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## Material Information- Title:
- Image interpretation for fields produced by high frequency line currents over finite conducting media
- Creator:
- Mathews, Bruce Eugene, 1929-
- Publication Date:
- 1964
- Copyright Date:
- 1964
- Language:
- English
## Subjects- Subjects / Keywords:
- Approximation ( jstor )
Electric current ( jstor ) Electric fields ( jstor ) Geometric lines ( jstor ) Image analysis ( jstor ) Line current ( jstor ) Magnetic field configurations ( jstor ) Magnetic fields ( jstor ) Mirrors ( jstor ) Vector currents ( jstor ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 022365842 ( AlephBibNum )
13660429 ( OCLC ) ACZ5181 ( NOTIS )
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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' 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 Dean, 3raduate School Supervisory comnittee: |

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PAGE 1 IMAGE INTERPRETATION FOR FIELDS PRODUCED BY HIGH FREQUENCY LINE CURRENTS OVER FINITE CONDUCTING MEDIA BRUCE EUGENE MATHEWS A DISSERTATION PRESENTED TO THE GRADUATE COUNQL OF THE UNI\'ERSITY OF FLORIDA IN PARTUL FULFILLMENT OF THE REQUIREMENTS FOR THE DECREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1964 PAGE 2 ACKNOWLEDGMENTS 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. B, Sm.ith for their help in the mathematical aspects of the dissertation. Special acknowledgment Is extended to Dr. M, H. Clarkson for his advice during the formulation of the problem. 11 PAGE 3 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OP ILLUSTRATIONS iv ABSTRACT v SECTION I. INTRODUCTION 1 General Problem 1 Previous Studies 6 II. FIELD EQUATIONS 7 Bifllar Conductor Configuration. ... 7 General Field Relationships 7 Equations Describing the Vector Kac^netlc Potential 10 Source Potential 11 Fourier Expansions of Potential Distributions 12 III. DERIVATION OF MULTIPOLE IMAGES 16 Asymptotic Approximation of Integral . 16 Potential Due to Single Line Source. . 20 Line Multipole Interpretation 2I4. Reflected Impedance Produced by Multlpoles 28 IV. SUMMARY AND CONCLUSIONS 32 BIBLIOGRAPHY 33 BIOGRAPHICAL SKETCH 2h iii PAGE 4 LIST Ox^ ILLUSTRATIONS Figure Page 1 Filamentary Conductor Parallel to SemiInfinite Conducting Medium 2 2 Image Conductor Produced by Perfect Reflection 5 3 Cross Section of Bifilar Fllam.entary Line Currents 8 i^. Complex Coordinate Systems at Image Positions 19 5 Single Filamentary Line Current 22 6 Line Dipole Current at Image Position ... 2^ 7 Line Quadrupole at Image Position 2? 8 Line Multlpole with Six Currents 29 Iv PAGE 5 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IMAGE INTERPRETATION FOR FIELDS PRODUCED BY HIGH FREQUENCY LINE CURRENTS OVER FINITE CONDUCTING MEDIA By Bruce Eugene Mathews April 18, 1961; Chairman: M. J. Larsen Major Department: Electrical Engineering The magnetic field distribution produced by a filamentary conductor in a nonconducting medium carrying sinusoldally time varying current parallel to a semiinfinite, 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 snail; that is, good reflection is obtained. If the medium has infinite conductivity, reflection Is perfect and the resulting PAGE 6 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 reflection 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 image position. These multipoles constitute a modification of the mirror image and can be used to describe nonperfect reflection. vi PAGE 7 SECTION I INTRODUCTION General Problem A magnetic field problem which has many practical applications involves an Infinitely long filamentary conductor in a nonconducting medium carrying sinusoidally time varying current parallel to the surface of a flat semi-infinite conducting medium. The configuration is illustrated in Figure 1 where the current is assumed to be leJ*^ . This is the commonly employed exponential notation used to represent sinusoidal time variation at a frequency of u radians per second, J being \^T. ?or this conf i?:uration to closely approximate a practical problem, it is not necessary that the medium extend to infinity, but only to distances large compared to the height of the conductor above the surface. The solution applies specifically to a conductor whose diameter is very small 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 basic to a large class of magnetic field problems. The author became interested in this problem while studying analytic methods of induction heating work-coil 1 PAGE 8 Figure 1. Filanentary Conductor Parallel to Seml-Inf Inlte Conducting Medium. PAGE 9 3 design. 1'^ Induction heating involves placing a conductor configuration (called a work coll) 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 heating to be effective, the item must be a fairly good electrical conductor. Practical applications involve heating both ferromagnetic and nonferromacrnetic materials. Designing a work coil for a specific test involves determining the conductor configuration which will give 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 filamentary conductor represents part of the work coil 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. <^ Superscripts refer to numbered bibliography. PAGE 10 k 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 heating by induction. The "perfect conductor" approximation merely assum.es the penetration is zero which im.plies that the surface is a perfect reflector. Having made this assumption, the field distribution above the surface can be easily calculated by replacing the effect of the reflector by a filamentary conductor located at the mirror imac^e position of the work-coil conductor. This image conductor is assumed to carry a current equal in maoinitude to the work-coil conductor but 180 de5:rees out of phase. This configuration, shown in Figure 2, produces the same boundary conditions as the perfect reflector. The procedure then is to base cslculatlons of inductance 9nd surface heating on the field distribution produced by the filamentary conductor and its image. An interesting question arises as to the criterion which makes this approximation a valid one. One would assumie that such a criterion would involve the relative magnitude of the penetration of the fields into the surface. Also, it would seem reasonable that some modification of the perfect image approach might be used to better the approximation for those cases which do not meet PAGE 11 Figure 2. Image Conductor Produced by Perfect Reflection, PAGE 12 6 the criterion, A study of a modified image approach is the subject of this investigation. Previous Studies A literature search revealed several studies in this general area. The reflection of nonperfect conductors has been studied by several inves tic^ators interested in the wave propagation from dipoles located above the finite conducting earth. Stratton summarizes the early work of Sommerfeld, Van Der Pol, 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 made to approximate the integral but no clear interpretation in terms of images resulted, Carson^ analyzed a configuration exactly as that of Figure 1. However, he limited his study to nonferromagnetic 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 but made no attempt to interpret these results in terms of Images. PAGE 13 SECTION II FIELD EQUATIONS Blfil'ir Conductor Conf ipjuration It 1.3 convenient Initially to set up the problem in terp-^s of txvfo lon;^ panllel f llnrrentary linos with currents of equal nan;nitude but 180 de.':crees out of phase. This could represent a conductor Tnd its return path Joined a Ion? distance avay to fcrm i closed circuit, A cross-section representation is shown in Fi.^ure 3. The lower half plane is occupied by a Triaterial with electrical conductivity op ^"^ reafrnetic permeability p.^. The upper half plane Is nonconductin.^ with permeability \i-^. General Field Relationships The analysis uses the concept of the vector magnetic potential, T, where the general relationships between this vector and the electric and magnetic fields are V^ A = tJLH, (1) V Â• A = -txe M, (2) E = -w M, (3) ^t PAGE 14 -le jwt r b 1_ M-1 -> PAGE 15 9 where H Is the map:netic field intensity vector, E is the electric field intensity vector, V is the scalar electric potential, and e is the dielectric constant of the medium. First note that due to the configuration we have assumed, only a z-component of magnetic potential, A , will result since all of the current is in the z-direction. Also, recalling that we are dealing 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, A^ is not a function of z. This means that the divergence of A is zero and from equation (2), iX is zero. What we are really saying is that the conductors are short enough so that no appreciable standing wave is produced. Now, if we neglect any static potential, V Is zero. Therefore, the equations (1), (2), and (3) reduce to Vx A = nH, (^) V . A = 0, (5) PAGE 16 10 Equations Describing the Vector Magnetic Potential Taking the curl of both sides of (I4.) and using (5), we obtain V^A = -M-V X H. (7) Prom Maxwell's equations for a general homogeneous medium, V X H = ol + e M, (3) ^t where the first term on the right represents the conduction current density and the second term the displacement current density. Now consider the upper half plane of cur configuration. Since it is nonconducting, the conduction current is zero except at the line currents. Also, if we neglect radiation we are in effect neglecting displacement current. Therefore, we are assuming that the curl of H is zero except at the singularities produced at the line currents and equation (?) for the nonconducting region becomes V^J^ = 0, (x > 0). (9) where the subscript "1" Is used to denote the field in the upper half plane. The medium in the lower half plane Is assumed to be a fairly good conductor. This means that the conduction current is much greater than the displacement current at PAGE 17 11 the frequency being considered. Therefore, the second term in (8) can be neglected and (7) for this region becomes where the subscript "2" refers to the lower half plane. Substituting (6) into (10) and assuming e-J time variation gives V^Ap = jk^Ap, (x < 0), (11) where ,2 ^ k*^ = ut\i202' (12) Source Potential Consider now the vector mac^netic potential produced in the upper half plane by the two filamentary conductors, neglecting for a moment the effect of the lower half plane. Equation (9) shows that the field distribution is quasisteady. 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 PAGE 18 12 z-component source potential, ^q^* S^ven by ti^Ie jut 'sz In i+it "(b-x)^ + (y-Ha)^ L(b-x)2 + (y-a)2j (13) Fourier Expansions of Potential Distributions The currents which are induced in the lower half plane will also contribute to the potential In the upper half plane. The resulting potential must also be a solution of (9). A(2;ain notins; that there is only a zcomponent of potential and that this component is only a function of x and y, the Laplaclan of A In rectangular coordinates is :h = (^A^) I^ = dx^ ^y T Uk) where a is the unit vector la the z-dlrection. Now (9) becomes a^ATÂ„ 5^A. br.' ^ + l_2ll= 0. (x > 0). (15) ^y A suitable solution of (15) can be written In terms of a Fourier expansion. For the symmetry in our case, a Fourier sine expansion is sufficient because the distribution is an odd function of y. Therefore, the potential distribution In the upper half plane can be written as g3_(n)e-''" sin(yn) dn + Ag^, (x > 0), (16) PAGE 19 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), where the Laplacian of A is defined by (II4.). Again using a Fourier sine expansion, we obtain as a generalized solution of (11) '2z = e^"^ j g2(n) e^/^^^ ^ 3in(yn) dn, (x < 0), (1?) where the root with the positive real part is used. The two functions gT(n) and g2(n) are determined by the boundary conditions at x = 0, These conditions require that the tangential electric and magnetic field intensity be continuous across the boundary. Prom (Ij.) and (6), this requires that 1_ ^^iz ^ 1_ ^Ag^ \x.\ ^x V-2 ^x (x = 0). (18) ^Iz " ^2z' (x = 0). (19) Before the boundary conditions are applied, it is necessary to obtain an integral representation of equation (13). From Bateman,'+ 00 e"P^ 3in(at) sin(pt) It = 1 In p^-t-(a^-p)^ L p2+(a-3)2j Re p Im (Â±aÂ±3) (20) PAGE 20 114. where Re and Im indicate "the real part of" and the "Imaginary part of" respectively. Therefore, (13) can be written Asz = ^.^leJ-^ r-(b.x)n 3in(yn) sin(an) ^, n (0 < X < b). (21) Substituting (21) into (16) and using (1?) and (18), we obtain ^1 M-iI -bn -ngT(n) + Â— iÂ— e sin(an) = pl.yn2+jk2 g2(n). (22) Applying the boundary condition (19) ~,lves g.(n) + -1e'^"^ sin(an) = gp(n). nit ^ Combining (22) and (23) and letting V-J^^ ' ^^ * 2n , . ^^l^ -bn , , X g. (n) = Â—=Â— e 3in(an} nit _ n4|i'vrr + Jk^ -1 (23) (2U) and (16) becomes ^Iz 2n -1 _n-Hi<'vh^+Jk^ .-(b+x)n s3_n(yn) sin(an) lÂ£ + A^,. n ^^ {2S) PAGE 21 15 Now note that by (20), the second term in the Integral of (25) is -U ]^f^r^-(b-^x)n 3in(yn) sin(an) ^ = :!iiÂ£!i!!in 2 2 (bH-x) ^^(y+a) L (b+x)^+(y-a)^ J (26) Comparing v/ith (13) Â» equation (26) describes the field produced by two filamentary currents located at x = -b and y = + a; that is, the mirror inage positions of the two source currents. Also, the negative sign indicates that the currents in the imatres are l80 degrees out of phase with the currents in the sources. These are exactly the mirror imaees produced with perfect reflection. Denoting this imaae potential described by (26) as Aj^^, we can rewrite (25) as 2niIeJ^Y " n -(b+x)n , . . ^Iz " Â— ^ 1 ii-------. e sin(yiO n Â° n+u'vh^+jlc^ Â• sin(an) ^ + A^^ + A^^, (x > 0), (2?) The first term in (2?) describes that potential which 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 multlpoles located at the image positions. PAGE 22 SECTION III DERIVATION OF MULTIPOLE IMAGES Asymptotic Approximation of Inte?,ral For purposes of discussion, denote the first term In (27) as A^^,. 2\i..lei^^ r^" -(b+x)n A^2 = -i [ ^ ' ^^ e ^"^^^^ sin(yn) n -^0 n+p.'v^n^7jk^ sin(an) ^ (28) The main task now is to find an interpretation of this Integral which will give insight to its significance and also make more practical the calculation of the field distributions which it describes. After much investigation 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 about n = 0; namely, 16 PAGE 23 17 f(n) = n+p,'vA2+ik^ = f(0) ^ f'(0)n -. ^"(Q^" . f"'(0)nf ^ 21 31 = Â«-JÂ«A -Ji_ -Â«-JÂ«/2 M, k _M-'k_ . e-J3Â«A Â•2n 1 i^ 2 J n L^x'k, (29) Note that f (n) has branch points at +^Ae^^^^^ and poles at +kiJL'e'^'^/V(l M.'^)"^^^, which means that (29) has a finite radius of convergence for finite k. The intec(;ral, on the other hand, extend to infinity. The result of replacing f(n) by the series in (29) and inte^^rating (28) term by tenn is an asymptotic series which will .^ive a good approximation for large values of k. Since this is the high frequency case, the series drives useful results for the problem being Investigated. A good discussion of asymptotic approximations is presented by Jeffreys,-^ The terms resulting from the expansion of f(n) and the subsequent integration can be Identified as repeated derivatives of the mirror image potential, ^]_zÂ» ^^ defined by equation (26), by noting that Sx(p) n Jo Â• sin(an) ^, n (p = 0, 1, 2, 3---) (30) PAGE 24 18 Therefore, equation (28) can be expanded as rz (n'k)^ &x3 (3i; To develop further the 3i~nlficance of A^.^, it Is necessary to derive neanlngful expressions for the repeated derivatives of Aj^g. This operation is simplified by using the theory of complex variables, V/ith each iraage position, we identify a complex variable coordinate system as shown in Figure I]., and define "l ^ ^1Â©-'Â®^ = x^ + jy;^, JÂ®2 U2 = r2e ^ = Xp + Jy^. (32) Equation (26) expressing the nirror inage potential can now be written Jut p.-! le 'iz = -^ ^1" ^1 ^^ ^2^en Noting that (33) In Uj_ = In r^ + je^^, In U2 = In r2 + ^e^* ^341 and observing that the two functions in (3l(.) are analytic PAGE 25 19 Figure I4.. Complex Coordinate Systems at Image Positions. PAGE 26 20 (except at the origin), we can write a^P^Clnr^^) = Re Sx2 (p) = Re d(P)(lnui)' duo(p) _ = (-DP-^^Cp-Di cos PÂ©1 = (.l)P-l(p.l),i2i2?2^ r^P (p = 1, 2, 3, Â•Â•Â•) , (35) where Re indicates that only the real part of the expression is used. Since the partial derivatives with respect to x. and x^ equal that with respect to x, equation (31) can be written Jut _ t^lle' rz Q-U/k ^ COS 9i _ cos 9; \i k Q-J7i/2/ C03 29i COS 29p ' (n'k)2 + (2'2n e"-^'^"^^ / cos 3Qi cos 39p (n'k)3 ^1^ r23 (36) Potential Due to Sint^le Line Source Up to this point, we have been dealing with two line current sources. This was convenient since it enabled the use of the integral representation of equation (21). Ive PAGE 27 21 can now investigate a single source by letting the distance between the two conductors become large compared to the distance each is above the surface. This in effect isolates the two conductors from each other. Assuming aÂ»b and limiting our attention to the fields in the vicinity of the source at y = a, r2 will become much larger then r^ and equation (36) can be written _ ^l^^ jut rz 'e-U/h ^ Â«-Jt/2 s. cos e, _e PL'kr, 1 -2 cos 2Q-. (^i'krl)2 ^ + (2-|i '^) -2 cos 39, (n'kri)3 (37 This, then, is the potential, in addition 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 by a single filament located at x = b, y = is M-lIe Hut 1Â« 27r mli. r (li'kr^) ^H cos e, Â£6 cos 2e 2 + 2(2-^,'2) e-J'^A ( tx'kr^) cos 30, (li'kr^) , (^ > 0), (38) where the first term represents the source and its mirror image and r, r^, and 0^, are defined in Figure 5. PAGE 28 22 Figure 5. Single Filamentary Line Current. PAGE 29 23 Equation (38) is the result desired since It produces a useful asynptotic approximation for high frequencies. Note that convergence is largely dependent on the product (^,'kr, ). Only a few terms are necessary to give 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 convergence requirement can be obtained by noting that where 62 ^^ ^^^ "depth of penetration," a factor commonly used to give a measure of the high frequency skin effect. Rapid convergence, then, requires UL'kr^ Â» 1 (I4.O) or, r, Â» 1. lii 62 . (UD Equation (i^l) 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. PAGE 30 2h Line I^ultipole Interpretation To give a physical interpretation of the terms in equation (38), assume the conducting medium in the lower half plane is removed and a pair of long filamentary conductors spaced a distance 2d apart with currents 180 degrees out of phase are placed near the image position as shown in Figure 6, Again, using the complex variable notation of equation (32), the vector magnetic potential produced by the pair is A = ^1^1^ 2. jut Re In u^+d u, -d _ ^^IH I.e^'^^ Re 2n In 1 + 2d un+d (1;2) Now, if we let d< PAGE 31 25 (x.y) -<5)Flgure 6, Line Dlpole Current at Image Position. PAGE 32 26 h = ikh) Note that the dipole current is not In phase v;ith 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 dipole, the resulting potential is AÂ„ = ne z 2n in f ('^l^d)(ui-d) (ui+jd)(ui-Jd) tiil2e j(jt Re 2n In 1 2d' u^2^d2 (US) Again letting d<< r, , ^^^1^2Â®*^'^^ / 2ci2 cos 29^ z ~ 2jt (U6) The third term in OS) then represents this type quadrupole with current Id = _ le -Jn/2 (dp, k) 'v^2 (1+7) By a siniliar approach, the third term in (35) represents a line multipole involving six currents as PAGE 33 27 ^Ige j(jt -l2e jut (x,y) Figure 7. Line Quadrupole at Image Position. PAGE 34 28 shown in ?lc!;ure 8 with current I3 = (2-n'2) le-J^^^ . (1,8) (dn'k)3 The multipole interpretation is more meaningful if we let d l/(n'k) = ^^ 52 . (1;9) \/2 M-i Using this spacing gives multipole currents of the same order of magnitude as the source current. Note that (I4.9) still satisfies the requirement dÂ«r]_ if the high frequency approximation is valid as stated by equation {kD Reflected Impedance Produced by Multipoles Another view of the significance of the asymptotic expansion can be obtained by viewing the effect of each multipole in terms of its contribution to the electrical impedance of the line source. From equations (6) and (37), the electric field produced by the multipoles, Ej,2{r,,Â©.), at the source is PAGE 35 29 Figure 8. Line Multlpole with Six Currents. PAGE 36 30 Ej.2(2b,0) = hk^^{2h,0) -jungle ,-J.A ^-JTr/2 + (2.u'2) ^l^h/^_n'k2b (|j,'k2b)2 3.A (n'k2b)^ (50) The impedance per unit length, Z^., due to this electric field can be defined in terms of the voltage drop produced, as Z.J. = r^ > ' . = R + JUL . (51) Ie>* ^ With this approach, we obtain .-lidL^l _n'k2bv^ (n'k2b)2 v/2(n'k2b)3 (52) J'jLr. = Uli]^ (2-M.'^) n'k2bv/^ ^(iJi'k2b)3 (53) It should be emphasized that the electric field produced by the source itself and its mirror image also contribute to the total impedance. This is the impedance which would result if perfect reflection were obtained. Using a coupled circuit analogy, the impedance given by equation (5l) can be considered an additional "reflected PAGE 37 31 impedance" due to the currents which actually flow in the conducting medium. The resistance term, R , is due to the power loss and the inductance, L^,, is due to the energy stored in the magnetic field inside the conducting medium. The first tern in equations (52) and (53) is supplied by the dipole image. It is interesting to note that this is exactly the 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 term in (52) but does not contribute to (53). The multipole involving six sources produces an impedance defined by the last term in (52) and (53) . PAGE 38 SECTION IV SUMMARY AND CONCLUSIONS Practical approximations to the fields produced by high frequency currents and good 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 m.edium. If perfect reflection were obtained, the effect of the nedium could be replaced by mirror images of the line currents. The analysis shows that nonperfect reflection can be approximated by multlpole images located at the mirror image positions. These multlpoles produce an asymptotic approximation which converges fairly rapidly when good 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 similiar modified image approximations for these cases. 32 PAGE 39 BIBLIOGRAPHY 1. B. E. Mathews, "Flat VJork-Coil Design," Transactions of the AIES . Vol. 7, Part II, pp. 2I4.9-256; November, 1957. 2. J. A. Stratton, "Electromagnetic Theory," McGraw-Hill Book Co., Inc., New York and London, pp. 573-587; 19i+l. 3Â« J. R. Carson, "Wave Propagation in Overhead Wires with Ground Return," Bell System Journal , Vol. 5, pp. 539-55!+; October, 1926. I4.. Bateman Manuscript Project, "Tables of Integral Transforms," Vol. 1, McGraw-Hill Book Co., Inc., New York, p. 159; 195U. 5. H. Jeffreys, "Asymptotic Approximations," Oxford University Press, London; 1962. 33 PAGE 40 BIOGRAPHICAL SKETCH Bruce Eugene Mathews was born June 1, 1929, at Peru, Illinois. After he was graduated from Duncan U, Fletcher High School at Jacksonville 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 fron 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 1956. Since then until the present time, he has pursued his work toward the degree of Doctor of Philosophy while 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 Beta PI. 3k PAGE 41 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 15, 196U Supervisory Connittee ; Chairman vA\ "^P Dean, College of EnglneeMng Deen, Greduate School PAGE 42 15 09 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EY79Z8EPM_0MJWVE INGEST_TIME 2017-07-19T20:37:48Z PACKAGE UF00097935_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |