LIBRARY MichiganSac '.: UM “Milli“will»Willem THIS" This is to certify that the thesis entitled INDUCED ELECTROMAGNETIC FIELDS IN CONDUCTING BODIES IRRADIATED BY MAGNETIC FIELDS AND MICROWAVES presented by ‘ Jen—Hwang Lee ‘ has been accepted towards fulfillment of the requirements for Ph. D. degree in Elect. Engr. & Sys. Sci. K 4% 6A f ‘ Major professor Date 7//7 /7? 0-7639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. in part Depa: INDUCED ELECTROMAGNETIC FIELDS IN CONDUCTING BODIES IRRADIATED BY MAGNETIC FIELDS AND MICROWAVES By Jen—Hwang Lee A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1979 IXDU :3 so: A near them current induced l magnetic fiel research. This biological resea irradiation of R efficient method induced electric ated by HF-VHF E1 is subdivided in‘ ridii and cross~s by the impressed ally, as induced associated with t Induced eddy curr electric field Wh field to yield th a ABSTRACT INDUCED ELECTROMAGNETIC FIELDS IN CONDUCTING BODIES IRRADIATED BY MAGNETIC FIELDS AND MICROWAVES BY Jen-Hwang Lee A new theoretical method for determining the eddy current induced by a uniform RF magnetic field or a beam of RF magnetic field in a biological body is developed in this research. This study was motivated by the fact that more biological research and medical applications utilize the irradiation of RF magnetic fields, and by the need for an efficient method of determining the magnetic mode of the induced electric field in a thick biological body irradi— ated by HF—VHF EM waves. The body of rotational symmetry is subdivided into a number of circular rings with various radii and cross-sectional areas. The eddy current induced by the impressed magnetic field can be considered, physic— ally, as induced by a circulatory impressed electric field associated with the uniform impressed magnetic field. Induced eddy currents in all the rings maintain a scattered electric field which can be added to the impressed electric field to yield the total induced electric field in the body. At the same tir complex conduct these relations simultaneous eq be obtained, an With the prese: magnetic fields bodies have bee: tionally used, c fails to yield a ZOiih'z, while tn accurately and e gical bodie Also include quantifi‘ing the : irradiated biolog integral equation the qUantificatio iiated bOdy has b dilation methOds, the human body is ‘5'. Jen-Hwang Lee At the same time, the eddy current is the product of the complex conductivity and the induced electric field. Using these relations and the point—matching method, a set of simultaneous equations for the induced electric fields can be obtained, and they are subsequently solved numerically. With the present method, the eddy currents induced by magnetic fields of l to 200 MHz in various biological bodies have been obtained. It was found that the conven— tionally used, quasi—static solution for the eddy current fails to yield accurate results for frequencies higher than 20 MHz, while the present method can be used to determine accurately and efficiently the induced eddy currents in biological bodies for frequencies higher than 20 MHz. Also included in this thesis is a numerical method for quantifying the induced EM field on the surface of an irradiated biological body based on two coupled, surface integral equations. In the field of theoretical dosimetry, the quantification of the induced EM field inside an irra— diated body has been performed mainly by volume integral equation methods. When a large biological body such as the human body is irradiated by an EM wave of microwave range, the body becomes electrically large and the induced EM field concentrates mainly in a thin layer of the body surface. A volume integral equation method then becomes inadequate or inefficient to handle this problem, and it is more efficient to quantify the induced EM field on the body surface be for this case. induced electric are determined, To check the acc bodies are consi surface integral obtained from * cc. integral eouatio cases, the surfa tages in accurac integral equatio: ‘_.._ —\ Jen Hwang—Lee body surface based on a surface integral equation method for this case. After the tangential components of the induced electric and magnetic fields on the body surface are determined, the internal EM field can be calculated. To check the accuracy of this method, electrically small bodies are considered first, and results obtained from the surface integral equation method are compared with that obtained from the tensor integral equation method, a volume integral equation method. It was learned that, in some cases, the surface integral equation method showed advan— tages in accuracy and computational cost over the volume integral equation method. To my mother, Kim-Chu Hong Lee, my wife, Ai—Chih, and my son, Yueh. I wish r' C) research adrisc and guidance t1". also wish. to th dents and encou.‘ for their val L132 excellent job in This resear Foundation under its. Army Resear< Finally, I i and love, which n ACKNOWLEDGMENTS I wish to express my sincere appreciation to my research advisor, Dr. K. M. Chen, for his constant support and guidance throughout the course of this research. I also wish to thank Dr. D. P. Nyquist for his helpful com— ments and encouragement about this study. Sincere appre— ciation is also extended to Dr. M. Siegel and Dr. D. Riska for their valuable help. I want to express my gratitude to Sue Cooley for her excellent job in typing the manuscript. This research was supported by the National Science Foundation under Grant ENG 74-12603, and, in part, by U.S. Army Research Office under Grant DAAG 29-76-G—0201. Finally, I thank my wife and my son for their patience and love, which made completion of this work possible. LIST OF TABLES .. use or FIGURES . 1- NERODUCTION II. EDDY Comm msroa A FIN Shimmy 2-1. Intro 2-2. Theorj 2'3~ Miner: 2~4~ Convei 2-5- Close C 7 2.6. as 2‘7' Bipen‘ Theore 2'8‘ Discus ITI SURFACE W 3'1- Intmd1 3-2. The Pre 3‘3' The Ger in an [ 34‘ DeriVai EqUatic 36' The SP6 Interfa '3'6' REView 3; The Nun TABLE OF CONTENTS LIST OF TABLES .................................... LIST OF FIGURES ................................... I. II. III. INTRODUCTION ................................. EDDY CURRENTS INDUCED BY RF MAGNETIC FIELDS INSIDE A FINITE CONDUCTING BODY WITH ROTATIONAL SYMMETRY ..................... . ....... . ....... 2.1. Introduction .......................... 2.2. Theory ......... . ...................... 2.3. Numerical Results ..................... 2.4. Convergence of Numerical Results ...... 2.5. Closed Form Solution for Sphere and Comparison with Present Numerical Results 2.6. Experimental Setup .................... 2.7. Experimental Results and Comparison with Theoretical Results ................... 2.8. Discussion ............................ SURFACE INTEGRAL EQUATION METHOD FOR INTERACTION OF MICROWAVE WITH BIOLOGICAL BODY ............ 3.1. Introduction .......................... 3.2. The Preliminary Theorems .............. 3.3. The General Solution of Maxwell's Equations in an Unbounded Honogeneous Space ..... 3.4. Derivation of the Coupled Surface Integral Equations . ............................ 3.5. The Special Case of an Infinite Plane Interface ............................. _3.6. Review of Moment—Method ............... 3.7. The Numerical Technique ............... 3.8. Numerical Results ..................... iv Page Vi 2O 29 31 38 4O 46 48 49 52 53 IV.PART1 AU pimzhu I—‘l—‘f—J MF-‘O V- some ,,,,, BBLIOGRQHY ‘ ‘ . . Page IV. PART 1 A USER'S GUIDE TO CQVIPUTER PROGRAM FOR INDUCED EDDY CURRENT INSIDE A FJNITE CONDUCTING BODY WITH ROTATIONAL SM’JETRY 124 4.1. Formulation of the Problem ..... 124 4.2. Description of Computer Program 127 4.3. Data Structure and Input Variables 128 4.4 An Example to Use the Program . . 132 4. 5 Printed Output . . . . . ...... . ..... 134 4.6. Listing of the Program . ........ 135 PARI‘ 2 A USER’S GUIDE TO COMPUTER PROGRAM FOR INDUCED EM FIELD ON THE SURFACE OF A FINI'I'E CONDUCTING BODY WITH ARBITRARY SHAPE ................................. 141 4. 7. Formulation of the Problem ..... 141 4.8. Description of Computer Program 144 4.9. Data Structure and Input Variables 146 4.10. An Example to Use the Program . . 152 4.11. Printed Output ........... . ..... 158 4.12. Listing of the Program ...... 164 V. SUMMARY ......................... . .......... . . . 230 BIBLIOGRAPHY ...................................... 233 we closer induce tivit§ of 50 intens tions nation 4'1- The 5y, corres files ' gram "1 Printg The syn Corresr files t gram “8 First 0 Second < 4'6. Third 01 FOUI‘th ( Printed Table 2.1. 4.1. 4.2. LIST OF TABLES Comparison of the present solutions and a closed form solution for the electric fields induced in a conducting sphere with a conduc— tivity of 8.0 S/m and a dielectric constant of 50 by a 300 MHz magnetic field with an intensity of l A/m. Also listed are the solu- tions obtained by the quasi—static approxi- mation . . . . . . .............................. The symbolic names of input variables and corresponding specifications for the data files used in data structure for the pro- gram "EDDY" ................................ Printed output of program "EDDY" . ..... . . . . . The symbolic names of input variables and corresponding specifications for the data files used in data structure for the pro— gram "SURFIDS" ...... . ...................... First output file of program "SURFLDS" . . . . . Second output file of program “SURFIDS" .. . . Third output file of program "SURFIDS" ..... Fourth output file of program "SURFIDS" .. .. Printed output of program "COMBINE" ........ vi Page 37 130 136 148 160 161 162 165 2.10. Figure 2.1. 2.2. 2.3. 2.4. 2.5. 2.10. LIST OF FIGURES A biological body is exposed to a uniformly impressed RF magnetic field . ............ A biological body is irradiated by a beam of RF magnetic field ............... . ....... A biological body of rotational symmetry irradiated by a uniform RF magnetic field is subdivided into a number of rings of various rahi..n .............................. Geometries of two rings in the subdivided biological body .................... ..... Geometries of a single ring inside the subdivided biological body .............. Distributions of amplitudes and phase angles of electric fields induced by a 100 MHz mag- netic field of 1 A/m in a cylindrical biologi— cal body with a diameter of 15 cm and a height of 30 cm ................................ Distributions of amplitudes and phase angles of electric fields induced by a 100 MHz mag- netic field of l A/m in a cylindrical bio— logical body with a diameter of 36 cm and a height of 90 cm ......................... Distributions of amplitudes and phase angles of electric fields induced by magnetic fields of various frequencies in a muscle disk with a diameter of 30 cm and a thickness of 1 cm Distributions of amplitudes and phase angles of electric fields induced by a 100 MHZ mag- netic field in the central section of a cylindrical biological body of diameter 36 cm with various heights . ................... Distributions of amplitudes and phase angles of electric fields induced in a cylindrical biological body with a diameter of 32 cm and a height of 20 cm.by a beam of 100 MHz nag- vii 12 13 14 16 21 23 24 26 Figure 2.11. 2.12. 2.14. 2.15. 2.16. 2.18. netic inten: Distri of ele netic (-3 = 0 Ccmpar soluti. a Cyli: ter of 100 MH; 1 A/m . A fine is inme rized 1 A finit. 2 on is Cted wi cross—5g t0 have Ctric (x MHZ OSci siti’ of Rheum fields 1 induced } Carparim results 1 induced j 4 an diam MHZ “39116 CQUParisO rESults f induced i With a di. and Virior Figure 2.11. 2.13. 2.14. 2.15. 2.16. netic field with a diameter of 8 cm and an intensity of l A/m ...................... Distributions of amplitudes and phase angles of electric fields induced by a uniform mag- netic field of 100 MHz in a model of man (0 = 0.889 S/m, Er = 71.7) .............. Ccmparison of 81-ring solutions and 9-ring solutions for the electric fields induced in a cylindrical biological body with a diame— ter of 36 cm and a height of 18 cm by a 100 MHz magnetic field with an intensity of l A/m ........... . ...................... A finite conducting sphere with radius R is irmtersed in a uniform magnetic field pola— rized in the +2 direction ......... . . . . . . A finite conducting sphere with a radius of 2 cm is simulated by a "ring sphere" constru— cted with 30 circular rings of two different cross-sectional areas. The sphere is assumed to have a conductivity of 8.0 S/m and a diele- ctric constant of 50, and is immersed in a 300 MHz oscillating magnetic field with an inten- sity of l A/m .................... . ...... Experimental setup for measuring the electric fields inside the phantom biological bodies induced by RF magnetic fields . . ..... . . . . Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom biological cylinder of 4 an diameter with various heights by a 750 MHz magnetic field ...................... Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom biological cylinder with a diameter of 3.8 cm, a height of 3.8 cm and various conductivities by a 750 MHz mag— netic field ........ . .................... Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom model of man (height = 22.2 cm, max. diameter = 3.8 an) by a 750 MHz magnetic field . . . . . . .................... viii Page 28 3O 32 34 36 39 41 43 45 3.1. Figure 3. l. 3.3. 3.4. 3.6.” Page A source region Gs enclosed by surface F s is charagterized by volume current densities (J , Jm) and equivalent surface current den- . . + . SlE‘LeS (je, 3m) . These current sources radiate into a homogeneous space with permeability ii apd 'ttivity_*s, and maintain an EM field (E, H) at point r ...................... . 54 A finite conducting body G, immersed in free space and enclosed by surface F, with a permea- bility u and a complex permittivity e +315 irradiated by the incident field (El, dH1) radi- ated from the source region G where the equiva— . . -r lent surface current denSities (3e, 3m) are defined on the surface F5. The EM fields scattered fgom and induc in the body are denoted as (E5 ) and (E , fid), respectively ..... 57 The scattered field (ES, RS) can be consider as originated from an equivalent current source existing inside the body region which has been replaced by free space. This equivalent current source consists of tum components, the conduction current and the polarization current .. . . 59 The scattered field (ES, Rs) in the exterior of the body and a zero field in the interior of the body are maintained by the equivalent surface currents (3S, 3;) , which are defined in terms of the scagtered field on surface F. The body region G has been replaced by free space . ........... . ...................... 62 The induced field (Ed, Ed) in the interior of the body and a zero field in the exterior of the body are maintained by the n tive of the equivalent surface currents (j , 3m) , which are defined in terms of the induc field on surface F. The region external to F is replaced by a medium with permeability no and complex permittivity e d ..... . . . ...... . .......... 64 The induced field (Ed, Ed) in the interior of the body and the scattered field (ES,H5) in the exterior of the body are maintained by the ne— gativgof the equivalent surface currents (3:, 3;!) , which are defined in terms of the incident field on surface F. The body G with permeability “o and complex permittivity Ed ix 3.10. 3.11. ill ill 3.14. With the i the t body qu defin‘ Surfa< An in: a find '40 an: a plar. The co interf Divisi< Square using 1 fUDCtic A Simul inciden directi The bod tiVT'Lty ‘ a “When e O —> e A = —— J —————T————— dv. mm 4W ./mth m Rmm o) ring After substituting the expression of Rmm and some straight— forward manipulations, we have the following form for the above equation: r _ “ (2.7) Amm _ d J s r K -jkORmm(0) where Kmm = g1 [Rmm(d) - Rmm (0)l/: COS¢ e d¢ . (2.8) + . . The total vector potential at rm maintained by the + I induced currents in all the rings, Am, can then be obtained as r _ _ A _9 K J (2.9) Am — 2 — ¢ 2 2w 5 r . I1 n ' u u n... —-.._.... At the ring can be in the nth i + — Jn— Equation (2. current and sidered as t] With eq. + — Am - .9 T ' . his Am 18 re the mth ring ( in all the ri1 Where f is th Since the indu sYmmetrical, n Wently, no so The total is the sum of 1 ring and the sc eq‘ (2.12): 18 At the same time, the induced current 3n in the nth ring can be related to the total induced electric field , . —> in the nth ring, En’ as —> __ , -+ _ ——-o Jn - [0n + j w (En — 60)] En — Tn En, (2.10) o + I Equation (2.10) expresses Jn as the sum of the conduction current and the polarization current, and Tn can be con- sidered as the complex conductivity of the nth ring. With eq. (2.10), Km can be rewritten as I: S r K T E (2.11) N :lo ”.3 s B :3 b 5 —> A Am=¢z n This am is related to the scattered electric field E; in the mth ring which is maintained by the induced currents in all the rings: =..ij =.—¢ jf u S r K T E (2.12) i o n n mn n n where f is the frequency. It should be emphasized that since the induced current in each ring is rotationally symmetrical, no electric charge is induced and, conse— quently, no scalar potential is maintained. The total induced electric field Em in the mth ring is the sum of the impressed electric field E; at the mth . ‘ . . +5 . . ring and the scattered electric field Em obtained in eq. (2.12): + . E = E + E (2.13) With eq . (2 Equation (2. of the N rin matrix form where Mm ‘ The total indu rings can be c inversion tech rent is needed (2.10). A compute method to calm finite conduct: cuSsed in the r 19 With eq. (2.12), eq. (2.13) can be rearranged as Em + z jf u s r K T E = E1 (2.14) Equation (2.14) can be point-matched at N reference points of the N rings to yield N simultaneous equations in a matrix form as 1 M11, M12, ........ , 1N El El 1 M21 M22, ........ , 2N E2 E2 .................. 9 . : . (2.15) . M E El lVJ-Nl’ IVLrJZI ooooooooo NN N N where Mnn = l + 3f U0 sn rn Knn Tn (2'16) an = 3f Uo 8n In Kmn Tn, (2‘17) The total induced electric fields, E to En’ in all the 1 rings can be determined from eq. (2.15) by the matrix inversion technique. Note that if the induced eddy cur— rent is needed, it can be obtained readily from eq. (2.10). A computer program has been developed based on this method to calculate induced electric fields in various finite conducting bodies. Some numerical results are dis— cussed in the next section. biological 30 cm being MHz with an . have a condm stant (er) 0: bOdY is subd: With a square The amplitude field E in ea tions of thes radial distan ferent sectio: is observed i1 induced elect radial distan Solution as c anElle of the ' ~110° and -15 field ii) whil be ~90° at any my. the ofte Estimate the 110t its phase drical body is 20 2.3. Numerical Results The first numerical example deals with a cylindrical biological body with a diameter of 15 cm and a height of 30 cm being exposed to a uniform magnetic field of 100 MHz with an intensity of l Amp/m. The body is assumed to have a conductivity (0) of 0.889 S/m and a dielectric con- stant (er) of 71.7. In the numerical calculation, the body is subdivided into 100 rings of various radii and with a square cross—sectional area of 1.5 cm x 1.5 cm. The amplitude and phase angle of the induced electric field E in each ring were calculated, and the distribu- tions of these quantities are plotted as functions of the radial distance from the cylindrical axis r for 10 dif— ferent sections of the body as shown in Figure 2.6. It is observed in Figure 2.6 that the amplitude of the induced electric field [El increases linearly with the radial distance; this result is close to the quasi—static solution as can be obtained from eq. (2.1). The phase angle of the induced electric field varies between around -llO° and -150° (with respect to the impressed magnetic field El) while the quasi-static solution predicts it to be —90° at any point in the body. For this cylindrical body, the often—used quasi-static solution can be used to estimate the amplitude of the induced electric field but not its phase angle. However, as the size of the cylin- drical body is increased, the quasi—static solution is Figure 2.6. Di 21 32 AMPLITUDE (E) I / 24 / (V/m) b 3-10 // 1 _ / +1 // 2 H // 16 — quasi—static _ approx. /// 8 b / / / l / o J 1 L l 1 1 J l J O 1 5 3.0 4 5 6 O 7 0 3 cm radial distance from center r (cm) .80 . . qua51—stat1c approx. r ____________________________ PHASE ANGLE —100 H1 = 1 A/m Phase f = 100MHz %f o = 0.889 S/m (degree)'120 er = 71.7 -140 9,10 —160 J l A l i L l_ 1 l 0 1.5 3.0 4.5 6.0 7, radial distance from center r (cm) Figure 2.6. Distributions of amplitudes and phase angles of electric fields induced by a 100 MHZ mag— netic field of 1 A/m in a cylindrical biologi- cal body with a diameter of 15 cm and a height of 30 cm. found to be the next ex In the biological b (3 = 0.889 S‘ with a dime in Figure 2.‘ an intensity distribution: induced elect Present resu] SOlutions: t is two or thr While the pha the quasiesta example, that field of VHF becomes compl The next the electric intensity (1 p 100 and 200 M11 and a .thicknes the disk are a Er = 160 to 56 The amplitudes 22 found to be completely inadequate, as can be observed in the next example. In the second example, we consider a cylindrical biological body with the same electric properties (0 = 0.889 S/m and er = 71.7) as the first example, but with a diameter of 36 cm and a height of 90 cm as shown in Figure 2.7. The same magnetic field of 100 MHz with an intensity of 1 Amp/m is impressed on the body. The distributions of the amplitudes and phase angles of the induced electric fields in Figure 2.7 show that the present results deviate greatly from the quasi—static solutions: the amplitude of the induced electric field is two or three times lower than the quasi—static solution, while the phase angles vary between -120° and -290° against the quasi—static solution of -90°. It is clear, from this example, that when a human body is exposed to a magnetic field of VHF range, the often—used, quasi—static solution becomes completely invalid. The next example as shown in Figure 2.8 deals with the electric fields induced by magnetic fields of unit intensity (1 Amp/m) and of various frequencies (10, 40, 100 and 200 MHz) in a muscle disk with a diameter of 30 cm and a thickness of 1 cm. The electrical properties of the disk are assumed to have 0 = 0.625 to 1.28 S/m and E = 160 to 56.5 for the frequency range of 10 to 200 MHz. r The amplitudes of the induced electric fields are found to H- 5 A. PI“ - V 23 6O // (El ,/ ~ AMPLITUDE / """"‘“ / (V/m) /// 45 r , / n n h // qu351—stat1c // approx. 30 >- // 1 / // 7 // h // /‘ ,// // /// 3 15 ~ ,// / / / § “ /’ 5—8 // 4 O ’ 1 1 1 1 1 1 4L in 1 1 L O 3.0 6.0 9.0 12.0 15.0 18.0 0 cm . . ’ 9 radial distance from center r (cm) -%)___ _ ___ .s- _ _— quasi—static 1+ . new approx- -140 H1 = 1 A/m Phase of f = 100MHz + E -190~ 0 = 0.889 S/m (degree) 5 = 71.7 1 O 3.0 6.0 9.0 12.0 15.0 18.0 radial distance from center r (cm) Figure 2.7. Distributions of amplitudes and phase angles of electric fields induced by a 100 MHZ mag— netic field of 1 A/m in a cylindrical bio- logical body with a diameter of 36 cm and a height of 90 cm. runny—c (V/m) 24 9O AMPLITUDE present theory ---- quasi—static approx. 1cm 0 3.0 6.0 9.0 12.0 15.0 i radial distance from center r (cm) '; K15 cm —80 I! quasi—static approx. for all fre<3 f ~__4________fi_,__, Phase 0 MHZ of ~100~ U if 100 MHz (degree) PHASE ANGLE -120— n1 = 1 Mn. _ 200 M112 0 = 0.625 to 1.28 S/m *140- m_,, present for 10-200 MHz theory — — ————— quasi-static Er = 160 to 56.5 approx. —16 1 1 1 1 1 L 1 1 for 10-200 MHz 0 3.0 6.0 9.0 12.0 15.0 Figure 2.8. radial distance from center r (em) Distributions of amplitudes and phase angles of electric fields induced by magnetic fields of various frequencies in a muscle disk with a diameter of 30 cm and a thickness of 1 cm. increase 1:? distance f1 slightly fr results for fields, how static solu In the drical body from 6 cm tc 11112 has an 1- ties of the 3= 0.889 s, at the Centr the diStribu are Plotted 12, 18, 30 a‘ as the CYlim the induCed ( CYlinder deCJ drastically 1 example indic magnetic fiel dependent On 25 increase linearly with the frequency and with the radial distance from the cylindrical axis; these results deviate slightly from the quasi—static solutions. The present results for the phase angles of the induced electric fields, however, deviate significantly from the quasi- static solutions, especially, for the high frequency cases. In the example shown in Figure 2.9, we study the effect of the cylindrical height on the induced electric field at the central section of the cylinder. The cylin— drical body has a diameter of 36 cm, but its height varies from 6 cm to 90 cm. The impressed magnetic field at 100 MHz has an intensity of 1 Amp/m. The electrical proper- ties of the body at this frequency are assumed to have 0 = 0.889 S/m and er = 71.7. The induced electric fields at the central section of the cylinder are determined and the distributions of their amplitudes and phase angles are plotted for the cases of cylindrical heights; H = 6, 12, 18, 30 and 90 cm. It is observed in Figure 2.9 that as the cylindrical height is increased, the amplitude of the induced electric field at the central section of the cylinder decreases greatly, and its phase angle varies drastically from the quasi—static solution of -90°. This example indicates that the eddy current induced by a RF magnetic field in a cylindrical biological body is strongly dependent on the cylindrical height. l l 18 Figure 1%! (V/m) p. Phase of .-> E (degree) 1 A/m 100 MHz 0.889 S/m 71.7 26 60F 40“ 20' AMPLITUDE 3.0 6.0 9.0 12.0 15.0 radial distance from center I (cm) ~190 —240 __... __._..__.._._d_ we. quasi—static approx. —290 3.0 6.0 9.0 12.0 15.0 18,0 radial distance from center r (cm) Distributions of amplitudes and phase angles of electric fields induced by a 100 MHZ mag— netic field in the central section of a cylindrical biological body of diameter 36 cm with various heights. The c magnetic f Figure 2.1 phase angl1 body with e a beam of j 8 cm and a: ties of : : body. The Culated frc 0f the ind: of the Cyli that the in rent is zer increases l beam, and t The deViati static SOlu Cdtlon Of a bean), hea- therapy is : can be 00mm The la: the electric with an inte man with a l". 27 The case of the eddy current induced by a beam of magnetic field in a biological body is studied next. Figure 2.10 shows the distributions of the amplitudes and phase angles of electric fields induced in a cylindrical body with a diameter of 32 cm and a height of 20 cm, by a beam of 100 MHz magnetic field with a beam diameter of 8 cm and an intensity of 1 Amp/m. The electrical proper— ties of o = 0.889 S/m and er = 71.7 are assumed for the body. The impressed electric field for this case is cal— culated from eq. (2.3). The amplitudes and phase angles of the induced electric fields in five different sections of the cylinder are plotted in Figure 2.10. It is observed that the induced electric field or the induced eddy cur— rent is zero at the center of the magnetic beam and it increases linearly to a maximum value at the edge of the beam, and then decays down toward the edge of the body. The deviations between the present results and the quasi— static solutions are indicated in Figure 2.10. The appli- cation of a beam of RF magnetic field for the purpose of locally heating a biological body in a hyperthermia cancer therapy is feasible if the low heating at the beam center can be compensated by other means of EM heating. The last numerical example is the quantification of the electric fields induced by a 100 MHz magnetic field with an intensity of 1 Amp/m in a cylindrical model of man with a height of 168 cm and a maximum diameter of 16 /\ A \ qua51—st atic approx. \ I W (V/m) 12 4 cm 8 -D(l§- (1:2 : . 4 1 1 it?! 4H .O 6.0 8.0 10.0 12.0 4 J 20 cm 14.0 16.0 3 i radial distance from center r (cm) 1 1 , ' l l 1 .MJ '80 l" quasi—static approx. 16 cm -120 w Phase of _> E (degree) —160 = l A/m — 100 MHz = 0.889 S/m = 71.7 m o H1x I I 1 1 I 1 1 1 l O 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 radial distance from center r (cm) Figure 2.10. Distributions of amplitudes and phase angles of electric fields induced in a cylindrical biological body with a diameter of 32 cm and a height of 20 cm by a beam of 100 MHZ mag— netic field with a diameter of 8 cm and an intensity of l A/m. 36 cm as s c : 7 and or cal calcul of various the induced that in the radius Of t The phase a. different f: bOdy' 2'4' M The aCc sented in th of numeriCal increaSed_ excellent CO1 demonstrated a Cylindrical of 18 m is e With an inten C 010 = 0‘889 29 36 cm as shown in Figure 2.11. Again, g = 0.889 S/m and Er = 71.7 are assumed for the body. In the numeri- cal calculation, the body was subdivided into 66 rings of various radii and with a square cross—sectional area of 6 cm x 6 cm. The amplitudes and phase angles of the induced electric fields inside the body are indicated in Figure 2.11. It is noted that these results deviate greatly from the quasi—static solutions. For example, the induced electric field in the head can be greater than that in the middle section of the body even though the radius of the latter is greater than that of the former. The phase angles of the induced electric fields are very different from —90° and they vary widely throughout the body. 2.4. Convergence of Numerical Results ______________________________ The accuracy and validity of numerical results pre— sented in this chapter depend greatly on the convergence Of numerical results as the number of subdivided rings is increased. Fortunately, the present theory gives an excellent convergence of numerical results. This fact is demonstrated in an example shOWn in Figure 2.12, in which a Cylindrical body with a diameter of 36 cm and a height 0f 18 cm is exposed to a uniform magnetic field of 100 MHz With an intensity of 1 Amp/m. The electrical properties 0f 0 = 0.889 S/m and Er = 71.7 are assumed for the body. 5 a“ + 13? Pha 5 of (degree) 30 E mom) ' aqéiflfie of O 3 J . Brimoiwiemnmififl . . ..2 fill—“1’ .llqllflTl ILIIITHI I Ext. 3.. T”. u. _ s..p.o...¢_u4n.4u+u_u+fl W14: llrl_l..P.-I/..5_5_5—Is.4_4_4. 4._4._5_5 _ .1 . 4|il1l I1ITI_...IFI_I! Til . _ . . fl _ _ _ _ _ . _ “I . 11.-.: 3.-....1-._I_L_-_r_7__:_- _ . _ i llnfilLlLll ‘JlLll—‘L‘lrl _ _ . Iq . _ _ _ . I“ _ _ _ _ _ _ _ _ _ Fl m —‘ 8 Ii 6 M..l—2..BM9..9..8.0.2.4 2 41—.3_5L6L_6.6_6_4._5_ .IN. _l2.. _fllufl. .IW. _l2.l“l8l.l.3.ql_l-.H.L.I4Ll. .3_Lm_.nhzn_ a. w. w. n. n. a 4... a._1m_m+w+w..-m+4frw I o o. o. o o. 6 4 i IIIHITII. L.ll~ Flt“. p.44. % «I .m _.L9l.lr l. _ . _ . _ . _ _ . _ inlunluuuwlwnwiilruwlw. . . _ _ ILIIMIITI+I+I+I¢ILILILIIL . . . _ _ _ _ _ _ _ w _ - _ — Pha§e angle of E + 180 (danee) gles mplitudes and phase an induced by a uniform mag— Distributions of a Figure 2.11. of electric fields model of man netic field of 100 MHZ in a = 71.7). E r 0.889 S/m, (o: In the fi: divided i1 cross-seci numerical 81 rings c sectional the induce divisions the agreen is excelle Subdivisio first SUbd cal rGSult results on induced by withOUt sul adVantage i (Mat Savi] T0 Cor analytiCal form timef In the first numerical calculation, the body was sub— divided into 9 rings of various radii and with a square cross—sectional area of 6 cm x 6 cm. In the second numerical calculation, the same body was subdivided into 81 rings of various radii and with a square cross- sectional area of 2 cm x 2 cm. The numerical results on the induced electric fields based on these two ring sub- divisions are compared in Figure 2.12. It is observed that the agreement between these two sets of numerical results is excellent even though the number of rings in the second subdivision increases by a factor of 9 from that of the first subdivision. This excellent convergence of numeri— cal results implies that it is possible to obtain accurate results on the electric fields or the eddy currents induced by a RF magnetic field in most biological bodies Without subdividing the body into too many rings. This advantage provided by the present method can lead to a great saving in computer time and cost. 2.5. Closed Form Solution for Sphere and Comparison With Present Numerical Results To compare our numerical results with the existing analytical solution [3], we consider the case of a con- ducting Sphere with radius R which is immersed in a uni— form time-harmonic magnetic field, ' ' ' A i . El = 2 H1 = f H1 cose — 6 H Sine H 1 t 1 A/m f‘ 100 MHz 3:0. 7 E 89 S/m -7 H H *7! i (1 .Ul‘e 2&2 (degree)-180 H = l A/m f = 100 MHZ 0 = 0.889 S/m Er: 71.7 Figure 2.12. Phase 80 1E! AMPLITUDE (V/m) quasi-static/z’ / approx./ dL/ /’ “———-'81—ring solution ,f/ / 6O A/ ———»——9-ring solutign/ I I l 9.0 I I 1 I 12.0 15.0 18.0 radial distance from center r (em) I 3.0 6.0 quasi—static approx. PHASE ANGLE ____________ —l30 /. / /’l‘ of // --——8l—ring solution //<;/// 1” I —-"-—9—ring solution —230 u- / :;::::;,,,/ 280 I I l l I l l l I I ' 0 3 0 6 o 9 o 12.0 15 o 18 o radial distance from center r (em) Comparison of 81—ring solutions and 9-ring_ solutions for the electric fields induced in a cylindrical biological body with a diame— ter of 36 cm and a height of 18 cm by.a 100 MHZ magnetic field with an intenSity of l A/m. where f 4 as shown tained b3 regions i 39* —) Aout wnere I3/ the CODdLu Can be de1 tions at I l) '1‘) 2) After Some SPhEriCal 33 where f and e are unit vectors in spherical coordinates, as shown in Figure 2.13. The vector potentials main- tained by the induced currents and the sources for the regions inside and outside the Sphere can be written as . C A X — (1/2) H1 sine —1 I [ ('w u o)% r]¢ in — 11o /f 3/2 3 0 when r < R (2.18) + i . C2 A Aout= (1/2) uo H Sine (r + —§)¢ when r > R (2.19) r where I3/2 is the spherical Bessel function, and O is the conductivity of the sphere. The constants Cl and C2 can be determined by use of the following boundary condi— tions at r = R, l) Ain = Aout a _ a 2) 5? (r Aout) — 6? (r Ain). After some manipulations and utilizing some identities of Spherical Bessel functions we have C ‘ -§-3::E-— (2.20) l — w I (w) 1/2 3 w I (w) — (3+w2) I (w) 3 c — ‘1/2 1/2 R (2.21) *2 — 2 w Il/2(W) 2 i . . where W = V2 j E, and 6 = (———-——42 15 the skin depth of 6 (DUO O the Sphere. Combining eqs. (2.18) and (2.20), we obtain Figure 34 ¢® Figure 2.13. A finite conducting sphere with radius R is immersed in a uniform magnetic field polarized in the +2 direction. the comp] in Then the given by To compare that the s; 8/m, and a a uDiform c 300 MHz, C radiUS of t Now for the SimUlated b fifferent r: Evaludted re circular rin fr“ eq. (2. The comparis Table 2'1 ari approximat i 01 It is aE cal SOlution the complete expression for Xi I —> _ i _ E I3/2 (Wit/R) A Ain — (3/2) “0 H Sine ———- ¢_ “E I (w) R 1/2 Then the electric field induced inside the sphere is given by . I (wr/R) + =_. + =__ . l . r—- R 3/2 A E jLUAin (3/2) jwlio H Slne Rr w_r W (I) (2.22) To compare eq. (2.22) with our numerical results, we assume that the sphere has a radius of 2 cm, a conductivity of 8.0 S/m, and a dielectric constant of 50, which is immersed in a uniform oscillating magnetic field with a frequency of 300 MHz. Under these assumptions the skin depth and the radius of the sphere are in the same order of magnitude. Now for the case of numerical evaluation, the sphere is simulated by a "ring sphere" which is constructed with 30 different rings, as shown in Figure 2.14. The numerically evaluated results of the induced electric fields in these Circular rings are then compared with those values obtained from eq. (2.22) at corresponding points inside the sphere. The comparison is made in Table 2.1. Also listed in Table 2.1 are the results obtained by the quasi—static approximation. It is apparent that the agreement between our numeri— cal solution and the existing closed form SOlUtion ls Figure ' 36 Figure 2.14. A finite conducting sphere with a radius of 2 cm is simulated by a "ring sphere" constructed with 30 circular rings of two different cross—sectional areas. The sphere is assumed to have a conductivity of 8.0 S/m and a dielectric constant of 50, and is immersed in a 300 MHZ oscillating magnetic field with an intensity of 1 A/m. . . i .( .rkap.c iiliiiiiliIiilIIiiliiiiiiiiliiiiiilliiiiiii I -COHUQEflXOHQQQ UHUM#@IHWQ3U QSU .E\< H M0 Nflflmfimuqfl C“ SUHE _ U UHHUUGHOflU Q UCQ E\W O.m n‘H UHHUUMVHQ COWHHMQEOU IHIN @Herfl UGUWHH OWH< . IMHO UCQUWCO . . @HOSQW WCHUUDQCOU m Cum MUQUUHUEH mmUHMUfl. WCOHUSHOW UCGWQHQ QHNU “HO %Q UQCHGUDO WCOHUJHOW USU 0M0 UHOHW UHHOCOGE N. QUHB COHUDHOW EVHOM HUGWCHU .m. MUG-m. i i h.le vmm.o 0.0m! mvm.o om! ovo.H ma o.mml mmm.o w.wml oow.o oml th.o vH H.owl mmN.o w.hmI HwN.o oml maN.o ma o.mml HBN.H w.hml wom.H om! oww.H NH m.mvl mmh.o H.hvl vow.o om! mww.o HH v.mvl mmN.o m.HVI Nmm.o oml mmN.o OH m.NmI wmw.H h.ool Nwm.H om! 0mm.H m N.mml hv©.H o.wml hmm.H om! omm.H m ¢.wvl va.H N.h¢l MNH.H oml om¢.H h m.>mt moh.o v.wml Nam.o om! mww.o w H.le mmN.o m.OMI omN.o oml mmN.o m 7 m.mml hwh.a ®.hml Hw©.H . om! Oho.N v 3 v.mvr mmN.H m.avu NwH.H om! omv.H m b.Hml Ohh.o O.Hml mw©.o Om! mmw.o N s.sm- mmm.o o.mm- omm.o om- omN.o a Amwumwov AE\> oav Amoumoav AE\> oav Amoummav AE\> oav mmmsm coopflcmoz mmwnm opspflcmoz Ilmmmmmllllwmwwmmmmm .02 8333. En: 5%.wame swmwwmmmm en .coflvmfiflxoummo oeumsmlflmosv on» we owQHMpQO mCOfluDaOm wsu mud Umpmfla Omam .E\< a MO muflmcmpsfl no cues cause venomous Ne: com a so om do usmumcoo oauuomamne m was E\m o.m mo mufi>fluosocoo w apes oumnmm msfluoopsoo m 2H UmOSpCfl mcamflm oeuummam wnu MOM COHuSHOm Show UGmOHo m cam mGOHHSHOm pemmmHm map MO cOmHHm EOU .H.N OHQME excelleni tions thl lent agre technique static sc cates the usually 1' the elect some expe filled wi 0f Substa] miCrOWaVe was Create reflect”. horizOntal ized' The of 500 to locatiOn 0 field‘ If 38 excellent for both the magnitude and the phase distribu- tions throughout the interior of the sphere. This excel— lent agreement confirms the accuracy of our numerical technique. Furthermore, the deviation of the quasi— static solution from the other two sets of solutions indi- cates that the conventional quasi—static solution is usually inadequate. 2.6. Experimental Setup A series of experiments has been conducted to measure the electric fields induced by UHF magnetic fields inside some experimental models constructed with plexiglass and filled with phantom biological materials for the purpose of substantiating the accuracy of theoretical results presented in this chapter. The experimental setup is depicted schematically in Figure 2.15. The experiment was conducted inside a large microwave anechoic chamber in which a standing EM wave was created by radiating an EM wave upon a metallic reflector. The electric field of the wave was polarized horizontally and the magnetic field was vertically polar- ized. The experiment was conducted in the frequency range of 500 to 750 MHz. The phantom body was placed at the location of a maximum magnetic field or a minimum electric field. If the body dimension in the direction of wave Propagation is small compared with the wavelength, the _\1>>>E\\\/\3 poosocfl weapon HMUHmOHOHQ Eoucmam 93 3:9: mcamflm canyowao on“ mcHHSmmmE How mduwm Hmucwfiauumxw .mH.N musmam .tcmuw fl i!\\Eu0m%Hom Emumhm cwpumuwv Cu m>m3 m>m3 cmuooamos ucmpaoca .uc< ” >won . . .mcouu «M II \ r. e *. m>m&. LI I was: mew—ocean Al N... Hobowammu\1. AI owaamuwe f meEmsu owozowcm m>msouowz impresse body can field in able elei wire Sys' The outpi tern outsi Phantom b The Elect to be app ment was at the 10' length (4. direCtiOn i CylindricE tral Sectj Cylindrice amplitudeS FigUre 2‘1 pending th lines. It field Prob tilde of th! . , 4O impressed magnetic field at any point inside the phantom body can be assumed to be uniform. The induced electric field inside the phantom body was measured by an implant- able electric field probe with an interference-free lead- wire system. This probe was described elsewhere [5]. The output of the probe was connected to a detecting sys— tem outside the anechoic chamber. 2.7. Experimental Results and Comparison With Theoretical Results The first experiment was conducted on a cylindrical phantom body with a diameter of 4 cm and a variable height. The electric properties of the phantom body were estimated to be approximately 0 = 5.0 S/m and Er = 50. The experi— ment was conducted at 750 MHz and the cylinder was placed at the location of a maximum magnetic field, one wave- length (40 cm) in front of the metallic reflector. The direction of the magnetic field was in parallel with the cylindrical axis. The induced electric field in the cen- tral section of the cylinder was probed for three cases of cylindrical heights; H = 2, 4 and 8 cm. The measured amplitudes of the induced electric fields are plotted in Figure 2.16 with dashed lines, in comparison with corres- ponding theoretical results which are indicated with solid lines. It is noted that with an implantable electric field probe loaded with a microwave diode, only the ampli— tude of the induced electric field could be measured. In 41 I /’ 4.131 = l A/m // 50 ~ ’ I 9 H iEl =3 1 (V/m) 40 ” j cit theory 30 _ -—-u-——exper1ment f = 750 MHz C = 5.0 S/m / Er: 50.0 // /// 20 " // / / / / / / / ,' 10 _ /// I :/ / " / .,;/ 0 1 ll 1 l O 0.4 0.8 1.2 1.6 2.0 Figure 2.16. radial distance from center I (cm) Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom biological cylinder of 4 cm diameter with various heights by a 750 MHz magnetic field. Figure 2. theoretic near the probe (ab significa: theory am ,. eifect 0 1‘1) severe if is about 0 in Figure Predicted - Was increag section of @1150 found Static solu The Sm Cylinder Wit to study the electriC fie conductiviti The EXPerime] was plaCed a1 in the Previc at the CeIltra e distribut Pique 2.17 i 42 Figure 2.16, the agreement between experimental and theoretical results is very good except in the region near the cylindrical axis where the perturbation of the probe (about 1 cm in size) on the induced current becomes significant. Another cause of discrepancy between the theory and the experiment can be attributed to the image effect of the metallic reflector; this effect is not severe if the distance between the body and the reflector is about one wavelength or greater. The important finding in Figure 2.16 is that it was observed experimentally and predicted theoretically that as the cylindrical height was increased, the induced electric field in the central section of the cylinder decreased. These results were also found to be significantly different from the quasi- static solutions. The second experiment was conducted on a phantom cylinder with a diameter of 3.8 cm and a height of 3.8 cm to study the effect of body conductivity on the induced electric field. Three kinds of phantom materials with conductivities of o = 2.2, 4.5 and 6.0 S/m were used. The experiment was conducted at 750 MHz, and the cylinder was placed at the location of a maximum magnetic field as in the previous experiment. The induced electric fields at the central section of the cylinder were probed, and the distributions of their amplitudes are plotted in Figure 2.17 in comparison with corresponding theoretical E1 20s ~>i H = l A 70, i ‘ h" 3 8C 0:2.2 4 S/m E 60 ~ I l 1 H (V/m) 1.9cm / / 50 l theory quasi—static / ___-.-——experiment ap rox.‘,/ X/ 40 ” f = 750 MHz '/ [x’ r = 50.0 /// ’/-./O—:}1: // l./ 30 L— // / ‘ // ’ c=6.0 / .. S/m " / --/ 20 ~ ' ,/’ /// / C/ a ’ /// / -v’ //// 10 _ / I L 0 I I O 0.4 0.8 1.2 1.6 2.0 Figure 2.17. radial distance from center r (cm) Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom biological cylinder with a diameter of 3.8 cm, a height of 3.8 cm and various conductivities by a 750 MHz mag- netic field. results. is increa effect wa by experi and exper quasi-Sta experimen The induced e. The Phant< and fille< 22,2 cm a1 Pi{lure 2.; and Elect} are aPPrO) Placed 11px field with the long c‘ fields at measured a indioated Pending th and experi rather Com mental and Very diffe 44 results. It was observed that as the body conductivity is increased, the induced electric field decreases. This effect was accurately predicted by theory and confirmed by experiment. It is also noted that both theoretical and experimental results deviate significantly from the quasi—static solutions. The agreement between theory and experiment as shown in Figure 2.17 is very good. The third experiment was conducted to measure the induced electric fields inside a phantom model of man. The phantom model of man was constructed with plexiglass and filled with phantom material, and has a height of 22.2 cm and a maximum diameter of 3.8 cm as shown in Figure 2.18. The experiment was conducted at 750 MHz, and electrical properties of the model at this frequency are approximately 0 = 5.0 S/m and Er = 50. The model was placed upright at the location of a maximum magnetic field with the impressed magnetic field in parallel with the long dimension of the body. The induced electric fields at 32 locations of the body were probed. The measured amplitudes of the induced electric fields are indicated in Figure 2.18 in comparison with the corres— Ponding theoretical results. The agreement between theory and experiment is considered to be very good in this rather complicated body. It is to be noted that experi- mental and theoretical results shown in Figure 2.18 are VGry different from the quasi—static solutions. . \ Figure 2'18 ......... 222 an 27. L-5--, {-7. 2il9.33 O —'- n -l I Ifill= 1 A/m f= 750 MHz o= 5J)Sfin er: 50.0 i Figure 2.18. Comparison of theoretical and experimental results for the amplitudes of electric fields induced in a phantom model of man (height = 22.2 cm, max. diameter = 3.8 cm) by a 750 MHz magnetic field. AVt as demon: the vali< sented ir tion of a Since the gation wa impressed be unite: electric as has bee Cal Calcu] body. From experiment standing E Was Where y t ‘ 46 A very good agreement between theory and experiment as demonstrated in Figures 2.16, 2.17 and 2.18 confirms the validity and accuracy of the theoretical method pre— sented in this chapter. 2.8. Discussion In the experiment, the body was placed at the loca— tion of a maximum magnetic field in a standing EM wave. Since the body dimension in the direction of wave propa— gation was small compared with the wavelength, the impressed magnetic field inside the body was assumed to be uniform. Under this approximation, the impressed electric field associated with the impressed magnetic field was __l i 2.2 E — 2 wuo H rd ( ) as has been given before. This Ei was used in the numeri— cal calculation of the induced electric field inside the body. From a different point of View, the body in the eXperiment was located at a minimum electric field of a Standing EM wave, and the actual impressed electric field . u . o A . i E1 = § j IE2 Hl sin ky = x jwuo H y (2.23) 0 Where y = 0 corresponds to the center of the body. was +1 Es in eq. (2 amplitude the y axi Iti Ei's can electric that, ind results 01 This phenc exPeriment Chapter. Ei given in eq. (2 in eq. (2.23) is linear amplitude of the latter the y axis. It is important to El's can yield the same .2) is circulatory and that given and antisymmetrical. Also the is twice that of the former along ask whether these two different theoretical values for the induced electric fields in the body? We have numerically proved that, indeed, these two +i . . . . E '3 give very Similar numerical results on the induced electric fields in the same body. This phenomenon resulted in a very good agreement between experimental and theoretical results reported in this chapter. Only Proposed energy be and has In TEthod in ITliSSiOn a Sical tOp Century, high‘Powe relating . teChHOlOg; effects 0; the elect] they are ; EXisi inducéd b3 Vollime int CHAPTER III SURFACE INTEGRAL EQUATION METHOD FOR INTERACTION OF MICROWAVE WITH BIOLOGICAL BODY Only in recent years has beamed microwave power been proposed as a new technique in transporting high—level energy between two remote transmitting-receiving stations, and has microwave heating been considered as an effective method in hyperthermia cancer therapy. Although EM trans— mission and scattering phenomena have been major and clas- sical topics in electrodynamics since the end of the last century, uncertainties about the potential hazards of these high-power microwave radiations are still major concerns relating to the acceptability of these newly developed technologies. In order to fully understand the biological effects of microwave radiation, it is necessary to quantify the electric fields induced in the biological bodies when they are irradiated by the incident EM waves. Existing methods of quantifying the electric fields induced by EM waves are mostly based on the solutions of volume integral equations. Unfortunately, when the body is large compared with the wavelength of the incident 48 wave, t1 ficient mainly t it is be nique is In two coup between volume i between hate SChE the Potez the beams fully cor A150, the are becon tiVe in it field int for the h to Obtain interdcti John WdVe irra. 49 wave, the volume integral equation method becomes inef- ficient. Since for this case, theinduced electricfield is mainly concentrated in the region near the body surface, it is believed that the surface integral equation tech— nique is potentially more efficient. In this chapter, we develop a new technique based on two coupled surface integral equations. A good agreement between our solutions and the existing results obtained by volume integral equation method has been verified. 3.1. Introduction High-level, beamed—power microwave transmission between satellites and earth has been proposed as an alter- nate scheme for providing energy in the future. However, the potential hazards which exist in the interface between the beamed-power microwave and personnel should be care— fully considered by the microwave systems design engineers. Also, the thermal—heating effects of intense microwaves are becoming well known, and have been demonstrated effec— tive in hyperthermia cancer therapy. But the allowable field intensity and the permissible level of absorbed power for the human body still remain unknown. It is desirable to obtain some quantitative information about microwave interactions with human or other biological bodies. Johnson and Durney [6] have investigated the plane wave irradiation of a prolate—spheroid model of man based on the l Their t'r compared approach the inte an ellip. develOpe( equation, be used t drbitrari This VOlu tiOns the Small Elet electriCa; inSide the face. Con becomeS in Gination m approach. emplOy/ed bf induced ins Ohly infinj Smack? int and Miller tives of th grands of t; 50 on the perturbation theory described by Van Bladel [7]. Their theory is valid only when the wavelength is long compared to the dimensions of the spheroid. The same approach has been adopted by Massoudi [8] to solve for the internal electric field induced by a plane wave inside an ellipsoidal model of man. Livesay and Chen [9] have developed a theoretical method based on a volume integral equation, the socalled‘Tensorlntegral.Equation,which.can be used to quantify the electric field induced inside an arbitrarily shaped biological body by an incident EM wave. This volume integral equation method removes the restric— tions that the irradiated bodies have simple shapes and small electrical dimensions. However, when the body is electrically large, the EM waves cannot penetrate deeply inside the body, and will concentrate near the body sur- face. Consequently, the volume integral equation method becomes inefficient. For this case, the surface integral equation method seems to be a reasonable alternative approach. A surface integral equation method has been employed by Wu et a1. [10] to solve for the EM fields induced inside arbitrary cylinders of biological tissue; only infinitely long cylinders were considered. Another surface integral formulation has been developed by Poggio and Miller [11]; however, in their formulation the deriva- tives of the surface unknowns are involved in the inte- grands of the surface integrals. This makes the numerical 4—_A solutior Wang [12 conducti divides cells, b: It is de: the fielc gral equa tries mot In t integral field on Conducting field can field, onc B8for Section 3. tiOn 0f Ma and 3.3, r‘ inte ' :: .— +. I fFUVO vdF fFv VOUdF with the surface divergence VO-v defined as lim 1 —> A —> vO-v a |sl+o s é(no.v)dl where [S] is the area of a small surface element S enclosed by contour C, and fio is a unit normal vector on C directed exterior to S. It can be shown that V0 = V-fi§%, where V represents the conventional three—dimensional differential del-operator. l Theorem 2. If the surface field 3(f‘) is continuous on F, then ++ 13(3) x IF_[§(§') x v'¢(?,‘£')] dF' = —2n3(g) + +1 ME) x IFIB’G') x v'mEE'H dF'. The sigr gration, Also The Sign gration, The are both limits wh SUI‘facG F 3'3- The Let l radiates i and iSOtro % as Show: re910m ‘ Gs _ densities a I n l + l u The Sign fEi implies that r is the field point in the inte— gration, and lying in the interior side of the surface F. Also :3) 3N X IFe[§(?') x V'q>(§,?')] dF' = 2n§(E) + _> Ni) x IFGG') x v'q>1dF' The sign fEe implies that f is the field point in the inte- gration, and lying in the exterior side of the surface F. The eventual field point E and the source point ;' are both on the surface F. We denote with IEi and fEe the limits which we obtain if ? approaches the point E of the surface F from the interior or exterior, respectively. 3.3. The General Solution of Maxwell's E uations W ______________________________Jl___ Let us consider the problem where a finite source region Gs, consisting of electric and magnetic sources With volume current densities of 3e and 3m, respectively, radiates into a lossless, unbounded, homogeneous, linear, and isotropic medium with permeability u and permittivity 8, as shown in Figure 3.1 where we assume that the source region Gs is enclosed by surface Fs. Based on the continuity equation, the volume charge densities are given as 54 .u usaom um Am _mv pamam Em cm :Hmwcamfi T. . + . . . pom rm wpfl>fluuflaumm mom 1 huflaflnmmEumm :uflz wommm msomcmmoaon m Opcfl mumflpmu mmoHDOm “couuso mmmse .AEm .mmv wmfluflmcwp pawn luau momMHSm ucwHw>stm pom REM .whv mwfluflwcmp ucwuuso wEDHo> >Q Uwufluouomummu mfl mm mommusm an bmmoaoco m0 coamou mOHSOm < .H.m Gunman where a the amt Frc the elec the abow Qel and rm 7:4 Hit: with o = ing the MT; is . Now the equix 55 . . . wt . where a time variation of e3 has been assumed, and w is the angular frequency of the sources. From the general solution of Maxwell's equations [14], the electromagnetic field at any point ; maintained by _ —> —> the above sources can be expressed in terms of Je’ ml Qe, and Qm as E - —£—f {-3 x v'o + gfiv'o — 'w E’o] dG' (3 l) " 4w Gs m e j u e ' + a — l f [ 3 x v'o + va-o ' 3 o] dG' (3 2) H(r) - Z? Gs e 7? gas m . —> —> -jklr-r'l with ® = E—T?:¥TT—, where E' is the position vector locat— ing the source point inside the source region Gs' and k = w/EE is the wavenumber in unbounded space. Now let's define the following source densities of the equivalent surface currents and charges on the closed surface FS Se E fi' X E 3m 2 —fi' x E/no qe = % Voge qm = % V0.31“ where fi' is the outward unit normal vector on Fs’ and no = Vu 78 is the intrinsic wave impedance in free space. 0 o Th can als‘ [14] where C The last as the b 3.4. Di To 1 first Cox that a f: irradiate lregion Gs aSSumed t mittivity With a ho Mo and co, Ed is def Where E I‘ J ductivity 56 The above electromagnetic field (E, E) at position 2 -+ can also be ex ressed in t f ? p erms 0 Je’ jm’ qe, and qm as [14] z + _ l f + , qe , . e , L(r) — '4—1; FS {—nojm X V Q + ?V q) — jwlljeQ] dF (3.3) fi(§)—if [l xv'o+qmv" ' lo ' s where ® has the same form as that of eqs. (3.1) and (3.2). The last two equations are very important, and will serve as the basic tool in the following derivation. 3.4. Derivation of the Coupled Surface Integral Equations To derive the coupled, surface integral equations, we first consider the geometry shown in Figure 3.2. We assume that a finite conducting body G enclosed by surface F is irradiated by the electromagnetic wave from the source region Gs' The medium in the region external to F is assumed to be free space with permeability “o and per- mittivity so. The finite conducting body is constructed with a homogeneous material characterized by permeability U and complex permittivity Ed' The complex permittivity 0 Ed is defined by where Er is the dielectric constant, and o is the con— ductivity of the body. The source region Gs is 57 .>Ho>fluommmoh rAUm rpmv can Amm rmmv mm pwuocop who xpon map as moonwcfl Ucw Eouw cmuwuumom mpaoflm Em one .mm mUMMMSm map so Umcflwoc mum AEn .omv moflpflm . . oh wonSOw may EOHM moanewmu Aam remv paoflm ucmpflocfl onu an Uwumwpmnufl we to hufl>flpuflfiuwm onmEoo m Uco 0: zuflaflnmeuwm m SpHB .m oommusm >9 pmmoaocm tam woman omnm CH pomHoEEH .0 moon mcflwuspcoo muflcflw m .m.m mosses mm+emum E u n mm.+flm m u mm QM? £< l K N A w hwy lem Hwy .m m .m charact ? 1 (1e: 1m Le by the 2 body by I llEldS 5 (id, id) Sider th I) TO the regi V X DEfinlng Current (3 V x Where 3% duction CL of Seq e as 58 characterized by the equivalent surface current densities ? + (je, 3m) on surface FS. Let us denote the incident field which is radiated by the sources in the absence of the finite conducting —>' —>' body by (El, H1). With the body present, we denote the fields scattered from and induced in it as (Es, fig _) (Ed ) and , g ), respectively. Now we are in a position to con— sider these three sets of EM fields separately. I) The Scattered Field (ES, fig): To begin with, let's consider Maxwell's equations in the region internal to F. They are +d . +d . +d V X H — jweOE + jw(ed to) E (3.5) v x Ed = —jwuofid. (3.6) Defining the last term of eq. (3.5) as an equivalent volume current density, then eq. (3.5) becomes v x fid = jweofia + EZq (3.5') where jeq _ . ) Ed (3.7) e : jw(€d _ Eo ' After substituting the expression for Ed into eq. (3.7), we may see that 3:q consists of two components: the con— duction current and the polarization current. We may think —> . . . I Of Jeq as a current eXisting in free Space as shown 1n e 59 luau coflpospcoo och .mpcmcomeoo oBu mo m ucmam>flsvm mach soon was weemcfl pmpMCHmHHO mm .m.m mosaflm ’ \\ II \ Ir \ o \ ow re_ a . . o u I u a; .s I QM. \\\ I ’8: ~\ . s I \ C v." . n s I m N ” Ami: wily \ o s ~ Figure and (3 include the see current Hence, written Ill 60 Figure 3.3, since only no and to appear in eqs. (3.5') and (3.6). It is to be noted that the source region GS is not included in Figure 3.3, because 3:q is the only source of the scattered field (ES, as). Furthermore, the equivalent current 3:q radiates into an unbounded, homogeneous space. 5 Hence, based on eqs. (3.1) and (3.2), E and ES can be written as Qeq +5 — i _6_ l _ ' eg I — 4n IGI E0v of jwpofie of] dG +5 _ l +eq ,. I — 4—TT fGiJe X V @f] dG where Q:q = V-3:q is the equivalent volumecharge density, 8 |k_1. -jk jf—f'l Qf = E—ngsTT——— is the Green's function for free space, r—r and k0 = w/uoso is the wavenumber in free space. Now let's define the equivalent surface current den- +8 . +5 sities in terms of the scattered field (E , H ) on the sur— face F as fiszfixfis e 3: a —fi x ES/n LO. shown 11 +‘S 1 n J ) raoi m We can e (3-3) an the EM f equivale 61 where n is the outward unit normal vector on F. From the equivalence theorem [15], [16], we know that the equivalent surface currents (3:, $2) on F can support (E5, is) external to F and zero field internal to F as shown in Figure 3.4. Since the surface currents (3:, 3:) radiate in an unbounded homogeneous space (free space), we can evaluate the EM fields supported by them with eqs. (3.3) and (3.4). From the uniqueness theorem, we know that the EM fields so calculated will be those postulated by the equivalence theorem. Hence we have s *5 _ 1 ts qe .- _ . ts , E — Z? fF[—nojm X V'®f + Egv of jwuoje¢f] dF (3.8) external qs to F *5 _ l is . m , _ . ?8 F' 3 9 h — I? fF[ je X V'Qf + KEV Qf jweonojm®fl d ( . ) as the scattered field at any point in the region external to F. In the region internal to F, the above two integrals vanish, i.e., s q _ l +5 u _2 v _ ' lso ] dF' (3.10) 0 ‘ Z? fF[_nojm X V ©f + 80V CI’f j“’“oje f internal 5 to F q es — _L +5 ! _IEV'Q) — 'we r] (I) ] (3.15” (3.11) O — 4h fF I 3e X V c1)f + no f 3 o ojm f because there is a null field internal to F. 62 .oommm omum hm pmomammu coon mm: 6 coflmms >UOQ one .. oommuom so pane“ pououumom can mo mEHou CH poCHmop can scan: .Amm .va mucmuuso wommnsw ucmam>flswo can an poCHmuCflmE mum xpoa map mo Hoanoucfl map :H macaw chow m can xcon och mo Hoaumpxo orb as Amm .mmv pfimflm pouwuumom ose .v.m mnsmflm o v I O O .I/ w a 1 .ss \ II .\\ ’v 5‘ <(fl' ~ s m . o . Ame me A: win .. L. + s ’v 1i‘c - w - a, \ II \\ I! o o rm II To conside region maintai of G. set O #41 terms 0 L—Jer :3 OJ 0 (do: Again, f that, Wi homogene complex SurfaCe field (E i“ the in him ‘(3 fields; a 63 II) The Induced Field (E d +d ,H): To investigate the induced field (Ed, fid), let us consider Figure 3.2 again. This time we treat the interior region of G as source free with the induced field (Ed, Ed) maintained by the sources existing in the exterior region of G. As for the case of scattered field, we may define a . . . e + . set of equivalent surface current denSities (3:, 3:) in terms of the surface value of (Ed, Ed): 3: E fixfid 3: E —n X Ed/nO new: Again, from the equivalence theorem [15], [16], we know that, with the exterior region of G being replaced by a homogeneous source free region with permeability no and complex permittivity Ed’ the negative of the equivalent surface currents (3:, 3:) will support the total induced field (Ed, Ed) in the region internal to F and a null field in the region external to F, as shown in Figure 3.5. Equa— tions'(3.3) and (3.4) can be applied to determine these . +d +d fields; a sign change is necessary since we define (je, 3m) 64 .Uo wufl>flpwHEMwm onmEoo can 0: xpflfiflnmeumm cuflz EDHUQE m wn cmomamwu ma M O# Hocuwuxm coflmou 039 .m wowmnsw co paoflm UmUDUCH wcu mo mEuwu as Umcflmwp mum scans .Amm .mmv mucwuuso mommusm pcoaw>fl5qm och wo ®>Hummwc osu mm UmcflmucflmE mum >603 ms» mo Moflumuxm wsa CH madam oumm m can >603 onu mo Hofluwpcfl Gnu ca Apm .Umv pawflw owospcfl one .m.m musmflm \ II E I ml // H . m. m , a A? an _ I U 0 a I, o . 1 \\ macaw ouwN Ir \\ \ It 0 .5 _ . . x / ‘I‘C m - < ml — .. 0+ / x If \\ Illu‘ m 0 lo tion for the comp; nal to p, beCause tr III) T0 in Cons ide r t :1 +i (L1H),\ rents 0n tr +i\A Je:n 65 in terms of the unit normal vector n which is pointing into the source region. Hence we have +d - ‘1 _'>d I‘ (g . ‘.*d I E _ Z? IF [nojm X V @d _ Engod + jwuoje¢d] dF (3.12) internal qd to F rd _ _l_ _+d ,- _ m ,- . ?d , H — 4” IF [ 3e X V (Pd EV (Pd + ngdnojmqud] dF (3.13) as the induced field at any point in the region internal - e‘jkd 73—va . to the surface F, where @d = ——T:—%TT—— is the Green'sfunc— r- tion for the conducting body region, and kd = quoed is the complex wavenumber in the body. In the region exter— nal to F, the above two integrals vanish, i.e., d _ l ?d ,- _ qe ,. . +d , O — Z? fF [nojm X V @d 55V Qd + jwuojeéd] dF (3.14) exuxnal qd to F — 1 —Id " — —fl " ' Id¢ dF' 3 i5 0 — Z? fF [ 3e X V @d “0V Qd + jwsdnojm d] ( . ) because there is a null field external to F. . +i +1 III) The Incident Field (E , H )z . . +i +i To investigate the inCident field (E , H ), let's consider the geometry shown in Figure 3.2 again. Based on (Eil fii), we define the following equivalent surface cur— rents on the body surface F: Fr the negg ing in t the tota Surface This is .- SUP; the Compl the IFree ‘ surfaCe c1 (ii: P71) 1' the region homogeneOu‘ Obstacle~rt fie1d_ Thi Again, Um EM fiel< \ ‘ Ei: 1 I 4n- F [ 66 JZ—nXE/n 1-: .n qe w Vo 3e i = i .Ii qm w Vo 3m' From the induction theorem [15], [16], we know that the negative of these equivalent surface currents, radiat— ing in the presence of the conducting body, will support the total induced field (Ed, Ed) in the region internal to surface F, and the scattered field (ES, 38) external to F. This is illustrated in Figure 3.6. Suppose that we are considering an extreme case with the complex permittivity of the body Ed being replaced by the free space permittivity so, then the negative of the surface currents (3:, 3;) will support the incident field (Ei, fii) in the region internal to F, and a null field in the region external to F. Because the entire space becomes homogeneous (free space), there exists no real scattering Obstacle—~the conducting body, so there is no scattered field. This situation is shown in Figure 3.7. Again, eqs. (3.3) and (3.4) can be used to express the EM fields shown in Figure 3.7. We have i q +i 3 x v'¢ — EEV'®f + jwuofié¢f1 dF' (3.16) O E =fifFM internal uDF 67 mam wwnm EH pmmeEEH we pm aufl>HHUHE HHHQMOEHmm nwflz w apon mSB .m wowmusm o mEku Ce Umcflmwp who SOHSB p fim twmv HM>H5W0 wsu mo ®>flummwc och >Q.Uw:wma umnxw man 2H Amm .mmv meme“ uwuwnumom muse man an new +emv Al weweu emoswce was .m.m museum A z. m P.m+v 68 CH emceuwe mum nouns .A .m momflusm :o pamflm ucmpflocfl mnu we mEumu Hm .wmv wucmuuso mommusm ucme>H5wm map mo m>flpmmwc msp wa pmcflmucfimE mum hpOQ mnu mo HOHkuxm may CH pamflw oumN m pcm >UOQ mnw wo uoflumuCfl map CH A ucmeeoce man u .Hmv pamflw tmowmm wmgm wn Umomamwu o coemmu Mpon.m:n snflz .>.m mudmflm III E ’ m \ ire . +, n . . . pawam OHwN o o s . cl w x 1 \ I \\ I/ \\ a O ‘5 . . h . z ’a «1 c . . mm I ~ I z \ 7r / \\ II\ o o b 21': J» II I as the to F. exteri field 1 tered f derive results Si rior re - ~> Point ; the lim‘ electrb may Wri1 H. raw, 57¢ ll WhEre f J, H1 theoreln _ internal i- - q1 to F .1 _ l _?1 , m . ?i. h _ ~4TT fF [ 3e X V (bf — EV'Qf + jweonojmupf] dF' (3.17) as the incident field at any point in the region internal to F. However, the above two integrals vanish in the exterior region of the body, i.e., i _ 1 ti ‘,. qe , . ti- 0 ‘ Z? fF [nojm X 7 9f ‘ 53V ®f + quojeof] dF' (3.18) eadernal qi to F _i _-.>i . _ m ,. . -.>i- ' 0 — 4w IF I 3e X V Qf EV (Pf + jweonojmof] d1? _ (3.19) Up to this point, we have considered the incident field (Ei, fii), the induced field (Ed, Ed), and the scat— tered field (Es, fis) individually. Now we are ready to derive the coupled surface integral equations based on the results we just obtained. Since eq. (3.16) is true at any point ? in the inte— rior region of G, if we let the field point 2 approach a point 3 on the surface F along the interior normal, then the limiting value of eq. (3.16) will be the incident electric field at the point 3 on the surface F. Hence we may write i . 3i x V'®F — EEV'Qf + jwuoiiéf] dF' m ._ E 51(3) = Z%'IF [n O .i O ‘ . 1 ' ‘ ' where fFi has the same meaning as that oescrlDed 1n _) theorem 2. After taking the CIOSS'PrOdUCt Of the above equatj at the :5) >< filte Applyi: I :3) ‘A’L—I F. The tinuOus <———---a=ttt———-:llllllllllllllllllllli"[ 7O equation with the unit normal vector fi(E), we will arrive at the following equation i _> . A l .1 , e , . +i n X 4” Iii [nojm X V 0f Egv Qf + jwuojle]dFu = 'U 3 . (3.20) Applying the same arguments to eq. (3.17), we have i . q . _A l _?1 , _ m , . +1 , n X Z? [E' [ 3e X V ©f E—V Qf + jweonojm®f] dF 1 o = _31 . (3.21) e Also, from eqs. (3.10) and (3.11), it is straightfor— ward to obtain 5 q A . l _z-S e l' _ ' +8. I = n X Z? fr. [_nojm X V'Qf + gtV Of JwUOJle] dF 0 i o (3.22) and s A t _ - m . ?S _ -n X i% IF. [3: X V”pf + E—v'®f _ jwaonoijf] dF' _ 0' —>l O (3.23) Before going any further, we should introduce the boundary conditions which must be satisfied at the surface F- The boundary conditions require that the tangential components of the electric and magnetic field must be con— tinuous across F, or equivalently we may write (3.22) :0 1x: i?!t~ Simila] Obtain 0n the tions f :1) >< as :7/l—“ I :‘> '5’: :T/I—l lowing 1 regular 71 ?s + +1 _ 1d 3e 3e — je +s + +1 _ +d 3m 3m _ Jm Now, take the difference between eqs. (3.20) and (3.22), and apply the above boundary conditions. We have qd A l -.>d , _ e ,- . #d , n X 47 IF. [nojm X V (bf av ‘Df + Wojeq’f] 0” +1 0 +i _ -nojm . (3.24) Similarly, subtracting eq. (3.23) from eq. (3.21), we obtain q A 1 i ,- _ m , . ?d F' —n X 47 fEi ['je X V (Pf TOV Qf + waOanm®f] d = _fii (3.25) e o On the other hand, we may derive the following two equa— tions from eqs. (3.14) and (3.15): d A +d e ' +d I = n X 4;,” f [nojm X V'Qd _ _€_Vl©d + jwuojqud] 6.]? 0 (3.26) Ee d qd d A #d _ m ~ - + I = -n X ZL'I [_je x V'Qd — fi_v'@d + jwednojméd] dF 0 TI Ee O (3.27) It can be shown that equalities exist between the fol— lowing limiting surface integrals and their corresponding regular surface integrals Subtract we find fiX However, hand side "U I—Q Applying ~ e(Nation , c' I. .qu It ShOUld ; beCaUSe V) 0 Then final; 72 /\ __ d d n A f v'@ dF' = A I - Ei qe f n X fF qe V Qf dF fix f Q V'Q, '= A d l' 1 Ee 9e O dF n X fF ge V @d dF . Subtracting the second equation from the first equation, we find that A' 'd' I.“ at 1.. n x fEi qe V Qf dF n X fie qe V Qd dF — fiqudV'(®-®)dF'=—fiXqud(-)dF' F e f ‘d F e f d ' However, since q: = % v5.33, the integral on the right— hand side of the above equation can be rewritten as d , _ 1 ,.?d _ , fF qe(®f - @d) dF — w fF (VO je)(®f 0d) dF . Applying theorem 1 to the right—hand side of the above equation, we can rewrite it as d _ _ l +d , _ , fF<1e(®f - @d) dF' — w IF e VO( —> vfi-E)=(A.W§+(B-WZ+XX(VXE)+Ex(vx$, it is easy to show that qd V’® dF' — n X f d V'© dF' Ei e f n X f 5e qe d _ _ 1 A 3d , . . - _ . — w n x IF Bye v ) v (@ ®d)]dF . (3.28) f Now, subtracting eq. (3.26) from eq. (3.24) with the latter multiplied by 60 and the former multiplied by ed, the result is 1 ed q: ,- . ad dF' so n X Z? fEi [nojm X V'®f - E;V Qf + jwuoje®fl - d edfi x Z%' f§e [no§g x v'ed - Eiv'ed + jmuofiied] dF' ii = -e onojm or Be becomes Similar.‘ derived Where K we norma of lengt: eQUatiom have als through ( 74 l A ‘?d —— { n X f e x I v _ A +d 4n [ Ki onojm V Qf dF n X IEe Ednojm X V'gi dF’] - [fi x f qd v'ef dF' — a x f qd v'a dF'] + Ei e Ee e d A . +d _->d [n X f jwu e j G dF' — n X f jwu e 3 © dF'J} Ki 0 o e f Ee o d e d = -e Xi onojm Based on eq. (3.28) and theorem 2, the above equation becomes rd 2 ti 1 A . 1d 2 = j - n X f [3k 3 (K (I) - q) ) + m K2+1 m 2W(K2+l) F o e d f _->d l 2 _ ' _3_ id . ' ' — ' 3m x v (K ed @f) + (3e V ) V (éd ®f)] dF . k0 Similarly, the other surface integral equation can be derived from eqs. (3.25) and (3.27), 1d _ ?i 1 A . 2d 2 _ _ ed , - _ 3e ‘ 3e + E n X (F [jkojm(K “Pa Ge) Je X ‘7 ”’d W + j ed , , _ , E;(3m v ) v (ad af)] dF where K = Ved7eo is the refractive index of the body. If we normalize all the quantities concerning the dimension Of length by l/ko, the above two coupled surface integral equations become (these two equations in similar forms have also been given by Miller [13] and Morita [19] through different derivations) the in( on the these t l/ko, t tions t< between With pel conduéti and 00m; 75 ‘?d 2 +1 1 A 2 —>d 3 3 ‘ an [3(an —<1>)' + m K2+l m 2n(K2+1) F d f 3e +d , . id - - 3m X V (K @d - Qf) + 3(je V') V'(q>d — of)] dF' (3.29) and +d +1 1 A ed ed Je—Je+4—flanF [3(K @d‘©f)jm—jeXV'(®d-®f) + . td - 3(3m ' V') V'(d — Qf)] dF' (3.30) Th . . +d +d e induced equivalent surface currents (3e, 3m) (or the induced surface field (Ed, fid» can be calculated based on the above two coupled, surface integral equations. In these two equations, due to normalization with respect to l/ko, the Green's functions @d and ©f have the forms of —'K)E—§'( e 3 I}?! 0d = ?_ ' _f 3 and Qf = if I) 3.5. The Special Case of an Infinite Plane Interface In this section, we apply the surface integral equa— tions to the special case of an infinite plane interface between two half spaces, a free-space in the region 2 < 0 with permeability “o and permittivity SO and a finite conducting medium in the region 2 > 0 with permeability “o and complex permittivity Ed’ as shown in Figure 3.8. We assume that the incident plane wave, which propagates in 76 .Qoflpomuflp N+ one he mCflbmmmmomm we to >hfl>flppflfihmm meQEoo cam 0: % )uosccoo wwflcflw m pcm mommm wmum so m>m3 wstQ m >9 mecHQEH uflaflhmwaawm hues Esflpwe mcfl thwQ wommuwuzfl ®#HQHMCH c< .w.m mudmflm mwmm «I N‘lllllllllucllltmw Nox. @I] \A we < 0: ¢[.% C % NOVHflI. Hm ( m m x Noxfl mm < m m x Nmuxml C < Noxhl H < Um.o: o .o the +2 the +x the +y and pr field and L'CH/ Q.) Where ( Wa conduCti prOblem, transmi t field (E Ed. 77 the +2 direction with its electric field E1 polarized in the +x direction and its magnetic field El polarized in the +y direction, is normally impinging on the interface, and produces a reflected field (E5, is) and a transmitted field (Ed, Ed) in the free space region and the conduct— ing medium region, respectively. Intuitively, these fields can be expressed as E1 = a El e-Jkoz +1 i —jk z . Ei -jk z , H = 9 H e o = y H— e o O in the region 2 <0 as _a E8 elk Z s . + A k 2 HS = y E— J 0 no Ed = x Ed e—JkdZ _ in the region 2 >0 a0 = 3 ad e-Jkdz y —— e d “a where nd = /u 75d is the complex wave impedance in the o conducting medium. By solving a simple boundary value problem, it is easy to show that, on the interface, the transmitted field (Ed, Hd) is related to the incident field (El, H1) by d 2 i E — 1 + K E 2K i (3.31) Hd = 1 + K H' eq. ( face previ( lowin< rent ( also Where ; in the N( currem last t» ing to thESQ t 78 It would be interesting to show that the results of eq. (3.31) can be deduced from the rather complicated sur- face integral equations which have been derived in the previous section. To do that, let's start from the fol- lowing definitions of the various equivalent surface cur— rent densities. . . . i + A +. A A A j: Z n X H1 - —z X y H1 = x g— o - H1 1 1 A A AE /\ 3; E -n X E— = z X x H_ = y 2— no 0 0 at z = 0 also +d_ 9d A.d jezfiXH—Xje -.>d:_fiXTE.d__A .d 3m — no - Y 3m where h is the unit normal vector on the interface pointing in the —z direction. Now, substituting the above expressions of surface current densities into eq. (3.29), we may show that the last two terms of the surface integration contribute noth— ing to the total surface integral. We should consider these two terms one by one as below. A -.>d 2_ , I) Let us consider the term n X fF 3m X V'(K 0d ©f)dF first. exPres 79 It is easy to show that + a x f jd x v'(K2® — e u _ A .d x v'(K2¢d - ef) dF' = o (3.32) is true, since the term V'(K2®d — ®f) has only two com- ponents, x and y components but not z component (nothing is z-dependent in this problem), both of which vanish after cross—producting with unit vectors 9 and 2 consecu— tively. A ?d II) Secondly, consider the term of n X IF (3e - V') I ' ._ G l V (@d -f) dF . + - Since the surface current j: is x-directed, we may express the first part of the integrand as ed , _ .d 3 (3e . V ) _ jex 8x' ‘ Because everything is z-independent, we may write the second part of the integrand as A 8 A 3 _ = X 5X—'(®d - (13f) + Y By'wd Qf)' Combining the above two equations, we have the following expression for the integrand: 2 +d ‘ A .d 8 _ (3e ‘ V') V'(d - @f) = x jex 5;72(©d éf) + 2 d 8 A ' —————— — 0 . y jex Bx'ay'(®d f) Hence] If we C we may where I algebra Ewe CO ~3 80 Hence, the integral considered can be rewritten as 8X f (3d . V') vl(q‘) _ Q ) dF' = A X f A .d 32 F e d f n F X Jex 8x'2 (Q " Q ) dF' + f1 X f A 'd 82 ( _ - ) dF' d f F y jex ax'ay' ®d Qf ' (3.33) If we choose the field point to be at origin; i.e., f = 0, we may write the Green's functions Qd and cf as —jKr' -jr' _ e - _ e Gd — _—ET_— , and @f — —_ET_ where r' = [f‘] = Vx'2+y'2. With some straightforward algebraic manipulations, it can be shown that 82 (e _ . e-jr'_Ke-jKr' + e-jr'_e'jKr'] + 3X'2 (1 Qf) _ [j r12 r13 _- n 2 —jKr' -jr' —jKr' 2 e 3r -K e . e —K e cos 6 [ r' 3] r'2 r'3 where e is the angle between the source-point position vector f’ and the x axis, as shown in Figure 3.9. Simi— larly, it can be shown that 82 e—jr'—K2 e_jKrl WMDd - (Pf) = Sine COSG [——E'—— Fl 8l J. we {r 8 +00 The coordinate system utilized in the infinite interface problem. Figure 3.9. In t) trari destr face unifo right last It is Side 0 have 82 - 3]. —2—— _ 3 _1, r r'3 In the above derivation, the field point has been arbi— trarily chosen at the origin. This assumption doesn't destroy the generality of this treatment, since the inter— face is infinitely large, and the field induced on it is uniformly distributed over it. Now, we must show that both of the integrals on the right—hand side of eq. (3.33) vanish. Let us consider the last term of that equation first. 32 2w f” -jr' ______ c — ' v = ' . _ fF Bx'ay'('d 9f) dF o Sine cose d6 0 [e _. , -jr'_ -jKr' —jr‘ _ -jKr' K28 jKr _ 33. e r)? e _ 3 I": ] drl It is clear that the first integral on the right-hand side of the above equation is equal to zero. Hence we have 2 F w(® " (I) ) dF' = 0. (3.34) f d f For the first term on the right—hand side of eq. (3.33), we may write that _jKr' 82 Q q) 00 I e_jrl_K e + — I : TT #— IF 3 ,:2( (i f) dF 2 0 [3 rl Then! SU We Obtai 83 'r' - KI"I 2 _ _' l e 3 —e j 1 d . c0328 d6 [e 3r _K2 e JKr ____.—TTT_——_ r + o r o -jr' ‘jKr' -jr -jKr' . e -K e ‘33 _“—‘_ET—‘———— - 3 e ?2 ] dr' m —jr' ‘jKr' w -jr' -jKr' _ _ e “K e _ e —e l 31/; ——————3?r—————— dr' Tyf r'2 dr co ' I ' - d e-Jr _e-jKr . I + 3K” ‘ J” = “j{ a;:-(-————Ej—————) dr' + ij — 3w ' l ' 1 1'1 -jKr _ -jr ' . rlfo [E—————;$————J + ij — 3w = o. (3.35) H Note that for the last step in the above derivation, the radiation condition is applied. After substitution of eqs. (3.34) and (3.35) into eq. (3.33), the result is .+ A - d g l _ I = n x [F (3e ~ v ) v (ed sf) dF 0. (3.36) Then, substituting eqs. (3.32) and (3.36) into eq. (3.29), we obtain .a _ 2 Bi ————¥1— -d 2 _ F' 3 37 3m - (K2+l) fi;-+ 2w(K2+1) je fF (K Qd ©f) d ° ( ' ) The integral term is easy to evaluate. We write down the result directly. _ 2 | _ ' _ IF (K éd -®f) dF — j 2fl(l K) Hence, eq. (3.37) becomes 84 .d 2 Bi K—l .d = —-— __ + . . Following a similar procedure, we can obtain another equa— tion from eq. (3.30), which is i . _ K-l .d J o + 2 3m . (3.39) Q: [Tl (D 3 Equations (3.29) and (3.30) have been simplified to become two linear equations with unknowns je and j:. The solutions of these two unknowns are e K+l no and .d_(2)§1_l jm — K+l n ’ 0 which also can be written as d _ 2 1 E _ (K+l) E and d _ 2K i H “Ia—1) which are exactly the same as eq. (3.31). We have shown that the coupled, surface integral equations can predict correct results for the simplified special case of an infinite two-dimensional interface between a finite conducting medium and free space. This .'1 example analytically proves the validity of the surface integral equations. In the following sections, we will develop a numerical technique for solving these surface integral equations for the cases of arbitrarily shaped, finite conducting bodies; after that, some numerical examples will be presented to discuss the accuracy of this numerical technique. 3.6. Review of Moment-Method Before actually solving the surface integral equa- tions, we should briefly review the basic steps involved in numerical implementation of such solutions, or the so—called Method of Moments (MOM) [17]. The idea of moment-method solutions is that the linear operator equation L(f) = g (3.40) where L is a linear operator, g is some known driving func- tion, and f is to be determined, may be solved by first expanding f in a series of basis functions fn with unknown weighting coefficients an, f = 2 an fn (3.41) n Next, a set of weighting or testing functions wm is chosen and an inner product (denoted by symbol < >) taken in eq. (3.40). After substitution of eq. (3.41) into eq. (3.40), the result is Z a < w , L fn> = (3.42) which may be written in matrix form as [lmn] [an] = [gm]. As illustrated in Figure 3.10, the idea when applied to the cubic box assumes that the box surface may be divided into square patches over each of which both com— ponents of surface current are assumed constant with unknown amplitudes (subdomain flat—top pulse-expansion functions). The testing functions chosen in this problem are 6 functions at the patch centers (collocation or point matching). Thus, by successively enforcing the surface integral equations to be satisfied at these match points, a system of linear algebraic equations is generated. Matrix inversion then yields the solution for the unknown coefficients of the current samples. 3.7. The Numerical Technique The validity of eqs. (3.29) and (3.30) has been checked in Section 3.5 through the consideration of a special case of a two—dimensional, infinite—plane inter— face between a finite conducting medium and free space. Since it is our goal to quantify the EM fields induced on the surfaces of arbitrarily shaped, finite conducting bodies, such as human bodies, it seems more realistic to consider a simulated human body with shape as shown in I I \ _L___L__ I \ 7. I ‘ i< \ I ‘ I \ rune—.- Figure 3.10. Division of the surface of a cubic body into square surface cells for moment-method solution using two— dimensional flat pulses as expan- sion functions and 6 functions as testing functions. 88 Figure 3.11. We assume that the body region G, being characterized by permeability no and complex permittivity 8d and enclosed by surface F, is illuminated by an incident plane wave which is propagating in the +2 direction with its electric field Bi polarized in the +x direction. The body is immersed in free space as shown in Figure 3.11. It is easy to see from Figure 3.11 that, for a simu— lated body, the body surface F consists of a number of plane surface elements. If the body is described in a rectangular coordinate system, which is the one used in Figure 3.11, the problem can be simplified by constructing the body surface with surface elements having normal vec— tors pointing in one of the six possible along-axis direc- tions, i.e., the in, the ty, and the i2 directions. For the purpose of clarity, we will name these surface elements according to the directions of their unit normal vectors. For example, the top surface element of the body head shown in Figure 3.11 will be considered as one of the +x surface elements, since its unit normal vector is pointing in the +x direction. It should be emphasized that a closed sur- face can consist of several surface elements with unit normal vectors pointing in the same direction, as can be seen from Figure 3.11. Before going any further, we should mention that we choose a plane wave as the incident wave, because it is a realistic approximation when the body is distant from a 89 fi lé" I if ” )--- w': I / 1JO, E:0 Lu- 1 I l l I '-----w-----}---- —>l ! j | r I I 7’ trivia). -- ’ 91 - y F G ' : r‘ _-_:..-.. --- 1J0, Ed'---,L-- __/ I l . f ,. __..r_..- _ Figure 3.11. A simulated human body is irradiated by an incident plane wave propagating in the +2 direction with a x—polarized electric field. The body is characterized by complex permit— tivity 5dr and its surface is subdivided into a number of square surface cells. Theinduced . -> -}- . equivalent surface currents (3e: 3m) at dif- ferent locations on the surface can be obtained by solving two coupled, surface integral equations. 90 radiating source, and it facilitates the analysis. Also, we choose the rectangular coordinate system since it is the most adaptable to arbitrarily shaped bodies. However, it is to be noted that the surface integral equations derived previously are so general that they can be applied to any form of incident field, as well as any orthogonal coordinate system. Now, let us consider Figure 3.11 again, where each surface element of F has been subdivided into several square subareas named surface cells. Each surface cell can have any arbitrary dimension which depends on the accu— racy desired. Furthermore, the center point of each sur— face cell is considered as a reference point which could be either a field point or a source point or both. The EM field induced inside each surface cell is assumed to be uniform. We will use the notation FX to represent the combined surface of all the +x surface elements, and use the symbol NX to denote the total number of surface cells included in Fx’ Similar definitions will be applied to the notations, F , N ; F , N ; F , N-x; F—y' N—y; and y y z z —x F—z’ N 2’ etc. In other words, we may say that the body surface F has been divided into totally Ncell surface cells, where Ncell is the sum of Nx’ Ny’ Nz’ N—x’ N—y’ and N . -z To solve the surface integral equations numerically by use of the moment-method, we must decompose the integral 91 equations first and then point-match them at the reference point of each surface cell. By doing this, a system of simultaneous, scalar, linear algebraic equations will be obtained. After transforming these linear equations into a matrix form, it can be solved by conventional matrix inversion techniques. To describe the procedures in more detail, let's start with the following definitions: m: the numbering index of the surface cells in each subsurface (FX, Fy’ FZ, F—x’ F—y’ F—z)’ when center points of the surface cells are considered as field points. n: the numbering index of the surface cells in each subsurface, when center points of the surface cells are considered as source points. n A : the area of the nth surface cell. _) v - rm: the position vector locating the mth field pOint, +m m m m where r = (x , y , Z )~ fn: the position vector locating the nth source . +n n n n pOint, where r = (X , y I Z ) Also, n m nm _ n m nm-In—E’m,xnmEx-x,y :y-y, m znmIZn-Z. and am = /Am/w: the equivalent radius of the mth surface cell. 92 We also like to introduce the following associated Green's functions; the reasons behind these definitions will be clear when the surface integral equations are decomposed: _ mn --Krmn _ r _.Krnm nm .e3r —Kej e] _e] G1 =3‘______77—“_‘- + 3 ' nm nm r r .nm . nm _ __ _ _. nm nm _ e'Jr _ K2 e jKr _ 3 e _ K e jKr _ 3 3r _ e jKr 62 3 3 4 T rm 3: r . nm nm _. nm nm e-jr _ K3 e—jKr e jr _ K2 e jKr G? "'3 ——‘—____TT~————__ + 3 J I‘nm rnm ' ' (3.43) Gnm _ — e-jr _ K2 e—jKr 4 rnm . . ja -jKa - . - Ka e - e H‘rln=w(:)eja—3KeJ + m a and _. In _-5“ H? = 2w(jKe JKa - je 3 - jK + j). Now, we are in a position to decompose the coupled, Surface integral equations, i.e., eqs. (3.29) and (3.30). For convenience, we repeat them here, . ' . 2 _ —)> + l = 2 +1 - ————i——— n X f [3(K e - @f) 3e + 3m X 3m 2 3m 2 F d K +1 2w(K +1) (3.29') V'()<2d - sf) + jGe - V') v'(¢d - ef)] dF' 93 + = ei 1 A _ . 2 X a 3e 3e + g; n X fF[3(K ed - ¢f) 3m — 3e x v'(@d — @f) . ‘T , I + 1(3m - V ) V (©d - ef)] dF'. (3.30') Note that, without any ambiguity, the superscript "d" for the induced equivalent surface current densities has been omitted in the above equations. Let us consider eq. (3.29') first. To numerically implement the various surface integrals of this equation, we have to consider each possible case about the field point locations. I) First of all, we consider the situation when the field point is located in one of the +x surface elements. Which means that the unit normal vector at field point, n, is the same as the X-direction unit vector X. For this case, we know that 1 S m S NX. Based on the previous defi— nitions, it is easy to see that the last integral of eq. (3.29') can be written as n x fF(je . vv) v-(cbd - ef) dF. = x x {[fFX + ny + fFZ + IF + IF + [F 1(3e . v') v'(©d — ef) dF'} (3.44) where we have decomposed the total surface integral [F into six different subintegrals, one for each subsurface. For example, IF represents the integration of the integrand x over all the surface elements having a as the unit normal 94 vector. The remaining five subintegrals in eq. (3.44) have similar meanings. Now, as an example, let's carry out the first subintegral of eq. (3.44). 0 _'> o ' l _ |=" . A - ' A a -- X fFXOe V ) V (‘Dd (Pf) dF X X fFXHjeY y + jeZ 2) (x —E)X' + a)1[(§<—3—+9—+23i)(® -)]ar' A_ A— ' l l Yayl +Zazl 3X 3y Z d f 2 2 2 2 A A . 8 . a A . 8 . 8 = X X fFX[X(Jey Bx'ay' + 3ez Bx'Bz') + y(jey y'2 + jez By'Bz') + 32 32 Z‘JeyW+3ezgp>1(©a’@f) dF 2 2 2 2 A . 8 . a A . 8 . 8 =A' —-+ —— +2 ——.+ )1 X k fo[y(3ey 8y'2 jez By'az') (jey By'az jez 82'2 ’ _ I (ed (bf) 61“ e—jK£ e-jE h + f' we ' — = = r — Since Qd — E and ®f — w ere r , may show that 2 8 _ = + . _ 2 G ———7(©d Qf) Gl (y y) 2 By' 2 2 a - — + ' — 2 G ———§(®d Qf) Gl (z ) 2 32' and ___—'°‘2 )( ' ) G - = ' - z - z By'az'md CIDf) (y Y 2 _l 95 where G1 and G2 have the same expressions as that of Gim nm . . and G2 defined in eq. (3.43) except rnm being replaced by r. Hence we have X - v') v'(q>d — @f) dF' =—§z IF {jey(y' — y) (z' — 2) G2 + X n ' _ 2 ' A u ' 2 jez[Gl + (z 2) G21} dF + z fFX {jey[Gl + (y - y) G2] + jeZ(Y' - y)(Z' - 2) G2} dF'. The integrals on the right-hand side of the above equation can be numerically implemented by summing up the products of integrands and areas of the corresponding surface cells with a special consideration being paid to the singular terms which occur when the source point coincides with the field point. This process is completely based on the moment—method technique with two—dimensional pulse func- tions as the basis functions, and 6 functions as the test— ing functions. So we may rewrite the above equation as d f :yzgxypm gunG€m_+ x n=l na‘m NX A '1» A o xfo (3e - V') v'(c1> -<1>)dF'=—y Z [J . Nx nm nm 'n Ana?“ + znm'znm (52% + 2 z [32y And qaf)dF yingl nzn An[fi ynm gmiGmn x ey 2 .n mm mm nm .m jez(Gl + z '2 GSm)] + jez HT} + A NX mm hm hm fin; An[j:y(Gm+y ‘-y r )+ 1 1 ”2 n#m .n Inn nm Im1 in Hm _ + , where 1.5 n15 N . (3.48) Jez y z 62 ] Jey l} x This concludes the evaluation of the first subintegral in eq. (3.44). The remaining five subintegrals in that equa— tion can be carried out similarly. Combining these results with eq. (3.44), the latter will become mnzmncgn+ ' NX N"? rfi.n A _.> o y 1 _ ' =—A Z + Z A j y x x fF(3e v ) v (cpd 2f) dF y{ (£25 IE1 ey g——_[ 98 N N- jn (Gnm + znm-znm GnHS] +( 2? + By) An [j; Xnm an Gnm + ez l 2 n=1 n=1 2 .n (Gnm nm an N—z erI DHI nm jez l + Z '2 nm)1 + (n E: + nE 1) An [j; z G2 + n rmI mn nm m . jey y G2 ] + jez H? } + Nx N Gmn mn nm 2 ( Z 4— Z An . dml (n=l n=l) [jey(G l + y y 2 ) + £1 fri mnGmw + (N _+NE) An [. Xnm nm nm + ez 2 n—l 11:1)32X y 2 N N. .n hm Zrmi Gmn 2 z n 11 nm nm + Z + Gm“ jez y G2 ] (n=l nil) A [jex X y 2 + .n mm mm an jey(ql + y -y @)]-+jey H; L where l S m S Nx' (3.49) This concludes the investigation of the last integral on the right-hand side of eq. (3.29'). If we repeat the whole process for the second surface integration of eq. (3.29'), we will find that it can be decomposed into the following form: A + ' 2 _ I _ _ _ XXfijXV(K d cpf)dF —y{(n_l+n£l)A x G3 jmy nfm (N3, N21)A An yannm n (1:2 + 22) An[jn yme + _ n=1 3 jm n 1 mx 3 99 . Nx N—X .n III-.1 nm A n mm m n x G } + 2{ Z 4- Z ' n¢m (éé’ + NEY) An[ n Xnm Gnm .n nm Gnm n=l n=l 31112 3 NIX Z 3 ] — NZ N_z n hm nm n Z +— Z A ' 1 < < (n=l n=l) 2 G3 jmx}, wqere l _ m _ NX, (3.50) and 63m is defined in eq. (3.43) . It is noted that the sin- gular terms have no contribution to eq. (3.50). Similarly, the first integral term on the right—hand side of eq. (3.29') can be transformed to become N N- A . 2 2. A X X n Im1.n 7, < Q - Q ' =-\ { Z + Z + X A F je( d f) dF / (n=l n=l) A G4 ez n¢m 11.n .m n A r fi§ N'X n ma J1 gm] +3 H’qu +Z)AG 3 + n l n=l 4 e2 ez 2 n=l n=l 4 ey n . NZ N.z n Ill“ 1'1 .m ' l < m < N I (nil + n21) A G4 JBy + jey HE }' Where ‘ _ X and 62m, H? are defined in eq. (3.43). If we enforce satisfaction of eq. (3.29') at each ref— erence point on the subsurface FX, with the help of eqs. (3.49), (3.50), and (3.51), we can obtain totally NX vector equations, i.e., one equation for each matching point. Furthermore, it is possible to decompose these NX vector equations into 2NX scalar, linear algebraic equations Since 100 each surface current possesses two tangential components. Based on eqs. (3.29'), (3.49), (3.50), and (3.51), it can be shown that the following are those 2Nx linear equations: NX N_X z + Z)-[—jn w (nm)-jn w (nm)+jn '1’(nm)]— { ( n=l ey 1X ' 62 22 ' my 3x ’ jm H(m) +jIn CLUE)“: (mm) +jn ‘9 (mm) + ez . jex le ez 22 qu N—z n .n .n ' - + Z W (nfin + W nJm + 3m 4’3y( n m) )1} {( nEl n_:) Dex 1y ) Jey lX( n .n .mi ' — , = m1 (3.52) jmx W3y(n,m) jmy kP3x(n H0] } jny NX N_X .n . .n .n + {( £1 + E1)'[jey v2y(n,m) + jez wlx(n,m) + jmz l1J3X(n,m)] N N‘Y n .m .n]. . 'n + ' \y nlrn _ jey H(no + jfim C } + {(nE: + nEl) [jex le(nlm) jez lx( ) N_z . .n . + 3:“ w3z— Jez le(n,m) - 3:12 I3X(n,m)1 } - HEY {(n=l + NEy l) [j; (n ,m) + jez WlX(n, m) - jmx W32(n,m) + .n sz LP,3X(n m)]} - {(nNEZl + :E:) [jex ‘l’lz(n m) + jey W2y(n,m) — niém .n .m . . ' jmx w3z(n,m)] - jey H(m) + 3:; c}: 4n 3:]; (3.56) also, N- n {( nEl + nEX l) [j jey le(n, m) + j; le(n,m) + jmy W3z(n,m) _ Ng n jmz W3y(n’)m ]} + {( (n_ l + NEY)- [jex L1’2X(n,m) + jez ‘i’ly(n,m) - NZ jmz W3y(n, ,m)] } + {( nEl + :E2 l) [jex W2X(n, m) + jeyw lz (n,m) + nfiu .n .m .m _ .mi jmy 1P3Z(n,m)] + jex H(m) + jmy C)— 4n jmy (3.57) where l 5 m 5 Nz' Similarly, we can obtain 2N_X, 2N_y, and 2N_z linear equations by enforcing eq. (3.29') at various reference points on the subsurfaces F_X, F—y’ and F_z, respectively. We list all these equations as follows: 104 IV) fi = -§: Nx -x .n .n .n m >: + z - — \y , - _ - _ “n=l r121) [ jey 1x(n m) jez W22(n,m) + jmy W3x(n,m)] jeZ H(m) n l N .m \I‘Y .n .n — + me C} { (11:2: n§1)' [jeX le(n,m) + jez W22 (n,m) + ij W3Y(n,m)]}- {(nEZl+ N22» [jn .n n—l ex ‘Ply(n,m) + jey Wlx(n,m) + .n .n __ .mi jmX WEE/(n,m) — jmy W3X(n,m)]} — 4n jmy (3.58) also, HEX + :E?) [jn (n,m)+j: zly‘l’lx(n,m)+jn (n,m)]+jm H(m) — ey L1123/ mz 3x ey mZC} + {( n=Zyl+N nZy-) [jex lPlz(n,m) + 3:312 ‘Plx(n,m) — N-z .n . .n jmx 1iJ3z(n' m) + jmzw 3X (n m H } + {( (mg: + nEl) [jex W12 (n,m) + mi =— (3.59 jey lPei/(n, ,m)- jmx l1’3z(n, m)] } 41T j1le ) where l s m g N—x' V) fi =-9: {(n EX + :2?) [jey ‘l’lxm, m) + jez \Pzzm, m) — jmy W3X(n,m)]} + 105 §¥ N.y {(n=l +n::)' [jEX lefl'l, m) + jez 9’22“}, m) + jmxw 3y (n ,m)] + n jEZH(m)-jm c}+{(nEZl + NE? [3“ ex ll’ly(n,m) + jey ‘ylx(,n m) + .n _ .n _ .mi jmx l1’3y(n,m) jmy ‘1’3X(n,m)]} —-4TT jmx (3.60) also, Nx NX n n E _ - _ . {(n=l + nEl) [— jey Mlz( m) jez le(n,m) jmy W3Z(n,m) + .n sz W3y(n m)] } - { (mi? + NEl) [jex W2X(n,m) + j z le(n,m) — NZ N_Z .n .m .m . sz w3y(n,m)] — flex H(m) - 3m c}—{ (nEl +nEl) hex “’2xm' m) + .n .n __ .mi 3ey W12 (n,m) + 3my w3z(n,m)]} — 4n 3mZ (3.61) where l g m < N {(E+Z)[3 (n,m)]}- jn .n — W ey W2y(n, ,m) jez ?lx(n’m) jmz 3x N N _y n .n _n C l - W ’ + {(nE: + nil) [jex le(n,m) + jeZ Wlx(n,m) me 3z(n m) .11 3'22 w3x(n,m>1 } — {(n E: + NE :-) [jex wlz]}+{ N22 N'Z ‘n -n 3H2 3y n,m (n=l + n21).[jex W2x(n,m) + 3ey le(n,m) + nsm .n .m _m _ .mi jmy W32 (n,m)] + jex H(m) - jmy C}——47T jmy (3.63) where l g m S N-z' Up to this point, we have obtained totally 2Ncell (Ncell = NX + Ny + NZ + N_X + N_y + N_Z) scalar, linear algebraic equations based on eq. (3.29'). Repeating the same procedure for eq. (3.30'), we can obtain another 2Ncell linear equations. We therefore have totally 4Ncell equa— tions, which are adequate to solve for the surface unknowns. For the purpose of saving space, we will not list the other 2Ncell equations; however, since eqs. (3.29') and (3.30') have very similar forms, they can be carefully deduced from eqs. (3.52)-(3.63). The combination of these 4Ncell linear equations can be transformed into a matrix form as shown in eq. (3.64), which then can be solved by a conventional 107 Avo.mv 2v 2v 2? =v NI~Z 2v >I... _ /z_ 2v _xl Nu: XI.>I _ E I» I _X x :_ KlaN _ E xv.> _ E x».z _ E XI 2? >.N| >.XI ZV 108 matrix inversion technique. The matrix elements in eq. (3.64) may be obtained from the above linear equations. It should be mentioned that the matrix [M] shown in eq. (3.64) possesses a four-fold symmetry, which is easy to see from the coefficients of the linear equations. These symmetry properties of the equations have been found very useful for the computer implementation of the numerical method. The initial version of the computer program is implemented directly based on eq. (3.64). Unfortunately, this program has only limited usefulness due to the pro— hibitive requirement of the matrix size needed to ade- quately sample current variations. However, if the incident plane wave is decomposed into four basic modes (cosine and sine variations of E and H fields), an eight— fold symmetrical property can be found for each type of surface current. These symmetrical properties are shown in Figure 3.12. A final version of the computer program, with the above symmetry conditions imposed, is then developed. By use of this computer program, it is possible to reduce the matrix size by a factor of 8 when an eight— fold symmetrical body is considered since it is sufficient to solve for the surface currents induced in only l/8 of the biological body. The induced surface currents for the rest of the body can be readily obtained by intuition. Of course, since the original incident plane wave has been +. -.> 3 of COSE 3m of COSE e node node Figure 3.12. The symmetrical properties of various modes of the induced equivalent surface currents (je, 3m) on the surface of a cubic body. llO separated into four different modes, the final solutions must be the combinations of results obtained for each basic mode. More details about the computer programs will be discussed in Chapter IV. 3.8. Numerical Results The numerical technique developed in the previous sec- tion has been applied to solve for the EM fields induced on the surfaces of several finite conducting bodies. These results of induced surface fields obtained by the surface integral equation method (STEM), accompanied with the results of internal fields induced in the same bodies obtained by the volume integral equation method (VIEM) [18], will be presented in this section. It should be mentioned that although SIEM gives solutions for both the induced electric field and induced magnetic field, for the purpose of simplifying the presentation we will only empha— size the induced electric fields, especially the component which predominates. Figure 3.13 shows the vertical component of the induced electric field, Ex' in a muscle layer of 6cm x 6cm x 1 cm irradiatedknran EM wave of 100 MHZ with a vertical incident electric field of l V/m at side-on incidence. The conductivity and dielectric constant of the body are assumed to be 0.889 S/m and 71.7. The top figure of Figure 3.l3 shows the results for EX (magnitude lll X f EX magnitude (0.1 V/m) V V i v I phase (deg.) C 56:0.u5 :o.uo:o.37 :o.3u;o.33 70 8§67 6 365.0i62.1 p58.7.53.6 ..... 4----_p_-__L__-_L_---i--___ o 78;o.6u ,o.54io.47 30.40;o.33 72.3168 5 :6u.9{61.o (55,5;46.O _____ 4----_i__--J_____i____4_____. o.87;o.72 io.61;0.52 ;o.u3;o.3u 72.6:68.8 (65.0;60.u :5u.2;43.1 z 1 J 1 4 L ‘ f.= 100 MHz fl = £1 V/m 0 = 0.889 S/m E = 71.7 r ; Ex magnitude (0.1 V/m) r h d . o.u6io.38 io.36:o.35 0.35:0.u1 1::ji ase( €92 71.6:69.2 :65.3;61.7 58.0;53.6 ____ _____ I _____ i _____________ o.6@70.57 ; 53:0 5 49.0.55 i I '— lO 75.6{72.2 i66.2:60.o I I ! I n i * T O. . 0 IO. 6 74.8:71.3 65.9:60.5 i5u.8;u7.9 an‘gfiggb Z J : : . f’ / —————————————— 1—--- —--—-1-——--1 0.77:0.65 .59{O.56 I .55{O.6l : 6 6cm 1 3_ Volume integral equation method i ‘i\ Y .‘lix 1cm fit Figure 3.13. The x-components of the induced electric fields determined based on the surface inte— gral equation method and the volume integral equation method. and phase angle) on the surface of the body obtained with the surface integral equation method. In this numerical calculation, 1/8 of the body surface is divided into 21 subareas of two different sizes. The bottom figure of Figure 3.13 shows the results for Ex (magnitude and phase angle) at the centers of the first layer cells, or at y = 0.25 cm plane, obtained with the tensor integral equa— tion method. In this numerical calculation, 1/8 of the body is divided into 36 volume cells. Comparison of these two sets of results, obtained with a surface integral equation method and a volume integral equation method, shows a qualitatively good agreement. This comparison is possible because the body is electrically thin in the y—direction, and the induced EM field should remain quite uniform in that direction. The disagreement between the two sets of results occurs mainly over the vertical edges of the body where the Circulatory magnetic mode of the induced electric field has a significant contribution; the volume integral equation method usually gives poor results for that circulatory magnetic mode of the induced electric field. Figure 3.14 shows the horizontal component of the induced electric field, E2, in the same body under the same irradiation as the case of Figure 3.13. EZ near the central portion of the body, or the region near the X-axis, can be considered as mainly consisting of the 113 X E i 1 I Z magnitude (0.] V/m) 0. 06 0.18 l0. 27i0. 31 10. 3210.27 h d , i189 795. 7 :87. 0'82. 6 178.3171.0 ””3856 (eg ) X _________ _i__ i____1____- f0. 03i0. 08—i"0 1510.173 17:0 13 3 8! 4 z 0.. 191 896. u :86. 2E81. :76. 70. l l l ___ ___ _ __ ___4____t___h 6 .0 7’ 0.0ui.. 005 05 .04 Tr/ J1 7 O7 2 01 i0 3 93. 996 .3 , 85 l l 1 66m Surface integral equation method Ei1_c uzc wcagn.o cm. —gouuaccaw. «onmumazcn. —caLa:;;_~. “cl daaupc;¢c¢. atlmnncr,n. yr: «Gnu: i.c—. _ra.u:.:_—. —;I ~rlurrr.:—. wzouunrr.—. ":1 welcfroict «12.5.5.7 wan 7o_w74:_:|~ rc_mrpz_:tx vr_z It.~. n.1un;;.m~. _ v_acsum~. m cc¢mct:r:c. w:ou:arr—>. —:u.::o:r—. N:n_aaco:r. c ”w cc¢n::;sn¢. «r¢u.o_~_n. Ncapcsuatr. n ntei:cvmo n 11 Amourpzrrx. «rom.::~.~. Nclpcnsram. «inuccaamu. acou: nFQLchh—Nw NcluaLLLme u mzlwc:tu:mo ~ .s\:Izv.tD;:ou .pmzou.taau_: z waywazv Lgtw Away—m: ~2Lux ncqtz 7_>_u Ly< c2_a IUdu uc m__p1Lu:;a _30: uI» Tali... ital: .a : _ -nharir_.- . .l maltrmcunc.. .u a H.cl.._:.r~...n.n .: n m. 712:..7- a 1i na-.¢:anau.- .r a , Us: grain H .zt__:\uh3a\. :suit onxhoosa manurzu r _._.._._:u Ma. .._ .CIC—L p_~;u:u._t 57.2.“.2» h—ZD Ih~zv CDC—u t~hL2ca.c—.a m manuscrim... TiLfZKH.‘ — Aaupu>\uhac>vo.dcuulzuuzaz_. z yruw_ u>__aguac.rpr. zcxcz< mrzz. up34cazc2ommg.s Z—QFLx :u>g:>z_ wzc_paaou»2~ w:h M7.»_uvcm»?~ gag—zutrz vr_z:: v.<>4L»:_;:m:— c~?_ D_:—>—: c2.mr ~; _— C:;N.N: “azrm.:- mu Dam: Drape >L7g20mzm “Th :tutr 2m>_: ti< guzaazv :U< wa34a> cbpcsmhum _:~ ozc—huax_: 4<:P::_N< z_ n c4_—L L~zhgu4u aug::;~ LT» If! // A.U.uc00v .N.v wHQMB N:o..r. gurus... 7.1 .L 9:1...“- ~ Iii: :1 mo4uqcnmwcol L cootmfu...2..a a .:._:....:....u. c m wettoeuowc: 7-19.5.4“. r, 11 «L.rae~r»r.- _:-at¢m_;_. m ouvutwnratcu (3 I .anccm/u. A .muucapo _.~v..mazrcéuzgu3nra~u 2 :r.:; 2m>_o Lz... was...“ a LCUSCZ_ cc waqu; 2:: muczr:_cqi my... A.©.pcoov .N.v wHQwB PART 2 A USER'S GUIDE TO COMPUTER PROGRAM FOR INDUCED EM FIELD ON THE SURFACE OF A FINITE CONDUCTING BODY WITH ARBITRARY SHAPE The purpose of Part 2 is to explain the computer pro- gram used for quantifying the EM field induced on the sur- face of a finite conducting body when illuminated by an incident plane wave. The theoretical derivation and the development of this numerical technique have been discussed in Chapter III. Besides explaining the usage of this pro- gram, we will also present an example accompanied with a sample print out for the purpose of better understanding. 4.7. Formulation of the Problem We are considering a finite conducting body illumi— nated by an incident wave which, for the purpose of sim— plicity, has been assumed to be a plane wave. Mathe— matically, the incident plane wave can be expressed in the following form: Ei(z) = xe_jz = § [cos(z)-j sin(z)] (4.1) +i A l -jz _ A l . . H (z) = y——e — y——[cos(z)-j Sln(Z)] no no We like to mention a few points concerning the above expressions: 141 142 1 v The wave is propagating in the +2 direction, and has a time factor exp(jwt) which is suppressed. 2) All quantities with the dimension of length are normalized by l/ko. 3) The incident electric field has unit intensity, I V/m. 4) no is the intrinsic wave impedance of free space. The first step in utilizing this program is the numerical formulation of the problem; hence, the first thing to do is to visualize the shape, dimensions and the orientation of the conducting body with respect to the incident electromagnetic wave. The incident wave may illuminate the body either at normal incidence or at end- on incidence. To begin with, the body surface is divided into NCELL surface cells with NX, NY, NZ, NNX, NNY, and NNZ cells facing the positive—x, positive—y, positive—z, negative—x, negative—y, and negative-z directions, respec- tively. Each surface cell is a square with suitable dimen- sion in order to obtain optimum results. The maximum number of unknowns that can be handled is about 150. Since there are four unknowns for each sur- face cell, the maximum number of surface cells, without any simplifications, can not be greater than 40. This imposes a restraint on the physical size of the body. But, due to the symmetrical properties which exist in most Of the bodies we are considering, it seems adequate to 143 apply these properties and reduce the number of unknowns by a factor of 4 or 8. It must be mentioned here that in order to apply symmetrical conditions the incident wave has to be decomposed into four different modes named COSE, COSH, SINE, and SINH, respectively, as can be under- stood from eq. (4.1). The above-mentioned symmetry condi- tions are justifiable since most of the bodies considered (either biological bodies or other finite conducting mate- rials encountered in engineering) possess four or eight similar—looking segments and hereafter called quadrants, such that it is sufficient merely to calculate the induced electromagnetic field at each surface cell of the first quadrant and then convert them into the induced field for the rest of the body. After determining the symmetry conditions, the next step in the numerical formulation of the problem is the specification of the location of each surface cell, its physical dimensions, and the electrical parameters of the body. The size of each cell can vary but the electrical properties are assumed to be uniform throughout the body. Note that the central location of each surface cell is predetermined by the user; then the incident field inten- sities are automatically evaluated for each cell. 144 4.8. Description of Computer Program This program is also coded in standard FORTRAN and can be compiled on either FTN or MNF compilers. The pro- gram is symbolically named as "SURFLDS" with input and output formats on any undefined logic units in conjunction with four magnetic tapes, "TAPEl," TAPEZ," "TAPE3," and "TAPE4." "TAPEl"—"TAPE4" are the names of the local files which are used to temporarily store the computed results due to COSE, COSH, SINE, and SINH components of incident field. Program "SURFLDS" makes use of the following sub— programs: "MNSB"--is a subroutine mainly used to print out the echo information which includes all the mes— sages just read in from the input data files by the main program. "MATRI," “LEE"——are subroutines which generate the matrix of the linear equations set. The matrix elements are related to the associated Green's functions evaluated at various loca- tions. "ELEMX," "ELEMY," and "ELEMZ"--are subroutines which evaluate the matrix elements when source points are in the ix-, the iy-, and the :2— directed surfaces, respectively. In these 145 subroutines, the symmetry conditions have been imposed. "EXIM," "CEIM," "CHIM," "SEIM," and "SHIM"--are sub— routines which calculate the incident electric and magnetic fields for EXPZ, COSE, COSH, SINE, and SINH modes, respectively. In these subroutines, the incident electric field is polarized in the +x direction and has unit intensity, 1 V/m. "T"-—is a subroutine which converts the equivalent surface currents to surface field quantities. "PRNT"——this subroutine, as the name suggests, is a program used to generate the print out. "COEFX," "COEFY," "COEFZ," "FCNXX"-"FCNZZ," and "CEFXX,"—"CEFZZ," etc.--are subprograms which evaluate the elements of the Tensor Green's function. "CMATP"-—is the program being used to solve a system of N equations in N unknowns. It is actually a Gauss-Seidel method of numerical technique. A listing of the program “SURFLDS,” including all the subprograms called, is given at the end of this chapter. In the next section, the structure of the input data files as well as the associated input variables are explained in detail. 146 4.9. Data Structure and Input Variables Figure 4.3 shows a sample body with finite conduc— tivity. With origin of the coordinate system being chosen at the center of the body, we divide it into eight differ— ent sections which are called quadrants. The numbering system used is also shown in Figure 4.3. Note that the symmetry conditions exist if the physical dimensions of each surface cell in the first quadrant are the same as their counterparts in other quadrants, which is the case for this sample body because we assume that all the sur— face cells have the same physical dimensions. Based on these existing symmetry conditions, we only intend to solve for the induced electromagnetic fields at various surface cells of the first quadrant and then convert them into the fields induced at the surface cells on the rest of the body. Now we like to introduce the input data files before we go any further in determining the induced sur— face field. First of all, the sequential structure of the data files, the format specifications and the symbolic names of the variables appearing on each file are outlined in Table 4.3. There are totally five input data files. Only the fourth data file contains NCELL data cards, while the rest Of the data files contain one data card each. The input Variables associated with these input data files are dis— cussed in detail as below: 147 4th_.u=azcu ~=oueae=.c. u .pmzeu u.z—uuau_e uaahc guatzawux=m n >: w =.=_ LLc_>_= w. aacu uuuuxaw zuc. .rcwoaaz .mz—u u:— we antenasw tr maauu : un:: :21. = h>=:. c "~27. c "N: . c c nu: ux‘ uxaxp nor—.m- caaocu— :Z..U.n. at..mNu N— beccr. Lvaeeo— Dwerwo ... ans. _- Hccer. cabano- :.:m». a I mu. .— sctcrn rccoco— rerrho econ». .- sec—um. enema“. aeaoeu— egrw. z casei- u¢=mnc scecea- n.:«th. n otter. Escrso _=:e=.— Ewan—u; . ccrrv. neers. 52.5.... 21...». . coccro =.cmn. ...=r- :._.uezo— c 5.7.}. acurho ccgrwu r..:.v..o— .. carom. czcmno stern. ocean..— H .¢.tr.o uérh. astrho L..yu=.— — .xu a—~ o .71.. .2. A :2 7—. > .79 2—- x z scar; ou—m.a L=¢ aaatu_uquz=w :cau u: zc_n:.1_r =z< zo_—au=e .:— . :m._z-nr=; _uuae .1.:..;=a :._3 sec: .;::»w m.: a :< c: p.c:.u:v at. 2° 0:4... eco;::_ c: r:__:aem m. m.:. =.wnqmmam= Emumonm mo OHHM usmuso umufim .v.v OHQMB 161 100.00.01.10‘OOO I00... 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Finally, the seventh and the last output files list the total surface fields for the fifth quadrant in real— imaginary and magnitude—phase forms, respectively. The above output files of the program "COMBINE" are shown in Table 4.8. 4.12. Listing of the Program ‘ For the purpose of reference, the programs "SURFLDS" and "COMBINE" are listed on page5183 -229. For about 150 surface unknowns, "SURFLDS" requires approximately 170000B words memory, while "COMBINE" requires about 3000OB words. When the number of unknowns is more than 150, the memory required for "SURFLDS" exceeds that obtainable from CDC 6500 at MSU. To overcome this difficulty, the job may be submitted through Merit Network System for execution at other computing facilities. In this case, due to differ- ent operating systems involved, apparently different con— trol commands must be employed. 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