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\'EPSI OF FLCOP.ID
IN PAIrTLL FULFILLMENT OF THE b.EQUIREM[ETS I O'. THE
DECLEE OF COCTOR OF PHILOSOPHY
UNlVER\]TY OF FLORJDA
April, iPA4
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
semninfinite 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 workcoil
1
serPiin'lnIte conductor
Figure 1. FL1arlentsry Conductor parallel to SemnInfil.te
'c.niuc tIrIg Iledium.
configuration (called a work coill. carrying radiofrequency
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 semiinfinite 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 filarrentr:r conductor
located st thie mirror Imr:e position cf t, wc.r'kcc.ll
con.ri.ctor. This lr.e e conductor Is assured to catry a
current equal In 'i n ltude to the workccll coriuctor
but 180 degrees out of phse. 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 surfce 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 aEornyition 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 convnient initially to set up the problem
in ter.s of two lon: zirnllel ftltrintnar 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 lnz distance a3ay to fcri, clcsep circuit. A
crosssectLon rsire:.er.tetLcn is shown in FPhure 3. The
lower half p1no 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
L2 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 zcomponent of magnetic potentlsl, A.,
will result since all of the current is in the sdirection.
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 zdirection
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 noncoinuctinr,, 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
zcomponent source potential, Asz, liven by
S IleJUt In (tbx2 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
ept 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 ebr 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 nt/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 iaes 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
2lIeJUtJ 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 .eje''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, wich 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 = 2eT4 (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 cocrdinate system 03 shown
in Figure L;, and define
ul = rleJ1 = *1 + Jyl'
u2' r= e J2 = 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(pl) 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 eJ_ / cos C 1 coS C
S11( 1L (COrb `
6j 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
+ (2a'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 2erj/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)( ld )
z 2n (ul+jd)(uljd)
12ejut 1i
 Re In
2.
(45)
1 2d2 2 i
 2 d2
Arsn letting d<
z
2n,
The third term n (3%) then
with current
2d2 co3 291
,l2
(46)
represents this type quadrupole
 eJ"/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 leJ3./ (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 14 l
I3e 1 Jy
Figure 8. Line Multipole witf Six Currents.
3Arz(2b,O)
Erz(2b,0) = z
1)t
jut 
=JauileJ*t ~ej.,/ ejJ'/2
S L''k2b ti'k2b)2
S(2 '2) e (50)
( k2hc I
The ImpeJdnce per unit length, Zr, due to tLls 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
semiinfinite, 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 '.orkColl Design,"
Transactions of the AIEE, Vol. 7, Part II,
pp. 249256; November, 1957.
2. J. A. Stratton, "Electromagnetic Theory,"
McGrawHIll Book Co., Inc., New York and
London, pp. 573587; 1941.
3. J. P. Carson, "Wave Propagation in Overhead
Wires with Ground Return," Bell System Journal,
Vol. 5, pp. 53955h; October, 1926.
h. Eateman Manuscript Project, "Tables of Integral
Transforms," Vol. 1, McGrawHill 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
Dean, 3raduate School
Supervisory comnittee:
