C [ 'cl b J cl 3 >

1 - F: [323 )*Ct b j 1 3

>. :t f,

J 1 3 < Y

( Z

- R [36 3 >*C[ 6 ji 1 3\ ct 1

CT

, 1 3 5 - c [ 2 4, 1 3 * C Y

-R t

•1; Cl 3 ) > c [ 1 0 J 1 3 ; c 1 1 CMUL'

3[ i 0 J 1 3 , c [ 1 0 J cl 3 , c 1 1 -'CMUL-' o: :[ i 0 1 3 , c t 1 0 J

‘cl 3, C [ 1 0 , 1 3*Rt

3

•j. c [ 1 0 , 1 3 |l c [ 1 0,

- c [ 4, 1 3*(Z--R t

"t b 3 > c [ 1 1 J 1 3

c l 1

- CMUL <C :[ i 1 5 1 3 , c [ 1 1 n cl 3,

c l 1 cmul o:;[ i 1 , 1 3 , c [ 1 1 J

'cl 3 , C t 1 1 , 1 3 *R[ 113 *}. c [ 1 1 , 1 3 n c [ 1 1. C [ 1 0 , 1 3 > R [ F , u j 1 3 n C [ 1 0 n 3 > fl[

[ 1 1 , 1 3 >R[F, IJ + 1 5 1 3 ; C [ l 1 J 2 3 >

i f f 1 g2; gt.

o

+ o

P p

r

L

t. "COEFFICI E NT o OF j Y R ND

P r t " CORRESF:'0 ND I N G T 0 E s IJ E P r t. " Y P = "

, R 1

3

2 3 f

II R ND ”7 P = II , R £ P c P r T C t 1 0 , 1 3 c [ 1 0 J 2 3 , c [ 1 1 J 1 3 £ P C c 1 1 - CMUL-' <-- c [ eJ 1 3 , - c [ cl

n c [ 1 0 , 1 3*R[ 1 1

> c [ 1 0 , 1 3 J c [ 1 0,

1 R [ 25 3 '•'2+0 R [

”l cl 3 > * c [ 8 J 1 3 > R [ 1 K 3 * R [ 2 5 3 -R [ cl

cr

._i 3 2 + c Y

- P [ 0 cl

l“ 1 1 -'CMUL " O :[ cL 8 1 3 , c [ cl 8 n

'cl 3 ,

1 1 -CMUL-' <i :t cl I;! j 1 3 , c [ cl 8 n cl 3 , c [

C;

, 1 3 * R [ 1 3 > c [ cl y , 1 3 J c [ 'cl 8 , c [ 1 0 , 1 3-Ct2fc3 , 1 3

.!• c t 10 n 1 3 j c [ 1 - c [ 2 4, 1 3*<Z--P [

•“> f 3 ) } c [ 1 1 1 1 3 ;

1 1

- CMUL-' O::[ 1 1 n 1 3 , c [ 1 1 n

ii. 3 ,

l“ 1 1 "CMUL" <i :c 1 1 9 1 3 , c [ 1 1 j 3 , c [ 1 1 , 1 3 * R [ 1 L 3 / c [ 1 1 , 1 3 J c [ 1 1 , c [ 1 0 , 1 3 >fl[F-H J LI

j 1 3 ; C [ 1 0 j 2 3 > r [ 1 1 , 1 3 > fi [ F Hi n IJ + 1 9 1 3 J c [ 1 1 n

i f f 1 g 2 ; g t o +8

spc

prt "COEFFICIENTS OF JY AND

prt "CORRESPONDING TOE SUE

prt 11 Y P = "

, R t 3 2 ] , "AND ZP=",R

s P : prt C C 1 0, 1 3

, C [ 1 0 , 2 ]

, C [ 1 1 , 13

spc

-CC24, 1 3*<Y-R[323 > >CC 10, 13

ell '' CMUL C C [ 1 O , 1 ] , C [ 1 0 , 2 3 , c 1 1 ' C M IJ L

' <CC 10, 1 ]

, CE 10, 2]

,

C C 1 0 , 13 *R[ 1 1 ] > C C 1 0 , 13 ; C [ 1 0

,

cl 1

'' CMUL ' C -C C 22 , 1 3 , - C C 22 ,

Ct 1 1 , 1 3*R[ 1 1 3 >Ct 1 1 , 1 3

; Ct 1 1

,

1 sR C 25 3 '-2+ C Z-R C 36 3 > *C C 1 5 , 1 3 R C 1 6 3 *R [ 25 3 /R L 25 3 '-2 + C Z-R C 36

ell ' CMUL ' < C C 28 , 1 3 , C C 28 , 2 3 , ell x CMUL •- < C [ 28 , 1 3 , C C 28 ,23,

C C 28 , 1 3 *R [ 1 1 3 > C C 28 , 1 3

; C C 28

,

Ct 1 1

, 1 3-CC28, 1 3 >CC 1 1 , 1 3

; Ct

C 1 1 0 , 1 3 > fit F + 2 , U , 1 3

; C C 1 0 , 2 3 > C C 1 1

, 1 3 > A t F + 2 , U+ 1 , 1 3

; C C 1 1 ,

if f 1 g 2 ; gt o +8

i: C 3 , 2 3 - F: t 3 2 3 > * C t 2 6 , 2 3 > C t 6 , 2 3 Z-R [ 36 3 > *C t 26 , 2 3 > C t 1 5 , 2 3 -C [ 24 , 2 3 * < Y-R C 32 3 > > C C 1 0 , 2 3 C C 3 , 1 3 , C C 3 , 2 3 , C t 1 O , 1 3 , C C 1 0 , 2 3 > C [ 2 1 , 1 3 , C [ 2 1 , 2 3 , C t 1 8 , 1 3 , C t 1 0 , 2 3 ) 23* R [113 >CC 10, 23

-C C 24 , 2 3 * < Z-R [ 36 3 > > C C 1 1

, 2 3 C [ 3 , 1 3 , C t 3 , 2 3 , C [ 1 1 , 1 3 , C t 1 1 , 2 ] > C t 2 1 , 1 3 , C t 2 1 , 2 3 , C t 1 1 , 1 3 , C C 1 1 , 2 3 j 2 3 *R [ 1 1 3 > C [ 1 1

, 2 3 F , U , 2 3 H t F ,

I.J + 1 ,23

J Z

" X Fit X P = 0

" ; fxd 4 [36 3 , "ARE" ; f 1 1 6

, C [ 1 1 , 2 3

3 , C [ 2 1 , 1 3 , C [ 2 1 , 2 3 , C [ 1 0 , 1 3 , C [ 1 0 , 2 3 2 3 * R [ 1 1 3 > C [ 1 0 , 2 3 C [28, 1 3

3 > * C [ 6 , 2 3 > C [28, 23

Ct 24 , 1 3, C [ 24 , 2 3 , C [ 28 , 1 3

, C [ 23 , 2 3 > C [ 2 1 , 1 3 , C t 2 1 , 2 3 , C t 28 , 1 3 , C [ 28 , 2 3 : 23* R [113 } C [ 2 8 , 2 3 0, 23- C [28, 23 > C t 1 0 , 2 3 -C [ 24 , 2 3 * < Z-R t 36 3 > > C L 1 1

, 2 3 C [ 6 , 1 3 , C t 6 , 2 3 ,: C [ 1 1 , 1 3 , C t 1 1 , 2 3 C t 2 1 , 1 3 , C [ 2 1 , 2 3 , C t 1 1 , 1 3 , C [ 1 1

, 2 3 > 2 3*R[ 1 1 3 >C[ 1 1

, 23

A t F + 1 , U , 2 3

3 > AtF+1 , U+l , 2 3

J Z

" Y FIT XF' = 0

" ; fxd 4 [363 , "ARE" ; f 1 t 6

, C [ 1 1 , 2 3

-C [24,23* ( Y-R C32 3 > > iZ [10,23

C [ 1 5 , 1 3 , C [ 1 5 , 2 3 , C [ 1 0 , 1 3 , c t : : C [ 2 1 , 1 3 , C [ 2 1 , 2 3 , C t 1 0 , 1 3 , C [ 1 0 . 2 3 2 3*R[ 1 1 3 >Ct 10, 23

3 , C [ 2 1 , 1 3 , C [ 2 1 , 2 3 , C [ 1 1 ,

1

'

. 23*R[ 1 1 3 >C[ 1

1

, 23

> C [ 28 , 13

3 >*C[ 15,23 > C 1 28 , 23

C [ 24 , 1 3 , C [ 24 , 2 3 , C [ 28 , 1 3 , C [ 2 : . Z : C [ 2 1

, 1 3

, C [ 2 1

, 2 3 , C [ 28 , 1] , C 1 28, 23

2 3 * R [ 1 1 3 > C [ 2 8 , 2 3

1 , 2 3 -C [ 28 , 2 3 > C 1 1 1

, 2 3 FltF + 2, U, 23

3 > At F + 2, U+l , 2 3

89
2 3 0

c. J b

spc

prt "COEFFICIENTS OF JY FIND JZ"

prt "CORRESPONDING TOE SUB Z RT XF' = 0

prt " YP= "

, R [ 32 ] , " AND ZP= ", R C 36 3 , " ARE

spc

prt C: C 10, 1 3 , C [ 1 0 , 2 3 , C [ 1 1 , 1 3 , C C 1 1 , 2 3 s p c n e x t N n e x t L for L = I C 1 1 3 to I [103

< L - 1 > * R [ 8 3 + R [ 3 7 3 > R[363

for M = I C 1 5 3 to I [14]

< fl - 1 > * R [ 6 3 + R [ 3 1 ] > R C293

IJ + 2 > I

J

fxd 4 f 1 t 6

:40

:4 l :4 2

: 4 3

:44

:45

:4 6

2 5 8

259

2 6 0

\ ( ( X — R [ 2 9 3 2 + Y 2 + Z - R [ 3 6 3

-• 2 } 1 R [ 2 3 3 c os ( R [ 1 6 3 *R [ 23 3 > xR [ 23 3 > C [ 1 9 , 1 3 ; -s i r. < R [ 1 6 3 *R [ 23 3 > /R C 23 3 > C [ 1

( < R [ 1 6 3 *R [23 3 > -2-3 > xR [23 3 4 l CC27, 1 3 ; -3*R [ 1 6 3 *R [ 23 3 xR [ 23 3 4

< X-R [ 29 3 > *C [ 27 , 1 3 > C [ 1 , 1 3 ; < X-R [ 29 3 )*C [ 27 , 2 3 > C [ 1

, 2 3 V * 0 [ 2 7 , 1 3

’.! C [ 4 , 1 3 ; Y * U C 2 7 , 2 3 u [ 4 , 2 3

< Z-R [ 3b 3 ) *C [ 27 , 13 > C [13, 13; CZ-R [ 36 3 > *C [ 27 , 2 3 > 0 [13,23

c 1 1 •- CMUL •- < -C [ 22 , 1 3 , -C [ 22 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C C8, 1 3 , 0 [ 8 , 2 3 CCS, 1 3 * R [ 1 0 3 > C [ 8 , 1 3 ; C [ 8 , 2 3 * R [ 1 0 3 > C C 8 , 2 3

9,23

} C [ 2 7 , 2 3

247 1 R [ 2 3 3 2 + < X - R [ 2 9 3 )+C[ 1 , 1 3 } C [28, 1 3 2 48 R [ 1 6 3 * R [ 2 3 3 x R [ 2 3 3 2 + ( X

- R [ 2 9 3 >*C[ 1,23 > C [28, 23

249 ell -'CMIJL-' C [ 28 , 13 , C [ 2 8 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C [ 2 8,13, C[28, 23

2 5 0 ell -'CMIJL- < C [28, 1 3 , C [ 2 8 , 2 3 , C [ 2 4 , 1 3 , C [ 2 4 , 2 3 , C [2 8,13, CC28, 23

251 C [28, 1 3*R[ 103 > C [28 , 1 3 ; C [28, 2 3 * R [ 1 0 3 } C [28, 23

CT “i C [ 8 , 1 3 - C [ 2 8 ,13} C [

8

, 1 3 ; C [8, 2 3 - C [ 2 8 , 2 3 >C[8, 2 3 2 5 3 - C [24, 1 3 * < Z -R [ 36 3 > > C [ 1 2 ,13; - C [ 2 4 , 2 3* (Z-R [36

C [ 1 2 , 2 3 25 4 ell -CMIJL-'':: C [ 1 2 , 1 3 , C [ 1 2 , 2 3 , C [ 1 , 1 3 , C [ 1 , 2 3 , C [ 1 *4- q 1 3 , C [ 1 2 , 2 3 )

CT CT

id- J _' ell 'CMIJL"':: C [ 1 2 , 1 3 , C [ 1 2 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C [ 1 2 , 1 3 , C [ 1 2 , 2 3 256 C [ 12, 1 3*R[ 103 >C[ 12 , 1 3 ; C [ 12, 2 3*R[ 103 > C C 12, 23

C [ 8 , 1 3 > R [ F , IJ

, 1 3 ; C [ 8 , 2 3 } A C F

C: E12, 1 3 >A[F, U+l , 1 3 ; C [ 1 2 , 2 3

i f f 1 g 2 ; g t o + 9

spc

U , 2 3 R [ F , U+l , 23

261

262

2 6 3

264

265

2 6 6

267

2 6 8

26 9

27 0

271

279

280

281

prt "COEFFICIENTS OF JX AND JZ"

p r t

" U U R F: E S F' U N D I N G T U E S U B X R T "

; f x d 4 prt " X P

=

"

, R [ 2 9 3 , " Y P = 0 , AND"

pr t. "ZP=" , R[363 , "ARE" ; f 1 t 6

spc

prt C [ 8 , 1 3 , C [ 8 , 2 3 , C [ 1 2 , 1 3 , C [ 1 2 , 2 3 spc

- C [ 2 4 , 1 3* < X-R [29 3 > * R [ 10] > C [

8

, 1 3

; -C [ 24 , 2 3 * < X-R [ 29 3 >*R [10 3 > C [ 8 , 2 3 c 1 1

' CMUL -' < C [8,13, C [ 8 , 2 3 , C [ 4 , 1 3 , C [ 4 , 2 3 , C [ 8 , 1 3 , C [ 8 , 2 3 > c 1 1 CMIJL " < C [ 8 , 1 3 , C [ 8 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C [ 8 , 1 3 , C t 8 , 2 3 > - C [ 2 4 , 1 3* (Z-R [36 3 > *R[ 10 3 > C [

2 , 1 3

; -C [ 24 , 2 3 * ( Z-R [ 36 3 > +R [ 1 0 3 > C [ 1 2 , 2 3 c 1 1 ' CMUL " < C [ 1 2 , 1 3 , C [ 1 2 , 2 3 , C [ 4 , 1 3 , C [ 4 , 2 3 , C [ 1 2 , 1 3 , C [ 1 2 , 2 3 > c 1 1 CMUL" CCC 12, 1

, C [ 1 2 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C [ 1 2 , 1 3 , C [ 1 2 , 2 3 > C [ 8

,

L 3 > R [ F + 1, U , 13; C [ 8 , 2 3 > R [ F + 1 , U , 2 3 C [ 1 2 1 3 > R [ F + 1 , U + 1 , 1 3 ; C [ 1 2 , 2 3 > A [ F + 1

, U+l , 2 3

i f f g 2 ; g t o +9

spc

prt 'COEFFICIENTS OF JX AND J Z

" prt 'CORRESPONDING TOE SUB Y AT"; f x d 4

p r t. ' X P

=

"

, RC293 , "Y P=0, AND"

prt 'ZP=“ , R [363 , "ARE" ; f 1 t 6

spc

|

90
283

prt C C 8 , 13 , C C S , 2 ]

, C C 1 2 , 1 ] , C C

2]

284: spc

285: —

c

C 2 4 , 1 X-R c 2 9 ] *R [ 1 0 ] > CC y ,135 - C C 2 4 9 ] * - RC 2 9 ] > *R c 10] >C Co, 2 ] 286 : c 1 1

' C M U L

' <CC 8 , i ] l CC Cj ,2], CC i 3,1] J CC 1 9 c- ] 9 CC 8 , 1 ] , c C 8, 2 ] >

i. O »' • c 1 1 CMUL <CC 8 , i ] , CC 8 5 2 ] , CC i 9,1] 9 CC 1 9 9 ^ ] , CC o

, 1 ] , c c 9 2 ] ) 1 3 3

c 1 1 CMUL ( -c C 2 c:

J 1 ] , - C C 2 2 9

*=- ] , C C 1 9 , 1 ] , CC 1 9 9 ] , c [ 1 2 , 1 ] , CC 1 2 , 2 ] > 289: CC 12 , 1 ]*R[ 10] > CC 1 , 1 ] ; C C 1 •i- 9 2 ]*RC 1 0 ] } c C 1 2 2]

29u : 1 RC 2 3] -N 2 h ( / — R C 3 t ] )

¥• c C 1 3 , 1 ] > C C 2 8 9 1 ] 291 : RC 16 3 * R C 2 3 ]

••-" R C cl • !

] a 2 + ( Z-RC 36 ] >*cc 1 9 ^ ] > c c ii! 8 ,

ill ]

~ Q ”•

• c 1 1 -CMUL <cc 2

C*

9 1 ] , c c q

”i ] , c c 19, 1 ] , C C 19 9 cL ] , c C 2 9 1 ] , c C 23, 2] > 293- c 1 1 CMUL' c c c 2 - n 1 ] , c c n ^ ] , c [24, 1 ],c C 24 9

ii! ] , c C 2 9 1 ] , C C 2 8 , 2] > 294 : CC •-> Q , 1 ]*RC 10] > CC 2 Q , 1 ] ; c c 2 O q ii! ] * R C 1 0 ] > c C 2 8 9 2]

295: CC 12 , 1 ] - C C 2 8 , 1 ] > c c 12 9 1 ] ; C C 1 L ,2]- c C 28 , 2] c [ 1 k 9 ] 296: CC 8 , 1 ] > ACF+2, L , 1 ] <1 CC y , 2 ] > RC F + 2,1 , 2]

297: CC 1 2 , 1 ] > A :f + 2 9 u+ 1 9 1 ] 9 c C 1 2 , =- ] > R C F + 2 ,L + 1,

~> ] 298: if f 1 g 2 ; g t o +9

? 8 9

spc

3 0 0

pr t "COEFFICIENTS OF JN AND J Z

301 : prt "CORRESPONDING TOE SUE Z AT";

302: prt "XP=" , RC29] , " YP=0, AND"

303: prt " Z P =

"

, R C 3 6 ]

, "ARE" ; f 1 1 6

304

spc

3 0 5 : p r t CCS, 1 ]

, C C 8 , 2 ]

, C C 1 2 , 1 ]

,C 12, 2]

306: spc

307

n ext- M

308

n e x t L 309

n ext- I 310: n e x t. J

311: n e x t K

312: U+ 1 > I C 1 6

]

313: F+2> I C 12D

314: i f f 1 g 3 ; g t o + 3

315: spc 2

316: fxd 0: prt " MR=

, I C 1 2 ] , " NR = "

, I C 1 6 ] 317: c 1 1 CQRDCT < I C 1 3 ] , I C 1 2 ] , I C 1 6 ] > 318: f 1 t 6

319: if f 1 g 4 ; g t o +7

3 2 0 : s P c 321 : prt "DIAGONAL"

• s p c

: 2 3 : f o r 1=1 t o I C 1 6

]

324 : prt DC I , 1

]

, DC I , 2]

“

r - * n e x t. I 326: . 3 7 6 6 8 1 > C C 1 6 , 1 ] ; - . 0 0 4 9 3 8 > C C 1

6

, 2 ] 327: F' x 1 8 0 > R C 5 ] 3 2 8 : . 30480 1 > R C 15]

3 2 9 : 177. 7 2 * R C 5 ] > R C 16]

330 : RC 15]*10>RC30]

331 : RC 15] *— 32. 5 > R C 3 3

]

332 : ldf 2

333 : e n d

334: "CQRDCT" : 335 : I >p4; J > p 5 ; K>p6

336 : f o r 1 = 1 t o p 3

337

0 > D C 1 , 1 ] > D C I , 2 ] ; I > J C I ] O O p . next I 339: f o r J =1 t o p 3

340 : p 2 - •J + 1 > p 7

91
341 O > p 1 2

34 2 f o r K = J t. o p 3

343 - VHCiRMR •' < p7 , J , K > > p 1 3

34 4 if p 1 3 > p 1 2 ; K > P 8 ; p 1 3 > p 1 345 n €' x t K 346 if' p 1 2 = 0 ; g t- g "CQRDCTXT II 34 7 p 8 > J [ J ] 3 4 8 f or 1 = 1 to p

3 4 9 R C I , J , 1 ] > p 1 0 ; R [ I , J , 2 ] > pll

350 R [ I , p 8 , 1 ] > R C I , J , 1 ] 5 fl [ I 5 P 8 , 2 ] 351 p 1 0 } fi [ I , p 8 , 1 ] 5 pl 1 > R C I

, P 8,2]

' -J S n e x t I

•*;. cr -z>

•" V NOR MR"' < 1 , J, J > > P 16

354 i f p 16# 0 ; g t c> " 5

,i 355 -pi 2 3 DC J, i ] ; 0 > D [ J ,2]

3 5 6 gt G "6"

357 “5" : -pl2*fi[ J, ,T

, 1 ] p 1 6 .!• D [ J , 1 ]

" 6 “ : R [ J , J , 1 ] - D [ J , 1 ] > fl [ J , J , 1 ] 359 ’ V N O R M R ' < p 7 , J , J > > P 1 4

360 f g r I = J t, g p 2

361 RC I , J, 1 ] p 1 4 > fit I , J , l ]

R [ I , J , 3 6 2 n e x t I 363 i f J = p 3 ; g % o

" CQRDCTXT"

364 J+l > p

9

365 f g r K = p 9 t g p ‘l! 366 c 1 1 Q T X R R R

•" < p7, J , K > 367 n e x t K 368 n e x t J

3 6 9 "CQRDCTXT " : 3 7 0 p 4 5 1; p 5 > J ; p 6 > K 37 1 r e t

3 ? 3

374

37 7

z> r o

3 7 '3 3 8 0

381

3 82

383

384

i- cr

i“i ._i j 8 6

3 8 7

" OMUL " : P 1 P 3 ~ p 2 p 4 > p 7 ; p 1 p 4 + p 2 p 3 > p 8 ; p 7 > p 5 ; p 8 > p 6 ; r e t

" C D I V "

: p 3 p 3 + p 4 p 4 } p 7 ; < p 1 p 3 + p 2 p 4 > 7 p 7 > p 1 0

<. p 2 p 3 - p 1 p 4 > p 7 > p 1 1 ; p 1 0 > p 5 ; p 1 1 > p 6 ; r e t

" QTXRRR "

0 > p 1 0 > p 1 1 f o r Q = p 2 t o p 2 + p 1 — 1 R C Q , p 2 , 1 ] J' p 1 2 ; - H [ Q , p 2,2] ) p 1 3

c 1 1 " C M U L

" ( p 1 2 , p 1 3 , R [ Q , p 3 , 1 ] , R [ Q , p 3 , 2 3 , p 1 2 , p 1 3 > p 1 0 + p 1 2 .! p 1 0 ; p 1 1 + p 1 3 > p 1 1 n €• x t Q

2p 1 0 > p 1 0 ; 2 p 1 1 > p 1

f o r Q = p 2 t o p 2 + p 1 - 1 c 1 1

•" OMUL < p 1 0 , p 1 1 , R [ Q , p2 , 1 ] , R [ Q , p2 , 2 ] , p 1 2 , p 1 3 > R [ Q , p 3 , 1 ] - p 1 2 } R [ Q , p 3 , 1 ]

; R[ Q , p 3,2]- p 1 3 > RCQ, p 3 , 2

]

next Q

389 r e t

3 90 " VHORMfl " : 391 p 2 > p 5 } p 6

392 0 > p 7

3 9 3 f g r V = 1 t 394 max < p7 , at

3 9 5 p 5 + 1 > p 5

3 9 6 n e x t V 39 7 if p 7 = 0 ;

398 0>p8

92
399 : for V = 1 to p

400

p 8 + 1 Fl C p 6 , p 3 , 1 ] 4 01: pt'+l }p6

402n e x t V

403: ret p 7 £

\

p

8

* 159

0 : " B u i 1 di ng C h ar ac t e r i z at l o n P r o q r .am s " :

1 : "EM T h e o r y d e v e 1 o p e d b y D r . M . T . M a " : 2 : " H P L P r o g r am £ £ u p e r v i s e d b y M . G . R r t h u r " : 3 : "Nat i o n a 1 B u r e au o f 8

an d ar d s "

: 4 : " B o u 1 d €' r , Col o r ad o " : 5: "CURRENT Program: Part 2, File 2":

6: rad

7: 2 4 > M > N

8: -2 > F

9: for K=ICU] to I [10]

10 <K-1>*R[8]+RC37] >Z

1 1 if K=2;gto "201"

12 . 7 5 * R [ 15] >R[35]

13 8 0 * R [ 5 ] > R [ 2 8 ] 14 q t o " 2 0 2

" 15 "201" : 2. 25*R[ 15] >R[35 ] 16 9 9 * R [ 5 ] > R [ 2 8 ] 17 "202" : for J= I [IS] to I [ 1 7]

18 < J-l >*R[7]+R[34] >V

19 for I = I C 1 5 ] to I [14]

20 <1-1 >*RC6]+RE31 ] >X

21 F + 3 > F

22 " 2 1 8 " : f x d 5

23 £pc 2

24 prt "EM FIELD DhTh RTF I ELD POINT C o

• i 25 - P 26 "ENTER"

27 beep ; ent "ENTER R M P

. R M P X = ? "

, R [ 1 ] 28 beep ; ent "ENTER PHSX. PHSX = ? "

, R [ 1 9 ] 29 b e e p ; e n t " E H TER R M p V

RMPY = ? "

? R [ 2 ] 30 beep; ent "ENTER PHSY. PHSY = ? "

, R [ 2 0 ] 31 beep; ent "ENTER RMPZ. RMPZ = ? "

, R [ 3 ] 32 beep; ent "ENTER PHSZ. PHSZ = ? "

, R [ 2 1 ] 33 "PRINT ENTRIES":

34 prt " R M P X =

"

, R[ 1 ]

, " PHS •./ _ li , R [ 1 9 ] , " R M P Y = "

, R [ 35 prt " RMPZ= "

, R C 3 ] , "PHS 7 = 11 , R [ 2 1 ] 36 spc 2

37 dsp "ARE THESE NUMBER s c OR R EOT? < Y/N )

38 ent "

" ,W$

39 if W$Cl]s"Y" or WtCl] = 11 "

; g t o +8

40 beep; ent "RE-ENTER RMPX. RMPX = ? "

, R [ 1 ]

P H 8 V

=

" , RC20]

93
41 : b € e p ; e n t

" R

E - E H TER PH 8

. P H S X = ? "

, R [ 1 9 ] 42: b € e p ; e n t " R E - E H TER A M P V

AMF'Y = ? "

, R [ 2 ] 43: b e e p ; e n t

" R E - E N TER P H S V

PHSY = ? "

, R [ 2 0 ] 44: beepjent "RE-ENTER flMPZ. AMPZ = ? "

, R [ 3 ] 45: beepjent "RE-ENTER PHSZ. PHSZ = ? "

, R [ 2 1 ] 46: gto "PRINT ENTRIES"

47: "CALCULATION"

48: R [ 1 I.-’. 5 6 > R [ 1 ] 4 9

RC2 ]••-'. 5 6 > R C 2 ] 50: R [ 3 ]

•'

. 5 6 > R C 3 ] 51 : (. R C 3 0 ] - X ) 2 + (. R [ 3 3 ] - Y ) 2 ) > R [ 2 6 ]

Cj . 9 0 * R [ 5

] - at n < R [35] / R [ 2 6 ] ) >

!' I c os < T > - C [16, 1 ] > C [ 23 , 1 ]

; -C [ 1 6 , 2 ] > C [

3,2]

54: c o s < T > + C [ 16,1] > C [28, 1 ] ; C [

6,2] > C [ 2 8 , 2]

cr tr . .j -J c 1 1 'CIUV-' ( C [23, 1 ] , C [23,2] , C [ 23 , 1 ] , C [28,2] , C [23, 1 ] , C [23,

56: < R [ 2 6 ] 2 + R [ 3 5 ] 2 > >R

57R [ 1 6 ] * R + R [ 2 8 ] > R [ 4 ] 5

C 1 +C [ 23 , 1 ] > * 1 8 . 5 15939 > C [ 1 8 , 1 ] 59: C [ 23 , 2 ] * 1 8 . 5 1 5939 > C [18,2]

6 0C[ 18, 1 ] / R > C [ 18, 1 ] ; C[ 18, 21/ R } C [18,2]

61 : c os < R [ 4 ] > > C C 28 , 1 ]

; -si n < R [ 4 ] > > C [28,2 ] 6 2

c 1 1 C M U L <C[ 18, 1 ] , C [ 1 8 , 2 ] , C [ 23 , 1 ] , C [28,2] , C [ 1 8 , 1 ] , C [ 1 3 , 6 3

f X d 6

64: i f f 1 g 5 ; g t o + 5 65: spc

6 6

prt "E SUE Z

=

"

; f 1 t 6

6 7

prt C[ 18, 1 ] , C[ 18, 2]

6 8

69

- F1 R [ 1 ] * c os < R [ 5 ] *R [ 1 9 ] > > E [ F , 1 ]

; R [ 1 ] *sn < R [ 5 ] * R [ 1 9 ] > E [ F , 2 ] 70

R [ 2 ] *c os ( R [ 5 ] *R [ 20 ] ) > E [ F+

, 1 ] ; R [ 2 ] * s i n < R [ 5 ]*R[£0] > E [ F + 1 71 : R [ 3 ] *c os < R [ 5 ] *R [ 2 1 ] >-C [18, 1 ] > E [ F + 2 , 1 ]

—> •”« . Ida R [ 3 ] *s i n < R [ 5 ] *R [ 2 1 ]

> -C [18, 2 ] > E [ F + 2 , 2]

“7 >• a n e x t I 74: n € x t J

-? cr .

1 a n e x t K 76: dsp "CALCULATION IN PROGRE SS "

i'O 7 9

80

81

8 2

8 3

" 2 1 9| : c 1 1 ' CQRSLT ' (I C 1 3 ] , I C 1 2 3 , I C 1 6 ] > R [ 2 £ ] I C 12 ] >RC22] ; fit 6

i f f 1 g 6 ; g

t, o + 3

SRC

prt " R E S I D U fl L

=

"

, RC22]

- 1 > U

for W = I [ 1 8 ] to I [17]

84: <N - 1 > * R [ 7 ] + R [ 3 4 ] > R [ 3 2]

35: f o r M = I [ 1 5 ] to I [14]

86

01-1 > * R [ 6 ] + R [ 3 1 ] > R [ 2 9]

87

IJ + 2 > IJ 88: i f f 1 g 7 ; g t o +6

89: spc ci 90

prt "CURRENT IN THE *7 P=0 PLANE HT" ; 91 : prt " X F' = "

, R[29] , " Y P = 9 R [ 32 ] j f 1

t

6

92

spc

9 3 prt " J SUE X",X[IJ,U [ IJ

, 2 ] , " J SUE Y 94: F'l 6 X t M 95: n e x t N 96

for L=I[U] to 1110]

97: <L-1 > * R [ 8 ] + R [ 3 7 ] } R [ 3 61

94
98 : 99 : 10O

101

102

103

1 04

105

1 06

107

1 08

109

1 10

1 1 1 112

113

1 14

115

1 16

1 17

118

1 19

120

121

122

123

124

125

126

127

129

130

131

132

133

134

135

136

137

1 39

140

141

142

143

144

145

146

147

148

1 49

150

151

152

153

154

1.55

for N = I [ 1 8 ] to I [ 1 7 3 < N- 1 > *R [7 ] +R C 34 ] > R [ 32 3 U + 2 > U

if f 1 g 7 ; g t- o +6

spc 2

prt "CURRENT IN THE XP =0 PLANE AT" ;fxd 4 prt " V P = "

, R [ 3 2 3 , " Z P = 11 , R [ 3 6 3 ; f 1 t 6

spc

prt.

1 J SUB V "

, X C LI

, 1 ] , X C IJ

, 2 3 , "

.J SUB Z "

, X [ IJ+ 1 , 1 ] , X C IJ+ 1 , 2 ] next- N next L for L= I [ 1 1 ] to I [103

( L - 1 )*R[83+R[373 > R [ 3 6 3 for M = I [ 15 3 to I [143

<M-1 )*R[63+R[31 3 >R[293

IJ + 2 > Li if f 1 g 7 ; g t o +6

spc 2

prt "CURRENT IN THE Y R = 0 PLANE HT "

; f xd 4 prt " XP= "

, R[293 , " ZR=

, R [ 3 6 3 ; f 1 t 6

spc

prt "J SUB X",X[ U,13, X [ U , 2 3 , " J S U B Z

"

, XUJ+1 ,13, XUJ+1 ,23

next M

n e x t L

r-cf 4,R[*3,X[*3

1 dp 3

e n d

"CQRSLT" :

I > p 4 ; J > p 5 ; K>p6; L > p 7

f o r 1 = 1 t o p 3

0 ; X [I,13> X [1,23

i f V N 0 R M D < 1 , I > # 0 ; I > p 1 3 n e x t I f o r J = 1 t o p 1 3 P 2 - J + 1 > p 1 4

c 1 1 Q T X A B E p 1 4 , J . J ) n e x t J

p2-p 1 3 } p 1 5

0 > R [22 3

i f p 1 5 > 0 ; - V N 0 R M E C p 1 5 , p 13+1 > > F: [223

c 1 1

•- CD I V < B [ p 1 3 , 1 3 , B [ p 1 3 , 2 3 , D [ p 1 3 , 1 3 , D [ p 1 3 , 2 3 , X [ p 1 3 , 1 3 , X [ p 1 3 , 2 3 p 1 3- 1 > p 1 6

f o r J = 1 t o p 1 6

p 1 3 - J > K

K + 1 } p 8

0 > p 2 0 > p 2 1 f o r L = p 8 t o p 1 3

c 1 1 CMIJL -

<: A [ K , L , 1 3 , A [ K , L , 2 3 , X [ L , 1 3 , X [ L , 2 3 , p22 , p23 > p 2 0 + p 2 2 / p 2 0

p 2 1 + p 2 3 > p 2 1 next L

B [ K , 1 3 - p 2 0 > p 2 2 ; B [ K , 2 3 - p 2 1 > p 2 3

c 1 1 - CD I V

•- < p 2 2 , p 2 3 , D [ K , 1 3 , D [ K , 2 3 , X [ K , 1 3 , X [ K , 2 3 > n e x t J

f o r 1 = 1 t o p 1

6

pi 3- I > J

J [ J 3 > p 1 2

X [ J , 1 3 > p 1 0 ; X [ J , 2 3 > p 1 1

95
156: X C p 1 2 , 1 ] > X [ J , 1 ]

; X [ p 1 2 , 2 ] > X C J , 2

]

157: pi 0 > X C p 1 2 , 1 ] 5 p 1 1 > X [ p 1 2 , 2

]

158: n e x t I 159: p 4 > I ; p 5 > J ; p 6 > K ; p 7 > L 160: r e t » 161: "OMUL":

16 2: p 1 p

3

- p 2 p 4 > p 7 ; p 1 p 4 + p 2 p 3 > p 8 ; p 7 > p 5 ; p 8 > p 6 ; r e

t

163: "CD IV":

164: p 3 p 3 + p 4 p 4 > p 7 ; < p 1 p 3 + p 2 p 4 >

••••' p 7 > p 1 0

165: ( p 2 p

3

- p 1 p 4 > / p 7 > p 1 1 ; p 1 0 > p 5 ; p 1 1 > p 6

r * t 166: “ QTXflEE " : 167: 0 > p 1 0 > p 1 1

1 6 8 : f o r Q = p 2 t o p 2 + p 1 - 1 169: iR C Q , p 3 , 1 ] > p 1 2 ; - R C Q , p 3,2] > p 1 3

170: ell C i'1 LI L

'' ( p 1 2 , p 1 3 , B [ Q , 1 ] , B C Q , 2 ] , p 1 2 , p 1 3 > 171: p 1 0 + p 1 2 > p 1 0 ; p 1 1 +p 1 3 > p 1 1 172: ne x t Q

173: 2p 1 0 > p 1 0 ; 2 p 1 1 > p 1 1 174: f o r Q = p 3 t a p 3 + p

1 - 1 175: c 1 1

•" C M IJ L C p 1 0 , p 1 1 , H C Q , p 3 , 1 ] , fi [ Q , p 3 , 2 D , p 1 2 , p 1 176: B C Q , 1

] -p 1 2 > E C Q , 1 ]

; E C 0 , 2

] -p 1 3 > E C Q , 2

]

177: ne x t Q

178: ret

179: "VNORMB":

180: p 2 > p 5 > p

6

181: 0 > p

182: for V=1 to pi

1 8 3 : m ax < p 7 , ab s < B C p 5 , 1 ] > , ab s ( B C p 5 , 2 ] > > > p 7 184: p5+ 1 > p5

185: next V

1 8 6 : i f p 7 = 0; r e t 0

18 7: 0 > p

8

188: f o r V = 1 t o p

189: p 8 + ( B C p 6 , 1 ] p 7 ) 2 + (

I' C p 6 , 2 ] p 7 ) 2 / p 8

190: p 6 + 1

> p

6

191: next V 192: ret p 7 * \ p

8

193: “VNORMB":

194: p 2 } p 5 > p

6

195: 6 > p

196: f o r V =1 t o fj 1 19 7: m ax ( p 7 , abs ( D C p 5 , 1 ] >

, ab s ( D [ p 5 , 2 ] > > > p

198: p5+l>p5

199: n e x t V 200: i f p 7=0; r e t 0

201: 0 } p

8

282: f o r V =1 t o p

2 0 3 : p8+ D [ p 6 , 1 ]

•••' p 7 ) 2 + ( D C p 6 , 2 ]

•••"

p

7 ) 2 } p 8

204: p6+l)p6

205: ne x t V

206: ret p7#\p8

*1745

96
0 : "Building C h ar ac t e r 1 z at i o n P r o g r am s " :

1 : "EM T h e o r y d e v e 1 o p € d b y B r . M . T . M a "

2: "HPL Programs supervised by M. G. Arthur":

3 : "National Bur e au o f S t an d ar d

s

"

4 : "Boulder, Col o r ad o

"

5: "FIELD Program: File 3":

6: "FLAG 1 CONTROLS PRINTING OF CONSTANTS ": sfg 1 ? : "FLAG 3 CONTROLS PRINTING OF MA AND NR": sfg 3

8: "FLAG 7 CONTROLS PRINTING OF CURRENTS ": sfg 7 9: r ad

10: 2 4 > M > N

1 1

dim A [ M , N , 2

]

, B C M , 2

]

, D [ N , 2 ], J [ N ] , I C 30] , C [ 50 , 2 ]

, R C 60 ]

,

X [ H , 2 ] 12: dim W S C 1

]

13: spc 5

14: prt "FIELD PROGRAM" ; spc 2

15: ldf 4,RC*],X[*]

16: "ENTER"

17: f xd 4

1 8

2 4 > M > N

19: prt "FIELD POINT POSITION:

20

spc

21 : beep; ent "ENTER X. X = ?",X

o • beep; ent "ENTER Y. Y = ? "

, Y

2 3

b e e p ; ent "ENTER Z

"7 — 11 "7 24 : "PRINT ENTRIES":

OC . C- -J • prt " X=

, X , " Y= " ,

, " Z = "

, Z 26: spc 2

27: dsp "ARE THESE NUMBERS CORRECT? < Y--N >

" 28: ent "

" , W$

28 : if W* [ 1 ] = " Y " or [ 1 ] = " y

" ; gt o +5

30: beep; ent "RE-ENTER X. X = ?",X

31: beep; ent " R

E - E N TER V . V = ?

"

,

V

32: beep; ent "RE-ENTER Z. Z = ?",Z

33: gt o "PRINT ENTRIES"

34: dsp "CALCULATION IN PROGRESS"

35: RC38D-RC31 ] >R[S]

36: R[39]-RC34]>RC7]

37: RC40]-RC37] >R[8]

38: R[6]*RC7] > F: [ 9 ] 39: R[63*R[8]>R[10]

40: R [ 7 D *R C 8 ] > R [ 1 1 ] 41: 1 . 481e8>R[ 14]

42: 4*atn<n>P

43: 2*P*R C 1 4 ] > R C 1 8

]

44: 4e-7*P>R[ 17]

45: le-9/<36*F'> >R[ 12]

46: 1 > I [ 1 1 ] 47: 2> I C 14] > I C 17] > I [ 10]

48 : R [ 1 8 ] *\ < R C 1 7 ] *R [ 1 2 ] > > R L 1 6

]

49 : 0 > C C 22 , 1 ]

; R C 1 8 ] *R C 1 7 ] < 4*P > > C C 22 , 2

]

50: 0 > C C 2 4 , 1 ]

; - . 25/CRC 1 8 ] *P*R L 1 2 ] > >CC24, 2]

51:ifflgl;gto+6

52: prt "THE VARIABLES OMEGA , MU , EPS I LONKAY

53: prt "ARE LISTED BELOW";flt 6

54: s p c 55: prt R C 1 8 ]

, R C 1 7 ]

, R [ 1 2 ]

, R C 1 6 ]

, C C 22 , 1 ] , C C 22

56: s p

c

0 M S 4 P , S 0 4 P E

2 ] , C C 2 4 , 1 ] , C C 2 4 ,

97
cj “7 -1 } IJ 58 f o r N= 1 to I [173

5 9 < H

- 1 ) R [ 7 3 + R [ 3 4 3 > R t 3 2 3 60 f o r M = 1 to I C14]

6 1 < M

- 1 > * R C 6 3 + R [ 3 1 3 > R C 2 9 3 62 U + 2 > IJ 63 i f f 1 g 7 ; g t o +7

6 4 s P c 65 prt "CURRENT IN THE ZP=0 PLANE AT" ; f xd 4

6 6 prt " X R

=

"

, RC29 3 , " Y P

=

" , RC323 ; f 1

t

6

h> i 6 8 pft ".J SUE X" , XCU, 1 3 , XCU, 2 3 , " J SUE Y" , X[U+1 1 ^ , X [ U + 1 , 2 3 69 spc

70 n e x t. M 71 n e- x t H

7 f o r L = I [ 1 1 3 to I [ 1 0 3 73 <L- 1 * R. [ 8 3 + R [ 3 7 3 > R [ 3 6 3 74 f o r N= 1 to I [173

*7 cr

i _> < N

- 1 > * R [ 7 3 + R [343 >R[323

7 b IJ + 2 > IJ

“7 “7

i f f 1 g 7 ; g t o + 7

»*' o spc

79 prt "CURRENT IN THE XR=0 PLANE AT" ; f xd 4 8 0 prt. " YF' = "

, R [ 32 3 , " ZP= ", R [ 36 3 ; f 1 t £

81 spc

8 2 prt "J SUB Y "

, X [ IJ , 13, X [ IJ, 2 3 , " J SUE Z" , X[IJ+1 1 3 , X[IJ+1 , 2 3

'I' "J spc

8 4 n e- x t N 85 n i x t. L

8 1 f o r L = I [ 1 1 3 to I [ 1 0 3

‘d> 7 (L- 1 > * R [ 8 3 + R [ 3 7 3 > R [363

8 8 f or M=1 to I [14 3 g 9 < M

- 1 ) * F: [ 6 3 + R [ 3 1 3 > R [ 2 9 3 90 U + 2 > u 91

Q

i f f 1 g 7 ; g t o + 7

9 3

-• P pr t "CURRENT IN THE YP=0 PLANE AT" ;fxd 4

94 prt " XF'= "

, R [ 29 3 , " ZP= ", R [ 36 3 ; f 1 t f. 95 spc

96 prt "J SUE X "

, X [ IJ

, 13, X [ IJ

, 2 3 , " J SUE Z" , X[IJ+1 1 ] , X[IJ+1 , 2 3 9 7 s p c 98 ri e x t M 99 n s- x t L 100: 1 > F 101: 0 > E [ F , 1 ] > E [ F , 2

]

102: 0 > £ [ F + 1 , 1 ] > E [ F + 1

, 2]

103: 0 > E [ F + 2 , 1 ] > E [ F + 2 , 2]

104: - 1 > U

105: for N=1 to I C 1 7

]

106: < N- 1 > *R [ 7 ] +R C 34 ] > R C 32

]

107: for M=1 to IC143

108: < M- 1 > * R C 6 ] +F: C 3 1 ] } R [29 3

1 0 9 : IJ + 2 > U 110: < < X -RC293 > 2 + < Y -RE323 > 2 + Z 2 ) JRC243

111: c os < R [ 1 6 ] *R C 24 3 ) /R [ 24 3 > C C 20 , 1 3

; -s i n < R C 1 6 3 *R [ 24 3 > /R C 24 3 > C [ 20 , 2 3 112: <<RC163*RC243> '- 2-3 > /R C243 --4 > CC25, 13; -3*R [ 1 6 3 *R C 24 3 RC243 -4 > C [ 25 , 2 3 113: (. X-R C 29 3 > *C C 25 , 1 3 > C C 2 , 1 3 ; C X-R C 29 3 > *C C 25 , 2 3 > C C 2 , 2 3 114: < Y-R [ 32 3 > *C C 25 ’ 1 3 > C C 5 , 1 3 ; ( Y-R C 32 3 >*C t 25 , 2 3 > C C 5 \ 2 3 115: Z*C C 25 , 1 3 > C [ 1 4 , 1 3 ; Z*C C 25 , 2 3 > C C 1 4 . 2 3

98
116

1 17

1 18

119

128

121

122

123

124

125

126

127

128

129

1 38

131

132

133

134

135

136

137

138

139

1 4 0

141

142

143

144

145

146

147

160

161

162

163

164

165

166

167

163

169

170

171

172

c 1 1 C C 7

,

1 R C RC 16

c 1 1 c 1 1 C C 2 8

CC 7,

-CC2

c 1 1 c 1 1 CC9,

e 1 1 c 1 1 BCF,

BCF,

-CC2

c 1 1 c 1 1 CC7,

c 1 1 CC9,

1 - • R C RC 16

c 1 1 c 1 1 C C 28

OMUL - C-CC22, 1 ]

,

1 3 *R C 9 3 > C C 7 , 1 ] ;

C

2 4 3 2 + ( X - R C 2 9 3 > *

] * R C 2 4 ]

••••' R [ 2 4 ] 2 + OMUL " (CC 24, 1 3 , C

'CMUL' < C C 2 8 , 13,

C

, 1 3*RC93 >CC28, 1

1 3-CC28, 1 3 >CC7, 1 4, 1 3*<Y-RC323 > >C

'CMUL (CC9, 1 ]

, CC

'CMUL' ( C C 9 , 13, CC

1 3*RC93 >CC9, 1 3 ;

C

CMUL " (CC7, 1 3 , CC

CMUL' (CC9, 13 , CC

1 3 +C C 2tf , 13+CC30,

23+CC28, 2 3+C C 30

,

4, 1 3*<X-RC293 ) >C

'CMUL' CCC7, 1 3 , C C 'CMUL' (CC7, 1 ]

, CC

C 2 8 , 1 ]

, CC23,

C 2 8 , 1 3 , C C 2 8 ,

> C C 9 , 2 ]

3 , c C 9 , 2 3 )

, 1 3 , C C 9 , 2 3 >

] >

3 >

] >

-c C 22 , 2 ]

,

c C 20 , 1 3

, C C 28 , 2 3 , C C 7 , 13 , C C 7

,

C 7 , 2 ] * R C 9 3 > C C 7 , 2 ] C C 2 , 1 3 > C C 28 , 1 3 <jf-RC293 )*CC2, 23 > C C28, 23

^ ^ ^ j ^ 3 , L C 2 8 , 1 3 , U C 2 8 , 2 3 , C

C 28 , 2 3 , C C 20 , 1 3 , C C 20 , 2 3 ,

C

; L [28, 2 3 * R C 9 3 } C [28, 2 3

3 ; CC7, 2 3-CC28, 23 > C C 7 , 23

C 9 , 1 3 s -C C 24 , 2 3 * < Y-R C 32 3 > 9 , 2 3 j L C 2 , 1 3 , C C 2 , 2 3 ,

IJ C 9 , 1 3 i i ] j t C 2 U j 1 3 j L C 2 y j 2 3 ( U C 9

C 9 , 2 3 * R C 9 3 > C C 9 , 2 3 7.23, X C IJ

, 13, X C U , 2 3 , C C 28 , 13, C C 28 , 2 3 > 9 . 2 3 , X C IJ + 1 , 13, X C U + 1 , 2 3 , C C 3 0 , 1 3 , C C 3 0 ,

1 3 > B C F , 1 3 2 3 > B C F , 2 3

C 7 , 1 3

; -C C 24 , 2 3 * < X-R C 29 3 > } C C 7 , 2 3 7 . 2 3 , C C 5 , 1 3 , C C 5 , 2 3 , C C 7 , 1 3 , C C 7 , 2 3 ) 7 . 2 3

,

C C 20 , 1 3

, C C 20 , 2 3 , C C 7 , 1 3 , C C 7 , 2 3 >

C 7 , 2 3 * R C 9 3 > C C 7 , 2 3 -C C 22 , 2 3 , C C 20 , 1 3 , C C 20 , 2 3 , C C 9 , 1 3 , C C 9 , 2 3

)

3 )

1 3*RC93 >CC7, 1 3;

C

'CMUL' (-CC22, 13,

1 3 *R C 9 3 > C C 9 , 1 3; C C 9 , 2 3 *R C 9 3 > C C 9 , 2 3 24 3 ~2+ ( Y-R C32 3 ) *C C 5 , 1 3 } C C 28 , 13

3 * R C 2 4 3 -' R C 2 4 3 ~2+ < Y-R C 32 3 > *C C 5 , 2 3 > C C 28 , 2 3 ' CMUL ' < C C 24 , 1 3

, C C 24 , 2 3 , C [28,13, C C 28 , 2 3 , C C 28 , 1 3

, C C 28

,

CMUL ' < C C 28 , 1 3, C C 28 , 2 3 , C C 20 , 1 3 , C C 20 , 2 3 , C C 28 , 13, C C 28

,

, 1 3 *R C 9 3 > C C 28 , 1 3

; C C 28 , 2 3 *R C 9 3 >C C 28 , 2 3 C C 9 , 1 3 -C C 28 , 1 3 > C C 9 , 1 3 ; C C 9 , 2 3 -C C 28 , 2 3 > C C 9 , 2 3 ell CMUL ' ( C C 7 , 1 3

, C C 7 , 2 3, X DU, 13, X C U , 2 3 , C C 28

,

ell ' CMUL ' ( C C 9 , 1 3

, C C 9 , 2 3 , X C U+ 1 , 1 3, X C U+ 1, 2 3 ,

C

B CF+ 1 , 1 3 +C C 28 , 1 3 +C C 30 , 1 3 > B C F + 1 , 1

B C F+ 1, 2 3 +C C 28 , 2 3 +C C 30 , 2 3 > B C F+ 1

, 2 3

1 3 , CC28, 23 ) C30, 13, C C 3 0

,

3 >

: 3 >

23 )

148 - c C 2 4 »—» l

l * -R c ill 9 3 ) } c C 7 y 1 5 -CC2 4 y cl 3 * f’ -R C 2 9 3 > > c C 7 , 3 149 c 1 1 ' i 3MUL'

(

CC 1* n i c c •’ y cL 3 C C 1 4 , 1 3 y c c 14 y

*=:- 3 , C C i'

-

y 1 3 , c c C- 3 > 150 c 1 1 i 3 M IJ L < CC "7

y i 3 , c c f

9 cL 3 C C20, 1 3y c c 20 q Cl 3, C C

«’ y 1 3 , c [

•*’ 3 ) 151 c c •’ y 1 ] * R C 9 3 > c C

—>

i y 1 3 ; c C 7 2 3 * R C 9 3 3 C c r y 2 3 152 - c C 2 4 1 3 * ( Y -R c 3 cL 3 ) > c C 9 y 1

; -CC2 4 y

ii! 3 * T -R C 3 2 3

;i y c C 9 ,

ii! 3 153 1 1 'i :mul '

(

CC 9 y 1 3 , c c 9, 3, C C 1 4 , 1 3 y c c 14 q

lil 3, c C 9, 1 3 , c c 9 3 ) 154 c 1 1 3 M U L ' < CC 9 y 1 3 , c [ 9 , cL 3 CC 20, 1 3 y c c 2 0 y ^ 3 , c C 9, 1 3, c c 9 •Si 3 ) 155 c c Q- 9 1 ] *R C 9 3 > c [ 9 » 1 3 5 c C 9 y 2 3 * R C 9 3 > c c 9 2 3 156 c 1 l x

i 3 M U L ' < CC y 1 3, c c i , cL 3 X C IJ

, 1 3 y X C u y ^ 3 , CC 2 a , 1 3 , C c 2 3 ) 157 c 1 i ' i 3MUL '

(

CC 9 y 1 3, c c 9, cl 3 XCU+1 y l 3 y cu + 1 y ^ 3 y CC 3 0, 1 3 ,

' :c 30

158 B c F + l

l + l.

.1 r

—

«i! o 9 1 3 + c c 3 0. 1 3 •BCF+2 y l 3 159 B [ F + 2 2 3+CC •_i r, cL O 9 cL 3 +

c

c 0, 2 3 BCF + 2 9 cl 3

3 >

n e x t M

n e- x t N

for L = I C 1 1 3 to I C 1 0

(L-l )*RC83+RC373 >RC363

for N=1 to I C 1 73

(N-l > * R C 7 3 + R C 3 4 3 >RC32 3 U + 2 > IJ

•• ( X

A 2 + ( Y - R C 3 2 3 >'• c o s ( R C 1 6 3 * F: C 2 5 3 )

( ( R C 1 6 3 * R C 2 5 3 )

• 2

X*CC26, 1 3 >CC3, 1

Y - RC323 )*CC26, 1 (2-RC363 )*CC26,

2 + < Z - R C 3 6 3 )

-• 2 > > R C 2 5 3 / R [ 25 3 > C C 2

, 1 3

-si

< RC 163*RC

- 3 ) / R [ 25 3 4 > CC26, 1 3 ; - 3 * R C 1 6 3

; X * C C 2 6 , 2 3 > U C 3 , 2 3

3 > C C6,13; ( Y-R C 32 3 >*CC26, 23 >C

3 > C C 1 5 , 1 3 ; ( Z - R C 3 6 3 > * C C 2 6 ,23 )

25 3 > RC253 C *RC253 RC253

C 6 , 2 3 C C 1 5 , 2 3

21.23

4 > C C 2 6

99
173

174

175

176

177

1 79

180

181

182

183

184

185

186

187

188

1 89

198

191

192

193

194

195

1 9 6

1 97

198

199

2 0 0

201

2 0 2

2 0 3

20 4 2 0 5 206

207

2 0 8

2 0 9

210

21 1

214

215

216

217

2 1 8

219

220

221

dd 2

2 2 3

229

230

-C [ 24 , 1 3 * < Y-R C 32 3 > > C C 1 0 , 1

]

; -C C 24 , 2 ] * < Y-R C 32 3 > > C C 1 8 , 2 3 c 1 1 CMUL "

< C C 1 0 , 1 3 , C C 1 0 , 2 3 , C C 3 , 1 3 , Ct 3 , 2 3 , C C 1 0 , 1 3 , C C 1 0 , 2 3 > c 1 1 CMUL < C [ 1 0 , 1 3 , C [ 1 0 , 2 3 , C C 2 1 , 1 3 , C C 2 1 , 2 3 , C C 1 0 , 1 3 , C [ 1 0 , 2 3 > C C 1 0 , 1 3 *R [ 1 1 3 > C C 1 0 , 1 3

; C C 1 0 , 2 3 *R C 1 1 3 > C C 1 0 , 2 3 -C C 24 , 1 3 * < Z-R C 36 3 > > C [ 1 1 , 1 3

; -C C 24 , 2 3 * < Z-R [ 36 3 > > C [ 1 1

, 2 3 c 1 1 CMUL < CCll, 1 3 , C [ 1 1 , 2 3 , C [ 3 , 1 3 , C C 3 , 2 3 , C C 1 1 , 1 3 , C C 1 1 , 2 3 ) c 1 1

•- CMUL < C [ 1 1 , 1 3 , C [ 1 1 , 2 3 , C [ 2 1 , 1 3 , C L 2 1 , 2 3 , C C 1 1 , 1 3 , C C 1 1 , 2 3 > C [ 1 1 , 1 3 *R C 1 1 3 > C C 1 1 , 1 3

; C C 1 1, 2 3 *R C 1 1 3 > C C 1 1, 2 3 c 1 1 - CMUL < C [ 1 0 , 1 3 , C E 1 0 , 2 3 , X C U , 1 3 , X C IJ

, 2 3 , C C 28 , 1 3 , C [ 28 , 2 3 > r 1 1 CMUL - C[ 1 1

9 1 3 , c [ 1 1, 23 , [IJ+1 9 1 3 9 [ IJ + 1 9 2 3 9 c [

:0 9 1 3 9 c [ 50 9

•:4 3 E [ F , 1 3 + C [ 2 o , l 3 + c [ 3 0 , 1 3 / B [F, 1 3 E [ F 9 ^ 3 + C [ 2 y , 4‘ 3 + c [ 30 , 3 3 > B CF, 3 r 1 1 CMUL" -c [ 2 JI 1 3 , - c [ 2 2,23 , C [ 2 1 9 1 3 9 c [2 1 9 cl 3 9 c [ 0 , 1 3 9 c [ 10 9 ^ 3 c [ 1 0, 1 3 *R[ 1 1 3 > f c 1 0 , 1 3 9 c [ 10, 3 *R [ 1 1 3 / C [ 10 , 2 ]

1 / R [2 5 3 •-•2 + £ Y

- R [ y cl 3 > fc c [ 6 , 1 3 : L [ 28 , 1 3 R [ 1 6 3 * R [25 ] /R [

•7 5 3 cl + <: Y R [32 3 >*C[ 6 ,

•“» C. 3 / c [2 O 9 cl 3 c 1 1 CMUL' »; C[ cl £ ji 1 3 , c [ cl £ 9 23 , c [24, 1 3 9 c [ LI 4 ,

'cl 3 9 c [ 'ci 1 ] , c [ ci O 9*=- 3 > c 1 1 CMUL' C[ cl 'r

ji 1 3 , c [ 2 £ 9 23, c [21, 1 3 9 c [ 2 1 , cl 3 9 c [ L' 1 3 9 c [ L! 8 9 ^ 3 > c [ cl 8

,

1 3*R[ 1 1 3 > c [ 2 8 , 1 3 9 c [ cl O ^ 3 * R [ 1 1 3 c [ d y 9 2 ] c [ 1 0, 1 3 - C [ 2 9 1 3 \ c [ 1 0 9 1 3 9 C[ 1 0 ,23- C [ d O 9 2 3 > c [ 1 0 9 ci 3 - c [ 24 .13*': 7 -R [

•“ b 3

|l

’[• c [ 1 1 9 1 3 ; C [24 9

'cl 3 * z -R c

• j b 3 :> > C[ 1 1 9 d 3 c 1 1 CMUL' ( C[ 1 1 ji 1 3 , c [ 1 1 9 23 , f [6,1 3 9 C [ 6 , 23 9 c [ 1 l , 13, c [ 1 1 , cl 3 ) r 1 1 CMUL' ( C[ 1 1 9 1 3, c [ 1 1 9 23 , c [21 , 1 3 9 c L ci 1 , 3 9 c [ 1 1 1 3 9 c [ 1 1 9 ^ 3 > c [ 1 1 , 1 3*R[ 1 1 3 } c [ 1 1, 1 3 9 c [ 1 1 , 3 *R [ 1 1 3 Jf c [ 1 1 9 cl 3 c 1 1 CMUL " ( C[ 1 0 , 1 3 , c [ 1 0, 23 ! [ IJ

, 1 3 9 X [ IJ, 23 9 c [ L y , 1 3 , c [ 'ci y , cl 3 >

1 1 CMUL" ( C[ 1 1 ji 1 3 , c [ 1 1 9 23 I- [IJ+1 9 1 3 9

I*'*! [ U + 1 9 2 3 9 c [::;! 0 9 1 3 9 c L 3 0 9 ^ 3

Ei [ F + 1 , 1 3 + C L 2 y 1 3 + CC 0 0 9 1 3 > E F + 1,1 3 E [ F + 1 ,23+C [ cl C> 9 "

3 + C [ o 0 9 3 > E F + 1,2 3 - c [ 24 , 13 *': Y -R [

•1 2 3 j 1 c [ 1 0, 13; C [24 , cl 3 * i; Y -R [ • cl 3 :> > C [ 1 0 9 ci 3

l” 1 1 CMUL ( C [ 1 0, 1 3 , c [ 1 0, 2 3 f [15, 1 3 9 c [ 1

cr

•- 9 cl 3 9 c [ 1 0 1 3 9 c [ 1 0 9 3 > c 1 1 CMUL ( C[ 1 0, 1 J, c [ 1 0, 2 3 c [21 , 1 3 9 c [ L 1, 'cl 3 9 c [ 1 0 1 3

9 c [ 1 0 9

l=- 3 J c [ 1 0, 1 3 * R [ i 1 3 > c [ 1 0 , 1 3 9 c [ 10, 3 *R [ 1 1 3 > c [ 10 9 2 3 r 1 1 CMUL " < - c [ cl 9 1 3 5 - c [ 2 2,2 3 , C [ 2 1 9 1 3 , c [2 1 9 cl 3 9 c [ 1 , 1 3 9 f [ 1 1 9 ^ 3 c [ 1 1 , 1 3 * R [ i 1 3 > c [ 1 1 , 1 3 9 c [ 1 1 3 *R [ 1 1 3 } c [ 1 1 9 2 3

1 R [2 5 3 2 4 7 — R [

“j b 3 > * c [ 1

ej ,131 C [28 9 1 3 R [ 1 6 3 * R [ 2 f 3 /R [ 5 3 cl + r z R [ 3

'.3 >*C[ 1 5 ,

ii! 3 > C[ 'cl 0 ,

L.' 3 c 1 1 CMUL' r C [ cl £ JI 1 3 , c [ 2 £ 9 23, c [24, 1 3 , c [ L 4, cl 3 , c [ 2 1 3 9 c [ L. y 9 cL 3 > r 1 1 CMUL " C [ cl £ 9 1 ], c [ 2 £ 9 23 c [21 , 1 3 9 c [ L.' 1 , 2 3 9 c [ L' 1 3 9 c [ L b

9 3 > c [ cl

9 1 3 * R [ i 1 3 > c L 1 3 9 f [ 28 3 * R [ 1 1 3 > c [ d y 9 2 3 c [ l 1 , 1 3 - C [

9 1 3 > c [ 1 1 9 1 3 9 CL 1 ,23- c [ cl 8 , 2 3 > C [ 1 1 , L 3

l“ 1 l CMUL " ( ci 1 0, 1 3 , c [ 1 0, 2 3 •' [ U , 1 3 9 [ IJ

, 23 9 c [ L!

C;

, 1 , c [ ci y , cl 3 ) c 1 l CMUL' ( CL 1 1 ji 1 3 , c [ 1 1 9 23 [IJ+1 9 1 3 , X [ LI + 1 , 2 3 9 c [: 30 9 1 3 9 c L 3 0 9 3 B [ F + 2 , 1 3 + C [ 28 Jl 1 3 + C [ Z 0 , 1 3 > E F + 2, 1 3 E [ F + 2 ,23+C;[ 28 9 c. 3 + C:[ 3 0 , 3 > B F + 2,2 3

n e x t L

f or L = I [ 1 1 3 t o 111 0 3

C L- 1> *R C 8 3 +R [ 37 3 > R [ 36 3 for M= 1 to I [143

<M-1 >*RC63+RC31 3 >R[293

U + 2 > U

( ( X - R [ 2 9 3 ) 2 + Y 2 + < Z - R [ 3 6 3 ) A d ) ) R [ 2 3 3 c os ( R [ 1 6 3 *R [ 23 3 > /R [ 23 3 > C [ 1 9 , 13; -si r. < R [ 1 6 3 *R [ 23 3 > /R [23 3 > C [19,23

' < R [ 1 6 3 *R [ 23 3 > •"•2-3 > /R [23 3 M) C [ 27 , 1 3

; -3*R [ 1 6 3 *R [ 23 3 /R [23 3 M> C [ 27 , 2 3

< X-R [ 29 3 > *C [ 27 , 1 3 > C [ 1 , 1 3 ; < X-R [ 29 3 > *C [ 27 , 2 3 > C [ 1

, 2 3 Y*C [ 27 , 1 3 > C [ 4 , 1 3

; Y * C [ 27 , 2 3 > C [ 4 , 2 3

< Z-R [ 36 3 ) *C [ 27 , 1 3 } C [ 1 3 , 13; < Z-R [ 36 3 > *C [ 27 , 2 3 > C [13,23

ell CMUL *

< -C [ 22 , 1 3 , -C [ 22 , 2 3 , C [ 1 9 , 1 3 , C [ 1 9 , 2 3 , C [ 8 , 1 3 , C [ 8 , 2 3 >

100
rv>

pj

pj

ro

po

po

ro

p

j

pj

pj

pj

pj

p-j

231 CE s $ l ]*RE 1 0 ] ) c [ 8 9 1 ] ; CE 8 9 2 ] * R E 2 32 is R E 2 3D A 2 + < X

- R E 9 ] ) * CE 1 9 i ] } c E 233 RE 1 6 ] *R E 23 ] R E 2 ] 2 + V - R E 2 9 ]

,'l 234 c 1 1 CMUL" c E o £ 9 1 ] 9 f E 2 8 9 y d 9 C E

tr c 1 1 CMUL" ( c E 2

O

9 1 ] 9 c E 2

•I; 9 cl ]

9 c E 236 CE k oo 9 1 ] *RE 1 0 ] ) c [ cl 8 9 1 ]; c E cl c 9 2 ] 237 CE 8 9 1 3-CE2 8 9 1 ] } c [ 8 9 1 ]

c E

;Zj 9 ] - Q -c E 2 4 , 1 ] * (

-7 - R E 6 ] > > c E 1 cl

9 1 ] 9 - c 239 c 1 1 CMUL" < c E l 9 l ] 9 c E

cl

9 cl ] 9 c E 24 0 c 1 1 CMUL" < c E l

•“> 9 l ] , c E 1 cl 9 cl ] 9 c E 24 1 CE 1 2 9 1 ]*R[ 1 0 ] ) C E l 9 1 c E 1 9 ] 242 c 1 1 CMUL" c E 8 9 1 ] 9 c [ y 9 cl ] 9

Ky E IJ

9 243 c 1 1 CMUL' < c E 1 cl 9 l ] 9 r E 1 cl

9 cl ] 9 X E 244 EE F 9 1 ] + C E 2 , 1 ] + c E y 0 9 1 ] > B E F 1 ] 245 EE F 9 ] + C [ 2 9 k ] + c [ 0 9 2] j E E ii! ] 246 —

c

E 4 , 1 ] * c X

- R E cl 9 ] > £ F E 1 0 ] ) f E y 9 247 c 1 1 CMUL" ( c [ 8 9 1 ] 9 c E 9 cl ] 9 C E 4 9 248 c 1 1 CMUL" f c [ 8 9 1 ] 9 c E 9 cl ] 9 c E 1 9

24 9 -C E 2 4 , 1 ]*< y - F: E j! 6 ] ) F E 1 0 ] } c E 1 2 250 c 1 1 CMUL" (. c [ 1 cl , l ] 9 c E 1 cl 9

'cl ] 9 c E 251 c 1 1 CMUL" (. c E 1 cl 9 l 3 9 c E 1 'cl 9 cl ] 9 c E e ' c 1 1 CMUL" ( c E 8 9 1 ] 9 c E 9

'cl ] 9 E IJ

9 c

c 1 1 CMUL" £ c E 1 cl , l ] 9 c E 1 'cl 9

'cl ] , E 254 BE F + 1 , 1 ] + C E k! 8 9 l ] + c E 0, 1 ] > l E F +

Cj C EE F + 1 , 2 ] + C [ k 8 9 cl ] + c E 0, 2 ] > E E F + 256 -C [ 2 4 , 1 ]*< vf'• - R E cl 9 ]

jl * F E 1 0 ] > c E 8 9 2

*7 c 1 1 CMUL" < c [ y 9 1 ] 9 c E 9 cl ] 9 c E 1 y

c 1 1 CMUL" ( c E y 9 1 ] 9 c E 9

'cl ] 9 c E 1 9

Cj Q c 1 1 CMUL" < - c e 9 1 ] 9 -C [ cl 'cl 9 2 ] 9 1 6 0 CE 1 2 j 1 ]*RE l 0 ] \ c [ 1 cl ) 1 c L 1 9 2 ] 261 is R e ii. 3 ] " 2

- R [ 6 ] )

-+ CE 1 9 1 ] c 262 RE 1 6 ] *R E 23 ] R E li. Z ] cl 4 r 2

- R E

1 6 ] ) 1 : j c 1 1 CMUL" t c E cl c

9 1 ] 9 c E 2

jl

9 2 ] 9 c E 264 c 1 1 CMUL" r r E cl 8 9 1 ] 9 c E 2 s 9 2 ] 9 c E 265 CE 2 z

9 1 ]*R[ 1 0 ] } f E cl y 9 1 ]

L E 2 !z J! ] 266 CE l 2 9 1 ] - C [ 2 £ 9 1 ] ) c e 1 2, 1 ] 9 f E k. 9 267 c 1 1 CMUL" ( c [

•I; > 1 ] c [ 9 cl ] 9 L IJ

9 2 6 8 c 1 1 CMUL" < c E 1 i. 9 1 ] s f:E 1 el

9 2 ]

,

y E 269 BE F + 2 , 1 ]+C [ k O 9 1 ] + c [ 30, 1 ] } I E F + 270 EE F + 2 , 2 ] + C C 2 y 9 2 ] + c [ ; 0 , 2' ] > E E F +

10])CE8,2]

28,1]

*CE 1

, 2D > C [28,2]

1 9 , 1 ]

, C [ 1 9 , 2 ]

, C C 28 , 1 ]

, C [ 28, 2]

24 , 1 ] , C C 24 , 2 ] , C C 28 , 1 ]

, C C 28 , 2 ] > *RC IQ] >CC28, 2]

C [ k! 8 , 2 ] > C t 8 , 2 ]

[ 24 , 2 ] * < Z-R C 36 ] > > C C 1 2 , 2

]

1 , 1 D , C [ 1 , 2 ] , C [ 1 2 , 1 ] , C [ 1 2 , 2 ] >

1 9 , 1 ] , C [ 1 9 , 2 ] , C [ 1 2 , 1 ] , C C 1 2 , 2 ] > * R [ 10] > C C 1 2 , 2

]

1 ]

, X C U , 2 ]

, C [ 28 , 1 ]

, C C 28 , 2 ]

)

U+ 1, 1

]

, X C U+ 1 , 2 ]

, C [ 30 , 1 ]

, C [ 38 , 2 ]

1 ] ; -C C 24 , 2] *< X-R [ 29 ] ) *R C 1 0 ] > C C 8 , 2 ]

1 ] , C [ 4 , 2 ] , C [ 8 , 1 ] , C C 8 , 2 ] >

, 1 ] , C C 1 9 , 2 ] , C C 8 , 1 ] , C [ 8 , 2 ] >

, 1 ]

; -C C 24 , 2 ] * < Z-R [ 36 ] > *R [ 1 0] > C [ 1 2,

4 , 1 ] , C [ 4 , 2 ] , C [ 1 2 , 1 ] , C [ 1 2 , 2 ] > 19, 1 ] , C C 1 9 , 2 ] , C [ 1 2 , 1 ] , C [ 1 2 , 2 ]

">

1 ] , XEU, 2] , C [28, 1 ] , C [28, 2D ) U+ 1 , 1 ] , X [ U+ 1 , 2 ] , C [ 30 , 1 ] , C [ 30 , 2 ] > 1,1]

1,2]

1 D

; -C [ 24 , 2 ] * < X-R C 29 ]

j -R C 1 0] > C [8,2]

, 1 ] , C C 1 3 , 2 ] , C L 8 , 1 ] , C E 8 , 2 ] >

, 1 ] , C [ 1 9 , 2 ] , C C 8 , 1 ] , C C 8 , 2 ] > C [19, 1 ] , C [ 1 9 , 2 ]

, C [ 1 2 , 1 ], C [ 1 2 , 2 ]

*R [ 1 0 ] > C [ 1 2 , 2

]

[28,1]

*C [ 1 3, 2 ] > C [ 28 , 2

]

1 9 , 1 ], C [ 1 9 , 2 ] , C [ 23 , 1 ]

, C [ 28 , 2 ] > 24 , 1 ]

, C [ 24 , 2 ]

, C [ 28 , 1 ]

, C [ 28 , 2 ] > *R[ 10] > C [ 2 8 , 2D

2]-C[28, 2D >C[ 12, 2]

1 ], X [ U , 2 ]

, C [28, 1 ]

, C [28, 2D)

IJ+ i , 1 ] , X [ U+ 1 , 2 i , C [ 30, 1 ] , CC 30, 2D

2, 1 ] 2,2]

71 n e x t M n ext L U+l > I [ 16]

74

c;

i -J

81

83

284

285

286

2 y i Opp i G'j

F + 2 > I [12]

if f 1 g 3 ; g t o + 2

fxd 0 ; prt " Mft= " , I [ 1 2 ] , " NR= " , I [ 1 6 ] f x d 6

. 376681 ) C E 16, 1 ]; 004938) UE 16, 2]

8/180) R [5D

. 304801 )R[ 15]

177. 72*R E 5 ] )R[ 16]

R [ 1 5 ] * 1 0 ) R [ 3 0 ] RE 15]*-32. 5)R[33D

epc

prt "E SUE X, E SUE Y ft N D E SUE Z AT THE FIELD POINT 1 Z-4*RE 15] )R[35]

< 4 1 . 557305*2+ 1 9 . 833333 > *R [ 5 ] ) R C 28

]

-••• < ( R [ 30 ] -X ) A 2+ <. R [ 33 ] - V > '-2 > ) R [26]

]

101
2 8 y 2 9 0

291

2 9 2

2 9 3

294

295

296

297

2 9 8

2 9 9

3 0 0

3 0 1 3 0 2 303

3 0 4 305

306

90*R [ 5 ] -.at. n < R [ 35 ] /R C 26 ] > > T ,13; -CC 16, 23 >CC23, 23

, 1 3 ; C[ 16, 23 >CC28, 2]

3,13 , C [ 23 , 2 3 , C [28,13, C C 28 , 2 3 , C C 23 , 1 3

, C [ 23 , 2 3

cosCD-CC 16, 13 } C [ 2 3

c o s < T ) + C [ 16, 13 > C E 2 8 ell -CD I V-' <CC2

\ < R [263 ' 2 + R C 3 5 3 2 > >R

R[ 16 3*R + R[28 3 >RC4 3

C 1 +C [ 23 , 1 3 > * 1 8 . 5 1 5939 > C [18,13

C [ 2 3 , 2 3 * 1 8 . 5 1 5939 > C [ 1 8 , 2 3 C [18,13 R > C [ 1 8 , 1 3

; C [ 1 8 , 2 3 /R > C [18,23

c o s ( R [ 4 3 > > c 1 1 - C M U L [28,13; -si r. < R [ 4 3 > >C[28, 23

C[ 18, 1 3, C [18,23, C [28, 1 3

, C[28

E [ F + 2 , 1 3 +C [ 1 8 , 1 3 > B C F + 2 , 1 3 ; E [ F + 2 , 2 3 +C [ 1

8

\ < E [ F , 13 2 + B [ F , 2 3 -•2 > > R C 1 3 \ CELF + l , 1 3 2 + E [ F + 1 , 23 A 2 ) > R [ 2 3 \ < B [ F + 2 , 1 3 2 + E [ F + 2 , 2 3 -2 > > R [ 3 3 ATAN2 < E [ F , 2 3 , E [ F , 1 3 > , R [ 5 3 > R [19 3 AT AH2 •- < E [ F+ 1 , 2 3 , E C F+ 1 , 1 3 > /R [ 5 3 > R [ 20 3

•' ATAH2 ' < E [ F + 2 , 2 3 , B [ F + 2 , 1 3 > /R [533 R [213

,23

,23

, C [ 1 8 , > B [ F + 2

1 3 , C [ 1

, 2 3 8 , 2 3

3 0 7 5 p c ; f x d 4 3 0 8 prt "X=",X, "V=" V " 7=" “7

9 <- 3 0 9 spe ; fxd 5

310 prt " A M P X

=

, R[ 1 ] , "PH S X = '* , R [ 1 9 3 31 1 prt " Ft M R V = "

, RC2] , "PH S

V

_ ii , R[203 , " R M P Z = 312 spe 2

313 b e e p ; d s p

" D 0 ANOTHER FIELD P 0 I N T ? < V

314 ent 11 ",W$

3 1 5 if W $ [ 1 3 = "

V

" or W $ [ 1 3 = " y

"

; g t o “ENTER"

3 1 6 beep; dsp "

C

HLCULHTION COMPLETED"

317 end

3 1 8 " ATAN2 " : 319 at. n < ib s < p 1 > /.ib S- < p2 > > > p

ij;! 320 if p2<0 arid pi >0; { - p 3

P 3

321 it" p2<0 ind p 1 < 0 ; p 3

- •( > p 3

322 if p2>0 ind p 1 < 0 ; p 3 * - 1 >P >!

-> cL • J r e t f

:' 3

32 4 "CMIJL " : 3 2 5 p 1 p 3 “ p 2 p 4 > p 7 ; p 1 p 4 + p 2 p 3 > P 8; p 7 > p 5 ; p8 >p6

326 " C D I V " : 3 2 7 p 3 p 3 + p 4 p 4 > p 7 ; ( p 1 p 3 + p 2 p

4::' P 7 > p 1 0

328 p 2 p 3 — p 1 p 4 ) p 7 > p 1 1 ; p 1 0 > P 5 ; p 1 1 } p 6 ; r e t

=" , R[33 , " P HSZ= " , RC21

r e t

*2682

102
B.2 Variable Assignments

Complex Variable Assignments

The complex variables are stored in arrays with one extra dimension.

Thus the complex variable can be stored in two parts: the real part, stored

in the top (X[X,1]), and the imaginary part, stored in the bottom (X[X,2]).

All complex arrays are assigned their own array variable while the rest of the

complex variables are assigned in the array variable "C".

FORTRAN VARIABLE

A ( M , N

)

B(M)

OIAG(N)

XS(N)

HPL VARIABLE

A[M,N,2]

B[M,2]

D[N ,2]

X[N,2]

C[50,2]pool of complex variables

AML3

AMN1

ANL2

BML3

CJX1

CJX3

BMN1

BNL2

CJZ3

CJV1

CJY2

CJZ2

CMN1

CNL2

Cl

C2

CML3

ESUBZ

C[l]

C[ 2]

C [3]

C[ 4]

C[5]

C[ 6]

C [ 7]

C[ 8]

C[ 9]

C[10]

C[11]

C[12]

C[ 1 3]

C[ 14]

C[15]

C[16]

C[ 1 7]

C[18]

103
GML3 C[ 19]

GMN1 C[20]

GNL2 C[21]

0MS4P C[22]

RSUBV C[23]

S04PE C[24]

SI C[ 25]

S2 C[26]

S3 C[ 27]

TEMP C[28]

TEMP C[29]

TEMP C[30]

104
Real Variable Assignments

The most commonly used real variables are assigned simple variables and

the rest of the real variables are assigned in the variable array "R".

FORTRAN VARIABLE

PI

RTHETA

X

Y

Z

AMPX

AMPY

AMPZ

ARG

DTOR

DX

DY

DZ

DXDY

DXDZ

DYDZ

EPS

ESUBZ

FREQ

FTOM

KAY

MU

OMEGA

PHSX

PHSY

PHSZ

RESID

RML3

RMN1

HPL VARIABLE

P

R

T

X

Y

ZR[60]pool

R[l]

R[2]

R[3]

R[4]

R[5]

R[6]

R[7]

R[8]

R[9]

R[10]

R[ 1 1]

R[12]

R[ 13]

R[ 14]

R[15]

R[16]

R[17]

R[18]

R[19]

R[20]

R[21]

R[ 22]

R[23]

R[24]

real variables

105
RNL2

RSUBT

TC

XPXPP

X START

YP

YPP

YSTART

ZD

ZP

ZSTART

X2

Y2

Z2

RC 25]

R[26]

R[28]

R[29]

R[ 30]

R[31]

R[32]

R[33]

R[34]

R[ 35]

R[36]

R[37]

R[ 38]

R[39]

R[ 40]

106
Integer Variable Assignments

The most commonly used integer variables, and variables used in "for"

loops, are assigned simple variables. The integer array "JPVT" is stored in

its own variable array "J" while the rest of the integer variables are stored

in the array "I".

FORTRAN VARIABLE

I, IX

J, IY

K , IZ

LM

N

IFJU

JPVT

II

IX

IYIYF

IZKX

KY

KYF

KZ

LF

LS

MA

MDIMA

MF

MS

NA

NF

NS

HPL VARIABLE

I

J

K

LM(also used to dimension arrays)

N(also used to dimension arrays)

F

UJ[N]

I(30)pool of integer variables

I[l]

I C 2 ]

I [3]

I[4]

I[5]

I[6]

I[7]

I [8]

I C 9]

i[io]

i[n]

I [ 1 2]

I[13]

I C 14]

I C 1 5]

I C 16]

I[17]

I [18]

107
B.3 Instructions for Running HPL Programs

To run the HPL CURRENT program, follow this procedure:

1. With the power to the computer turned on, insert the magnetic program

tape in accordance with the computer's operating instructions.

2. Press LOAD and type in 1, or type ldf 1, and press EXECUTE. This will

load CURRENT program. Part 1, into the computer.

3. Make any program changes, such as flags, as needed or desired. These are discussed in section 6.3. The procedure listed here is based on no changes to the program as shown in section B.l.

4. Press RUN. This will cause the computer to print the heading "CURRENT

PROGRAM" on the paper tape. Next, the computer will print "COORDINATES

OF FIELD POINTS:" on the tape.

5. The computer will beep and display "ENTER XI. XI = ?" as a prompt.

6. Type in the first X-coordinate value and press CONTINUE. The computer

will beep and display "ENTER X2. X2 = ?."

7. Type in the second X-coordinate value and press CONTINUE.

8. Continue in this manner until XI, X2, Yl, Y2, Zl, and Z2 have all been

entered.

9. After Z2 has been entered, the computer will print out the values of XI through Z2 that were entered. Then it will display "ARE THESE NUMBERS

CORRECT? (Y/N)."

10. If all are correct, type in either y or Y and press CONTINUE. If any are

not correct, type in either n or N and press CONTINUE.

108
11. If you have typed n or N, the computer will beep and display "RE-ENTER

XI. XI?" as a prompt. Repeat steps 6 through 10 until the numbers

printed out in step 9 are correct.

12. When you type y or V, the computer will display "THIS CALCULATION TAKES 3

MINUTES." Take a break while the computer does its thing.

13. When the computer completes this calculation of the A coefficients and

has performed the QR factorization on the [A] matrix, it automatically

loads CURRENT program. Part 2 from file #2.

14. The computer then prints "EM FIELD DATA AT FIELD POINTS:", beeps, and

displays "ENTER AMPX. AMPZ = ?" as a prompt.

15. Type in the amplitude of the x-direction component of the measured field

at data point (XI, Y1 , Zl) and press CONTINUE. The computer will beep

and display "ENTER PHSX. PHSX = ?" as a prompt.

16. Type in the phase of the X-direction component of the measured field at

data point (XI, Yl, Zl) and press CONTINUE. The computer will beep and

display "ENTER AMPY. AMPY = ?" as a prompt.

17 Continue in this manner until AMPX, PHSX, AMPY, PHSY, AMPZ, AND PHSZ for

data point (XI, Yl , Zl) have been entered.

18. After PHSZ has been entered, the computer will print out the values of

AMPX through PHSZ that were entered. Then it will display "ARE THESE

NUMBERS CORRECT? (Y/N)."

19. If correct, type in either y or Y and press CONTINUE. If not ^o r, o^.,

type in either n or N and press CONTINUE.

20. If you have typed n or N, the computer will beep and display R

AMPX. AMPX = ?" as a prompt. Repeat steps 15 through 19 until tue

numbers printed out in step 18 are correct.

109
21. When you type y or Y, the computer again prints, "EM FIELD DATA AT FIELD

POINTS:", beeps, and displays "ENTER AMPX. AMPX = ?" as a prompt.

22. Repeat steps 15 through 21 for measured field data at data points (X2,

Yl, Zl), (XI, Y2 , Zl), ( X2, Y2, Zl), (XI, Yl, Z2), (X2, Yl, Z2), (XI, Y2

Z2), and (X2,Y2,Z2) in that order.

23. After the measured field data have all been entered correctly for data

point (X2, Y2, Z2) and you have pressed y or Y, the computer will display

"CALCULATION IN PROGRESS." This calculation requires several seconds.

24. When the computer completes this calculation of the currents J, it

automatically loads and runs the FIELD program stored in file #3.

To run the HPL FIELD program, follow this procedure:

1. If the FIELD program is to be run immediately following the CURRENT

program, the computer will have printed the heading "FIELD PROGRAM" on the paper tape immediately after loading the FIELD program in step 24,

above. Procede to step 4, below.

If the field program is to be run at some later time following the

running of the CURRENT program, and the computer memory has been erased

so that it no longer contains the results of running the CURRENT program,

begin with step 2, below.

2. With the power to the computer turned on, insert the magnetic program

tape in accordance with the computer's operating instructions.

3. Press LOAD and type 3, or type Idf 3, and press EXECUTE. This will load

the FIELD program into the computer.

4. Make any program changes as needed or desired. These are discussed in

section 6.4. The procedure listed here is based on no changes to the

program as shown in section B.l.

110
5. Press RUN. This will cause the computer to print the heading "FIELD

PROGRAM" on the paper tape. Next, the computer prints "FIELD POINT

POSITION:" on the tape.

6. The computer will beep and display "ENTER X. X = ?" as a prompt.

7. Type in the X-coordinate of the arbitrary field point at which the field

strength is to be computed and press CONTINUE. The computer will beep

and display "ENTER Y. Y = ?."

8. Type in the Y-coordinate of the arbitrary field point and press

CONTINUE. The computer will print out the values of X, Y, and Z that were entered. Then it will display "ARE THESE NUMBERS CORRECT? (Y/N).'

10. If correct, type in either y or Y and press CONTINUE. If not correct,

type in either n or N and press CONTINUE.

11. If you have typed n or N, the computer will beep and display "RE-ENTER

X. X = ?" as a prompt. Repeat steps 7 through 10 until the numbers

printed out in step 9 are correct.

12. When you type y or Y, the computer will display "CALCULATION IN PROGRESS." This calculation requires several seconds.

13. When the computer completes this calculation of the field at point (X,

Z), it prints "E SUB X, E SUB Y, and E SUB Z at the FIELD POINT." Then

it prints out the values of X, Y, Z, AMPX, PHSX, AMPY, PHSY , AMPZ, AN? - v -

14. Then the computer displays "DO ANOTHER FIELD POINT? (Y/N)."

15. If yes, type in either y or Y and press CONTINUE. If no, type in e't'V”

n or N and press continue.

16. If you have tyDed y or Y, the computer will print "FIELD POINT -DSI li i:

on the tape, and continue from step 6, above. If you have Typed

the computer displays "CALCULATION COMPLETED" and ends the ru”.

Ill
y

/

Ei

Figure 1. A theoretical model of penetration of

electromagnetic waves into a cavity

with apertures.

/

4- E

i

Figure 2. A solid perfectly conducting box without

apertures in free space.

112
Figure 3 Normalized induced currents on Face I of Figure 2 O for a (0.5X) cube illuminated by a normal incidence,

plane wave H- toward +y -direction (a) |Jj, and

(») Uz |. [s]

113
Figure 4 Normalized |J | on Face

for a (O.IX)^ cube with

I of Figure 2 normal incidence.

114
Z/A

x A

Figure 5. Normalized |J

Z

| on Face I of Figure 2

for a (0. IX)

3 cube with normal incidence.

115
z

1

Figure 6. A plane wave incident normally on a conducting

cavity with a slot aperture (a) side view,

and (b) front view.

A

1 d

Figure 7. A network representation for the model in

Figure 6.

116
E|

|t

/E

0

Figure 8. Electric field distribution for TE^ mode in a copper cavity at natural resonance, (x * z * 0).

117
Figure

9.

Idealized

rectangular

building.

coo

o

-N

118

(H‘0‘M)
set-up.

uac

-5fDo

CDQJCOc

“5

fD

fD

13r+

x

/ir rs id rt -h

fD fD

=3 -S

33 fD

Qi 13n

fD

30 TJ

fD =3- -h QJ

fD CO

-S (D

fD

33O

fD

119

NBS

Isotropic

Antenna
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBSIR 82-1659

2. Performing Organ. Report No. 3. Publication Date

February 1 982

4. TITLE AND SUBTITLE

A Study of the Electromagnetic Fields Distribution Inside Buildings with

Apertures Excited by an External Source

5. AUTHOR(S)

Mark T. Ma and M. G. Arthur

6. PERFORMING ORGANIZATION (If joint or other than N BS. see in struct ion s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Supported in part by: U.S. Army Communications-El ectronics Engineering Installation

Agency (USACEEIA)

Fort Huachuca, Arizona 85613

10.

SUPPLEMENTARY NOTES

j Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) Two special cases of the penetration of electromagnetic fields into a cavity,

building or box are formulated and analyzed. One is to consider the case of a lossy cavity with small apertures in free space, based on an application of the

equivalence principle and the use of a generalized network formulation. It is

found that the field strength at the aperture center is approximately inversely

proportional to the square-root of the conductivity of the cavity walls and that

high field levels can exist inside the cavity under certain physical conditions.

The second case is to treat the problem of large buildings with large apertures

on a practical lossy ground by a combination of theoretical approach and measurement data. Field levels inside the building for this latter case depend on the

transmitter power, the transmitter- to-building distance, the ground conductivity,

and the measurement antenna height relative to that of the transmitter.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s)

aperture; cavity; equivalence principle; field distribution; slot.

13. AVAILABILITY

[K] Unlimited

(yy| For Official Distribution. Do Not Release to NTIS

1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C

20402.

[JXj Order From National Technical Information Service (NTIS), Springfield, VA. 22161

USCOMM-DC 6043-P 80

14. NO. OF

PRINTED PAGES

123

15. Price

$ 12.00

U.S. GOVERNMENT PRINTING OFFICE: 1982-577-844 Region No. 8