.

Sensor and simulation No&es

Note 254

August 1978

SotirceExcitation of an Open,Parallel-Plate Waveguide, Numerical Results

Ackriowledgement

The author is thankful to Professor R. Mittra for discussion of the

problem and for the helpful suggestions during the course of this work.

V. KrichevskyUniversity of Illinois at Urbana-Champaign

Urbana; Illinois 61801

Abstract

In this work we investigate numerically the problem of the source exci-

tation of ah open, parallel-plate waveguide. The following assumptions are

made for the source current 1) the current is oriented in the y-direction,

2) it is located at x = O, 3) there is no variation in the y-directin:.,

4) and the current has exp (if3z)behavior along the longitudinal z-di~ection.

We provide graphical output for the EM-field components as functions zf a

longitudinal propagation constant and transverse coordinates and then discuss

these results.

The financial support of AFOSR Grant 76-3066B is gratefully acknowle?.gdd.

254-6 EM? 1-27

I. INTRODUCTION

In the previous report [1], we derived analytical expressions for

the source excitation of an open parallel-plate waveguide. However,

these formulas were very complicated, and it became necessary to evaluate

them numerically. The purpose of this report is to present the numerical

results. The computer progran contained in Appendix A was written and

used to obtain the field distribution as a function of the longitudinal

propagation constant and the transverse coordinates. The numerical out-

puts arc presented in graphical forms. The Cyber 175 at the University

of Illinois was used for all of the numerical studies.

The organization of the report is as follows: Section II contains

a statement of the problem and the basic formulation. Section III

presents the real and imaginary parts and the amplitude of the component

field distribution as functions of several

and a detailed discussion of the numerical

is the conclusion.

parameters in graphical form

results . Finally, Section IV

—

—

—.

381

EMP 1-27●

. . .. <

254-7

11. STATEMENT OF THE PROBLEM AND BASIC FORMULATION

In this section the fields due to a vertical current located inside

an open, finite waveguide are investigated. The geometry of the problem

considered is shown in Figure 1. This structure consists of two perfectly

conducting plates with separation 2H located in a homogeneous and isotropic

medium. A Cartesian coordinate system with its y-axis normal to the plates

is erected. Both plates are infinite in the z-direction and finite in the

x-direction with length 2L as shown in Figure 1. All figures appear at the

end of Chapter II. The current is oriented in the y-direction and is

defined as

AJ = Y6(x) exp (iBz) , (1)

where @ is the propagation constant in the z-direction, and 8(x) is the

delta function. In [1] using the vector-potential approach and the Wiener-

Hopf technique, we obtained a solution for the problem at hand in a general

form for any pa%neters with orierestriction: kL must be much greater ●than 1, i.e.,

l&>>l , (2)

where

k=- (3)

and &, v are the homogeneous media parameters. Using the solution which

was obtained in [1], we will perform a numerical calculation for the case:

w=:= 0.16670 ,

L5—.

‘o ‘

where

(4)

(5)

(6)

382 0

a

o“

—

—

.

254-8 EMP 1-27

Figure 10 Geometry of the problem of source excitation o: aparallel-?late waveguide.

- - —---

383

~.

EM? 1-27

Ls the free-space

(2), (3), and (5)

2.54-9

wave length. Because of the limitations of Equations

we calculated numerical results for

~< f3<0.93 , (7)n

●

where 6 is the normalized propagation constant $ = a. It can beU6

readily proved frmn Equations (3), (4), and (5) that if 6 iS in the region

[PO, 0.931, where

;* =0.80008997 , (8)

then only the first mode can propagate in the x-direction; therefore, the

electromagnetic field has only three components [Ey, Hx, Hz]. When 3

intersects the point ‘dO and goes to zero, the second mode begins to

propagate, and the EM-field consists of five components [Ex, E , Ez, Hxs Hz]+Y

Rewriting the field solution from [1] for the case,+en no more than two

modes can propagate, we arrive at the EM field:

Z’(x,y,z) =X(x,y) ● exp (i13z) (9)●

Z’(x,y,z) =Ii(x,y) ● exp (i3z) (lo)

E(x,y) = ‘fix + ;E + ;EY ‘z

. AZ(x,y) =-XHX + ZHZ

(11)

(12)

(13)

.,. - -. . . .—.

384 0’—

254-10EMP 1-27

[q+(k) 12exp (i2a)

{

2b [~+(al) 12 exp (i2~ al)-’:,(18)

‘1 = (1 + Tl)a“1+

e

2bM (k)M (Q) exp [ia ~l+~;1+ 1+1 ~ \ b ]j

‘2 = a“ Q7

I/’

(19)

,/ ,_

(CYl+ b)2-;

al J-1 , (20)

+ ~lz (: - arc sin :) ,=

(;1;

/

-.

(22)

(23)

(24)

(2.5)

m2a=+ , (26)

.— . .

385

~.EMP 1-27 254-11

. . .

T1 = (M1+O+]2 r bfi IT-1 ●,z

exp(i2a) l+~exp(-i~)l . (27)L J

It should be mentioned that we investigated the lossless medium case;

therefore, in the region @o < $ < 0.93, the propagation constant for

the second mode,has only an imaginary part. Because we neglect terms

which decrease exponentially, our results for the above mentioned region

of B reduce to:

l]E=E=O andx z

2) more simple expressions for the other three components of the field.

We apply numerical analysis only over the regions O ~y ~ H, O c x ~ L.

For the remainder of the waveguide, one can obtain results using the

correlations:

Ex(x,y) = - Ex(-x,y) ; EX(X,Y) = - EX(X,-Y)

Ey(x,y) = Ey{-x,y) ; Ey(x,y) = Ey(x,-y)

Ez(x,y) = Ez(-x,y) ; Ez(x,y) = - EZ(X>-Y)

Hx(x,y) = Hx(-x,y) ; Hx(x,y) = ‘Ax(x,-y)

HZ(x,y) = - Hz(-x,y) ; Hz(X,y) = Hz(x,-y) . “ (28) ,

It is interesting to note that Ex, Hz are continuous and that E , Ez, ‘dXY.-

are discontinuous when ~ crosses BO (or more exactly: they are exponentially

decreasing). It is also of interest to determine the character of the.

behavior of the x-component of PoynCing’s vector. , As one can easily see

from the previous expressions for the EM fields, the x-component of

Poynting’s,vec~or for the second mode is proportional to al and goes to-.

zero when B + !3o“

From this, one finds that the ener~ flow in the

x-direction is continuous when 9 intersects the point 6..

-—. —.. .—. ● ’386

t

cI. ..

—

1.

1.~i~ure 2. The real and imaginary parts of an x-cOmpOnenL of the electric field as., functions

of a longitudinal propaga LiOn constant fOr pOintS of View:x

i= 0.1; * = 0.0, 0.5

!.3W.f.

1-W

,-

, *B t-

1 t-i

u, *’ .4

,-,,3

,-. ~: P.:

i.5- -

Fi&ure 3. TIIe rea 1 and imaginary parts of an x-component of the electr~ field asyfunctiona

of a longlcudinal propagation constant for points of view: – = 0.4; ~ = 0.0, 0.5.L

●

1

—

Fi~are 4. The real aud imaginary pacts nf an x-component of che electric field as., functions

of a iongcitadi,niii propagation constant for points of view:x

i= 0.6; ~= 0.0, 0.5.

~.

.4,:,

. ,_r.> ,,- P. -b. .G

Figure 5. l’he real and inuaginauy pares of an x–component of the eleccr+c field asyfunctions

of a lon~itufli. nal propagation constant for points of view: –= 0.9; fi = 0.0, 0.5.L

●

“:.?= .l~:~L,,c,.. .

i3 .’$,>,-,

Figure 6. ‘(’he reul aI)d imuginary purts of a y-component of the electric field as,rfunctions

oi a Io}igi Ludin,il propagation cons Lant for points of view:x

x= 0.1; ; = 0.0, 0.5.

..

t

l.!jt

t I

Figure 7. The real and imaginary parts of a y-component of the eleccriq field ae.,, functions

of a longicudinul propagaciou constant for points of view: ~ = 0.4; ;= 0.0, 0.5.L

:..5

;E:”

-L

FigIIre 8. “ilIe reul and imugin.iry parts. of a y-component of the electri$ field asyfunc~ions

of a longitudinal propagation constant for paints of view:i

= 0.6; ~= 0.0, 0.5.

.I

&’+-

1

. .

1

Figure 9. ‘1’lIe real and imaginary parrs of a y-component of the electric field as.. functions

of a longitudinal propagation conscant for points of view:?

= 0.9; # 0.0, 0.5.

r.TI-Jw

●

wwIn

i

t

I/fr.;:~~,:l . e~: .... ...i F. ....

?

c-...1

t

T

Figure 10. ‘1’lIe real and imaginary parts of a z-component of the electri: ~ield as functions~

of a 10ngitLldi1121] propagation constant for points of view: ~ = O. 1; ~i = 0.0, 0.5.

,;‘j

i-

.l .5-f-

Figure 11. ‘TIIe re.a~ and imaginary parts of a z-componenc DE the electri$ field asytunctions

of a h.mgitudinal proptigacion constanr for poinzs of View:L

= 0.4; ~ = 0.0, 0.5.

● ● ●,,.

Nu!f.

ivw

I

‘“‘“w’

TL

q-.

. L-”

..( q. ..,. , ,<...

I II

,-,i:l P

Figure 13. TIIIJ real and imaginary parts of a z–component of rhe electri$ field asyfunc t.ions

oti a longitudinal propagation constant for pOints of view: – = 0.9;L ii

= 0.0, 0.5.

1

i

.-, .- B.L . ~.

l:ig(kre 14. l’he rea J and lmaglnary par Ls nf an x–componenu of the magnetic field asyfunctions

Oi a longitul in~l prupagacinn constant I“or points of view:i

= 0.1;i

= 0.0, 0.5.

s,0,

-1. st

I

Figurc 15. The real aad imaginary parts of an x-component of the magnet+c field +unctions

of n lon~ltudinal propaga L ion consr-anc for points of view: – = 0.4; ~ = 0.0, 0.5.L

.

● ✌

$-0P

1., _i’8- . ~Q:$L

Y=L3 .Ciil:tl-i

m 1

H ,- e-f .-... ,~,.-

11

Figure 16. ‘1’he real and imaginary parts of an x-component of the magnet+c field asyfunc~ions

uf a lnngitudinu] propagation constant for points of view: – = 0.6;L ii

= 0.0, 0.5.

E-1CJ

,5

/.5 -.

I

Figure 17. Tile ctial and im~ig.inary parts of an x-component of rhe magnet~ field asyfunctions

of a Lonpitudinil propagation constant for points of view: – = 0.9; fi = 0.0, 0.5.L

t t I I 1 I

,..—

Figut-l! Iii. ‘1’hc rezil and imaginary parts of a z-component of the magneti~ field asyfunctions

of 4 lnngi twdinal propagation constant for points of view: –= 0.1; fi = 0.0> 0.5.L

wWI

f-

Wco

I

t

Figure 19. The real and imaginary parts of a z-component of the magneti$ field asyf unctions ~

of u longitud Lnal propagation constant for poin~s of view: – = 0.4; == 0.0, 0.5.L

Q

● @ ●

1

Vigrrt-c 20. ‘Tlie real and im<]gi nary pares of a z-component of the magneti; Eield asyfunc Lio))s

Oi d lungitudlnal I)ropagdtion constant for points of view:i

= 0.6; ~= 0.0, 0.5.

Ef.u0

‘F=om‘1

I

-L

t 1 II

II

O.:,,+*

.-, ,.-.s c>

t

Figllrc 21. TIN? reu L and Lmuginary parts of a z-cm)iponent of ‘rh& magneti$ Eield asyfunctions

uf a longitudinal propztgntion constanr for points of view: –= 0.9; fi = 0.0, 0.5.L

I ●

$-0-4

t-

++

c-._!

/

—-..---~’

E-t-–-t’--t--k-t-+-t--iEl .s

.2 “; ,: Bi-:il;i$LJrC 22. ‘Ihc IIIUIJUICCIC a y-componenc of the electric field as a functioo of a

lol~giludin:]l propagation constanl fur poinls of view: ~ = 0.1, 0.4; ~ = 0.0, 0.5.

.

T

Figure 23. The lWLilllCIJ~ a y-r(mponenc of che electric field as a fync~ion of a

longitudinal prnpagathu constant for points of view: i = 0.6, 0.9; ~ = 0.0, 0.5.

i ●

.-

:,D

L

.-,i.

,,-K:

~. .—,—

Figure 25. ‘f’he mOdulPOC an x-ccmlpI,I)enL ot the magnetic ~Lelcf aS a ~nction Of a

Iollgi tud inal prupdga LioII constant for po{nts of view: – = 0.6, 0.9; ~ = 0.0, 0.5.L

● e

. .

.

“r_.+_+..+._

G. .4,..,,.,,

Pi:,..:

l:iguuc 26. Tlw IIIC)LI(Ile of .1 z- I,ollIpOnen L OE [lie magnetic field as a fynct Jon Of a

l.mgiLlklinal propogiition constilnt for points of view: ~ = O. 1, 0.4; ~ = 0.0, 0.5.

b,”

i-NI

I — —.

. . . . .,

.:1

-1 1 t I 1 I I +—+-+--+-4Q .4 .% P

.2 .6

Figul-e 27. The muluie Or a z–component Of the magnetic field as a f:nccion of a

lungirudln~l pruptixiJLIoDujust~nt foc points of view: — = 0.6, 0.9; ~ = 0.0, 0.5.L

4=.

:,’

,

t

Flgul-c 28. ‘Tile real and imaginary parts of a.,y-cxmponent of the eLecLrie field as functions-.01. an x–coordina~e [or(3

= 0.4; ;“= 0.0, 0.5.

I

“I%w–t i--i

0. .4 .8.2 4 ?

*,5..

I 1

.+

z, 1

0

‘i.5

_——— ———. .

Fig(Ire 29. The retl I dncl imag Lntiry pares ur ayy–c,)wponent oF rhe elecrric fie].d as functionsh~

of an x-coordinate Cur - = 0.75;6 n

= 0.0, 0.5.%

●

Ilotibow”—

& .4 .8 J!_

.2 6 L

-J.5

-.—. ... ...— ---- .—. —_ ——. —.. _._. — ___________ ——

i

“1

,.

~= .75 ‘.5 y=~ p,fi#-~-----

0.–

1.5- -

RHW3 f p= .75–.5

t II t

,Nj!&i

y= .5t3:#l+

-i@.

.6 -t

!. 5-””

I

Flgurc 3] . The rea 1 ald imaginary purcs Of an,, x-ccunpcment of the magnetic field as funccions ~

of m x–coordlnace {or;

= 0.75; ;= 0.0, 0.5.

● ●

.F-t-Ju

5

i

.5

“l’wvo’.-...—t–––1----1

0 .8

.6 t

1.5 “

I’igaru “IZ. “1’lw rua 1 and imagiimry parts OF ayz-compojlent oi Lhe nmgnecic ~ield as funt. LiOas

Of dll X–CLNJ~dili,l LIJ [0,- - = ().4;B Ii

= 0.0, 0.5.

1.5t

Nl-nf-

.PL.J

I

254-44

111. YL~RICAL STUDY

Because the analytical expressions

are rather complicated and difficult to

the solutions using a digital computer.

section in grap’nical form. This section

first we discuss the field components as

propagation constant; in the second - as

coordinates.

()F THE PROBLEM’

ierived in the seco~d section

analyze, we numerically evaluated

The results are presentsd in this

consists of tvo parts: in the

functions of the longitudinal

functions of the transverse

A. Real and Imaginary Parts and Amplitudes of the Field Components

AS Functions of the Longitudinal Propagation Constant

The graphics that are supplied in this section were plotted with a.

step for 6 equal to 0.005. The point, i= 0.80009, also was used. The

figures were plotted using 188 points. The output for the real and

imaginary parts and the amplitudes of all five field components for eight

x Yobservation Feints: ~ = 0.1, 0.4, 0.6, 0.9 and ~ = 0.0, 0.5 are presented.

From the figures it is observed that S Hx, Hz are dominant components.Y’

In Figures 6-9 the real and imaginary parts of E are given. By compariagY

Ythese results, one can observe for points ~ = 0,0 and ~ = 0.5 that the

F H;<,Hz -~.>Y

field behaviors as functions of 3 are ~Lore complicated in

the middle of the waveguice. This is hardly surprising in view of the

largest contribution af the seccnd mode Eor the above-mentioned com-

ponents for $ = 0.0. When che point of observation approaches $ = 0.5,L

the contribution of the second mode for those components decreases and

has a Limiting value equal to zero, In Figures 6-9, 14-17, 22-25 we-.

observe the step c’hanging for the E Hx components at 3 = 3..Y’

The

—

— 419

EMP 1-272.54-45

second node is responsible for this misbehavior. When~+O .5, the

..contribuz<on of zhat step decreases, and for $ = 0.5, it equals zero.

-.

One can observe that the curves are smooth at 3 = 30. It should be

mentioned

(a)

neglected

tha~:

The discontinuous

the exponentially

(b) In a lossy medium,

behavior is observed

decreasing terms;

the rate of decrease

because we have

would be less. The

H.-component (see Figures 18-21, 26) doesn’t have the step behavior&

because the contribution of the second mode is proportional to Zl,-..

which goes to O, when $ + 3o“

The other two components, Ex and Ez, are

equal to zero on the x-axis and have their largest contribution for

~ = 0.5 [see Figures 2-5, 10-13). E:<is smooch and Ez exhibits the step

behavior at ?o“

The figures demon-strate-that the complexity of the

curves accurs approximately in the region 0.7 < 3 < 0.9, where we

observe a sharp peak, which is due to the resonance. In the region

O < 6 < 0.5 the figures demonstrate the very smooth character of the

curves .

B. Real and Inaginary Parts of the Field Components as Functions

of Transverse Coordiaaces

In this section we present the rsal and imaginar~ par~s of the

dominant field components Ey, ‘dx, Hz ES functions of the x-coordinate

for cwo values of ~ = 0.0; 0.5 and two vzlues of 3 = 0.4; 0.75. The

graphic output, shown in Figures 28-33, was abcainecl using the results

of calculations for 81 points of s in the region [0.1, 0.9] (step = O.Oi!)).L

.,.4LL the graphs have very smooth charac~eristics. ~or 3 . 0.75 they have

slightly more complicated Form than for 3 = o.~. As mentioned in tb.e

.=

420 ‘“ “

●

254-46

previous section, for $ = 0.5 the contribution cf the second mode squals

zero. We see that the amplitudes of the cumes are constant for the

entire region of view. For the;= 0.0, the field components are sums of

the contributions of two modes. One can obse.mre that the amplitudes

of :he curves are changing along the x-direction.

421

.EMP 1-27 2.54-47

r?, CONCLUSIONS

In this report the problem of a source excitation of an open

parallel-plate waveguide was developed. Extensive numerical results for

the field components in the waveguide as functions of several parameters

of the waveguide and propagation constant were supplied.

..,. .—_ ._

422

254-48

v. REFERENCES

.L

EMP 1-27

[I] V. Krichevsky and R. Mittra, “Source excitation of an open parallel-plate waveguide,” Sensor and Simulation Note 253, October 1977.

—

— 423

FM? 1-27254-49

v-l. APPEYDIX A

SOURCE EXCITATION OF AllOPEN, P.ARALLEL-pLATE WAWGUIDE PROGRM

..

●✝

A complete program for source excitation of an open, parallel-plate wave-

guide program 1s presented. The computer program provides chree-

dimensional data-s~orage for the real and imaginary parts of five components

of the EM field. Data were obtained for ~between 0.1 - 0.9 with step 0.1;

~between 0.0H

- 0.9 wtth step 0,1; and 5 - propagation constant between

6.0 - 0.93 with step 0.005 plus (o) 0.80009. These data were used to

plot EM Field components as functions of the propagation constant. The

program can be readily modified to obtain data for plotting the E&f-field

component as a ‘functionof the x-coordinate.

J.

. .

. .424

FrioII

mxoII

HIIi-=

xII>.:toxH

T?r

cdP-1-+DII

J%2m

CJC3z-iHz

i%

(7Gz

mim+3a).-..

H-J-t--

(-3C3zIIr.3

-..-f.1%-.&

o-Ji-+Gz

-iHz

i

I +o. f.n

nn-r

zII

HIIw“

I-J’0.0

(n-i P

“

l-’cf.

></-.0

.-.

/!c!zx.-..0+..w

x.,A●

tJj i--

. .

-i-i&&.-a

mncl

cd.mD-Jwr-

\.\E4’.

i’+Kx.c1

-fJF’-*

.coc)(J1

.-.w

.

.,..

..-.+o.

i-lCJU-J----

.

“

-g

H‘a

x

,.,,>2D03+m

:<0

= ><

n0z.8

... ..*= UJ

m

.

EMP 1-27254-51

. .

426

‘7iin3.’7>

r

i=l

0

a----[J.

m-.,,Ll-J

c-l(3x~;—

J-H

zc-ii-ii%

?.@moc-iH-,

i%

-i,.%-i“\/..1-.

-1-

><.,-.w’●

$,P.

i-Ql-’

–{.-.-iJ

H

*mx.m(3x-1

,-.c?x

F.,A

P‘...

1F’.

~. -H X.

i

x-mi--iz/-.u!\H

‘..(J-J23m-+

,-.

xxH

..

tJ.+$..-.W-x----

,...r.).

g

-iJ-.

.-. 1-●

+-

M.3’C~<+Jtr~i

I

if’(-laz.

(’30z

Ar

%*FJ‘..

0m/-.JJ2

.-.

;-..s.w

EM? 1-27254-53

VII. APPENDIX B“

ADDITIONAL NUMERICAL RESULTS

In Sec Lion 111 of this r~port, we have

calculated the electromagnetic fields as functions of 3, the

normalized propagation constant in the z-direction. The para-

meters chosen for the computation in the report were:

H–= 0.16670, ~

2n

L= 5, where

o‘o=— .&~ “

At the request of Dr. D. Giri of SAI~ we have now derived

additional numerical results for the following choice of parameters,

which correspond to Chose GE the experimental paraliel-plate structure

being investigated at IIamard.

L = 12.5 III

H= 12.75 m

f = 25 Mny (XO = 12m)

*presently with LuTech, Inc.* BerkeleY~ CA

-. -.—.

428

254-54EM? 1-27

The propagation constant ia the x-direction csn be written in the form:

7 kH 2a =——

Hm-m

n

m

1/2

9 m= 0,1,2, ...

~J ii“o

LkL~koL = 27 ~= 6.544985,

●

The asymptotic analysis presented in this report was based on the

assumption (kL>>l). Consequently, great care should be exercised ~~hen

the range of application of these formulas is extended below kL = 10.

It is not difficult to prove that in the range O ~ ~ < 0.337916,

only three modes are above cut-off in the x-direction. Furthermore,

two modes are propagating in the range 0.337916 < 3 ~ 0.882353 and only

one”mode can propagste in the range ~ ~ 0.882353. The application of

the formulas and computer programs developed in this report, though

not the theory itself, is restricted to the range where tk-anodes can

propagate in the x-direction. For this reason, we dsvelop tunenumerical

results only for tlieregion 5 > 0.34, and specifically for the range

.We would like to mention that it is possible to develop the necessary

formulas and numerical results for tuneregion 13< 5 < 0.34 using the

theor,rgiven in a:previous report [Z];

429

$-LJ0,

I

. .1’5.

I 1 I # I, 1

II

I1 I

E , .- ,4 ,:,. ,J ii

.. .

.d .U

I:i gure ] . Real And imaginary parts of E , as functions 01 ?! fOL- parameters given in the lower half ofPage 1. TIIC x ,y valIIfis of tlI& observation point are shown in clie inset.

I

L.!w1-

1

.—

L.

Figure 2. Real and imaginary parts Of E 0s func’ciOns of ~ for parameters given in che lower half of

Page 1. The x,y values of ch~ observation point are shown in the inset.

“‘mI

E’jgure 3. [kill and imnginary parts OE E as functions of ~ for parameters given in the lower half of

●Page 1. TIie x,y vahLes Of th% observa~ion point are shown in the inset.

e ●

1,5 L-—.—.-—-.-—--.——._.,_______

I 1 I-t-

1 i 1

,.1c .4

.>.& ,6

Figurti b. Real and iumglnary parLS of E as functions of ~ for paramews given in the lower half of

Page 1. The x,y values of thz observation point are shown in che inset.

5

-t-

I

j :#.1-i

H

G

J~~”j.’~ ~ ..>-... -

P-

,!, ~~:$L

c, LJ - X.-.&l\- ..!*J.~..-- d

I I I I * I B I 1 I

@

:“’%’I ~- .. . ..-

——. — .-

Figure 5. Real aad imaglndry parts of E as funct ions of ~ for paranw Lers given in Lhe lower lla~f of

Page ]. TIIe x,y values nf thZ observacioa point are shown in the inset.

1

—--.. _L

Vigurc 7. Real aIvJ imagintiry par Ls of E as functions of ~ for parameters given in the lower half of

I’LlgL! 1. l’he x,y vdlues of Cl)% obscrvacion point are shown in the inset.

_L----- —

c. --!

!.

Fi$T,LIL-e 8. Real and imag i nary pares of II as fun(:Llons of ~ for parameters given in the lower half a~

I’age 1. ‘1’lle x, y val ses OF cI13 observation point are sh,own .in Che inset.

i..

Fig!lrfi 9. Rml nod imaginary pnrcs of E as funcrions of ~ for parameters given in the lower half of

PMgL I . Tile x,y values of tl~& observation point are shown in the inset.

● ●

.c-Ldw

I

t

Figure 10. Rtal und imag[nary parts of E as fullc L ions of 6 for

l’+.c 1. “~lle x,y values of lh~ observation poit)t are

parameters given in the lower half of

shown in thti inset.

0

●

Pigure 11. Iteal and imaginary parts of Ii as functions of ~ for parameters given in the 10wer half of

Page 1. The x,y values of cl): observation point are showm in the inset.

●1,

—

11 .2--

———.—... JVigure 12. Real ad in,aginary parts of E as functions nf ~ for parameters ~iven in the lower half of

Ptigc 1. ‘1’lIe x,y values of tl12 observation point ace shown in tlle inset.

P$-iv

I

●

I

Figure 1.3. Real and imaginary pares of E m functions of B for parame’cers given in the lower half OLC

Page 1. TIIC x.y values of tli~ ob.serva~ion point are shown in the inset.

●

1

-L

t

t i I --1” I

l.3j-

+-+

e.

I

T

Figure 14. Real and imaginary parLs Of L as functions of B for parameters given in the lower llalf of

Page I . ‘~lle x, y values 01’ LII& observation poinl are shown in the inset .u

—

4-4“

t

.,.~

k’igllr~ ] 5.

%

RCMI and hag inary parts Of E.. as funcc’ionsof B for parameters given in :che lower half OF

Page 1. ‘L’lIe x,y values of thg observation point are shown in tlIe inset.

&f.m

@

.

.— 1-

-1.5

?

.U

I —.

Fig~[re 16. l<eal and imaginary pares of E as funclions of ~ for parameters given in the lower half of

I’ilge 1, ‘1’[1{> x, y vti) nes (>f t]l> observal jon poin L arc shown in the inSe L.

t

J______Figure 17. kleal and imaginary parts of E as functions of ~ for parameters given in the lower half of

l’age! 1. The x,y values uf th: ohs-e

@

t~ion point are shown in the inset.

e

1 I 1. 1 b t 1 1 I

i

Q ., .*-, ii,-

.&.. .ij

I

.—— 1

Fi.gul-e 18. Real and imaginary [lar LS of E as Eunctinlls of ~ fOr parzmleLers given in the ]CmJer half of

Page 1. ‘flke x,y values of Lh~ observation point are shown in che inse L.

8+7?

.w

Mlm

-T-%?+

-c xII II

.=

Th]

>;

“ ..,.--:. . . ,.

II II

. .

M *1~1 12!

*++

“x r-

LZ-T W

1 I I 1 t t I 1I i

g . ,- .+,:,‘-, ;

,, ..-

.- . 0

-t

I

lpi gill-e 2[). Real and ilt)agillary pauus OF 11 as functiol~s of IK COT parameters given in the lower half of

l’a~u J . Tile x, y Val ucs of t 113 observation point are sl]own in tile inse L .

.2--

.L . ~,

1.5--

l~igul-e 21. Ilt?al and imaginary par Ls nf E aS func~ions of B for parameters given in the lower half of

●Page 1. ‘1’IIc x,y values of thi observation point are shown in ‘the inse~.

●*

AFWL-TR-80-401

DISTRIBUTION

e AUL/LDE/Maxwell AFBDTIC/DDA/AlexandrfaAFSC/DLt4/Andrews

AFTJL, Kirtland AFB

(SUL)

(HO)

Official Record Copy (AFWL/NTllT/Dr Baum)

-.

451/452

. ..—