cLEARED

Xote 160

December, 1972

Fields at the Center of a Full Circular TOR~Sand a Vertically Griented TCRUSon a Perfectly Conducting Earth

C. E. BaumAir Force Weapons Laboratory

H. ChangThe Dikewood CorporationAlbuquerque, New Mexico

Abstract’

In this note a special toroidal coordinate system is introduced in

order to derive general solutions for the electric and magnetic fields in

space and on the toroidal surface. These solutions depend on the current

distribution on the toroid and are in integral forms. !50me Fourier expan -

sion tec.hnique~ have been used in order to simplify these integral equations.

The surface current is found under the assumption that the thickness of the

toroid is thin compared to the wavelength, which leads to an analytic SOIU:

tion”for cne iieids at the center. The frequencj response with a delta-gap

source is shown and the transient behavior of the fields with a s.~ep-func~ion

excitation is also discussed. Some ‘kinds of uniform loadi’ng impedances

are used with the purpose of producing a plane -wave-like field at the

center of the toroid.

..

Sensor and Simulation Notes

Note 160

December, 1972

Fields at the Center of a Full Circular TORUSand a Vertically Oriented TORUSon a Perfectly Conducting Earth

-.

C. E. BaumAir Force Weapons Laboratory

H. ChangThe Dikewood CorporationAlbuquerque, New Mexico

Abstract

In this note a special toroidal coordinate system is introduced in

order to derive general solutions for the electric and magnetic fields in

space and on the toroidal surface. These solutions depend on the current

distribution on the toroid and are in integral forms. Some Fourier expan-

sion techniques have been used in order to simplify these integral equations.

The surface current is found under the assumption that the thickness of the

toroid is thin compared to the wavelength, which leads to an analytic solu -

tionfor the fields at the center. The frequency response with a delta-gap

source is shown and the transient behavior of the fields with a step-function

excitation is also discussed. Some kinds of uniform loading impedances

are used with the purpose of producing a plane -wave -like field at the

center of the toroid. f3J, J 9,

$ Jl%_? ‘~y.

~)

‘ /l/co‘ /

Q ?& 42 r-’

2$:2CV,~ ‘$>.G %o:e+@;;:,+ &

‘%<t.~

>i \

$’7$, J

Acknowledgements

We would like to thank Mr. J. P. Martinez of Dikewood for the

numerical calculations and comments. We also wish to thank

Dr,. P. Castillo of AFW”L for his interest in this study; and Mary Carroll

of Dikewood for typing the manuscript.

TABLE OF CONTENTS

Chapter =

●

-h’

Figure

1

I.

II.

III.

Iv.

v.

VI.

VII,

VIII.

IX.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction. .,...,.. . . . . . . . . . . . . . . . . . . . . . . . .

Special Toroidal Coordinate System. . . . . . . . . . . . . . . . .

Fields and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution of the Integral Equations . . . . . . . . . . . . , . . . , .

Current-on TORUS Loaded with Uniform Impedances . . . . .

Fields at the Center . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TORUS in a Full Toroid Configuration . . . . . . . . . . . . , . .

2

3

4

5

6

‘7

Geometry of TCRUS Simulator for Use with Systemonthe Earth Surface.. . . . . . . . . . . . . . . . . . . . . . . . . .

Relation of a Vertically Oriented TORUS on a PerfectlyConducting Ground Plane to a Full Circular TORUSwith Three Coordinate Systems. . . . . , . . . . . . . . . , . . , ,

Ideal Impedance .,,..... . . . . . . . . . . . . . . . . . . . . . . .

.ilfor~=[ I

O.lwith Z/Zo= Xln(8~) - 2 for Various X. . . .

bA. for:=

[ I0.01 with Z/Zo = X ln(8~) - 2 for Various X. . . ,

Afor$=[ I

0.001 with Z/Z. = X ln(8~) - 2 for Various X. . .

-1-

i

ii

4

7

16

25

35

52

64

70

72

74

75

76

77

78

79

80

—

E@E8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Afor~’O. lwith Z/Zo= ln(8:) - 2 + ikaa for

Various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A for:= 0.01 with Z/Zo= ln(8 ~) - 2 + ikaa for

Various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

bAfor–= 0.001 with Z/Z. = ln(8~) - 2 -1-ikaa for

aVarious ~.,,.....,.. . . . . . . . . . . . . . . . . . . . . . .

Mangitude of (A -1).,..,. . . . . . . . . . . . . . . . . . . . .

Maximum A - 11 for “Best” Impedances . . . , . , . , . . . ,

Fields and Asymptotic Forms for Z/Z. = ln(8 ~) - 2b

and —=O. l.,...... . . . . . . . . . . . . . . . . . . . . . . . .a

Fields and Asymptotic Forms for Z/Z. = ln(8 ~ ) - 2

and ~ = 0. 02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fields and Asymptotic Forms for Z/Z. = ln(8 ~ ) - 2l-.

and~=O. 01,,..... . . . . . . . . . . . . . . . . . . . . . . . .

Fields and Asymptotic Forms for Z/Z. = ln(8 ~ ) -2b

an.d — = 0. 001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a

bE-Field for Z/Z. = ln(8 ~) - 2 with : as a Parameter . . . .

bH-Field for Z/Z. = ln(8~) - 2 with ~ as a Parameter. . , .

Step-Function Response for the E-Field in the Time

Domain with Z/Z. = In ( 8 ~ ) -2 and ~ as a Parameter . . .

Step-Function Response for the H-Field in the Timeb

Domain with Z/Z. = ln(8 ~) - 2 and : as a Parameter . . . .

bE-Field for – = O.lwith Xas a Parameter. . . . . . . . . . .

;H-Field for– = O.lwith Xas a Parameter . . -:, . . . . . . .

ab

E-Field for – = 0.02 with Xas a Parameter. . . . . . . . . . .a

H-Field for ~ = 0.02 with Xas a Parameter . , . . . . . . , .

Page

—.o--81

82

83

84

85

86

87

88

89

90

91

*

—

92

93

94

95

96

970.

a

-2-

Page

25

26

27

28

29

30

31

32

33

34

35

36

bE-Field for – = O. Olwith Xas a Parameter . . . . . . , . . . 98

;H-Field for ~ = O. Olwith Xas a Parameter . . ,. . . . , . . 99

E-Field for ~ = 0.001 with Xas a Parameter. . . . . , . . , . 100

bH-Field for ~ = 0.001 with Xas a Parameter. . , . . , . . . . 101

Step-Function Response for the E-Field for ~ = O. 1with Xas a Parameter . . . . . . . . . . . . . . . . ,.. ,., ,,O 102

Step-Function Response for the H-Field for ~ = O, 1with Xas a Parameter. , . . . . . . . . . . . ..a . . . . . . . . . 103

Step-Function Response for the E-Field for ~ = 0.02with Xas a Parameter. .,. .,. ,.. ,.. ,O. . . . . . . . . . 104

Step-Function Response for the H-Field for ~ = 0.02with Xas a Parameter. . . . . . . . . . . . . . . . . ... .,, .O 105

Step-Function Response for the E-Field for ~ = 0,01with Xas a Parameter. . . . . . . . . . . . ...?.. . . . . . . . 106

Step-Function Response for the H-Fie Id for ~ = 0.01with Xas a Parameter. . . . . . . . . . . . . . . . . . . . . . . . . 107

Step-’Function Response for the E-Field for ~ = O. 001with Xas a Parameter. . . . . . . . . . . . . . . . . . . . . . . . . 108

Step-Function Response for the H-Field for ~ = O. 001with Xas a Parameter. . . . . . . . . . . . . . . . . . . . . . . . . 109

●

‘4

-3-

I. Introdu ct ion

There are two types of TORUS (Transient— Omnidirectional _&diating

Unidis tant and Static) simulators. One of them is shown in figure 1 in a— —

full toroidl configuration as appropriate for simulating an incident plane

wave without reflections from the earth surface. Such a simulator will

be held by some kind of dielectric stand (such as a trestle) high above the

earth surface in order to reduce the effect of earth reflections. Another

type of TORUS includes a ground reflection as part of the simulator.2

Figure 2 shows an example of this class. It consists of only a half toroid

connected to a ground or water surface. The plane of the toroid can be

angled with respect to the vertical plane. Both types of TORUS simula -

tors would have some impedance per unit length Z’ distributed around the

antenna for

netic pulse

wave shaping purposes in order to simulate the electromag-

(EMP) from a nuclear burst.

It has been suggested that a section of bicone3 or a distributed

4source be inserted in the toroid as a generator such that the field of

bicone electric field distribution or

distribution tangential to an appro -

radiation is a biconical wave at early time. The distributed source is

assumed configured to reproduce the

some other appropriate electric field

priate surface, such as a circular cylinder.

In this note, we will consider the field from a full circular toroid

with a pulse generator configured to initiate electromagnetic fields from

-

a,..T.

—

*—

b

-4-

,,

.

.

.

w

a position on the circumference. A

a perfectly conducting eart_h, by the

half toroid on a vertical plane above

method of images, also .forrns a full

toroid. Therefore, the derivations in this note are applicable to both

types of simulators. It is assumed that the toroid has a mean major

diameter of 2a and a minor diameterof 2b, and a2>> b2 in order to simplify

the analys is. ~-he system under test lies roughly in the center of the

toroid so that the distance of the system from the center of the generator

is “a”, and is independent of the orientation of the toroid and generator.

In the study of the frequency response of TORUS an impulse genera-

tor will be used for the purpose of mathematical convenience. However,

an impulse generator is not appropriate for simulating the strong electro -

magnetic pulse of a nuclear burst. A high power generator may be

achieved by charging a large capacitance to an initial voltage V.. If the

source decay time is sufficiently long it may be considered as a step

function generator in the time range of interest. In any event, the step

response will illustrate features of both early time and late time response.

The source, in this note, will be regarded as occupying a delta-gap

only, and located at @ = O. For a vertically oriented half toroid, the total

field at the test volume then is the superposition of the field radiation

from the source and its image. Techniques developed in this note may

be extended to more complicated sources, such as an approximation of

a biconical source, in the future.

-5-

A toroidal coordinate system is convenient in performing calcula-

tions for fields on the surface of TORUS. However, rectangular and

cylindrical coordinate systems have some advantage in calculating the

fields in the test volume. So, some switching back and forth in the three

coordinate systems will be done in order to simplify the problem. The

relationships among the coordinate systems are given in the following

chapter.

Very general solutions of the fields in space and on the toroid are

derived in Chapter III. These solutions involve some integral equations

which can be s implified by Fourier expansion techniques, discussed in

Chapter IV. For uniform impedance loading the Fourier coefficients of

the current are solved in Chapter V. The fields at the center of the loop

are calculated using simpler techniques in Chapter VI. An important

part of this chapter is the search for a loading impedance such that the

ratio of E to H is that of free space in order to simulate plane waves at

the center of the toroid. Extensive numerical results for the fields at the

center with various impedances are shown in Chapter VII. Finally, a

discussion is given on extensions of this work for future studies.

.

.

-6-

II. Special Toroidal Coordinate

A special right-handed coordinate sys tern

System

based on the center of the

toroid’s minor circle is defined as a toroidal coordinate system and will

be represented by the symbol (k, ~, ~). Its relations to the rectangular

(x, y, Z) and cylindrical (Q7,~, z) coordinate systems are shown in figure 3.

A unit vector ?of a particular coordinate in terms of the others may be

formulated, as well as the vector identities in terms of these three coor-

dinate systems. The relation between the rectangular and cylindrical

coordinate systems are well known,

x = Vcos ($)

y =@ sin(@)

@~ = Eqcos (@) -@dsin ($)

z = F&sin (~) +@+cOs (d)Y

Q2=x2+y2

7s~= =Xcos (4) +Fysiw

z=-4

@xSin(d) +Eycos (4)

where ~ with a subscript is a unit vector in the appropriate coordinate

direction.

The toroidal coordinates in terms of them are,

(2. 1)

-7-

A2=(V-a)2+z2=(’=-a)2+z2

tan(~)=~~=-z

x2+y2-a

The other coordinate systems can be written in terms of toroidal coor-

dinates as given below,

Note that the restriction Q~O is made.

Using these formulas, one obtains the scalar product of two unit

vectors as follows,

(2.2)

—

(2.3)

.

s

-8-

.

.

~! “z? =3! “s = ~!x

“7s=1x YYZZ

(F--)l, = -a “ F = Cos (~ - (j’)X& w

0S’)4=S’ “F = -sin(@ -@’)w

~.

(F’)Z = o

@j)w = sin (~ - ~’)

(&j$ = Cos ((j - ~’)

(?$2 = 0

(’%)A= cos($)cos(~’)cos (4- fJ’)+ sin(~) sin(5’)

(F~)g = -cos(E’) sin(E) cos(~ - ~’)+ sin($’)cos ($)

GS~)4 = cos (&’’) sin(@’ - ~)

‘w =- cos(%’) sin(g’)cos(~ - +’)+ sin(~) cos (g’)

(~~)~ = sin(g) sin(g’)cos(~ - $’) + cos(~)cos ($’)

R&)d = sin(~’)sin(~ - +’)

(=~)$ = sin ($)sin (gj’ - ~)

‘ah=Cos‘4- “)

(2.4)

Another interesting set of components of the unit vectors is for the

relationship between the toroidal (A’, ~‘, ~’) and cylindrical (v, @, z)

-9-

coordinate systems.

(2.5)

(E’Jw = -F’ “ 7s2-Q

= Cos (’S’ )COS ((j! - (j))

(F~)d = cos ($ ’)sin (~’ - +)

F~)z = - sin (~’)

(@~)W= - sin($’)cos($ - ~’)

%)$ = ‘in(&’)sin(@ - ‘$’)

‘k)z = - Cos(g’)

@j)w= sin(@ - @I)

@~)4 = Cos (~ - ~’)

@$)z = 0

The distance between r and r’ in terms of rectangular coordinates is,

+r- 7’ =(x- X’)F”X+ (y - y’)e’+ (z - Z’)az (2.6)

There is a particular surface of interest on the toroid itself described by

(b, 5, $) in toroidal coordinates or (a + b cos (~), ~, - b sin (~)) in cylindrical

coordinates. With r’ taken on this surface S’ and using eq. (2. 3),

F- ;1 = [(a + kcos ($)) cos (~) - (a + b cos (&’’ ))cos (O’)]E’X

[ I+ (a+ kcos (g)) sin(@) - (a+ bcos(&’))sin (@’) ~

Y (2. 7)

+ [- ~sin(~)+bsin (~’)]=z

By simple algebraic manipulations,

.

*

.

“

●

-1o-

D2 = 1~-@12 = ~2+b2-[

2bA COS (&’)COS (5’)COS ((j - ~’)

( )[24-4’+sin(&’) sin(~’)l+4 asin —

2a+kcos(~)+bcos (~’)

I(2.8)

If, instead, the cylindrical coordinate system is used with (b, ~’, d’)

on S’, then,

[ I+ Usin (@) - (a+ bcos(&T ))sin(~l )]~y-l- [z+ bsin(~’)~z

(2.9)and

ID2= Z-a- bcos (g’)

112

2+ z+ bsin (~’)1 ( )[

+41p~in2 0- 0’ a+ bcos (~’)2 1

(2. 10)

The vector ~ - ?’ can also be written by using various unit vectors.

In the cylindrical coordinates, this is,

T- @. Qz+ z<-g I 1

a+ bcos(~’)@’&+b sin (~’)=’z z

I‘Q- (a + bcos (s’))cos (4 - 4’)]EQ

[ 1 [ 1+ a+ bcos (~’) sin ($- ~’)= + z+ bsin($’)=z

@(2.11)

In the toroidal coordinate system (k, g, ~),

-11-

—

;- ~! . [(a + ~cos (g)) - (a + bcos (~’))cos (~ - ~’)]=W

[ 1+ a+ bcos ($’) sin (~-@’)@ -4

[Asin(!il) - bsin(~’)]~z

{- k+2acos($) sin— 2 (%#) - b[cos(%os(gr)cos(~ -$)

(

II+ sin($) sin($’)] %x+ - 2asin(&) sin 2 (+).

[ II+b sin(&) cos(~’)cOs(@ - ~’) - cos(~)sin (~’) ~~

A line element may be represented as,

d~= hldul~l + h2du2~2 + h3du3~3

= dx~x + dy~ + dzaY z

The magnitude is,

(2. 12)

(2. 13)

Idllz = (h1du1)2 + (h2du2)2 + (h3du3)2 = (dx)2 + (dy)2 + (dz)2 (2. 14)

In this expression Ul, U2, and U3 are orthogonal such that,

dui—= 6..du. l,J

J

and the hi are scale factors which can be found from eqs. (2. 14) and

(2. 15) as follows,

(2. 15)

(2.16)

●

-12-

.

.

.

.

Using the relationship among the coordinate systems it can be shown

that

hx=hy=hz=h~=hx=l

ho=tp= a+lcos (g)

h=~E

(2. 17)

With these scale factors defined, the vector operators may be determined.

The Gradient Operator is defined as,

. 4 -

‘1 a ‘2 a ‘3 a.— —— —_‘= hl W1 + h2 i3u2 + h3 au

3

‘ ;~vl + ‘Zvz + ‘Svs

The Divergence Operator is given in toroidal coordinates by,

v. x= 1[&2h3AJ +*

h1h2h3 WI(h h A. ) + &(h1h2A3)

23123 1

(2. 18)

(2.19)

-13-

(2.20)

The Curl Operator is defined as,

z1.—

‘2h3

‘2e2

~ ~

au2 au3

I hlA1 hA22 ‘3*3

a(h3A3) a(h2A2)

au2 - au3

-..

‘3

[

a(h2.q2) a(hlA1)

+ h1h2 aul - 8U21

.

(2.21)

So, in cylindrical coordinates this becomes,

The stale factors can also be used to determine the surface integral.

Using primes for variables on the toroidal surface one has $‘, $’ and the

+three unit vectors %!~, ~~, and~~ on the surface. Here e; is the surface

.

.

-14-

normal (pointing outward) and the other two unit vectors are parallel to

the surface. The scaling factors on the toroid surface, according to

eq. (2. 17) are,

h’=bE

h’ = a+ bcos (~’)d

The increment on this surface is then,

(2. 24)

IdS’ = b a + bcos (g’) d$’d~’ (2.25)

The formulas derived in this chapter will be used in the calculations

in succeeding chapters.

.

.

-15-

—

III. Fields and Currents

It is well known that the vector potential ~ and the scalar potential

4@ can be calculated from the current density J and the charge density p.

-m

_71F-?’!m) = JQ- - %?’ )e

4?T - ~, (p. ylj ‘v’

(3. 1)

Here, the

two-sided

frequency dependent factor elut is omitted (or equivalently the

Laplace transform of the quantities is used) and

-y. iuu(a + iuc ) (3.2)

In general, an antenna is made from an impedance loaded toroidal

surface (including conductors ) which makes the current

centrate on the surface. Actually, knowing the current

be sufficient in determining the potentials since, by the

and charge con -

distribution will

continuity equation,

the surface charge density p can be obtained from the surface currents

dens ity ~s by,

P~ = +“ 7s

Define Green’s function as,

(3.3)

-16-

—

.

Then, the potentials become,

~(r) = - 1.J/[

VsI

“ ~s@ ) G(T’,TF’ ) dS’~+ ‘we s!

The electric field can be found from the potentials by,

(3. 4)

(3. 5)

On the toroid there are only &‘ and ~’ components of surface current,

i.e. , J~ = 0. From eq. (2. 20),

1T“ “ pf) =

s b[a + bcos (~’)] [1~ (a + bcos(g’))Ja$’

(%~’)]

‘E’

(g’, +’)]I

(3. 7)

Using the scalar product relationships in eq. (2. 4) and the vector

identities of eq. (2. 19), one obtains,

-17-

72 //~E=2-G(F, 3? )

‘k = Z. k ah S,b[a+bcos(g’)_J [[& (a+ bcos(&’’))J (E’,(j)’)

i up %’

2

I-~ f~G(~,3?) [-cos(g) sin(g’)cos(~ - $’)+ sin(~) cos(~’)]Js (~’,@’)

.

s’ ~’.

I

+ cos(&)sin(@ - @’)Js I(g’,+’)b[a + bcos(~’)]dg’d~’ (3.8)

4’

~ = ~up;(o + iu .) is the characteristic impedance. Similarly,——

where Z

2 la- G(F”,~’ )‘E =–-

ilp ‘~ = ZO & k a~ // bra + bcos (g’)] [[--&- (a+ bcos(?’))J (~’,@’)]

s’ %’

a+—

I(8’, ()’)

II I Ib a+ bcos (~’) d$’d~’

ad’ bJS, ,Q

2- v -(~ G(F, 7? )

[

sin(?) sin(&’)cos(# - +’)+ cos(~)cos (~’) Js (~’,~’)

s’ ~’

}1 1+ sin($) sin (~’ - ~)Js (~’,~[) b a +bcos (~’) d$’do’ (3.9)

d’

and,.

.

-18-

b[a + bcos (~ ’)]dg’d~’ - v2J~ G(~l?)[[

sin(g’)sin(~ - ~’)J (g’, ~’)s’ s ~’

I

[+ cos (~ - @)J I(g’,~’)]b[a+bcos (g’)] dg’d~’ (3. 10)

‘d

Rearranging these equations,

.

1+ sin(~)cos (g’) Js

I(~’,~r) + cos($)sin(~ - @’)Js (g’, ~’) dg’d~’

5’ d’(3.11)

[[ Isin(~) sin($’)cos(~ - @!) + cos(~)cos (~’) Js (~f,$l)

E’

- sin(g) sin(+ - ~’)J

t

(~’,~’) d~’d~’

‘d’

(3. 12)

-19-

and,

~E =1 JJ G (F,l? )

@ a+hcos(~)$ S, [1-& (a + bcos (~’))J (E’, +’)

‘o ‘E’I

aJ+b—[

(E’, +’)1}

d~’d~’ - ~2b ~IG(F,7’)[a + bcos(~’)]a~’

‘d ‘St

\lsin(g’)sin($ - $’)Js1}

(&’,$’)] + [cos (~ - ~’)Js (:’, ~’) d~’d~’g,

4’(3.13)

Equations (3. 11) to (3. 13) are the exact solutions for the electric

field in terms of the surface currents on the toroid.

WU5 obtained equations (3. 12 ) and (3. 13) in the same form as above.

But there exists a discrepancy in the results. There is an opposite sign

in the sin(~) sin(~ - @’)Js ,, (&’,@’) and in the sin(g’)sin(~ - @’)Js E,(~’, $’)~

terms. This difference has been carefully checked and it has been con-

cluded that the two signs in Wu’s results are in error. However, this

error does not affect his calculations on current distribution. This is

because Wu considered only eq. (3. 13) and neglected the JsE, component.

Eq. (3. 12) in some cases does not contribute new information; this point

will be clarified in the next chapter. Without the Jsg, terms, eqs. (3. 11)

to (3. 13) have precisely the opposite sign as obtained by Fante et a16 for

all terms.

The electric field components are expressed in cylindrical coor-

dinates, but with the surface current density components still expressed

in toroidal coordinates, are,

.

.

-20-

.

- -y2b1? [

G@,3?’) a+ bcos ($’)1[

sin(~’)sin($ - @’)J (5’, f#)’)

s’ %

- y2b~fG(~,~’)[a + bcos ($ ’)1 [- cos (~’)JI

(5’, ~’) dg’d~’s’ ‘c’ (3. 14)

The magnetic field can be derived from the vector potential of

eq. (3. 5), which gives,

(3.15)

-21-

Using eq. (2. 23) and the scalar products of eq. (2. 4),

Gk

I

t)[a + kcos(~)J

‘x A’a+Acos(~7 Aag [sin(~’)sin(+ - ~’)A&, + cos (+ - $’)A ,

4 I

a-~

(I Isin(&) sin(~’)cos(@ - $’)+ cos(~)cos (~’) Ag,

+ sin (5) sin ($’ - @)A~jl+a+~os(g)l$([-cos(~)sin(g)cos($-~;

+ sin(~) cos(~’)]A$l

)(

+ cos($)sin(~ - @’)A+, - & [a+ Ices (?)]

Isin (~’)sin(~ - $’)A~, + COS($ - 4’)A $! 1)]

G

[(

48“X- m

k~sin(~) sin(~’)cos($ - ~’) + cos (&)cos($’)]AE,

+ ksin(&)sin (~’ - $)A

)([

~, _; - cos(g)sin(~’)cos(~ - 4’)

1+ sin(g) cos (~’) A ~, + cos(g)sin(~ - ~l)A )14’ (3.16)

and thus, the magnetic field components are obtained as follows,

-22-

.

+ cos(~)cos($’)]J~ (~ ’j@”) +sin(~)sin (~’ 1-@Js (~’,~’) d~’d$’g, ~!

1+- sin (s) cos (~ ‘) Js ($ ’,~’)+cos(g)sin (~- ~’)J

I(?’, $’) d~’d($’

g!%’

a-—JI [ II

G(~3?’) a+~cos (g) a+ bcos (~’)ah 1[

sin($’)sin(~ - $’)Js (~1,$’)s! g!

+ cos(~ - (j)’)J I(~’,~’)d~’d~’

%’

[ I.Tba

[ 1[‘o= I m ~:@’7’)~ a+ bcos (g’) ‘in(g) sin(g’)cos (d- ~’)

1+ cocos Js (~’,+’) - sin(~) sin($ - @’)J

t

($’, ~’) d~’d~’p s~,

a-— I{f G(~,7’)[a + bcos (g’)] [- cos(~)sin(~’)cos(~ - ~’)ag S,

1+ sin(g) cos (~’) Js (~’,~’)+ cos(%)sin(~ - $’)J

I(g’,@’) d~’d~’

g!%’

(3.17)

●

-23-

I

The magnetic field components expressed in cylindrical (v, $, z ) coordin-

ates, but with toroidal coordinates (X’, &‘, @‘ ) for the surface current

components, are,

b~-H=–— J!J [ 1[

IG(~,3?) a +bcos (~’) - cos (~’)J~ (~’ J@’) ‘~’d$’

w Wa($ ~, ~’

I- b&~~-G(F,? )[a+bcos (&’)] sin($’)sin (o- d’)J ($’~d’)

.,s’ ‘$’

I+ COS ((j)- ~’)Js (~’,~’) d~’d@

$!

H =b;~f@ I

Gm,~’)[a+bcos (?”)] -sin(~’)cos (b- d)J (~’~@’)

s’ ‘~’

+ sin(@ - @’)J I J/(~’,($’)d&’d(#)’ -b&[

‘-G(~,~’) a + bcos (~’)1

% ‘ s’

1-cos (~’)J

I(~’,~’) d&’d&

‘E’

baHz=–—

I/“”’ G(~, ~) W[a+bcos (~’)] sin(g’)sin($ - @’)Js (~’,~’)

‘Vaw s’$,

I+ COS (~ - cj’)Js (~’, @’) d~’dd’ - ‘&fl [

‘G(F,7’) a + b cos (~’)]

4’*8$. S,

1-sin(g’)cos (~ - ~’)Js I(8’,~’)+ sin(~- ~’)J (~’, $’) ds’d+’ (3 18)

$’ s 6’.

Therefore, knowing the current distributions on this special config-

uration of TORUS, the electric and magnetic field in space can be calcu-

lated either in the toroidal or in the cylindrical coordinate systems.

Unfortunately, the current in

next chapter how to solve for

conditions.

general is unknown. It will be shown in

the current distribution from boundary

-24-

the

0

IV. Solution of the Integral Equations

.

.

E the sheet impedance ~ is given, the electric field

on the to ro idal surface can be formulated in terms of the source field

Eg and the current distribution as follows:

~J’a@)= Eg(w+%w “ 7s(’%$) (4. 1)

It is also known from eqs. (3. 12) and (3. 13) that the surface electric field

is,

and

[[sin (g) sin (&’)cos (~ - 1

(#)’)+ cos(~)cos (g’) J (~’, (j’)

%

(4. 3)

-25-

5?’ ) is the Green’s function in eq.where Gs(F~, s (3.4) with~s = (b, g, ~) and

-1r = (b, ~’,~ ‘ ) on the toroid surface.s

Integral equations (4. 2) and (4. 3) can be reduced to a simpler form

by Fourier expansions of the integrand. The surface current on the

toroid may be expressed in terms of a Fourier series expansion as follows,

.

where,

1

-/

2~Azn(~) = ~ ~ Js(~, @)cos (n~)d$ fern =0,1,2 ,.. .

1

/

2~_

%(g) ‘ z oJs(&, ~)sin(n~)i@ forn=l,2 ,-. .

(4. 4)

(4. 5)

The coefficients @n and ~~ are vectors (with E and ~ components) and, in

general, are functions of ~. Similarly, the tangential electric field on the

toroid surface,

can be expanded in the form

n. 1

(4. 6)

(4. 7)

-26-

.

with,

~n($) = 1 ~2m~~(~,@)cos (n@d@g fern =0,1,2,0..

o

(4. 8)

The idea, of course, is to relate the ~ =1 ~n, and ~’ Now, the Greenfsn’ n’ n“

function on the toroid surface is defined by,

(4.9)

where D = r - r!s s

can be obtained from eq. (2. 8) by letting 1 = b.s For

this case

D: = 2b2 1 - cos($)cos(~’)cos(~ - ~’) - sin(~) sin(~’)1

+ 4asin2 (~) I a+ bcos(,g)+ bcos(~f)1 (4. 10)

Note that Gs is even in ~ - ~’. Thus, Gs can be Fourier expanded with

respect to ~ - 41 as follows,

CcJ

GS=GO+2 ~ Gncos [n(~ - ~!)]

n. 1

-27-

(4.11)

Here the Gn are functions of ~ and ~’ but not of $ or ~’. The coefficients

are given by,

1J

2%-

[ 1Gs cos n(~ - 0’) d(d - d’)

‘n=GO

Substituting these Fourier expansions into eq. (4. 2), one obtains,

(4. 12)

●

+b ~J1

(g’, 4’) d~’d~’84’ s ,

4

- ~2b[a+ bcos(~)]~f[a + bcos($’)]s! ●

I

sin(g’)J~ (5’,4’)[Gosin(@ - 0’) + 2$ Gmcos(m(@ - @’))sin(~ - d’)I

~’ m. 1

ccl

[+ Js ($ ’,$’) Go COS(~ - (#)’)+2 I

Gmcos(m($ - ~’))cos($ -I

(p)] d~’d~’

$’ m= 1(4. 13)

Hereafter, for simplicity, Js& and Js instead of JsE, and ‘S4, ‘U1 be

4

used to represent the surface current.

Multiplying both sides of eq. (4. 13) by ~ cos (n+) and using the

o rtho gonality relations,

.

.

●

-28-

.

.

(4. 14)

as well as the geometric rules,

[ 1cos m(~ - ~’) = cos (m$) cos (m~’) + sin (m$)sin(m~’)

[aces m($ -a(j

~’)] = - resin (m+) cos (m~’) + mcos (m@) sin(m@’)

[cos m($ - @’)]sin($ - $1) = ~[- sin((m - l)~)cos ((m - 1)~’)

+ cos ((m - l)~)sin ((m - 1)~’) + sin((m +-l)$)cos ((m + 1)4’)

- cos ((m+ l)~)sin ((m+ 1)4’)]

[cos m($ - $’)]cos(~ - 4’) = ~[cos ((m - I)@)cos ((m - 1)4’)

+ sin ((m - l)+)sin ((m - 1)~’) + cos ((m+ I)@) cos ((m + 1)+’)

+ sin((m + l)~)sin ((m + 1)4’)] (4. 15)

.

.

it can be shown after integrating over O < @ s 27r that

-29-

Ising’JI [(~’, @’)[Gn+l - Gn_l sin(n$’) + Js@(~’, @) Gn+l(l + ao, n)

‘~

nl(l-tion+G

I)]COS (n@’) dg’d+’ for n = 0, 1,2, . . . (4. 16)

>

Similarly, multiplying by & sin (no) and integrating over O S d ~ 27T,

..

.

●

I- sing’Js (~1, @’)[Gn+l - ‘n-l1 [

cos (n@’) + Js (~’, d’) Gn+l

e 4

+G1 Isin(n~’) d$’d+’ forn=l,2, . . . (4. 17)

n-1

Now, using eq. (4. 4) and integrating with respect to ~’, eq. (4. 16) becomes

I 1~a+bcos(~)~o =

‘o2m~TGnn [*[(a ‘bcos(g’))a$ ‘“)]

n n.

I1

27T

–bna+ (~’) dg’ - I~~2b[a + bcos (~) Jo [a +bcos (~’)

n.

[[ 1sin (g’) Gn+l - Gn_l ~~ (~) + [Gn+l(l + 60, n)n

+Gnl(l-60n I)]a@n(E)dE’ (4. 18)>

@

-30-

Eq. (4. 17) becomes,

[[~[a + bcos (~)]~~ = - 2r~2mGnn -& (a+ bcos(g’))ag (~~)]‘o n n

1+bna~ (~’) dg’ - my2b[a + bcos(&)]~2m[a +-bcos (g’)]n

II- sing’ G

1- Gn_l Q

I I~ (~’)+ [Gn+l + Gn_l cY~,(S’) d$’ (4. 19)

n+ 1n n

Here, ~~n is linked to Qk and Qdn, with pin linked to CYg and Q$n, in an n

somewhat complementary fashion between ~~n and ~$n for n Z 1. The

~~n equation can be obtained from the ~$n equation by substituting for 13$~,

‘k;and ~$n by ~~n, -a$ , and a~n respectively,

n

In the same way, the Es equation in (4. 3) expanded in $ isE

+ 2$ Gmcos (m(@ - ~!))]II--& (a+ bcos(~’ )) J~&(g’, @’)]

m. 1

+ cOs(g)cOs(g’)Jsg(g’3 +’)lG0+2$‘mcos(m(@ - “))1- ‘in(g)Js~g’$’)m. 1

co

[Gosin(~ - ~’) + 21 Gmcos (m(~ - ~’))sin(~ - ~’)] (4. 20)

m. 1

-31-

Multiply both sides by & cos (n+) or ~sin (n+) and integrate over

O < # S 27r; one gets,

~~g = &/fGncos (n~!) [[& (a+ bcos(g’))J (&’,$’) ]

n s’ %

+b

I

y2b2~J~ (g’, (j’) d$’do’ -~a+’ I

~~[a+bcos ($’)} sin(g) sin(?’)J (E’,@’)

4 s’ ‘E

[Gn+l(l +60 n)+ Gn_l(l -6 0 n)~cos(n$’)+ 2cos(&)cos(~’)J (g’,+’)

> > ‘E

[ 1 IGncos (n@’) - sin($)Js (~’, d’) Gn+l - Gn_l sin(n~’) d~’d~’

4

fern = 0,1,2,... (4.21)

●

and,

Tb .—J

G sin(n~’)[[

--&- (a+ bcos(~’))J (E’, 4’)]~op~ - ~& S,n

n ‘E o

[

22

+b ‘J (g’,+’) 1}d~ld+l-7?.ad’ s J/[

[

a+ bcos (~’)] sin(~) sin(~’)J ($ ’,4’)

4 s’ %

[G +G

1n ~ sin(n$’) + 2 cos (~)COS (~’)J (~’, @’)Gnsin (n@’)

n+l -‘5

[ 1 I+ sin(~)Js (~’, 0’) Gn+l - Gn_l cos (n@’) dg’d+’ for n=l,2, . . .

4(4, 22)

Now, integrate over +’,

-32-

‘ ‘ .27T-ryb

/[ 1( [a + bcos(~t) sin(~) sin(~r)ag (&’) Gn+l(l +/j. n)

-o n >

+G n_lo - a o n)]+ ‘COS(~)COS(~’)Qg (~’)G> n

n

sin (g)a’[ l\@n ‘n+l - ‘n-l ‘g’

and,

(4. ‘3)

[ II+ 2 cos ($)COS (g’)a~ ($ ’)Gn + sin(~)ad (~’) Gn+l - Gn_l d&’

n n J

(4. 24)

Again, ~~n, linked to ag and ah , is somewhat complementary to ~~n n n

which linked to CJ’ and aEn On” ‘he @kn

equation can be obtained from the

~~ equation if ~g , Q& , and Qi are substituted by ~~ , CY~ , and -adn n n n n n n

respectively. This is not true for n = O since ~~ = O.n

Equation (4. 1) can also be expanded by Fourier series techniques.

Equating it with eqs. (4. 18), (4. 19) and (4. 23), (4. 24), the current, in

principle, can be solved. This should be a general solution since no

assumptions or restrictions have been made so far. Unfortunately, these

-33-

equations are still in the integral form and, for techniques familiar to the

authors, Gn can be obtained only by numerical methods. It is rather com-

plicated to solve for an integral equation with numerical parameters.

This study will be left for future investigation. In a later part of this note

we will assume that “a” is much greater than “b”. This way, the integral

equation can be reduced to an algebraic equation and thus an analytical

approach would be possible. This assumption in reality is true for most

cases. Also, an analytical result could give a check with the existing

formulas for the extremely high and low frequencies. It also provides a

general feeling of the TORUS behavior for future complete investigations.

.–

.

.

-34-

v. Current on TORUS Loaded with Uniform Impedance

.

.

.

According to eqs. (2. 8) and (3. 4) the surface Green’s function

depends on Ds in such a way that

.(7i-

‘G (~)d~ = G (r) - Gs(-T) = Oa~ s

-li- S

Therefore,

(5. 1)

[+lk[+[ H1

(a + bcos(~’))Js (g’,@’) + b~J (~’,~’)] d$’d~’dg = Oa(#)’ s-n f @ (5. 2)

Also from eq. (2.8), D(~, $’) = D(-g, -&’), which makes

G(~,~’) = G(-$, -g’)

From this relation, it can be shown that,

JT 7i-

In other words, G(~, ~’)d~ and f G($, ~’) cos ($j)d~ are even, and-r ‘-r

J7r

G(~, ~’) sin(~)d~ is odd with respect to ~’.-T

-35-

(5. 3)

(5. 4)

For a uniform impedance loading (Es reducing to a constant scalar

oZ~), with the source at @ = O, symmetry” implies that

(5. 5)

Thus ,

J

T

Js (g,d)d~ = O-IT E

●(5. 6)

Using these relations it can be easily shown that,

-T s

- ?2bJ~Gs[a +bcos(S’)] cos(g)cos (f ’) Js&(S’, @’)d~’dd’d~ = O

-r

.

Each term in the integration of eq. (4. 3) with respect to ~ and g’ from

-m to ~ is zero before any $ ‘ integration is taken. Thus, eq. (4. 3) does

-36-

not give any useful information and therefore can be discarded if the load

.

.

is uniform. In this case eq. (4. 2) is the only fundamental equation for

solving the current distribution. This equation can be simplified by

assuming that2

a2 >> b (5. 8)

and

kb <<1 (5. 9)

The fields calculated under this assumption therefore are valid only

for kb less than one. For higher frequencies, the charges will be accumu -

lated near the source. The known8’ ‘ far field formula of an infinite

cylindrical antenna can be used to represent the high frequency field near

the center of TORUS. Under these assumptions of eqs. (5. 8) and (5. 9) the

current on the TORUS can be considered as uniformly distributed with

respect to ~ and we may assume,

Js (@,~)sO

E

J (d,g)~~)

%

If the TORUS is located in free space where

(“ )

2

72’: . (ik)z

(5. 10)

Z. = ~po/co s 377 ohms (5. 11)

-37-

then eq. (4. 2) is reduced to

(5. 12)

-1/2

K(4) = 2~7Gsd~’ = z+ ~r [(2asin (~))2 + (2bsin ($))2]-T -T

IIexp -ik ~2a~in @))2

2

Instead of using eqs. (4. 4), (4. 7),

1/2

+ (2bsin(~))2 ] d~ (5. 13)

and (4. 11) we may expand the current,

Green’s function, and surface electric field in the following ways in order

to compare our derivations with some of the previous works.

m. -m

where

(5. 14)

o

.

.

.

!

-38-

.

.

.

The Fourier coefficients defined in the last chapter are related to them

as follows,

10‘ 2Tb%o; (I -t-I )=2rbQ

-m m $m’m#O

$(1 -Im) = 2rba’-m $m’

m#O

Eo=pdo

~ (E +Em)=~-m dm ‘

m#O

; (E - Em) = PI-m dm ‘

m#O

(5, 15)

(5. 16)

Substituting the above equations into eq. (5. 13)

_ 47faik m—x

E e-imo (5. 17)‘o

mm= -m

1 in~Multiplying by ~ e and integrating with respect to ~ from O to 27i- gives,

-39-

I

This reduces to

a— I(@’)e

in~’ad’

(-i~n) d+’ +*~;p@’)leind’Kn-l + ‘in+’Kn+lld”

Now, substitute eq. (3. 15)

2n

-Jo mI. e-ire+ 1‘ein+’(-imn) dd’

m

[

2227r m+ka

[1 Ime (-im$’ein~r K + K

2 ‘on-1 )]n+l ‘d’

m= -~

i4r ak=—En

‘o

.

-40-

It becomes

[

~2a2In (-n2Kn) + y @n_l + Kn+~

)). yEn

Let

[

(-n2Kn)kn=a ~a

+ ka~ (Kn_l + Kn+l) 1

then

i2aInXn = ~ En

o

(5.21)

(5. 22)

(5. 23)

Thus, knowing the Fourier coefficients of the surface electric field, the

current can be found. Here En is also a function of In if the toroid is

loaded.

From eqs. (5. 13) and (5. 15),

.27r . -1/2Kn = ~ ( e-lnd

2 .()~))2 + (2bsin(~))2]d+fm [(2a sin ( z

4T -’n

Now, let,

.

.

[[ Iexp -ik (2a sin(~))2 + (2bsin (~))2 de

A = 2bsin (8/2)

(5. 24)

(5. 25)

such that,

r-

2

fi- sin2(~)d6 = b 1 -~do = ~=d8 (5.26)dA=bcos(; )de=b2

4b

and define a new function

-41-

.271 . -1/2Mn(A) = J eln@[(2asin(~))2 + A2 ]

II~xp -ik ~2asin(4))2 + A2 1’2 ~d

o 2 I

(5. 27)

then,

.e=~ -1/2 -1/2Kn=~~ (4b2

g.n

2T2 .- A2) Mn(A)dA = ~ ~ (4b2 - A2) Mn(A)dA

O.. n r O.()

1

[

2b -1/2=—

~z.()(4b2 - A2) Mn(A)dA

Mn(A) can be simplified for small A. If O < A2<< b2 and kA <<1, then

exp[-ik(2asin (~))]Mn(A) = l-27ein@

dd$

o 2a sin ( ~)

The differences are,

I 4

/

2m exp -2ika sin(Z)An- M n+l(A) - M (A) =

I[ei(n+l)d _ ein$

@ 1d+n -o 2asin (~)

[

@2m exp -2ikasin (~)=

(“o 42a sin(~) ‘@T2-e-i$ld’/

[.2n iexp -2ikasin ($) + i(n+~)~]

—— dd‘o a

-17T zi

— ~ exp [-2ikasin(e)+i(2n + l)~]d$‘o

(5. 28)

(5. 29)

(5. 30)

.

-42-

Now, let n = O,

.

[

421/2

.2m exp -ik (2a sin(z)) + A2MO(A) = I

II1/2

d~-o

[(2asin(~))2 +A2]

.2~ exp[

4

/I

-ik(2a sin (Z)) - 1 2n

dd + ~d+=

4J

(5.31)‘o 2asin(~) 2asin(~)]2 +A2

The first term is

[

@

/1

.2Z exp -ik(2asin(~)) - 1

@d~ = ~o”

o 2asin(~)

exp c-i2ka sin (91a sin (0 )

-i ‘z//

- 2ka

I 1

.2ka -n=— de

a.exp -ix sin (o) dx = -$

Jdx

/ exp00 0 “o

_in .2ka=—

I [J(x)+ i E )](X dx

a .00 0

The second term is10,11

, for A/a~O,

‘2”*= :1n(2)+’ermHere,

~ do

I-ix sin (d) de

(5. 32)

J ;/ cosb 1~(z)=l ‘= -z sin(d) de

o

(5. 33)

(5. 34)

-43-

is the Anger 1s function which reduces to the Bessel function Jv (z) if ~ is

an integer;

E (z)= ~/msin[~~ - zsin(tl)]dtlv o

(5.35)

is called Weber’s function. This is also written as -$2(z) where C?(z) is

the Lommel-Weber function referred by Wu.5

The integral of the Anger-

Weber function, Sin(z) is defined as

O’z[J~.)+i~~x)]dxSin(z) = ;j”

12and has been discussed in detail in Mathematics Note 25. Therefore,

()MO(A) = ~ln(~) - ~So(ka) + O ~’a

From eq. (5. 30),

o

2i ‘n/

.Mo+:. o exp

(5. 36)

(5. 37)

n-1 n-1

Mn(A)=Mo +~Am=Mo+~JoT ~exp1[ 1-2ika sin (e) exp i(2m + l)LI de

o

n-1

1[1

-2ika sin(6) ~ ei(2m+1)o do

o

(5. 38)

—-

.

-44-

Here

.

n-1 ~i(2m+l)fli2n6

I

ig l-e 1i2n0

=e-e=

1 - ei2e- 2isin (6)

o

Therefore,

/

--nMn(A) = M. -.

0

(5. 39)

i2ngexp[-2ika sin (6 )][1 - e J

dga sin(o )

+I

‘n exp~-i2ka sin(e )] - 1 de _ “ ~ exp[-2ika sin (O ~[1 - ei2n~ de

o a sin (0 ) /o a sin (6 )

+ n Eexp(-i2kasin (e )) - lJei2n6 do + n [ei2nd - 1] do/-o a sin (6 ) [-o a sin (9 )

= ~ln(~) - ~~kadx/nexp[i2no - ixsin(6)]d6-o

+3 -Tn-1

1 [1e

i(2m+l)6 1doa.

00

= %(%) - %Jo2ka [J2;x) + iE2n(x~

_2n-1

()- ~ln ~ - ~ S2n(ka) -$ ~ Zm ~_l

~. o

Define,

[

z 1[2

1}Nn(A) =~ m eino ‘Xp ‘ik a2@2 ‘1:2 d~-m

[ a2@2 + A21

n-1

dx-$~1

2m+lm. ()

(5. 40)

(5. 41)

-45-

Letq5=~ sinh(t) such that d@ = ~ dcosh(t)dt = @2 + ($)2 dt,

then,

Nn(A) = -$ (mexpI

-ikA cosh(t) + inI

$ sinh(t) dt-m

= H::=++-(’)-:-hdtd’Change variables by letting k = B cosh(x) and n/a = B sinh(x), then,

Nn(A) = ~~~ exp[[ 1}-iAB cosh(x) cosh(t) - sinh(x) sinh(t) dt

-w

= +r’=’l-i--(t -‘Odt-CQ

Where B2 = B2 cosh2(x) - B2 sinh2(x) = k2 - (n/a)2. Letv=t-x such

that dv = dt, then,

Nn(A) = ~ -(~ exp 1-iAB”cosh(v) ] dv

Referring to Gradshteyn and Ryzhik13

(5. 42)

(5. 43)

(5.44)

(5. 45)

O

-46-

.

where $) -m the zero order of the Hankel function of the second kind and

KO is the zero order of the modified Bessel function of the second kind.

For small A,

(5. 46)

14where -y is Euler’s constant, numerically about O. 5772157.

At least for n not too large, the difference between Mn(A) and Nn(A)

is approximately independent of A. In other words,

Mn(A) - Nn(A) I zMn(A) - Nn(A) I (5. 47)for any A for small A

The right-hand side can be obtained from eqs. (5. 40) and (5. 46). The

exact Nn(A) is given in eq. (5. 45). Thus, the general solution of Mn(A)

for any A is approximately

If we define,

n-1cn= ’y-2

I

12m+l

+ In (4n)

(5. 48)

(5. 49)

m. ()

-47-

we obtain

aMn(A) = 2K,[~A]+l.[l -&)2] +2cn-2.s2n(ka)

By the same argument, we may assume that the difference of,

Thus ,

aMn(A) = 2K0 (~) + 2C - 2mS2n(ka)n

Now,15, 16

1

1

2b dA .$~ () &2_A2

1/

2b Ko(~)dA _ I 7r12

J[~-o &2 _ A2 ~2 O- sin (6 )

‘O a

‘390%’%%’)

(5. 50)

(5. 51)

(5. 52)

de

(5, 53)

where Iois the zero order of the modified Bessel function of the first kind

A = 2b sin (O ) is used so that do =dA

and2bcos (()) =

h“

-48-

.

.

●

.

-—

.

.

.

The Fourier coefficients of the Green’s function can be calculated from

eq. (5. 28), which becomes,

IKn=+ K (nb0 ~)Io(~) ‘C - mS2n(ka)n I(5. 54)

Now, consider the case for n = O. Eq. (5. 37) shows that for A2<< a2

MO(A) = $ln(~) - ~So(ka)

Referring to Dwight,16

—— 22

7ra“1

1

+

o

_2[

-y In(%) sin-lxl~ +~ln(2)]7fa

=~ln( y)

Using eq. (5. 28), (5. 55), (5. 53), and (5. 56), we obtain

IK = L ln(~) - nSo(ka)O a7r

1

Equations (5. 54) and (5. 57) agree with Coiling and Zucker. 17

(5. 55)

(5. 56)

(5. 57)

-49-,

Current distributions on TORUS can be found from eq. (5. 23) if the

surface electric field is known. This equation is valid only if the antenna

is loaded with a uniform impedance. For a load of Z‘ ohms per unit length

the surface electric field is the sum of the potential drop in Z‘ and the

source field Eg which is assumed a function of @. That is,

E = Eg(4) + Z’1(~ )

%

Note that both Eg and the drop Z ‘I are taken in the positive ~ direction

and add to give the net E on the toroi.d. The Es expanded in Fouriers

series is,

co

E=I

(gn + Z’In) e- in@

‘+ n=-m

and so,

En= gn+Z’In

From eq. (5. 23),

27raInAn = i(~)(gn+Z’In)

o

(5. 58)

(5. 59)o

(5. 60)

(5. 61)

Thus we obtain the Fourier coefficients of the current distribution in

terms of the total impedance on the TORUS,

Z = 27raZ’

.

(5. 62)

-50-

and the source of excitation gn’

.

.

.

- 2~agn

In = Z + i*ZO~n

If the source is a delta-gap voltage VO located at ~ = O, the source field

17on the surface of the loop is,

Eg = -Vo6(0)/a

(5. 63)

(5. 64)

n. -m

Therefore,

V.In = (5. 65)

z + ‘nz OAn

This is the Fourier coefficient for the surface current shown in eq. (5. 14).

Since K = K_nn

given in eq. (5. 24), it can be easily shown that A = kn -n

from eq. (5. 22) and In = I from eq. (5. 62). one may also express the-n

current in the form of eq. (4. 4) whose Fourier coefficients can be found

from eq. (5. 16). They are,

and

‘in= 0

Clearly, the surface current is symmetry with respect to @ . We have

neglected the current in ~ direction so that both a ~n

and a~ are zero.n

(5. 66)

(5. 67)

-51-

VI. Fields at the Center

In order to minimize the testing volume, equipment will be located

near the center of the TORUS for testing the nuclear EMP effects. In this

chapter we will study the possibility of simulating a plane-wave at the

center of the TORUS using a uniform loading impedance. The source of

excitation will be a delta-gap generator of

in the time domain.

Fields in space can be calculated by

constant voltage VO, an impulse

the integral equations in

Chapter III. They, in general, can only be computed numerically. Here,

an analytical approach will be tried to find the field at a particular point

of interest, namely the center of TORUS.

The current on the toroid has two components; the @ component

‘s+and the ~ component Js . As was discussed in Chapter V, J may

5 ‘~

be neglected if kb <<1. However, as long as kb is not too large, Js stillE

is small compared to Js4

in most portions of the toroid and J is‘4

approximately independent of @ . The contribution to the fields at the

center under the condition b2<< a2 would be mainly due to the Js com-4

ponent even if kb is close to 1. The ~ surface current density thus can

be regarded as uniformly distributed in such a way that the total current

I(4) = 2~bJ (d). Omitting the time dependent factor eiwt,‘4

the vector

potential may be calculated as follows,

●

.

-52-

F-? ‘I

.

1.

= ZM7(i) ::itj’~‘“R”‘@’dE’CJf

Here the continuity relation for the charge accumulation PS and current

J has been used as follows,

‘d

a

m % (1?) = iups(F’) = - V’” Fs(l?) = - ‘[W)]aa~’ 2z-b

A vector identity gives,

(6. 1)

(6. 2)

(6. 3)

Vx(-:’:;’=,)=v(:::’) x=,+(;::’)vx=,(6. 4)

Here, V x $’+

= O for a fixed F1 which is not a function of Y.

-53-

In rectangular coordinates,

v

so,

1/2

(x - X’)2 + (y - y’)z + (z - ~!)z I

-F(x- X1) +F (y - y’) +Sz(z - z’)r-rrl= x.4

#-pi

-+17-7’1 -&j’(~ xl) +-F (Y

[.

e x Y II-y ’)+%(z -z’) ]F- =’1 (+) - 1]

l~_~,, =

.

At the center where r = O,

-*IF--PIve, --,l =

]y +’

3X Cos (If):

The fields at the center are now obtained.

(6. 5)

(6. 6)

.

●

-54-

.

.

(6. 8)

It has been shown in eqs. (2. 10) and (2. 3) that 1711 is even and z’ is odd

with respect to g’. The terms involving the integration of z‘ with respect

to ~’ are therefore zero.

With a uniform load, the current will be evenly distributed with

respect to the

eq. (5. 14) can

location of source, ~ = O. The Fourier

now be written as

m

1(+’) = 10 -I-2I

Incos (no’)

3J4#=_2~nIn sin (n~’)

n. 1

expansion in

(6. 9)

-55-

and therefore,

Substituting eqs. (6. 9) and (6. 10) into (6. 8),

Since a2 >> b2, kb<<lj and Ir’1 z a we finally obtain,

IZ

( )

iua-—10 iua+l+c

Gc=-s —— —c

ey2ac iua

With k = u/c, the fields at the center of the toroid are,

(6. 10)

(6. 12)

(6. 11)

-56-

—

-E Ic 1 (1 + ika - k2a2) . ~, (1 + ika - k2a2)

‘~ ZOHC ‘~O (ika - k2a2 ) (ika - k2a2)

llZOEc=-c — ( )1 ~-ikaika+l+—

y 2a ika

10

~c=~zx (I+ika)e-ika

Instead, one may let 1~’[~ a + b sin (~). Then,

(6. 13)

That is, we should multiply ()eq. (6. 12) by a factor of Jo $ . However,

numerical computations of the field strength show larger errors with the

Bessel function than without this factor. It is decided, therefore, to use

eq. (6. 12) or eq. (6. 13) for the field calculation at low frequencies.

The ratio of electric to magnetic field can now be calculated,

(6. 14)

With the total loading impedance Z, it is shown in eq. (5. 65) that,

11~1=—=

z+‘VZOAO

10 z+‘nzokl

Here k. and A.l are derived in Chapter V. For n = 0, nb/a<<l. The

18asymptotic approximations for the K. and 10 can be used. These are,

(6. 15)

-57-

Io(x) = 1

KO(X) N -ln(~)-~

From eq. (5. 54)

[[K = ~ In(%) - 2] - mS2(ka)

1 a7fI

K.2 = $II

in ( ~)- $] - mS4(ka)I

(6. 16)

(6. 17)

These equations in association with eqs. (5. 57) and (5.22) give

lo = ka1[* ln(~ ) - 2] - S2(ka)

I

[[~ .kal

1– 21n(

>77~) -$ ] - SO(ka) - S4(ka) I

IIlla i in(%) -21- ‘2(ka)-—

I

(6. 18)

The main object of this study is to find a loading impedance such

that the ratio of electric to magnetic field at the center is that of free

space, i. e. , A = 1. According to eqs. (6. 14), (6. 16), and (6. 18) this

impedance Zi is related to frequenciess in the following manner,

zi

[8a) -2 ] - mS2(ka) - [(ka)2 + i(ka)3 ]—= ln(-j-j-

‘0

\[

2F

-$ So(ka) - 2S2(ka) + S4(ka)1}

(6. 19)

-58-

.

.

.

This impedance contains not only passive elements but also active ele-

ments. It is rather impractical and therefore Zi will be referred to as

an ideal impedance. However, it is possible that some kind of combina-

tion of the passive elements might come close to this value. For example,

at extremely low frequencies the condition of h = 1 can be accomplished

by a loading resistance of

[RO=ZO ln(~) - 21

Actually, when ka is large,

(6. 20)

as can be seen from eqs. (6. 14) and (6. 15). Using the asymptotic expan-

12sion of Sm(ka) and ecj. “(6. 18), one can show

Therefore, A is always one at large ka and is independent of load. Thus,

a load of R. could produce plane waves at the center of the toroidal loop

antenna at low and high frequencies. At the intermediate frequencies the

ratio of E to H fields fluctuates around one. As “a” increases the center

is farther away from the current on the loop, which is the source of the

field; the field is closer to a plane wave.

(6. 21)

-59-

With load RO,

The fields at low frequencies

VozoEo’-G—

y2aRo

now become

.

(6. 22)

‘oFTo’F’-&-0

This magnetic field agrees with the static field19

at the center of a loop

since V./R. is the D. C. current on the loop. At very high frequencies,

it can also be shown that eqs. (6. 13) agree with the fields from an

infinitely long cylindrical ant enna excited by a delta-gap voltage.

Let’s assume that ka >>1 >>kb. The asymptotic formula for the

12integral of the Lommel-Weber function at large ka gives,

(6. 23)

.

-60-

+

.

18From the psi function relation ,

*($) =-?- 21n2

@(z+ l)=@ (z)+Lz

one gets @(5/2) = @ (1/2) + 8/30 Therefore, from eqo (6, 18),

[[~~m Xl = ~ ~ 21n(~) -~]- SO(ka) - S4(ka)I

=[[

‘~ ~ln(-— ~)+~]+iI

Therfore, under the condition that ka >>1 >>kb,

V.=

nZ Oka

[[Z+ z l-qln(~)+~l

lrI

(6. 24)

(6. 25)

(6. 26)

(6. 27)

and from eq. (6. 13)

-61-

For kb<< 1, the Hankel function of the second kind can be represented as

follows :14

(2)‘o

(kb) = Jo(kb) - iyo(kb) z 1 -~i[ln(~)-l-yl

H(2)~ (kb) = Jl(kb) - iyl(kb) x ~

So, eq. (6. 28) can now be written in terms of Hankel functions as

(6. 28)

(6. 29)

which agrees with the radiation field from an infinitely long cylindrical

8,9ant enna. This is certainly understandable. At very high frequencies,

the currents are largest near the generator. The other part of the antenna,

whether it is curved or straight, has very little effect on the far field

radiation. It is found numerically that eq. (6. 30) agrees very well with

eq. (6. 13) somewhere around kb = O. 3 for b/a from O. 01 to O. 001.

Therefore, we may use eq. (6. 13) for fields at kb <0,3 and then switch

o(6.30)

.

-62-

“

●

to eq. (6. 30) for higher frequencies. This way, we obtain

fields at the center of TORUS at all frequencies. Although the as sump-

tions were made that not only az >>b2, but also that kb <<1

tions to eq. (6. 13), the results indicate that this formula

good for kb close to one.

If kb is not very small, then as long as b2<< a2, the

in the deriva -

actually is still

local geometry

around the source delta-gap closely approximates that near a delta-gap

in an infinite cylindrical antenna of radius b.

(almost uniformly distributed in ~) will result

The gap electric field

in an almost uniformly

distributed J with a small “J by comparison.‘4

The infinite cylindrical‘E

antenna then provides the approximate fields at the center of TORUS for

ka >>1 provided b2<< a2.

-63-

VII. Numerical Results

We have obtained an ideal loading impedance Zi for generating a

plane -wave at the center of the toroid at all frequencies. The ratio of

Zi to the characteristic impedance of free space ZO is given in eq. (6. 19)

which is,

ZiI— = In (~) - 2! - mS2(ka) - [(ka)2 + i(ka)3 ]

‘o

II

2~ So(ka) - 2S2(ka) + S4(ka)!—-—

3 I

and is plotted in figure 4 with respect to ka. This impedance contains

not only passive elements but also active elements as shown in the phase

plot . It is shown in eq. (6. 20)’ that Z = RO at low frequencY, wherei

(7. 1)

‘o = ZO(ln(8a/b) - 2) (7. 2)

is independent of frequency.

From eqs. (6. 14) and (6. 15) we have already obtained the ratio Of

‘c ‘0 ‘oHc’

(7. 3)

.

.

●

which is plotted in figures 5 through 7 with a purely resistive load

Z=XRo“

When X = O, A has a peak near ka = 1. The resonance is

-64-

.

sharper for a thinner toroid. This is what one would expect from a cir -

cular loop antenna. The peak decreases as resistances are loaded to the

ant enna. In order to simulate a plane-wave-like field, the amplitude of

A must be close to one. From these figures, one finds that the optimum

choice for X is one for which A is in general about one at all frequencies,

although A may be closer to one at some

values of X. For X = 1, the field at the

low and high frequencies as shown in the

particular frequency with other

center is near a plane wave at

plots . This characteristic is

also proved analytically in Chapter VI. In the intermediate frequencies

the maximum fluctuation of 1A - II is about 50% for b/a = O. 1. This

fluctuation drops to 25~0 for b/a = O. 01 and becomes only 17% at b/a = O. 001.

One can not make the exact ideal impedance Zi with passive elements.

But some other kind of combination of the passive elements might be able

to produce an impedance close to Z.. As shown in1

resistive at low frequencies and becomes inductive

figure 4, Zi is purely

at intermediate

frequencies. So,

Z = R. + ikaaZo.

to 10 for various

one may choose a load Z in the following form:

The amplitude and phase of A are shown in figures 8

b over a ratios. It appears that the maximum flu ctua -

tion of 1A - 1 has a minimum for a between 1 and 2.

given in an expanded scale for 1A - l! in figure llb.

maximum flu ctuation of jA - 1 I is about 367’owith a =

The optimum a is

For b/a= 0.1, the

1.556. For

b/a = 0.01, ~= 1.811 and b/a = 0.001, Q= 1.951, the maximum fluctua-

tion is about 22~0 and 16% respectively. This shows some improvement

-65-

in the intermediate frequency as compared to the pure resistive case

(figure ha). Figure 12 gives the comparison for peak 1A - II for a load

Z = RO and Z = RO + ikaaZO with optimum Q. One should bear in mind that

EMP usually has a very wide frequency range. Since an inductance

becomes an open circuit at very high frequencies, this kind of load may

not simulate EMP properly at high frequencies.

This optimum Q is a function of b /a, and it only applies to the E/H

ratio. For good high frequency efficiency a series inductance is undesir-

able. However, other more reasonable impedance functions can also be

investigated.

There is an interesting phenomenon in the pure resistive cases.

The magnitude of A.is one at ka z O. 7, 3, 5.7, 6.3, etc. At these fre-

quencies it is independent of load as well as b/a ratio. The impedance

Z appears in the formula for A. If it is real then at certain frequencies

the other terms are pure imaginary but conjugate with respect to numera-

tor and denominator, This gives magnitude one but does shift phase, the

amount of phase shift being proportional to the imaginary part.

The electric and magnetic fields will be shown in terms of normal-

ized factors E. and Ho given in eq. (6. 23). As was discussed in the last

chapter, one should choose eq. (6. 13) for the fields at low frequencies

and then switch to eq. (6. 30) at high frequencies. That is, for low

frequencies,

.

.

-66-

.

.

:0=-%(2:1.)(,,.+,++)EN=-c

(1 + ika) (70 4)

and for high frequencies,

-2R e-ikb r 1

E~=O

1

1Ny

1

= HNE’Zi7rZ

o(2)

Ho (kb) - i($ )(Q) H(2)(kb)oal

(7, 5)

-bThe source is turned on at t = - ~ .

Equation (7. 4) and (7. 5) are plotted in figures (13) to (16) for com-

bparison. For ; = O. 001, they agree very well, although the current

os collations along the loop do not appear in the asymptotic formula. The

bportion of agreement decreases as ; increases. Since Ikbl <<1 and

22a >>b were assumed in the derivations of eq. (7. 4) and eq. (7. 5) is

good for ~ ~0, one would expect some disagreements in these two formulas

bfor large ; ratios. A typical example is shown in figure (13) for ~ = O. 1.

From these figures, the best switching points seem to be at ka = 9 for

b–=O. landkb=O.3for~ <0.02. This way, we obtain a complete fre-.

quency response for fields at the center as shown in figures (17) and (18).

bAlthough, there is a clear discontinuity in the swit thing point for ~ = O. 1,

it does not appear too bad really. So, the above formulas give us the

-67-

fields at all frequencies for ~b

<O. landz= O. 1 can be considered the

limiting case. More accurate studies are proposed in the next chapter

for extending the range of validity to a thicker toroidal structure.

Since the field strength increases at large ka, one needs to con-

sider a very wide frequency range in order to obtain an impulse response.

However, as it is discussed in the Introduction that a unit step function

is a more adequate simple function for giving an EMP-like response.

The frequency response from a unit step source is that from impulse

-1multiplying (ika) . Thus, the field will now drop to zero at high fre -

quencies and we need only some finite frequency ranges in computing the

inverse Fourier transform. The transient behavior is shown in figures

(1.9) and (20). The decay is extremely fast in the early time and

approaches one at later time. There is no great variation in the results

for various b/a except b/a = O. 1 which is not quite accurate in the fre -

quency domain.

With various loading resistances, the frequency and time behaviors

of E and H are shown~ in figures (21) to (36) for some typical values of

b/a, namely O. 1, 0.02, 0.01 and O. 001. Without a load, the electric

field has a resonance at ka = 1 and becomes sharper for a thinner loop.

At this frequency, the circumference equals the wavelength. The phase

will not be changed even with some resistive loads. Therefore the field

at the center is enhanced. At low frequencies, the electric field is

mainly due to the voltage drop from the source spread out over the toroid

0..

.

—

.

-68-

to give a charge separation. The normalized E

independent of the load and will approach one at

hand, the magnetic field depends on the current

field at the center is

late time. On the other

on the loop at low fre -

quencies. Therefore the normalized H field shown in these figures is a

function of load and be comes infinity when X = O. The transient behavior

of the E field shows os collations when the loading resistance is zero or

smaller. The os collations are damped as the load increases and finally

approach one at later time. For the transient H field, it shows a rise

after an initial decay. This increase of field strength is more significant

for small loading resistances. The field finally approaches a constant

steady state value equal to 1/X.

.

-69-

VIII.

This note is one of a series

Summary

which discuss and study the electro -

magnetic performance of the TORUS type of simulator. In this note, a

special coordinate system was introduced to depict t oroidal geometry

and thereby fa cilit sting

magnetic fields. Some

Fourier expansions, to

the derivation of solutions for the electric and

special techniques were employed, such as

simplify the integral equations. Some assump-

tions and restrictions were made, like a2 >>b2, to further simplify the

expressions. These assumptions may be considered valid for most cases.

been

As a result of this study a fairly extensive investigation of the Anger-

Weber function was done and documented.12

General solutions have

derived for fields in space and on the toroidal structure. Extensive

numerical results for fields at the center of TORUS in both the frequency

and time domains are given.

This work can be extended in the following order according to the

authors’ preferences.

1. Considering

terms in the Fourier Expansion with respect to $, the field near the

center can be found.

the first together with the next higher order

of

by

volution of the source function with the impulse response.

2. Using eq. (5. 62) the fields at the center from various types

source can be calculated. The transient behavior can be obtained either

inverse Laplace transforms of the frequency response or by the con-

.

.

.

-70-

3. Instead of using eq. (5. 10), one may assume that

“

(8. 1)

to account for the decrease of current density and electric field with increas-

ing cylindrical radius. This way, one may be able to obtain solutions for

larger b over a ratio.

4. For any value of a and b (but a/b > 1) one may try to solve

the integral formula in Chapter IV analytically or numerically to obtain

the surface current density not only in the @ direction but also the ~

component.

5. By numerical methods the fields anywhere in space can be

computed from eqs. (3. 14) and (3. 18).

6. Non-uniform loading impedances can be considered for

improving the field distribution in some sense in various frequency bands.

7. A tilted TORUS above a perfectly conducting ground is an

interesting topic for future study.

8. One may investigate the

the ground has finite conduct ivity.

behavior of TORUS for the case that

.

-71-

IX-; References

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11,

C. E. Baum, “EMP Simulators for Various Types of Nuclear EMPEnvironments: An Interim Categorization, “ Sensor and SimulationNote 151, July 13, 1972.

C. E. Baum, “Some Considerations Concerning a Simulator withthe Geometry of a Half Toroid Joined to a Ground or Water Surface, “Sensor and Simulation Note 94, Nov. 17, 1969.

C. E. Baum, “Design of a Pulse -Radiating Dipole Antenna asRelated to High-Frequency and Low-Frequency Limits, “ Sensorand Simulation Note 69, Jan. 13, 1969.

C. E. Baurn, “The Distributed Source for Launching SphericalWaves, “ Sensor and Simulation Note 84, May 2, 1969.

T. T. Wu, “Theory of the Thin Circular Loop Antenna, ” Journalof Mathematical Physics, Vol. 3, No. 6, 1962, pp. 1301-1304.

R. L. 17ante, J. J. Otazo, and J. T. Mayhan, “The Near Field ofthe Loop Antenna, “ Radio Science, Vol. 4, No. 8, Aug. 1969,pp. 697-701.

C. E. Baum, “Interaction of Electromagnetic Fields with an Objectwhich has an Electromagnetic Symmetry Plane, “ InteractionNote 63, March 3, 1971.

R. W. Latham and K. S. H. Lee, “Radiation of an Infinite Cylindri-cal Antenna with Uniform Resistive Loading, “ Sensor andSimulation Note 83, April 1969.

C. H. Papas, “On the Infinitely Long Cylindrical 1Antenna, ‘‘Journal of Applied Physics, Vol. 20, May 1949, pp. 437-440.

J. E. Storer, “Impedance of Thin-Wire Loop Antenna s,” AmericanInstitute of Electrical Engineers Transactions on Communicationsand Electronics, Vol. 75, No. 27, NOV. 1956, pp. 606-619.

E. HaHen, “Theoretical Investigations into the Transmitting andReceiving Qualities of Antennas, ‘‘ Nova Acts Regiae SocietatisScientiarum Upsaliensis, Series IV, Vol. II, No. 4, 1938,pp. 2-44.

●✎

-

.

0

.--

-72-

12.

13.

14.

15.

16.

17.

18.

19.

C. E. Baum, H. Chang, and J. P. Martinez, “Analytical Approxi-mations and Numerical Techniques for the Integral of the Anger-Weber Function, “ Mathematics Note 25, Aug. 1972.

I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series,and Productsj “ Academic Press, New York, 1965, pp. 955 and 958.

J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc. ,New York, 1965, pp. 71-75.

G. N. Watson, A Treatise on the Theory of Bessel Functions,Cambridge University Press, Second Edition Reprinted, 1948,p. 441.

H. B. Dwight, Tables of Integrals and Other Mathematical Data,The Macmillan Company, New York, 1965, p. 241, eq. 863.41.

R.. E. Collin and F. J. Zucker, Antenna Theory, Part 1, McGrawHill, New York, 1969, Chapter 11.

M. Abramowitz and 1. A. Stegun, Editors, Handbook of MathematicalFunctions, AMS 55, National Bureau of Standards, 1964,

Eqs. 9.6.12 and 9.6.13.

S. Ramo, J. R. Whinnery, and T. VanDuzer, Fields and Waves inCommunication Electronics, John Wiley & Sons, Inc. , 1965, p. 110.

.

-73-

Simulator structure plus

pulser supporfed by

various means such as

dielectric poles,

/’

balloons, etc.

System

d

full toroid

~ Ground surface

FIGURE 1. TORUS IN A FCLL TOliOIT) ~O~FIGITRATION

101!I IL

.

HALF

/

‘\ \~CC

—1— -’y

\

HORfZONTAL~ ~REFERENCE AXIS

FOR SYSTEM

\ ROW OFGROUND RODS

FIGURE 2. GEC)METRY Ol? TORUS SIMULATOR FOR USE WITH SYSTEMON TIIE EARTH SURFACE

-75-

d’

b

/ ///////” ‘///////

(A) Front view

tA

---- --

T( $0

z /4///v

‘//////

a

L( ]—--- .---

(B) Side view

.

.

Y“- ‘@intopage)

.

.

(C) Cross-section

FIGURE 3. RELATION OF A VERTICALLY ORIENTED TORUS ON APERFECTLY CONDUCTING GROUND PLANE TO AFULL CIRCULAR TORUS WITH THREE COORDINATE SYSTEMS

-76-

●

.

.

103

102

Zilzo I

101

10°

30

20

2n-arg(Zi/Zo)

109

8

7

6

5

4

3

b,005

—=. 001 .002a + . .

1 I I I 1 1 I I I 1 1 I I i I I i I 1

10-1 10° 101ka

A. MAGNITUDE OF Zi/Zo FOR VA R1OUS &

—.

.1

4I I I I 1 I I I I I I I 1 I I I

10-1

B . PHASE

FIGURZ 4,

10°ka

OF Zi/Zo FOR VARIOUS br

IDEA L IMPEDANCE

-7’7-

101

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

2.00

1.50

1.00

.50

Arg (A)

0.00

-.50

-1.00

-1.50

-2.00

0

10-1 10°

ka

A. MAGNITUDE OF A

I I I 1 I I I I I I 1 I 1 I I I 1X’o I

.1

.5

10

10-1 10°

101

110’

ka

E. PHASE OF h

FIGURE 5. AFOR~= 0.1 WITH Z/Z. = X[

ln(8 ;) -21FOR

VAR1OUS X -78-

8.00

5.25

4.50

3.75

3.00

2.25

1.50

:\

)(=, 10

IIA.

1

.2

n

.75

0.00i1

inL. .-w Lu

10-L lUka

A. MAGNITUDE OF AI 1 1

I ( I I I 1 I I 1 I I I I I

X.o2.00

1.50

1.00

Arg (A).50

0.00

.

///

-.50

-1.00

-1.50

-2.00

I II 1 1 I I I I L i I 1 I I t

-1 10° 10110

ka

[WITH Z/ZO = X ln(8~) ‘21FOB

I.

B . PiTASE

FIGURE ~.

OF A

AFOR~’ 0.01

VAR1OUS X-79-

8.00

7.00

6.00 AX=lo5.00

IAI4.00

3.00

2.00

1.00

0.00

2.00

1.50

1.00

.50

Arg (A)0.00

-.50

-1.00

-1.50

-2.00

IIo

k.1.2A

1

.5

A. MAGNITUDE OF Aka

x.()

.1

10-1 10° 101

10-1 10° 10*

kaB . PHASE CIF A

FIGURE 7. AFOR ~ = 0.001 WITH 2/2 =o

FOR VARIOUS X.

r

-80-

x rln(8~ )-2]

.

.

4.80

4.20

3.60

3.00

IAI

2.40

.

1.80

1.20

.60

0.00 I

10-1

1.00

.75

.50

.25

Arg (A)

0.00

-.25

-.50

-.75

-1.00

10°ka

A. MAGNITUDE OF A●

I I I I I I I I I I I I I I 1 I

a=lo

I-/ 2

0 A.

—

\ /10

10-1 10°ka

B. PHASE OF A

bFIGURE 8. A FOR ~ = 0.1

FOR VARIOUS d .

1

10’

WITH Z /ZO = ln(8& )-2 +

101

ikaa

-81-

3.20

2.80

2.40

2.00

A1.60

1.20

.80

.40

I I I I I I I I I I I I I I 1 I (

.40

.20

0.00

.Arg (A)

-.20

-.40

-.60

-.80

I I I I I I I I I I I I I 1 I I I0.00

10-1 10°ka

A. MAGNITUDE OF A

101

10 /

vi 1 1 1 I 1 I I I I 1 I I I J I J

161 10°ka

B. PHASE OF A

FIGURE 9. A FOR % = 0.01 WITH Z/ZO = ln(8 %)-2 + ‘kaa

.

●

“

o.—

FOR VA”RIOUS d .

-82-

.

.

2.40

2.10

1.90

1.50

1.20

.90

.60

.30

0.00

.40

.20

0.00

-.20

Arg (A)

-.40

-.60

-.80

I I I i I 1 I I I I I I I I I )

I I I I I 1 I I I I I I I I I I I

10-1 10°A. MAGNITUDE OF A

ka

I I I 1 I I I I 1 I I 1 I I Ia= 10

I

I I I I I I I 1 I I i I I I I 10-1

10°B. PHASE OF A ka

?IGURE 10. A FOR b/a = O. 001 WITH Z/ZO =

FOR VARIOUS CI. .

101

.101

ln(8 ,~)-2 + i.kaa

-83-

. 6(

.5[

.4(

I 1[A-

.3(

.2(

. 1(

O.O(

b—= .1a

10-1

ka 10°

A. MAGNITUDE OF (A-1) FOR Z/Z. = in (8 ~) - 2 FOR VARIOUS ~

101

1 I I I I I I I I I I I I I I I I

.:,:::=n A -

i I 1 I 1 I I II I I I I I 1 I I10-1 ka 10° 10

B . MAGNITUDE

WITH THEIR

.4C

.35

..3C

.25

IA-11

.20

.15

.10

.05

0.00

OF (A-1) FOR Z/Z. = In (8 &) - 2 + ikacy FOR VARIOUS:

OPTIMUM ~

.

FIGURE 11. MAGNITUDE OF (A-1)

-84-

7.0

6.0

5.0(

ln (8&)-2

4.01

3,0(

2.0(

2.10

2.00

1.90

Optimum a

1.80

1.70

1.60

1.50

.

Max IA- 1 I

I 1 I I I I I I I I I I I I I I

10-3

Ib

10-2 10-1T

A. IIMAXIMUM A-1 WITH Z/ZO = ln(8 ~) -2 AS A FUNCTION OF ~

+.3 L—

.2

.1

I 1 I I I 1 I i I I 1 I 1 I I I

10-3 b

10-2— 10-1

B.

a

MAXIM.UM‘1A- 1 ~ITH Z/Z. = ln(8~ ) -2 + ikact FOR OPTIMI!M a

AS A. FUNCTION O ~

FIGURE 12. MAXIMUM [A-l! FOR “BEST”

-85-

IMPEDA~C ES .

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

IE

as.ym

‘/

4.(.)0

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00

10° 101ka

A. MAGNITUDE AND PHASE OF E

102

I

Hasym

/

.—— ——— .—. —.— . .

/z Arg (H)q

. . 0

.

.

103

●

10U 101 10’ 103ka

B. MAGNITUDE AND PHASE OF 11

FIGURE 13. FIELDS AND ASYMPTOTIC FORMS FOR

Z/Zo=ln(8; )-2 AND; =0.1.

-86-

4.0

3.5(

3.0[

2.50

2’00

1.50

1.00

.50

1 I I I 1

.

0.00

.“

lo- ?in.

ka Lu

A. MAGNITUDE AND PHASE OF E

4.0

3.5(

3.0(

2.50

2.00

1.50

1,00

.50

0.00

I

IH

Ar.g (Has ,ym )

——. ——. . —

// 1 I I “kI I I f II 1 I t 1 h

10°1 I

101I 1 I t I I I (

ka 102 Tn 3

B. lUMAGNITUDE AND PHASE OF 13

FIGURE 14. FIELDS AND ASYMPTOTIC FORMS FOR

Z/Z O=ln(8~)-2AND~= 0.02.a-87-

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

.

.

. Ar,q (E )asym

\ __— — —— ——,----

n 1 9

1o“ 101 10” 10’ka

A. MAGNITUDE .fi.N1) PHASE (2F E

.

.

I I I 1 I 1 III I I I I I 1Ill I I I I I It

Il-l

asvm I

J10’ ka

13. MAGNITUDE ANT) PHASE OF H.

FIGURE 15. FIELDS AND ASymptOtiC

102

FORMS FOR

03

●Z/Zo= ln(8~) -2 AND R’ 0.01.a

-88-

.

.

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

A

4.00

3.50

3.00

2.50

2.00

1.50

1.00

.50

0.00

B.

.— 1--1 I I I I Ill I I I I 1 1111 1 1 I I I II10° 101

ka

. MAGNITUDE AND PHASE OF E

102 103

0 I I 1 I I Ill I I 1 I I 1111 I I I I I II10U 101

ka 102 103MAGNITUDE AND PHASE OF H

.FIGURE 16.;” FIELDS AND ASYMPTOTIC FORMS FOR

Z/Zo=ln(8~)-2AND~= 0.001.a

-89-

4.00

3.50

3.00

2,50

IE I

2.00

.

1.50

1.00

b.1—=

I

.1.50

0.00 9 1 I I I 11111 1 I 1 II1111 1 I I II1111 I I I IIll10-1 10° 101 102 1

ka

A. lVAGNITUDE

1.50

1.25

1.00

.75

Arg (E)

.50

.25

0.00

-.25

b—. 1

-.50 I I I I t 11111 I I I 1 I Ill I I I I I 111 I I I I I 1

●

10-1 10° 101 102 103B . PHASE ka

FIGURE 17. E-FIELD FOR Z/Z. = In (~ & ) -2 ITITIT b AS A PA RAMETI:Rz

-90-

.

.

4.00

3.50

3.00

2.50

I H I2.00

1.50

1.00

b—=a

.50

0.00 I 1 I 1 1111 I 1 I I I Ill I I I I 11111 I i I I I I

10-1 10° 101 102 1ka

A. MAGNITUDE

1.50

0.

1.25

1.00

.75

Arg (H)

.50

.25

0.00

-.25

-.50

B

F

1 I I I 1111! I I I IIIll I I I I IIII I I I I IIll+

#

10-1 10° 101 102 103

. PHASEka

IGURE 18. H-FIELD FOR Z/Z. = in (8 & ) -2 WITH ~AS A PARAMETER

-91-

1.20

1.10

1.00

.90

.80

1.20

1.10

1.00

.90

.80

I I I I I 1 I 1 I

o .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Ct

A . E-FIELDT

.

I I I I I I I I I1 n 7.+ “ “

.1 .ffl L,L ..s .+

FB . SCALE OF A EXPANDED

FIGURE 19. STEP-FUNCTION RESPONSE FOR THE E-FIELD IN THE TIME

D-ONIAIN WITH Z/ZO = In (8 >) -2 AND ~ AS A PARAMETER

-92-

.

.

0

.

1.10

1.00

.90

.

.80

.70

(

1.20

1.10

1.00

.90

.80

m

.

.

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 :P+

A. H-FIELDa

I

b = .001r

.1

.1 .2 Ct .3 .4T

)

B . SCALE OF A EXPANDED

FIGURE 20. STEP-FUNCTION RESPONSE FOR THE H-FIELD IN THE TIME

DOMAIN WITH z/zO=l n (8 ~) -2 AND ~ As A PARAMETER.

-93-

IIE

6.00

5.00

4.00

3.00

2.00

1.00

0.00

1.50

1.00

.50

Arg (E)

0.00

-.50

-1.00

-1.50

1 I 1 I I I Ill I I I I I Ill I I I I I II

~

x= o

10-1 10° ka ~ol.102

A . MAGNITUDE OF E

I I I I I 1111 I I I I I 1111 1 1 I I I If

I I I I I I I II I I 1 I I Ill I I I I I 1(. “

10-1 10U ka 101 10Z

.

.

,

.

El. PHASE OF E

FIGURE 21. E-FIELD FOR ~ = O. 1 WITH XAS A PARAMETER

-94-

6.00 1 \ \l I I I I 1111 I I I i 1 IIII 1 I I I I IIi

.

.

5.00

4.00

3.00c.

2.00

1.00

0.00

A.

1.50

1.00

.50

Arg (H)

0.00

-.50

-1.00

-1.50

B.

.1

25

10-1 10U ka 101 1(

MAGNITUDE OF H

I I I I I 1111 I I I I I 1111 I I I I I I

X=lo

.

01 I I I Id I I I I I 1111 I I I I I I

-110 - 10” ka 10’

PHASE’ OF H

FIGURE 22. H-FIELD FOR ~ = O. 1 WITH X AS A PARAMETER

-95-

6.00

5.00

4.00

HE3.00

2.00

1.00

0.00

A.

1.50

1.00

.50

Arg (E)

0.00

-.50

-1.00

-1.50

I I I I I I Ill I I I I I Ill I I I I I II

10-1 10° ka 101

MA.GNITUIIE OF E

I I I I I I II I I I I I 1111 I I I I I II

B.

“

10’

I I 1 I 1 I Ill I 1 I I I 1111 I I I I I 1[

10-1

10° ka 101 102

PHASE OF E

b - 0 02 WITH XAS A PARAMETERFIGURE 23. E-FIELD FOR ~ - .

-96-

.

.

.

.

6.00

5.00

4.00

If{]

3.00

2.00

1.00

0.00

I I I I I I II I I I I I 1111 I I I I I II

x. ()

.5I I 1 I * ,

.10-1

A. MAGNITUDE OF H

1.50

1.00

.50

Arg (H)

0.00

-.50

-1.00

-1.50

B.

10° ka 101

I I I I I i II I I I I I 1111 I I I i I Ii

.

x= 10

0

.10-1

PHASE OF H

10° ka 101

102

10

FIGURE 24. H-FIELD FOR !?- = 0.02 WITH X AS A PARAMETERa

-97-

6.OC

5.00

4.00

HE3.00

2.00

1.00

0.00

102A. MAGNITUDEOFE

1.50I I I I I

.-1.0(

.5C

Arg (E)

0.00

-.50

-1.00

tx= o ~L!A .

/-

/42:2

\

-1.50 i ) I 1 1 I Ill10-1

1 I I I I II II I i I I I I I.a 0 1lU ka 10’ ,JB. PHASEOPE

FIGURE 25. E-FIELD FOR & = 0.01 WITH XAS A PARAMETER

-98-

.

.

.—

0

,

.

6.00

5.00

4.00

3.00

2.00

1.00

0.00

1.50

1.00

.50

Arg (H)

0.00

-.50

-1.00

-1.

.- AA1

2/

5I 1 1 , t 11 t I I I 1 I 1 II

. . i10-1 10” 10’ 1

ka

A. MAGNITUDE OF H

50

I I I I I IIII I I I I I IIII I I 1 I I II

x= 10

— /fi\ ,

<,,,,, I I 1 I I 1 III 1 I I I I II* n 1

10- ‘ 10U ka 10’ lo-

B. PHASE OF H

FIGURE 26. H-FIELD FOR ~ = 0.01 WITH XAS A PARAMETER

-99-

6.00

5.00

4.00

~E{

3.00

2.00

1.00

0.00

I I I I If

.

I I I I II1111 I I I 1 IIll] I I I I Ilt

1 x= o

.

I 1

1.50

1.00

.50

Arg (E)

0.00

-.50

-1.00

-1.50

10-1 10U 101 10L 1

ka

A . MAGNITUDE OF E

I 1 I I 1111 I I I iI1111 I I 1 I 1111 I I I I Ill

I I I i 1111 I I I 1 1 1 1 I 1 I I I 11[1 Illr>

1 Ill. n

10-1 10” 101 10’

ka

B. PHASE OF E

FIGURE 27. E-FIELD FOR ~ = 0.001 WITH X AS A PARAMETER

-1oo-

.

.

6.0(

5.0(

4.0(

IHI

3.0(

2.0(

1.0(

0.0(

.2

.5

1

2

5

1.

Arg (H)

o.

-.J

-1.

10-1

10° 101 102ka

A . MAGNITUDE OF H

1.50~ I 1 I I 1111 I I I I 11111 1 I I I I Ill I I I I Ill

-1.

00 -X’lo

50 -

00

50 -

00 “

o

50.

10-1 10U 101 10Zka

B. PHASE OF H

FIGURE 28. H-FIELD FOR + = 0.001 WITH XAS A PARAMETER

-1o1-

103

10

5

2

1

E

.5

.2

.1

1-

)(.(J

—

.050 2 4 Ct 6 8 10

T

A. E-FIELD IN TIIE TIME l) OMAIN

20

10

5

2

E

1

5. .

.2

.1

h I I I 1 i I I I I I

.2 .1

,3—

—

/ I I I 1 I I II

) .1 9. . (’t ?., .4 .5—a

I-3. SCALE OF A EXPA WIET)

o

●FIGURE 29. STEP-FUNCTION R-ESPONSE FOR THE E-FIELD FOR

b = o 1 WITH XAS A PARAMETER.z.

-1o2-

10

.

5

2

1

11

5. .1.

.2

.1 —

o 2 4 (.;t 6 8 10—a

A. Ii-l? IELr) IN TIJE TIME I>OMAIIY

20

10

5

2

11

1

.5

.2

.1

—

I,, ~o .1 .2 Ct 3. . .4— i.,

a

B. SCALE OF A EXPANDED

FIGURE 30. STEP-FUNCTION RESPONSE FOR THE H-FIELD FORb = o 1 wITH x AS A PARAMETER.z“

-1o3-

10

5

2

1

E

.5

.2

~

.1

Lx=

“1 ~=

o

.

.05

0 2 4 Ct 6 8 10

a

4. I?- FIEIJI IN THE TIME DOMAIN

E

20

10

5

.2

.1

.1 .2

1 m

—

5

—

I I 1 I I I I I II .1 .2 Ct ?. . .4 .5—

a

R . SCALE (’)F A EXPANDED

FIGURE 31. ~~EP--FUNCTION RESPONSE FOR THE E-FIELD FOR

- 0.02 WITH X AS A PAR.AMETER;---:

-1o4-

.

.

●

.

10

5

2

1

.5

.2

.1

‘2

5

-\ --l—

1= d

o 2 4 Ct 6 8 10—a

A. 11- FIEI,T) IN TIIE TIME DOMAIN

20

10

5

2

If1

.5

.2

.1

.1)(.

—

—

2 .1 .2 (!t .3 .4—a

I-3. SCAI,E 01+1A RXPANDEIJ

FIGURE 32. ~TEP-FUNCTION RESPONSE FOR THE H-FIELD

– = O. 02 WITH X AS A PARAMETER.a

.5

FOR

-1o5-

10

5

2

1

E

.5

.2

.1

.05

20

10

5

2E

1

.5

.2

.1

I I I I I I I I I .

)(=()

m 7

—

o 2 4 ct

-i-

A . E-E’IEI. T) IN THFI TIME DOMAIN

6 8 10

—

—

I .1 .2 Ct 3. . .4 .5

a

B . SCALE CIF A EXPANDED

FIGURE 33.B

TEP-FUNCTION RESPONSE FOR THE E-FIELD FOR- 0.01 WITH X AS A PARAMETER.

z–

-106-

“

..7

●

●

El

k

10

5

2

1

.5

.2

.1’

.05

20

10

5

2

H

1

.5

.2

.1

—

o 2 4 Ct 6 8 10

T

A . H-FIELD IN THE TIME DOMAIN

!kA X=.1

—

—.

I .1 .2 Ct .32i--

B . SCALE OF A EXPANDED

FIGURE 34. ~TEP-FUNCTION RESPONSE FOR THE H-FIELD FORu– = 0.01 WITH XAS A PARAMETER.a

-1o7-

E

E

10

5

2

1

.5

.2

.1

1

5

10

.00 2 4 6 8 10

Ct

20

10

5

2

1

.5

.2

.1

a

A. E-FIELD IN THE TIME DOMAIN

●

—

—

I I I I

o .1 .2 Ct .3 .4 .5

a

B . SCALE OF A EXPANDED

FIGURE 35. STEP-FUNCTION RESPONSE FOR THE E-FIELD FORb—. O. 001 WITH X AS A PARAMETER.a

-108-

b

9v

10

5

2

1

H

.5

.2

.1

m- 1 I I 1 I 1 I 1

-+

1

.5 1

1

5T

0 2 4 6 8 10Ctzi-

A . H-FIELD IN THE TIME DOMAIN

20

10

5

2

H1

.5

.2

.1

X=.1

~

o .1 .2 Ct ?.L .4 .5

B.a

SCALE OF A EXPANDED

FIGURE 36. ~TEP-FUNCTION RESPONSE FOR THE H-FIELD FOR

– = 0, 001 WITH X AS A PARAMETER.a

-109-