Title: Image interpretation for fields produced by high frequency line currents over finite conducting media
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Title: Image interpretation for fields produced by high frequency line currents over finite conducting media
Physical Description: Book
Language: English
Creator: Mathews, Bruce Eugene, 1929-
Publication Date: 1964
Copyright Date: 1964
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Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Bibliographic ID: UF00097935
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000568447
oclc - 13660429
notis - ACZ5181

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IMAGE INTERPRETATION FOR FIELDS
PRODUCED BY HIGH FREQUENCY
LINE CURRENTS OVER FINITE

CONDUCTING MEDIA








By
BRUCE EUGENE MATHEWS


A DISSLT.% .ION PPESENTED TO THE (PAlL'U.L COUNCIL OF
THt LINI\'EP-SI OF FLCOP.ID
IN PAIrTLL FULFILLMENT OF THE b.EQUIREM[ETS I O'. THE
DECLEE OF COCTOR OF PHILOSOPHY











UNlVER\]TY OF FLORJDA
April, iP-A4














ACKNOWLELGMENTS


The author wishes to acknowledge his gratitude

to the members of his supervisory committee for their

guidance. Particular thanks are given to the chairman,

Dr. M. J. Larsen, for the constant encouragement and

personal Interest given during the author's entire

graduate program.

Thanks are also given to Dr. T. S. George and

Dr. C. E. Smith for their help In the mathematical

aspects of the dissertation. Special acknowledgment

is extended to Dr. H. H. Clarkson for his advice during

the formulation of the problem.














TABLE OF CONTENTS


Page

ACl KOWLE GCOKE TS ............... ii

LIST OF ILLUSTRATIONS. .. . . . . .. Iv

ABSTRACT . . . . . . . . . .

SECTION

I. INTRODUCTION . . . . . ... .1

General Problem. . . . ... 1
Previous Studies . . . . .

II. FIELD EC'ATIONS. . . . . .. 7

Elfilar Conductor Conficuration. . 7
General Field Relationships. . . 7
Equations Describing the Vector
ra:snetic FotEntial . . . ... 10
Source potential . . . . .. 11
Fourier Expanslons of potential
Distributions. . . . . . 12

III. DERIVATION OF r'ULTIPOLE IMAGES . . 16

Asymptotic Approximation of Integral 16
Potential Due to Sin7le Line Source. 20
Line Multlpole Interpretation ... 24
Reflected Impedance Produced by
Multlpoles . . . . . . 28

IV. SUMMARY AND CONCLUSIONS. .. . . 32

BIBLIOGRAPHY . . . . . . . ... 33

BIOGRAPHICAL SKETCH. ... . . . . .. 34














LIST 07 ILLUSTPATION[S


FLgure Page

1 Filamentsry Conductor Pqarllel to Semi-
Infinite Conducting Medium. . . . 2

2 Image Conductor Produced by Ferfect
Reflection. .............. .

3 Cross Section of Sifilar Fllsr.entary
Line Currents . . . .... . ..

4 Complex Coordinate S.'sterrs at Ima:ge
Positions . . . . . . ... 19

5 Single Filgmerntr;: Line Current . . . 22

6 Line Dipole Current at Image Position . 25

7 Line Quadrupole at Image Position .... 27

? Line Multlpole with Six Currents. ... . 29


















IMAGE INTERPRETATION FOR FIELDS PRODUCED BY
HIGH FREQUENCY LINE CURRENTS OVER FINITE
CONDUCTING MEDIA


By
Bruce Eugene Mathews


April 18, 196k

Chs.inrTan: M. J. Larsen
Major Department: Electrical Engineering


The magnetic field distribution produced by a

filamentsry conductor In a nonconducting medium carryLing

sinusoidally time varying current parallel to a semi-

infinite, finite conducting medium Is considered. It is

assumed that the wave length of the fields in the medium

surrounding the filamentary conductor Is large compared

to the dimensions of the configuration so that retardation

can be neglected. The study Is devoted particularly to

the high frequency case where the field penetration Into

the conducting medium is relatively small; that is,

good reflection is obtained. If the medium has infinite

conductivity, reflection is perfect and the resulting








field distribution can be calculated using a mirror Image

approach. The purpose of this study is to derive a

modified image approach which can be used when the reflec-

tion is not perfect; that is, the medium has finite

conductivity. The analysis is generalized with respect

to the magnetic permeability of the medium.

An equation is derived for the vector magnetic

potential produced in the region above the conducting

medium. The resulting expression is interpreted as that

potential produced by the source and its mirror image

plus a potential described by an integral. An asymptotic

series approximation to the integral reveals terms which

can be interpreted as line multipoles of increasing order

located at the mirror Imare position. These multipoles

constitute a modification of the mirror Imsee and can be

used to describe nonperfect reflection.













SECTION I

I UTRODUJCTIOI


General Problem

A magnetic field problem which has many practical

applications involves an infinitely long filamentary

conductor in a nonconducting medium carrying sinusoidally

tine varying current parallel to the surface of a flat

semn-infinite conducting medium. The configuration is

illustrated In Figure 1 where the current is assumed to

be Iejut. This is the commonly employed exponential

notation used to represent sinusoldal time variation at a

frequency of a radians per second, j being -1.

For tiis :onfi.-uration to closely approximate a

practical problem, 't 13 not necessAry that tith medlum

extend to Infinity, hut only to distances large compared

ts the height of the conductor saove the surface. The

solution applies specifically to a conductor whose diameter

is very snall compared to the height. Or, if a larger

conductor is involved, the result can be considered a

superposition of many filamentary conductors. In summary,

this specific problem is basct to a large class of magnetic

field problems.

The author became interested In this problem while

studyLng analytic methods of Induction heating work-coil

1










































serPi-in'lnIte conductor


Figure 1. FL1arlentsry Conductor parallel to Semn-Infil.te
'c.niuc tIrIg Iledium.










configuration (called a work coill. carrying radio-frequency

currents near the item to be heated. The time varying

magnetic fields induce eddy currents in the Item and

produce heat. For the heatlnr to be effective, the item

must be a fairly good electrical conductor. Practical

9ppllcitions involve heating both ferromagnetic and non-

f rromaenetic materials.

Desilning a work coil for a specific test involves

determining the conductor conf'!uration which will pive

the desired heating pattern. The heating pattern Is

determined by the magnetic field distribution. It should

be mentioned that the interest is in the induction field

produced by the current since the dimensions of the coils

are generally small compared with the free space wave

length of the frequencies Involved. Thus, retardation and

the radiation field can be neglected in any analysis.

In Figure 1, the filanentary conductor represents

part of the work coll and the semi-infinite conductor

represents the item to be heated. The normal method of

analyzing this type configuration is to first assume that

the item is a perfect conductor; that is, its electrical

conductivity is infinite. This is usually considered a

good approximation since due to skin effect the magnetic

field penetrates only a very small distance into the item.




i' Superscripts refer to numbered bibliography.






14

This, of course, depends on the magnitude of the frequency

and the electrical conductivity. However, for most

applications the penetration is quite small, since this

Is generally what Is desired when heatLnz by induction.

The "perfect conductor" spprox.imati:n merely issues the

penetration is z-:ro which Implies that the surface is a

perfect reflector.

Having nade this assLtption, the field distribution

above the suLrface cinr be easily calculate- tv replsc ng

the effect cf the reflc tor b' as filarrent-r:r conductor

located st thie mirror Imr:e position cf t-, wc.r'k-cc.ll

con.ri.ctor. This lr.e e conductor Is assured to catry a

current equal In 'i n ltude to the work-ccll cor-iuctor

but 180 degrees out of ph-se. This configuration, shown

in F'cure 2, pr',duces the same boundary conditions as

the perfect reflector. The procedure then 1r tI base

cslculetions of inductance and surf-ce heating on the

field distribution produced by the filamentary conductor

and its Image.

An interesting question arises -s to the criterion

which makes this apprc Ionmr.on a v lid one. One would

assiur e that such a criterion would involve the relative

mrniltude of the Fenetratton of the fields into the

surface. Also, it would seem reasonable that some Iodlft-

catlon of the perfect imane ap.proach minht be i3ed to

bettsi the aE-ornyition fc.r those cases whi'ich io not meet






























Imagejut


image
/^


Figure 2. Image Conductor Produced by Perfect Reflection.


lejut


~I



j





6

the criterion. A study of a modified image approach is

the subject of this Investiration.


Previous Studies

A literature search revealed several studies in

this general areas. The reflection of nonperfect conductors

has been studied by several investigators interested in

the wave propagation from dipoles lIcated above the

finite conducting earth. Stratton2 summarizes the early

work of Sornerfeld, Van Der Fol, Weyl, and others on this

problem. The analysis showed that part of the field

above the earth could be interpreted as that produced by

a perfect image. However, there remained an additional

term Involving an integral which was difficult to

interpret. Attempts were nade to approximate the integral

but no clear Interpretation in terms of images resulted.

Carson3 analyzed a configuration exactly as that

of Figure 1. However, he limited his study to nonferro-

macnetic conductors since his interest was in the effect

of the earth on overhead transmission lines. He was able

to obtain approximations to the field distributions in

the form of an asymptotic series hut made no attempt to

interpret these results in terms of Images.













SECTION II

FIELD EQUATIONS


?ifillr Conductor Confiruratlon

It Is conv-nient initially to set up the problem

in ter.s of two lon: zirnllel ftltr-intnar lne3 W:ith

crre-.ts of equal n~in'ttude but 1IC devroes out of phase.

This could: rerescent conductor n3n its return c th

jcined c l-nz distance a3ay to fcri, clcsep circuit. A

cross-sectLon rsire:.er.tetLcn is shown in FPhure 3. The

lower half p1-no i~ occupl i: by i aterial with electrical

conductivity n2 and rr.an.etic perrsirlbility L2. The a:per

half plane is nonconductinz ilth penrmeabilLty gi.


General Field Relationships

The analysis uses the concept of the vector magnetic

potential, A, where the general relationships between this

vector and the electric qnd magnetic fields are


.7 x T. = Q7., (1)


A = 'e (2)
at



6t





























LLI
-1 0



-Xe-


aleJUt

b


L-2 p 02


Figure 3. Cross Section of Bifilar Filamentary Line
Currents.













Lu II [ri'tU 1 Lurl

L'Irst note thrit due to the corLi'luraticn we have

assumed, only a z-component of magnetic potentlsl, A.,

will result since all of the current is in the s-direction.

Also, recalling that we are deallng with frequencies

whose free space wave length is very long compared with

the dimensions of the configuration, we can assume that

the current in the lines is uniform along the z-direction

and in turn, Az is not a function of z. This neans that

the divergence of A is zero and from equation (2), --
(t
Is zero. What we are really sairq is that the conductors

are short enough so that no appreciable standing :save is

produced. Now, if we neglect any static potential, V

is zero. Therefore, the equations (1), (2), and (3)

reduce to


x K =A (4)


7 K 0, (5)


at






10

Equstiona Describing the Vector Magnetic Potential

Takin- the curl of both sides of (4) and using (5),

we obtain


rCA = .UT x H. (7)


From Maxwell's equations for a general homogeneous medium,


r.H =E + e (
at

where the first term on wie right represents the conduction

current density and the second term the displacement

cu:rernt density.

lou; consider the upper half pi nr.o of cur

confiuraticn. Since Lt Is nonco-inuctinr,, the conduction

current Is zero except Ft t.e line z:rrents. Also, if we

neglect r:dlstion we arc In effect neglectin- displacement

current. Therefore, w:e ar. airsur.in that the curl of H

Is zero eYcept at the sInrularitlos produced &t the line

currents and eqi.i' ton (7) for the nonconductinr reclon

br.cor.es


-2K1 = 0, ( > 0), (9)


whert the subscript "1" Is use- tc denote the fleld In

the upper half pl;ne.

The medium in the lower half plane Is asSume to

be 2 fairly c ocd conductor. This rieans that the conduction

current Is much Creator than the Jizplaccnent current at





11

the frequency being considered. Therefore, the second

term In (8) can be neglected and (7) for this region

becomes


-222 -t22 Eg, (x < 0), (10)

where the subscript "2" refers to the lower half plane.

Substituting (6) Into (10) and assuming ejut time variation

: ves

V2W2 = jk22, (2 0), (11)


where


k2= 2 j202. (12)



Source Potential

Consider now the vector magnetic potential produced

in the upper half plane by the two filsmentary conductors,

neglecting for a mcnaent the effect of the lower half plane.

Equation (?) shows that the field distribution is quasi-

steady. That is, the spatial distribution is the same as

that produced by a steady current. The resulting field is

then merely the product of the time variation of the current

and the steady spatial distribution. For line currents,

the vector magnetic potential has the familiar logarithmic

distribution. Therefore, the two lines alone produce a






12

z-component source potential, Asz, liven by

S IleJUt In (tb-x2 s)2 2 (131
4(-x)2 + (;-)_j


Fourier Eyosnsions of Potent'!l Distributions

The currents which are induced In the lower half

plans will also contribute to the potential In the upper

half olcRne. The reAultlnr potential must also be a

solution of (-). Again noting tint there is only a z-

component oi potential and tast this component is only a

functlcn of x and y, the Laplacisn of A In rectangular

coordinates is

F2 2lz 2A,-
A_ ( = A ) z z (1)



here 9, Is the unit vector lr' te :-direction. :ow (9)

becomes


S- 0, (x > 0). (15)
6X2 2

A suitable solution of (15) can be written In terms of a

Fourler e. pan3 io. For t'h s r. ~ar 11t in our ca3s, a oourler

cIrne epsnelvin is sufficient necu3use the diatributIon is

an odd function of y. Therefore, the potential 3dstribution

In th. upper half plane can be written as


lz = Jt l( gn)e-' sinrynl dn + A z, (. > 0), (16)
-0






13
where the first term represents the contribution from the

current in the lower half plane and the second term is

due to the source currents and Is given by Equation (13).

The potential distribution in the lower half plane

must be a solution of (11), 'were the Laplacian of A is

defined by (14). Again usin.i a Fourier sine expansion,

we obtain as a generalized solution of (11)


A2z 2 et g (n) e"1 + x sin(yn) dn, (x 4 0), (17)

where the root with the positive real part s1 used.

The two functions gl(n) and g2(n) are determined by

the boundary conditions at x = 0. These cond'itons
require that the tsnventlal electric and magnetic field
intensity be continuous across the boundary. From (4) and

(6), this requires that

__ bAlz A2zz
1 A1 = A22, (x = 01, (18)
LN l.\ U2 'r'


Alz = A2z, (x = G). (19)

Before the boundary conditions are applied, It is

necessary to obtain an integral representation of equation

(13). From Bateman,4

e-pt sin(ct) sin(lt) dt 1 In P2+(a)2]
0 t p2+ (.) 2


Re p > I m (at) I ,


(20)





14
where Re and Im Indicate "the real part of" and the
"imaginary part of" r.spectlvoly. Therefore, (13) can
be written

A I t -0 (b- )n r.(yn) sn(an)
x 0 n
0


( C < x b).


Substituting (21) Into (16) and using (17) and (l'),
obtain

C1 [-nglrfn) + I e-br 3n(an)1 = 2 2+jk2 (n).

Applying the boundary condition I:I) -.ves

gl(ril + e"-n rin(anl = .(n).
nn

Combining (22) and (23) and letting UliM2 2 |

I I -bn n 2n
ql1(n) e -b e 31n(sn) n -
rtvL n-t/2n+ I


we


(21)






(22)




(23)




(24)


and (16) becomess

A LlIJUt2n -1i e-(bxy)n sr,(yn)
n 1 .n+ .n2++jk2


(25)


* sjnfan) 2 + A .
n 3"


we






15

Now note that by (20), the second tern In the

integral of (25) is

"-~IeJUt e-(b+x)n sinrfn) sin(an) dn
4 n

ej n (b+x)2 ( )+a 2 .(26)
S(b+x)2+(,-a)2


Comparing with (13), equation (26) describes the field

produced by two filamentary currents located at x = -b

and y = + a; that Is, the mirror Ir.ane positions of the

two source currents. Also, the negative siLn Indicates

that the currents in the -ia-es are 180 degrees out of

phase with the currents In the sources. These are

exactly the mirror images produced with perfect reflection.

Denoting this Imnae potential described by (26) as Aiz,

we can rewrite (25) as

2-lIeJUtJ n -(btx)n
A = e sin(yii)
S -'0 nu ', 2


Ssln(an) dn + + A, (x > 0). (27)
n
n 3Z Lz,

The first term in (27) describes that potential wnlch is

not accounted for by the mirror images. The purpose of

the next discussion Is to show that this remaining

potential can be interpreted as being produced by line

multipoles located at the image positions.













SECTION III

DERIVATION OF MWLTIPOLE IMAGES


Asyr.ptctic A porolmiation of Integral

For purposes of discussion, denote the first term

in (27) as Arz.

S.Ilejut -tbx)n
Ar 21 '' e s0 in(yn)
0 0 n+u T,,-jk;


3* ln(an) dn (28)
n


The main task now is to find an interpretation of this

integral which will .rie insight to its stgnLficance and

also make more practical the calculation of the field

distributions which it describes.

After much Investi~gtion Involving various methods,

the most useful approach appears to be one involving an

asymptotic approximation to the Integral. To form this

approximation, part of the Integral is expanded in a

Taylor's series aloat n = 0: nearly,








f(n) =
n+u .Jn +jk2


(O 2 '- 3
= f(o + f'i)n + 2 ()n .
21 31

= -"/ n .-e-je''2 rFn 2
u'k Ll]J

Se l l 3- * (29)



Note that f(n) has branch points at +kej' and poles at

+ku'e '/l1 '2 1.'2, w-ich neans that (29) has a finite

radius of convergence for i1nlte k. The integral, on the

other hand, extend to infinity. The result of replacing

f(n) by the series in (29) and Interating (28) term by

term is an asymptotic series which will lIve a good

approximation for large values of k. Since this is the

high frequency case, the series fives useful results for

the problem being investirated. A good discussion of

asymptotic approximations Is presented by Jeffreys.

The terms resulting from the expansion of f(nJ and

the subsequent integration can be Identified as repeated

derivatives of the mirror Image potential, Aiz, as

defined by equation (26), by noting that

P(AL (-l)p l .nPe-'b+x)n 3in(yn)
ax!P n Jo


(p = 0, 1, 2, 3***) (30)


Ssln(an) d"n,
n





18
Therefore, equation (12) can be expanded as


A = 2-eT4 (A- I e' -jn2 2(Atz)
rz
Sx ( ) x2

(1- Q ,-3kj3, .-'4+ 1 ( 1)

.(" k)3 Sx3 .


To develop further 'he !.-nl'ifcance of Arz, It is necessary

to derive nertningful ieprcs3son.s for the repeated reriv-

atives of AIz. This operation .z sir:plifled by using

the theory of corrmpler. varlablc. 'ilth each imame -o-'iton,
we identify .~ complex v: 'rlble coc-rdinate system 03 shown

in Figure L;, and define


ul = rleJ1 = *1 + Jyl'

u2' r= e J-2 = x2 Y2 (32)


Equaton (26) expreEsinn the r rror Iname potential can
now be written


Ait = Ie fl n r1 In r2) (33)
2i

Noting that

In r = in r- j@,

In u2 = In r2 + Je2, (3B)

and observing that the two functions in (34) are analytic




















































Filure .. Complex CoordLnate Systems at Image Positions.







(except at

a(P) lnrl)


a(P)(nr2)


a32(P)


20
the origin), we can write


d(P) (nul)
= Re =
Re dul(P)



= Re d (Pnu2)
L du2 (P) -


(- )P) cos pa
r P


(-l)p1(p-l) cos pe2
r2


(p = 1, 2, 3, ***) ,


where e ndiaestes that only the real part of the
expression is used. Since the partial derivatives with
respect to x1 and x2 equal that with respect to x,
equation (31) can be written


A = Ille jut e-J_ / cos C 1 coS C
S11( 1L (COrb `



6-j a'2 cos 2 1 cos 2i9
S2 ,



+ (2 3) cos i cs j .... (36)
'l)3 r 3 "J



Potential Due to Single Line Source
Up to this point, we have been dealinE with two line
current sources. This was convenient since It enabled the
use of the Integral representation of equstlon (21). We


(35)










to the distance each Ir nhove the surfrcn. This in
effect Isolates the two conductors from erch other.
Assuming a >> b and linlting our attention to the fields
in the vicinity of the source at y = a, r2 will become
much larger than rI and equation (36) can be written


iIejt -Jn/t -, /2
Arz = ---- cos r d e- ,2 cos 261
n 'kr ( 1, :r )2


+ (2-a'2) -3 co- 31 *** (37)
( 'krl)3


This, then, is the potential, in nodtion to the source
and Its mirror image, which is produced In the upper half
plane by a single filamentary current. The total potential
in the upper half plane produced oy a single filament
located et x = b, y = I is


llejut rl 2e- J c 2er-j/2
Az = n - + cos 91 2 cos 261
2 r ('krl) (a'krl)2


2(2- ) 8 cos 3 i ... (x 01, (38)
( u'krl )3


where the first terry. represents the source and Its mirror
image and r, r and 61, are defined in Figure 5.





























Ijt r (x,y)

7T r
I /



1/
b
/7


Figure 5. Single Filamentary Line Current.






23

Equation (39) Is the result desired since It

produces a useful asymptotic approximation for high

frequencies. Note that convergence is largely dependent

on the product (l.'krl). Only a few terms are necessary

to zive a good approximation if this product is much

greater than unity. As the product approaches Infinity,

the fields approach a distribution produced by perfect

reflection. A useful physical interpretation of this

ccnverpence requirement can be obtained by noting that


1: = ./C,2i =% /62, (39)


where 62 is the "depth of penetration," a factor commonly

used to cive a measure of the high frequency skin effect.

RapLd convergence, then, requires


L' kri 1 (40)


or,


r, L 2 62 (41)



Equation (bl) is interesting since it shows that perfect

reflection is not justified entirely by a penetration

small compared to the other dimensions. The ratio of

magnetic permeabilities also enter into the criterion.

However, this is entirely reasonable when one considers

how the field distribution is dependent on this ratio even

with steady currents.








Line T'ultlpole Interpretation
To zlve a physical interpretation of the terms

in equation (33), assure the conJuctina medium In the
lower half plane Is removed and a pair of long fila-
.entary conductors spaced a distance 2d apart with
currents 1i.3 degrees out of phase are placed near the
Lmace position as slho.:n in 'i'ure 6. Again, using the

complex variable notation of equation (32), the vector
magnetic potential produced by the pair is

U jut ul tLd
A le u
A =l- Re In --
Z 2n u_ -d


LL eut Re In ( 1 + 2d (42)
Rel ul+d l)
2w L \. Ul+d _

'lowr, if we let d<

Az = I1le2 Re E2d
SC '


= llle jut 2d cos (43)
2n r /


With the restriction that d' Fiure 6 forms a line dipole and equation (43) describes

the resultlne field. Therefore, the second term in
equation (3?) represents such a line dipole with a current





























(xy)


S11lejut

. -IleJ~


Figure 6. Line Dipole Current at Image Position.









(44)


Note thnt the dipole current is not in phase with the

source current.

To interpret the third term in (38), consider a

line quadrupole as shown in figure 7. Using the same

approach as with the d1pole, the resulting potential is


A = e in (u +d)( l-d )
z 2n (ul+jd)(ul-jd)


12ejut 1i
- Re In
2.


(45)


1 2d2 2 i
- 2 d2


Ar-sn letting d<

z
2n,


The third term n (3%) then
with current


2d2 co3 291

,l2


(46)


represents this type quadrupole


- e-J"/2
(I_= )2


(47)


Ey srinillla, approach, tre third term In (3.)

represents 9 line multlp:.le rivolvinr six currents as


I =- leJ ,i
1 d ad k


)\























x
X





(x,y)
/
r1 */




I -2eJut b,

* -I2eJut j
2r- '


Figure 7. Line Quadrupole at Image Position.





28

shown In ?Fiure 8 with current


I, = (2- a21 le-J3./ (-4)
-,i I i



The multlpole interpretAtion is more -meaningful

if we let

d 1/(u'k)

1 2 2
2 2 (49)
"2 1I


Using this spacing gives multipole currents of the some

order of magnitude as the source current. Note that

(4'L) still satlifies the requirement d'
frequency approximrtLon is valid as stated by equation (41).


Reflected Impedance Produced b. Multlpoles

Another view of the sLnificance of the asymptotic

expansion can be obtained by viewing the effect of each

multiple In terrs of its contribution to the electrical

Impedance of the line source. From equations (6) and

(37), the electric field produced ov the nultipoles,

Erz(rl', ), at the source is





























,(x,y)


rl /


x 1 /
J13e t b,
Jut 1-4-- l
-I3e 1 Jy


Figure 8. Line Multipole witf Six Currents.








3Arz(2b,O)
Erz(2b,0) = z
1)t

jut -
=-JauileJ*t ~e-j.,/ e-jJ'/2
S L''k2b ti'k2b)2

S(2- '2) e- (50)
( k2hc I


The ImpeJdnce per unit length, Zr, due to tL-ls electric

field can be defined in terms of the volteae drop

produced, as

E,,(2h,0) .
"r + jLr (51)
leJut r r

With this approach we obtain


Rp Fp. .. "" (12)
r 1 p l:2.b t 2.2 )2 s' .' l.:2t.)-




J Lr 2b/ /2(u k2b)3 +(


It should be emphasized that the electric field

produced by the source itself end its mirror image also

contribute to the total impedance. This is the impedance

which would result if perfect reflection were obtained.

Usinr a coupled circuit analo:y, tr.e impedance given by

equation (51) can be considered an additional "reflected






31
impedance" due to the currents which actually flow in
the conducting medium. The resistance term, R rv is

due to the power loss and the inductance, L rF is due to

the energy stored in the magnetic field inside the

conducting medium.

The first term in equations (52) and (53) is

supplied by the dipole imnage. It is Interesting to

note that thiis is exactly thle approximation obtained if

one first calculates the field distribution at the

surface of the conducting medium based on perfect

reflection, and then uses this surface field distribution

to calculate the losses and stored energy In the medium.

This approach is often used in problems of this type.

The quadrupole supplies the second terrn In (52)
but does not contribute to (53). The multiple involving

six sources produces an impedance defined by the last

term in (52) and (53).













SECTION IV

SUMFMARY AND CONCLUSIONS


Practical approximations to the fields produced

by hi:h frequency currents and .;ood conducting media

can often be obtained by assuming perfect reflection.

This analysis has attempted to shed some light on the

basis for this assumption by considering the specific

case of filamentary line currents parallel to a flat

semi-infinite, finite conducting medium. If perfect

reflection were obtained, the effect of the medium could

be replaced by mirror images of the line currents. The

analysis shows that nonperfect reflection can be

approximated by multiple images located at the mirror

image positions. These multipoles produce an asymptotic

approximation which converges fairly rapidly when Food

reflection is obtained.

It would be interesting to extend this analysis

to finite conductors which are not parallel to the

surface, other boundary shapes, and cases involving

retardation effects. One might expect to obtain

similar modified image approximations for these cases.













BIBLIOGRAPHY


1. B. E. I'athews, "Flat '.-ork-Coll Design,"

Transactions of the AIEE, Vol. 7, Part II,

pp. 249-256; November, 1957.

2. J. A. Stratton, "Electromagnetic Theory,"

McGraw-HIll Book Co., Inc., New York and

London, pp. 573-587; 1941.

3. J. P. Carson, "Wave Propagation in Overhead

Wires with Ground Return," Bell System Journal,

Vol. 5, pp. 539-55h; October, 1926.

h. Eateman Manuscript Project, "Tables of Integral

Transforms," Vol. 1, McGraw-Hill Book Co., Inc.,

New York, p. 15Q; 195k.

5. H. Jeffreys, "Asymptotic Approximations,"

Oxford University Press, London; 1962.













BICORAFHICAL SKETCH


Sruce Eurene Mathews was bonr June 1, 1929, at

Peru, Illinois. After he was graduated from Duncan C.

Fletcher High School at JacksonvLlle Beach, Florida,

he attended the University of Florida where he received

the degrees Bachelor of Electrical Engineering and

Master of Science in Engineering in 1952 and 1953,

respectively. He then served in the United States Air

Force front 1953 until 1955. After working for one year

with North American Aviation Corporation in Los Angeles,

California, he returned to the University of Florida in

1r56. Since then until the present time, he ha3 pursued

his work toward the degree of Doctor of Pillosophy wnile

teaching and conducting research in the Department of

Electrical Engineering as a Research Associate.

Bruce Eugene Mathews is married to the former

Donna Lee Breazeale and is the father of three children.

He is a member of the Institute of Electrical and

Electronics Engineers, Sigma Tau, Phi Kappa Phi, and

Tau Eeta PI.








This dissertation was prepared under the direction

of the chairman of the candidate's supervisory committee

and has been approved by all members of that committee.

It was submitted to the Dean of the College of Engineering

and to the 'Graduate Council, and was approved as partial

fulfillment of the requirements for the degree of Doctor

of Philosophy.


April li, 1964








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