)VISSI_J RETURNING MATERIALS: Place in book drop to nan/Ines remove this checkout from —:——. your record. FINES will be charged-if book is returned after the date stamped below. INTERACTION OF ELECTROMAGNETIC WAVES WITH HETEROGENEOUS BODIES OF ARBITRARY SHAPE AND PARAMETERS by Huei Wang 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 1987 ABSTRACT INTERACTION OF ELECTROMAGNETIC WAVES WITH HETEROGENEOUS BODIES OF ARBITRARY SHAPE AND PARAMETERS By Huei Wang This thesis consists of two parts: The first part deals with the quantification of interaction of electromagnetic fields with finite heterogeneous bodies, and the second part presents an application of electromagnetic waves for detecting a small movement of a biological body behind a barrier. In the study on the quantification of interaction of electromagnetic fields with finite non-magnetic lossy bodies, some new numerical methods have been developed. These new methods improve the numerical accuracy of the existing tensor integral equation method. The induced electric field of electric mode which is excited by the symmetrical part of incident electric field can be solved accurately from the tensor electric field integral equation with the method of moment and pulse basis expansion. However, the induced electric field of the magnetic mode excited by the antisymmetri- cal part of incident electric field can not be determined accurately by the same method. The solution of magnetic mode can be improved by using an iterative loop-EMF method which is designed to calculate the induced electric field of magnetic mode by the incident magnetic field. Another alternative is to introduce an equivalent magnetic current to compensate for the discontinuity of the tangential component of electric field at the boundary created by using the pulse-basis expansion. The EM fields with finite lossy magnetic bodies has also been investigated in this research. A set of coupled tensor integral equations has been derived to solve for the induced EM field in a finite heterogeneous body which is irradiated by an incident EM field. This set of equations can be decoupled into a separate tensor electric field integral equation (EFIE) and a tensor magnetic integral equation (MFIE). Numerical solutions of the coupled integral equations and decoupled EFIE are compared. The procedures for calculating numerical solutions of these integral equations are also included. In the study of an application of electromagnetic waves for detecting a small movement of a biological body behind a barrier, a series of experiments were con- ducted to measure the breathing and heart signals of a human subject behind a thick layer of bricks with microwave life detection systems. A theory was developed to predict the transmission of a non-uniform plane wave passing through a wall. Experi- mental results on the detection of breathing and heart signals of human subjects behind brick walls of various thicknesses are presented. The basic principle of the microwave life—detection system is also included for completeness. To My Family ii ACKNOWLEDGMENTS I offer my most grateful thanks to many persons who helped me on this thesis. Special thanks are extended to Dr. Kun-Mu Chen for his sincere guidance and finan- cial support during my graduate study, under whose supervision this research was con- ducted. Thanks are extended to the members of my guidance committee, Dr. Dennis P. Nyquist, Dr. Hassan Khalil and Dr. Byron Drachman for their comments and sugges- tions. Thanks go to Mr. Y. Q. Liang for his beneficial discussions in the research work, also to Dr. D. Misra and Dr. H. R. Chuang for their guidance and help in conducting experiments. P I I sincerely thank my family members for their encouragement, especially my wife Yun, whose patience and understanding are truly appreciated. Finally, I owe many thanks to my departed father. iii Table of Contents LIST OF TABLES ..................................................................................................... vi LIST OF FIGURES ................................................................................................... vii I. INTRODUCTION ................................................................................................. 1 11. NEW NIETHODS FOR QUANTIFICATION OF INDUCED EM FIELDS IN FINITE, NON-MAGNETIC AND LOSSY BODIES ................... 6 2.1 Tensor Electric Field Integral Equation (Tensor EFIE) ........................ 6 2.2 The Limitation of the Existing Tensor Integral Equation Method .................................................................................................... 12 2.3 Iterative Loop-EMF Method .................................................................. 16 2.4 Equivalent Magnetic Current Compensation Method ........................... 21 2.5 Numerical Comparison between the New Methods and Existing Method .................................................................................................... 28 III. INTERACTION OF EM FIELDS WITH FIN ITE, HETEROGENEOUS, DIELECTRIC, MAGNETIC AND LOSSY BODIES ...................................... 45 3.1 Coupled Tensor Integral Equations ....................................................... 46 3.2 Decoupled Electric Field Integral Equation (EFIE) and Magnetic Field Integral Equation (MFIE) ............................................ 52 3.3 Coupled and Decoupled Integral Equations in Terms of Free Space Scalar Green’s Function .............................................................. 58 3.4 Numerical Solutions of Various Integral Equations with Pulse- Basis Expansion and Point—matching .................................................... 67 3.5 Discussions ............................................................................................. 79 IV. PLANE WAVE SPECTRUM ANALYSIS OF A NONUNIFORM PLANE WAVE PASSING THROUGH A LAYER OF LOSSY MATERIAL ........................................................................................................ 128 4.1 Plane Wave Spectrum Analysis ............................................................. 129 iv 4.2 Predictions of Field Distributions for a Nonuniforrn Plane Wave Passing through a Layer of Lossy Material ............................... 135 4.3 Brief Description of Microwave Life Detection Systems ..................... 152 4.4 Experimental Results of Detecting Movements of a Human Body behind a Wall by Using Microwave Life Detection Systems ................................................................................................... 155 V. SUMMARY ......................................................................................................... 172 APPENDICES ............................................................................................................ 175 COMPUTER PROGRAMS ....................................................................................... 186 BIBLIOGRAPHY ...................................................................................................... 208 3.1 3.1; List of Tables 3.1 Summary of agreement and convergence properties for the coupled tensor IE and tensor EFIE (I) ......................................................................... 123 3.1a Summary of agreement and convergence properties for the coupled tensor IE and tensor EFIE (II) ........................................................................ 124 vi 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 List of Figures An arbitrarily shaped non-magnetic lossy body in free space illuminated by an incident plane wave. ......................................................... Electric mode distribution of the x- component of the induced electric fields in the body based on 216-cell division and 512-cell division. ............................................................................................ Magnetic mode distribution of the x- component of the induced electric fields in the body based on 216-ce11 division and 512-cell division. ............................................................................................ A single layer of non-magnetic lossy body regarded as an impedance network system with N loops. ....................................................................... Illustration of the equivalent surface charge and the equivalent magnetic current created by the discontinuity of the electric field between neighboring cells. .................................................................... A homogeneous rectangular biological model with dimensions 6x6x1 cm illuminated by an EM wave at end-on incidence. .................................. Conventional EFIE solutions based on 36-cell division. (electric mode, f = 100 MHz) ......................................................................... Conventional EFIE solutions based on 288-cell division. (electric mode, f = 100 MHz) ......................................................................... Conventional EFIE solutions based on 36-cell division. (magnetic mode, f = 100 MHz) ...................................................................... Conventional EFIE solutions based on 288-ce11 division. (magnetic mode, f = 100 MHz) ...................................................................... Iterative loop-EMF solutions based on 36-cell division. (magnetic mode, f = 100 MHz) ...................................................................... Equivalent magnetic current method solutions based on 36-cell division. (magnetic mode, f = 100 MHz) ...................................................................... Conventional EFIE solutions based on 36-cell division. (magnetic mode, f = 300 MHz) ...................................................................... Conventional EFIE solutions based on 288-cell division. (magnetic mode, f = 300 MHz) ...................................................................... vii 18 24 26 31 31 32 32 33 33 35 35 I bx) I) 3.3 3.4 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 3.1 3.2 3.3 3.4 Iterative loop—EMF solutions based on 36-cell division. (magnetic mode, f = 300 MHz) ...................................................................... Equivalent magnetic current method solutions based on 36-cell division. (magnetic mode, f = 300 MHz) ...................................................................... Conventional EFIE solutions based on 36-cell division. (magnetic mode, f = 500 MHz) ...................................................................... Conventional EFIE solutions based on 288-cell division. (magnetic mode, f = 500 MHz) ...................................................................... Iterative loopoEMF solutions based on 36-cell division. (magnetic mode, f = 500 MHz) ...................................................................... Equivalent magnetic current method solutions based on 36cell division. (magnetic mode, f = 500 MHz) ...................................................................... Conventional EFIE solutions based on 36-cell division. (magnetic mode, f = 700 MHz) ...................................................................... Conventional EFIE solutions based on 288—cell division. (magnetic mode, f = 700 MHz) ...................................................................... Iterative loop-EMF solutions based on 36-cell division. (magnetic mode, f = 700 MHz) ...................................................................... Equivalent magnetic current method solutions based on 36-cell division. (magnetic mode, f = 700 MHz) ...................................................................... A heterogeneous body of dimensions 6x6xl cm with the parameters specified in the shaded and unshaded regions. ............................................. Iterative loop-EMF solutions in the heterogeneous body with the impedances determined by the complex conductivity of each cell. ............. Iterative loop-EMF solutions in the heterogeneous body with the impedances determined by the average of the complex conductivities between two adjacent cells. ........................................................................... An arbitrarily shaped magnetic lossy body in free space illuminated by an incident EM wave. ........................................................... Illustration of the equivalent sources induced by the incident EM wave in the volume region v and on the enclosing surface s. ............. A symmetric body partitioned into symmetrical octants, denoted by Roman numerals. The origin of the coordinate system is located at the center of the body. .................................................................. A homogeneous cubic body of dimensions 6x6x6 cm illuminated by an viii 36 36 37 37 38 38 39 39 4O 40 41 43 77 incident plane EM wave. ............................................................................... 80 3.5 Solutions for the symmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the magnetic body. (x- component of electric field) ...................................................................... 82 3.6 Solutions for the symmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-ce11 division in the non-magnetic body. (x- component of electric field) ...................................................................... 83 3.7 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-ce11 division in the magnetic body. (x- component of electric field) ........................................................................ 85 3.8 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the magnetic body. (z- component of electric field) ........................................................................ 86 3.9 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the non-magnetic body. (x- component of electric field) ...................................................................... 87 3.10 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the non-magnetic body. (z— component of electric field) ...................................................................... 88 3.11 Solutions for the symmetric mode of induced electric field obtained from the tensor EFIE based on 216—cell division in the magnetic body. (x- component of electric field) ........................................................................ 89 3.12 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 216-cell division in the magnetic body. (x- component of electric field) ...................................................................... 90 3.13 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 216-ce11 division in the magnetic body. (z- component of electric field) ...................................................................... 91 3.14 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on ix 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 512-ce11 division in the magnetic body. (x- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 512-cell division in the magnetic body. (z- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 1000-ce11 division in the magnetic body. (x- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on lOOO—cell division in the magnetic body. (2- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 512-cell division in the magnetic body. (x- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 512-cell division in the magnetic body. (2- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 1000-ce11 division in the magnetic body. (x- component of electric field) ...................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 1000-cell division in the magnetic body. (2- component of electric field) ...................................................................... A homogeneous cubic body of dimensions 12x12x6 cm illuminated by an incident plane EM wave. .......................................................................... Solutions for the symmetric mode of induced electric field obtained from the coupled tensor integral equations in the rectangular magnetic body. (x- component of electric field) ...................................................................... Solutions for the symmetric mode of induced electric field obtained from the tensor EFIE in the rectangular magnetic body. (x- component of electric field) ...................................................................... 93 94 95 97 99 100 101 103 106 107 108 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations in the rectangular magnetic body. (x- component of electric field) ................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations in the rectangular magnetic body. (z- component of electric field) ........................................................................ Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular magnetic body. (x- component of electric field) ................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular magnetic body. (2- component of electric field) ................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular non-magnetic body. (x- component of electric field) ................................................................... Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular non-magnetic body. (z- component of electric field) ................................................................... A heterogeneous cubic body of dimensions 6x6xl cm with different parameters specified in the shaded and unshaded regions. ....................... Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations in the heterogeneous body. .................................................................................... ' Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations in the non-magnetic homogeneous body. ............................................................. Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations based on 512 cell division in the heterogeneous body. (x- component of electric field) ................................................................... Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations based on 512 cell division in the heterogeneous body. (z- component of electric field) ........................................................................ Solutions for the antisymmetric mode of induced EM field xi 109 110 111 112 113 114 117 118 120 121 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 obtained from the coupled tensor integral equations based on 512 cell division in the heterogeneous body. (y- component of magnetic field) ................................................................... An incident plane EM wave on a lossy layer of thinkness d. ...................... Equivalent circuit for a lossy dielectric layer. .............................................. The front view and side view of the antenna aperture located next to the lossy layer. ................................................................................... The magnitude of the x- component of plane wave spectrum behind the barrier (d = 0.2 m). ...................................................................... The magnitude of the x- component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.2 m. ................................. The magnitude of the 2- component of plane wave spectrum behind the barrier (d = 0.2 m). .................................................................... The magnitude of the z- component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.4 m. ................................. The magnitude of the x- component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.2 m. ................................. The magnitude of the z- component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.2 m. ................................. The magnitude of the x— component of plane wave spectrum behind the barrier (d = 0.5 m). ...................................................................... The magnitude of the x- component of electric field distribution behind the barrier (d = 0.5 m) at a distance of 0.2 m. ................................. The magnitude of the 2- component of plane wave spectrum behind the barrier (d = 0.5 m). ...................................................................... The magnitude of the z— component of electric field distribution behind the barrier (d = 0.5 m) at a distance of 0.2 m. ................................. The magnitude of the x- component of electric field distribution behind the barrier (d = 0.5 m) at a distance of 0.4 m. ................................. The magnitude of the z- component of electric field distribution behind the barrier (d = 0.5 m) at a distance of 0.4 m. ................................. Circuit diagram of the microwave life detection system. .............................. Experimental set up for the measurement of heart and breathing signals of a human object through brick wall, using microwave life detection system. ..................................................................................... Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection xii 122 132 132 136 139 141 143 144 145 146 147 148 149 150 151 154 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 A.1 system. ............................................................................................................ Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection system. ............................................................................................................ Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. ............................................................................................................ Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. ............................................................................................................ Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection system. One layer of wet bricks severely hampered the penetration of the microwave beam. ................................................................................. Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. One layer of wet bricks had a considerable effect on the penetration of the microwave beam. ............................................................. Experimental set up for the measurement of heart and breathing signals of a human object under layers of bricks, using microwave life detection system. ..................................................................................... Heart and breathing signals of a human subject, lying with face-up or face-down position under 3 layers of bricks, measured by the 10 GHz life detection system. ................................................. Heart and breathing signals of a human subject, lying with face-up or face-down position under 4 layers of bricks, measured by the 10 GHz life detection system. ............................................................................. Heart and breathing signals of a human subject, lying with face-up or face-down position under 5 layers of bricks, measured by the 2 GHz life detection system. ............................................................................. Heart and breathing signals of a human subject, lying with face-up or face-down position under 6 layers of bricks, measured by the 2 GHz life detection system. ............................................................................. Heart and breathing signals of a human subject, lying with face-up or face-down position under 7 layers of bricks, measured by the 2 GHz life detection system. ............................................................................. Different notations of the parameters and variables in a permeable body used by Tai and this thesis when it is illuminated by a plane xiii 167 168 169 170 171 wave. ............................................................................................................... 178 A2 Illustration of the exclusion of a small region which contains the singularity point when evaluating the principal integration. ........................ 184 xiv in fiej eds; this and the r COnst ‘0 ex Dirac. 110” O CHAPT ER I INTRODUCTION The interaction of an electromagnetic (EM) field with a heterogeneous material body, with arbitrary electric and magnetic parameters and finite physical dimensions, is a fundamental and important research subject. This subject has received attentions of many researchers because of its relevance in medical and engineering applications. For example, the induced EM field in a human body when it is irradiated by an. incident EM wave is an important piece of in formation in the assessment of potential health hazard, or in some medical applications such as the hypertherrnia cancer therapy using EM radiation. In the field of engineering, the knowledge of interaction of EM waves with a finite body of arbitrary compositions has direct applications in the modification of radar scattering from space vehicles. This knowledge is also relevant in other electronic fields whenever a material body of a device is exposed to an EM field. To quantify the induced EM field in an irradiated heterogeneous body is not an easy task. The mathematical complexity of three-dimension computation exacerbates this problem. A tensor electric field integral equation (EFIE) was derived by Livesay and Chen to quantify the induced fields in a non-magnetic and lossy body [1], where the method of moment (MOM) was used for the numerical calculation, the piecewise constant functions (or pulse functions) were chosen as basis functions (trial functions) to expand the components of unknown induced electric field distribution and the Dirac-Delta functions were selected as weighting functions (it is the so called colloca- tion or point-matching testing procedure). Some researchers [26], [27] solved this problem by an equivalent EFIE in terms of free space scalar Green’s function, with the choice of some other basis and weighting functions for the method of moment. The results of the latter approach, when compared with the analytical solution of the induced electric field distribution in a homogeneous non-magnetic lossy sphere, seem to improve over the results of Livesay and Chen [1]. However, to use the pulse-basis expansion and point-matching in the method of moment is still advantageous over other choices of basis and weighting functions from the viewpoint of the simplicity in the procedures of calculation. There are some limitation of solving this EFIE with pulse-basis expansion and point-matching which have been reported [5], [6]. A reported problem [6] states that induced electric field of electric mode which is excited by symmetrical part of incident electric field can be solved accurately but the induced electric field of magnetic mode which is excited by the antisymmetrical part of incident electric field can not be determined accurately unless a large number of parti- tion of the body is used. This thesis presents some methods, namely, the iterative loop-EMF method and the equivalent magnetic current compensation method, to improve the efficiency and accuracy of the existing method (solving tensor EFIE with pulse—basis expansion and point-matching). The results of these new methods in the frequency range of several hundred MHz are compared with the results of Livesay and Chen [1]. To study the interaction of EM waves with magnetic and lossy bodies, a set of coupled tensor integral equations [3], [14] have been derived based on the method of equivalent polarized currents to relate the induced EM fields with incident EM fields. Since the unknown induced electric and magnetic fields are coupled together in this set of equations, we need to solve them simultaneously. In order to reduce the number of unknowns, we can decouple this set of equations into a separated tensor EFIE and a separated tensor magnetic field integral equation (MFIE) in which case the electric and magnetic field distributions can be solved separatedly and only one half of unknowns need to be handled in the procedure of numerical calculation. The method of moment solutions of coupled and decoupled tensor integral equations are compared and the effect of induced fields due to the magnetic material is investigated. These tensor integral equations are constructed in terms of free space dyadic Green’s functions and can be transformed into integral equations in terms of free space scalar Green’s func- tions. Similar numerical results can be obtained by solving the integral equations in terms of the scalar Green’s functions. A study of an application of electromagnetic waves for detecting a small move- ment of a biological body located at a distance or behind a barrier is also presented in this thesis. The so called microwave life detection systems are developed to measure the small movements due to breathing or heartbeats of a human subject at a distant away or behind a barrier. The original purpose of the systems was that Navy needed a system to indicate whether a wounded soldier on the battlefield is still alive or not. Such a system can prevent fellow soldiers from taking risks attempting to help a dead combatant. It turns out that there are many other applications. For example, this sys- tem can be used to find out whether there are people under a collapsed building after an earthquake or it can be used as an alarm system to monitor intruders into a room. The basic principle of the systems is to illuminate the human subject with a low intensity microwave beam and then extract the body movement of the subject from the modulated back-scattered wave with a detecting system. Two different operating fre- quencies have been selected to construct the systems, one is 10 GHz (X-band) and the other is 2 GHz (L-band). The L-band system is specially designed for detecting small body vibrations behind a thick wall because of the better penetration ability of the EM wave of the lower frequency. Plane wave spectrum theory [17], [18] is used to analyze the nonuniform plane wave from the antenna passing through a layer of lossy material and to predict the electric field distribution on the other side of the barrier. The predicted electric field distributions and experimental results for the detection of breathing and heart signals of a human subject behind brick wall of various thicknesses are presented. In chapter II, a brief outline of the derivation of the tensor EFIE which can be used to quantify the induced electric field distribution in a non-magnetic body is given in the beginning and then an example is given to show the limitation of the MOM solution with pulse-basis expansion and point-matching of this equation. We introduce two new methods to improve the accuracy and efficiency of the tensor EFIE method. The iterative loop-EMF method and the equivalent magnetic current methods are dis- cussed in sections 2.3 and 2.4, respectively. For the iterative loop-EMF method, we use the concepts of impedance networks and Faraday’s law to relate the induced currents and the induced EM fields, then apply a coupled tensor integral equation to perform the iterative process. For the equivalent magnetic current compensation method, an equivalent magnetic current on each adjacent cell boundary is introduced to compensate the discontinuity of the tangential component of the induced electric field distribution which is artificially produced by the pulse-basis expansion. Discussions and numerical results are included at the end of chapter. Chapter III discusses the interaction of EM fields with magnetic bodies. An out- line of the derivation of the coupled tensor integral equations which can be used to quantify the induced EM field distribution in a magnetic body is given in section 3.1. The transformation from this set of coupled equations into a separate tensor EFIE and a separate tensor MFIE in various forms [13], [16] are presented next These sets of integral equations in terms of the free space scalar Green’s function, rather than the dyadic Green’s function, can be derived from the concept of potential functions main- tained by equivalent current and charge densities. The derivations are given in section 3.3. Numerical solutions of the coupled equations and decoupled equations are com- pared and the effects of the induced EM fields due to the magnetic material are inves- tigated in this chapter. Both the advantages and disadvantages of each set of the integral equations are also discussed. Chapter IV is devoted to the study of the application of electromagnetic wave for detecting a small movement of a biological body located at a distance or behind a bar- rier. Plane wave spectrum analysis is outlined in the beginning of chapter and fol- lowed by some numerical simulations of the predicted electric field distributions for a nonuniform plane wave passing through a layer of lossy barrier. The description of the microwave life detection systems, along with the circuit diagram and the principle of operation, is included in section 4.3. Finally, a series of experimental results for the detection of the breathing and heartbeats of a human subject behind a brick wall are presented. CHAPTER II NEW METHODS FOR QUANTIFICATION OF INDUCED EM FIELDS IN FINITE, NON-MAGNETIC AND LOSSY BODIES Some new numerical methods for quantifying the induced EM field in a finite, heterogeneous, non-magnetic body irradiated by an incident EM field are investigated in this chapter. The numerical solutions of the tensor electric field integral equation (tensor EFIE) [1] by using conventional pulse-basis expansion with point-matching are compared with the results of the new methods. In the first two sections, we introduce the derivation of the tensor EFIE and discuss some limitation of this method. Section 2.3 and 2.4 illustrate the new methods, i.e., the iterative loop-EMF method and the equivalent magnetic current method, to improve the accuracy of the numerical solu- tions of the induced EM field distribution. For the iterative loop-EMF method, we use the concepts of impedance networks and Faraday’s law to relate the induced currents and the induced EM fields, then apply a coupled tensor integral equation to perform the iterative process. In the equivalent magnetic current method, an equivalent mag- netic current is introduced to compensate the discontinuity of the tangential component of the induced electric field distribution which is produced by the pulse-basis expan- sion. The comparisons and comments between various methods are presented in sec- tion 2.5. 2.1 Tensor Electric Field Integral Equation (Tensor EF IE) A brief outline of derivation of the well known tensor EFIE is given in this sec- tion. Fig-2.1 shows a finite heterogeneous system composed of dielectric, and lossy medium with arbitrary shape, being irritated by an incident EM field E’ and Hi of angular frequency to. The conductivity 0’ and permittivity e of the medium consisting this system are both functions of location, i.e., o: o(r) (2.1.1) 8 = E(r) (2.1.2) where r is the position vector in R3. The incident fields E’,H’ must satisfy Maxwell’s equations in free space: V x E’(r) = 1'qu H’(r) (2.1.3) V x H‘(r) = —icoeo E’(r) (2.1.4) where i = «I: , (2.1.5) the time harmonic factor 42"“ is assumed and so, 110 are the permittivity and permea- bility of the free space, respectively. The total fields E and H inside the system which is induced by incident fields should also satisfy Maxwell’s equations: V x E(r) = icouo H(r) (2.1.6) V x H(r) = c(r)E(r) - ime(r)E(r) . (2.1.7) The scattered fields are defined as the total fields subtracted by the incident fields, that is, E‘(r) = E(r) — E’(r) (2.1.8) E(r).H(r) E(r),r10.o(r) Fig-2.1 An arbitraily shaped non-magnetic lossy body in free space illuminated by an incident plane wave. H‘(r) = H(r) — H’(r) . (2.1.9) From the set of equations (2.1.3), (2.1.4) and (2.1.6), (2.1.7), and using (2.1.8), (2.1.9), the following equations are easily obtained: V x E‘(r) = icouo H‘(r) (2.1.10) V x H‘(r) = {6(r) - i0) [E(r) - 80]} E(r) - icoeo E‘(r) . (2.1.11) Suppose we define an equivalent volume current density as Jeq(r) = 1(r)E(r) . (2.1.12) where t(r) = o(r) - in) [E(r) - 80] (2.1.13) is the equivalent complex conductivity of the medium, then we can rewrite (2.1.10) and (2.1.11) as V x E‘(r) = 1'qu H‘(r) (2.1.14) V x H‘(r) = Jeq(r) — imeo E‘(r) . (2.1.15) We can determine E‘ in terms of Jeq from (2.1.14) and (2.1.15) now. By taking curl operation in both side of (2.1.14) and making use of (2.1.15), a differential equa- tion is obtained: V x V x E‘(r) — k3 E‘(r) = £qu Jeq(r) for all r , (2.1.16) where k0 = romeo (2.1.17) is the wave number of free space. 10 To find the solution of this equation, we can find the Green’s function first. The Green’s function 80 is a tensor quantity and must satisfy the following dyadic differential equation in free space: V x V x 80(r,r’) - k3 800,1“) = 7’5(r-r’) (2.1.18) where 7’: xx + yy + zz (2.1.19) is the unit dyad and x, y, z, are the unit vectors in x—, y—, z- direction, respectively. The free space dyadic Green’s function 80 must also satisfy the radiation condition at infinity. 60 can be determined as [4]: _1. 6003]") = ( r+ k3 VV)¢(r,r’) for r¢r’ (2.1.20) where I eikOR f I ¢(r,r) - m or r¢r (2.1.21) is the free space scalar Green’s function and R = lr—r’l . (2.1.22) Thus, the solution of (2.1.16) is E’(r) = j , itoqueq(r)-80(r,r’) dv’ (2.1.23) where v is the volume of the body. Since the integrand of (2.1.23) is not in the L1 space, i.e., not integrable, this integration is valid only in the sense of Cauchy principal value and a correction term is needed to overcome the singularity problem, [1], [2], i.e., 11 s . . , , J. (r) E (r) = P.V. [ , rmttoJeqfl ):G’0(r,r) dv + 7120—8; . (2.1.24) From (2.1.8) and (2.1.24), we can easily obtain the well known tensor EFIE: [I + :t—OEZTJE“) - P.V. I v icou01(r’)E(r’)-Go(r,r’) dv’ = E’(r) . (2.1.25) This integral equation can be transformed into a system of linear algebraic equa- tions by applying pulse-basis expansion and point-matching and then solved numeri- cally. The details of the transformation are shown in section 2.4 where we will illus- trate the differences between this method and the equivalent magnetic current method. Another integral equation, which will be used in the iterative loop-EMF method in section 2.3, in terms of incident magnetic field can be derived by taking curl opera- tion of (2.1.23) and utilizing (2.1.9) and (2.1.14): V x E-‘(n = V x j , imnoz(r')E(r')-8o(r,r') dv’ . (2.1.26) Using equations (2.1.14), we have H‘(r) = -$ V x E’(r) , (2.1.27) then (2. 1.27) becomes mm = - j , t(r’)E(r’)-[V x 80(r,r’)] dv’ . (2.1.28) By applying (2.1.9) and (2.1.28), we can obtain a coupled tensor integral equation in terms of H’: H(r) + j , 1:(r’)E(r’)-[V x 60(r,r')] dv' = H’(r) . (2.1.29) We will make use of this equation to establish the iterative process in the iterative loop«EMF method since the induced magnetic field H(r) is easily obtained via 12 numerical integration provided the induced electric field distribution E(r) and the incident magnetic field H’(r) are known. More details are presented in section 2.3. 2.2 The Limitation of the Existing Tensor Integral Equation Method The limitation of the Existing Tensor Integral Equation Method has been dis- cussed by several researchers [5], [6]. One limitation is concerned with the specific absorption rate of energy (SAR). It was reported that the tensor EFIE method with pulse-basis expansion and point-matching gives good values for whole-body average SAR, but the convergence of the solutions for the electric field distribution and the SAR distributions is questionable. The convergence problem has also been indicated by Chen and Rukspollmuang [6] by using a cubic body as an example. This example is given here to illustrate the limitation of the tensor EFIE. An incident plane elec- tromagnetic wave of frequency to, E‘ = x 50‘3"“ (2.2.1) Hi = y Hoe’k“ , (2.2.2) where E0 and H0 are some complex constants, can be decomposed into two standing waves as 1;:i = E; + Ef, (2.2.3) Hi = Hf, + Hf, , (2.2.4) where E; = x Eocos(koz) (2.2.5) 13 151;, = x iEosin(koz) (2.2.6) . E0 . H; = y i-C—srnacoz) (2.2.7) . E0 H; = y -C—cos(koz) (2.2.8) and 1 I 110 = — 2.2.9 C 80 ( ) is the free space wave impedance. The induced electric field E due to E‘, is called the electric mode solution, while the induced electric field E due to BL, is called the mag- netic mode solution. Two sets of numerical solutions of electric mode and magnetic mode field distri- butions in a cubic body obtained by solving the tensor EFIE with pulse-basis expan- sion and point-matching are shown in Fig-2.2 and Fig-2.3, respectively. Fig-2.2 shows the x- components of the amplitude of the induced electric fields IEXI inside a 4 x 4 x 4 cm cube with the conductivity 0' = 4.5 S/m and the relative permittivity e, = 50, excited by a symmetrically impressed electric field E’ = x cos(koz) of fre- quency 750 MHz. The middle part of Fig-2.2 shows the distribution of the amplitude of Ex within one eighth of the cube, obtained when the cube was subdivided into 216 cubic cells. In the lower part of Fig-2.2 the distribution of the amplitude of 1'3Jr within one eighth of the cube obtained with 512-cell subdivision is shown. Fig-2.3 shows the distribution the amplitude of Ex inside the same cube excited by an antisymmetrically impressed electric field Ei = x isin(koz) of frequency 750 MHz. The numerical results for the amplitude of E, are given in Fig-2.3 for the cases of 216-cell subdivision and the 512-cell subdivision. It is noticed that the electric 14 4— y z r a 750 MHz 8: a 50 e 4cm .I o- : 4.5 S/m ( 216-cell subdivision ) 1 ( 512-cell subdivision ) E a xcosk z 0 I E" in(mV/m) 2 1 m7 19.1 19.1 25.3 r17 14.5 20.0 15.1 14.9 20.3 22.8 22.8 27.9 20.1 19.8 24.3 21.1 20.7 25.1 24.3 24.1 29.0 22.1 21.7 26.0 23.5 22.9 27.0 ( 216-cell subdivision ) | E‘l in(mV/m) 4 3 2 1 . 4. 444 4 4222.7 19.2 19. 19.6 26.6/14.414.5 14.4 20.5 14.9150 14.9 20.8 15.1152 15.1 20.8 1.8 21.8 21.9 27.4 19.4192 18.9 23.6U 20.9 20.6 20.2 24.5 21.6 21.4 20.8 25.0 2301229228 27.7 21.8 21.5 20.8 2501/ 24.0 23.6 22.8 26.5/ 25.2 24.7 23.8 27.3 23.5 23.4 23.4 27.7 23.0 22.6 21.8 25.6I/ 25.6 25.0 24.0 zfi 26.9 26.3 25.2 28.4 ( 512-cell subdivision) F ig-2.2 Electric mode disuibution of the x-component of the induced electric fields in the body based on 216-cell division and 512-cell division. 15 A x 2 y +— ——— / y ‘— z r a 750MHz t .I Er :- 50 4cm (5 = 4.5 S/m 4cm (216-cell subdivision) Ei . . . _ _ ‘3’“ srn koz (512-cell subdrvrsron) | Ex] in(mV/m) 2 1 r a 14.4 14.2 15.3 7.0 6.9 7.6 2.4 2.3 2.6 28.9 28.4 29.2 28.9 28.4 29.2 5.3 5.2 5.3 36.2 35.6 36.2 36.2 35.6 36.2 6.9 6.7 6.7 (216-cell subdivision ) I Exl in(mV/m) 4 3 2 I 19.6 19.41189 19.8 12.2120 11.6 12.2/ 7.6 7.4 7.2 7.4 / 2.5 2.5 2.4 2.5 42.5 42.3 41.1 40.8 29.2 28.8 27.72773 18.2 17917.2 ‘16".7 6.1 6.0 5.8 5.6 58.7 58.0 56.3 55.1 41.1 40. 38.8 37.7 25.7 25.2 24.1 23.2/ 8.7 8.5 8.1 7.8 66.6 66.0 64.0 62.4 47.2 46.4[446 43L 29.5 28.9 27_.7'26.'4' 10.0 9.8 9.4 8.9 ( 512-cell subdivision ) Fig-2.3 Magnetic mode distribution of the x-component of the induced electric fields in the body based on 216-cell division and 512-cell division. 16 mode solution exhibits a good convergence but the magnetic mode results for the 512- cell subdivision deviate significantly from those of the 216.cell subdivision, especially at the outer layer of the cube. A simple physical explanation is that the electric mode solution is induced by a symmetrically impressed electric field so that it is linear in nature while the magnetic mode solution is induced by an antisymmetrically impressed electric field so that it is circulatory in nature. Intuitively, for the latter case, we do not expect good results with the pulse-basis expansion which is linear in nature. To produce accurate results for the magnetic mode of induced electric field field, the cell size should be smaller in comparison with the case of the electric mode shown in Fig- 2.2. 2.3 Iterative loop-EMF method A finite body can be considered as an impedance network system from the viewpoint of electric voltage and current [7], [8]. The body is subdivided into a number of cells, each of which is then replaced by an equivalent impedance loop. One can find the induced current distribution of this network when it is exposed to an incident field with the applications of Faraday’s law and Kirchhoff’s circuit theory to the network. The electric field distribution of the body can thus be determined by the relation: J.q(r) = t(r)E(r) . (2. 1.12) An iterative process can be added to improve the accuracy of the induced currents in the resulting network and hence the induced electric fields inside the body by using the coupled integral equation (2.1.29). The iterative loop-EMF method is based on the 17 concept of impedance network combined with the iterative process. The procedures of this method are given in this section. Fig-2.4 is an example of a single layer of body illuminated by an antisymmetric incident electric field Efn of Ejn = x iEosin(koz) , (2.2.6) which has the associated magnetic field Hf, of 50 C The single layer body is cut into N cubic cells of width d while we regard each cell as Hf" = y cos(koz) . (2.2.8) a loop in the whole system. In each loop, say the n-th one, we define a loop current, 1,, = IRE, (2.3.1) where En can be considered as a loop electric field in the n—th cell and 1,, is the equivalent complex conductivity of this cell. This loop current can be calculated in terms of magnetic field by Faraday’s law: I , E-dl = 161110] A, H-ds . (2.3.2) Let us apply (2.3.2) in the i-th cell for illustration. From (2.1.12) we have L (l') E = —"——, 2. . (r) 1:(r) ( 33) then (2.3.2) and (2.3.3) give 1’44 11d Jkd J’d—' (H d?- (234) T,- rj 1,, t, _ 103110 i)’ ’ ° ° or in other words, 451° - Ej -Ek - E1 = 10.)“.0(Hi)yd , (2.15) 18 N Fig-2.4 A single layer of non-magnetic lossy body regarded as an impedance network system with N loops. 19 where (HR), is the y- component of the H field for the n-th cell. Eventually, (2.3.4) can be thought as the application of Kirchhoff’s voltage law while Ji’s are correspon- d . . dent to mesh currents and ? correspondent to branch 1mpedances along the srdes of 1' the cell. The application of (2.3.5) to all the cells will give rise to . ‘F . r . 011 012 . . . 011v 51 (Hl)y “21 022 - - - 021v E2 (”fly . . . . . . . = . (2.3.6) [am am ' ' ' aNN‘ [Ev _(HN)y. where am = c Lmn for m=1,2,...,N n=1,2,...,N (2.3.7) C: ,d (2.3.8) 1031-10 and Ln”, belonging to the set {—4,-3,—2,—1,0,l,2,3,4}, is dependent on the relative loca- tion of the loop in the network system. We can obtain 5,, in terms of (Ha), by solving the system of simultaneous linear algebraic equations (2.3.6). After that the loop current density of each cell 1,, is sim- ply obtained as 1,, = r 5,, . (2.3.9) 7: The actual current density along the side of each cell can be found by subtracting those loop current densities of the adjacent cells. For example, in Fig-2.4, Jxl’ 1,1, Jfl and 1,2 are the actual current density of four sides along the i-th cell respectively, and can be expressed as follows: distn' whic.‘ (2.1.2 in the is the moth- are the lively. Till (2.3.10) fiCld at l [he Sides 2O er = - J; (2.3.10) 1,, = 1,. - J,- (2.3.11) 1,2 = 1,. - 1,, (2.3.12) 1,, = J,- - J, (2.3.13) Once we know the actual current distribution, we can easily find the electric field distribution by (2.3.3). The remaining problem is that we still do not have the magnetic field distribution which is needed to apply (2.3.2). Thus, an iterative process is developed by using (2.1.29) to estimate the H field. We can choose the initial guess as 11“” = Hi (2.3.14) in the zeroth-iteration, where H“) , for k=0,1,2,..., (2.3.15) is the estimated magnetic field distribution in the k-th iteration. From (2.3.6), we are able to calculate for the loop electric field 15:?) in this zeroth-iteration, and thus for the loop current 1):”, where ESP and 131‘), for k=0,1,2,..., (2.3.16) are the loop electric field and loop current of the n-th cell in the k—th iteration, respec- tively. The actual electric field distribution can be easily constructed similarly to that in (2.3.10) to (2.3.13) in this zeroth-iteration. In order to make use of the actual electric field at the center in each cell, we can approximate that by averaging the values along the sides of each cell, for example, _ (5,, + 5,2) ,x 2 (2.3.17) .— 21 _ (£21 + 522) i2 2 (2.3.18) for the i-th cell as shown in Fig-2.4, where Eb, and E], are x— and 2— components of electric field at the center in the cell. So EU”, the zeroth E field distribution, is con- structed automatically. Now we are on the way to proceed the iteration process. For the first-iteration, Hm can be calculated from (2.1.29): 11mm = H‘(r) - j , t(r’)E(°)(r’)-[V x 80031-0] dv’ , (2.3.19) where the integration in (2.3.19) should be carried out numerically. Again, we can obtain 5),” from (2.3.6) by using H(l). Similarly, for the kth-iteration: 110%) = H’(r) - j , t(r’)E("'l)(r’)-[V x 80(r,r')] dv’ , (2.3.20) and then ESP can also be obtain from (2.3.6) with HU‘) calculated from the (k—l)-th iteration. This iteration process should continue until both 5,, and H converge. As we can observe, a nearly uniform, incident magnetic field will enable us to apply the Faraday’s law accurately and thus will give better approximation in this method. This process, as described above, will be more accurate and efficient if we applied it on an electrically small size body with an antisymmetric incident electric field. 2.4 Equivalent Magnetic Current Compensation Method 22 When (2.1.25) is solved numerically with pulse-basis expansion in a finite body, this body is partitioned into a finite number of cells and the induced field in each cell is assumed to be uniform. In this way, we allow the induced electric fields to vary from cell to cell. Obviously, this assumption creates a discontinuity of E field between adjacent cells. In a homogeneous region, the induced E fields need to be continuous everywhere but the consequence of the above assumption clearly violates this fact. When point- matching is applied as the testing procedure, there is a physical explanation for the discontinuity of the normal component of the E field [9] and a brief discussion is given here. In (2.1.18), the dyadic Green’s function can be regarded as the E field generated by a current element and a pair of opposite charges at the ends of the element. A source region with conventional current and charge density can be considered as an ensemble of small cells each containing a current element or an electric dipole of cer- tain magnitude and orientation. Based on this picture, two adjoining cells containing current elements of different magnitudes or orientations will result in a net charge at the boundary of these cells. On the other hand, the discontinuity of the normal com- ponent of the E field between two adjacent cells as we mentioned earlier will also yield a net charge at the interface by applying the boundary condition of the Maxwell’s equations. From this argument, we say the discontinuity of the normal component of E field is compensated by the equivalent surface charge at the boundary which effect is taken into account by the free space tensor Green’s function 80. The discontinuity of the tangential components of E fields, however, still exists between adjacent cells. An equivalent magnetic surface current Km is proposed at the cell of ti neigl we h. where adjoin maintz Thus (. Therefc Where 3 face Cnc modifies In a matrix 23 cell boundary to compensate for this discontinuity. Fig-2.5 illustrates a rough picture of the equivalent surface charge and the equivalent magnetic surface current between neighboring cells. By definition of magnetic surface current and the principle of equivalence [10], we have: K =-nXE=(n1XEl+n2XE2) (2.4.1) 03 where n is the unit normal vector and n1, n2, are the unit normal vectors on the adjoining surface of cell 1 and cell 2 respectively, as shown in Fig-2.5. This Km will maintain another scattering electric field: E"(r) = - j s (n x E(r’)) x V¢(r,r’) ds’ . (2.4.2) Thus (2.1.8) becomes E = E’ + E‘ + E" . (2.4.3) Therefore the tensor EFIE (2.1.25), is then modified as follows: [I + %JE(r) - PM] v imuo‘t(r’)E(r’)-Go(r,r’) dv’ + j for x E(r’)) x Vq>(r,r’) d5" = E’(r) , (2.4.4) where s-represents all the six surfaces of each cell except the outrnost boundary sur- face enclosing v. Equation (2.4.4) can be solved numerically in a similar way as before with a modification in the matrix elements, as described below. In the tensor EFIE (2.1.25), the inner product E(r)-Cour) can be represented as a matrix product as E(r’)-Go(r,r’) 24 —/l Fig-2.5 Illustration of the equivalent suface charge and the equivalent magnetic current created by the discontinuity of the electric field between neighboring cells. for m then 25 Era-r) + —‘-¢..(r.r’) -1-¢.,(r.r’) irntrr’) k3 k5 "3 5,0) = —’-¢,., (2.4.12) where 3V 1/3 an = [41‘ J (2.4.13) is the radius of the approximated sphere and V" stands for the volume of the n-th cell. The rest of elements in GM can be carried out numerically. Now in the modified integral equation (2.4.4), the integrand of the surface integration (n x E(r’)) x V¢(r,r’) can also be represented in another matrix product form: (n X E(r’)) >< V¢(I‘,r') 27 n2¢y(r7r’) + n3¢z(r7r,) ‘nl¢y(rrr’) -n1¢z(rrr’) Ex(r’) = _n2¢x(rrr’) nl¢x(rrr,) + n3¢z(r7r’) -n2¢z(rrr’) Ey(r’) -n3¢x(r.r') -n3¢,(r.r’) n1¢x(r.r’) + n2¢,(r,r’) Eztr’) (2.4.14) where n1, n2 and n3 are x—, y- and 2— components of the unit normal vector 11, i.e., n = nlx + nzy + n32 (2.4.15) n? + 11% + n3 = 1 , (2.4.16) and , _ a I _ 5“ (xp-xp’) . 9x,(r.r) - 3390;) _ 4M2 R (1 - szR) (2.4.17) for p=1,2,3. Suppose we apply pulse-basis expansion and point-matching at the center of each cell, the system of linear algebraic equations will become P_ — _ q F .1 (:11 G_12 913 El E1 021 622 623 E2 = 55 (2.418) G31 G32 G33 E3 5% where (TM is again a N x N matrix, the elements of the matrices for p,q=l,2,3 are i1:(r,,) 30380 ((34% = 812480141 "' i ' + icouo P.V. [ ,_ r(r’)[8pq¢(rmr’) + kg 0 ¢xpx'(rn,r')] dv’ 3 +1 3;, [5m )3 n.9,...(rmr’) — (1 — 8,,1n,¢.,(r...r’n 44', I34 r=l m,n=1,2,...,N (2.4.19) 28 where 57,, represents the six surfaces of vm, except for those surfaces which belong to the outrnost surface s, and of course, up depends on s}. The surface integration in (2.4.19) should also be carried out numerically. Now we can find the electric field at the center of each cell by solving the above matrix equation. Numerical examples will be shown in section 2.5. 2.5 Numerical Comparison between the New Methods and the Existing Method Several sets of numerical solutions of induced electromagnetic fields inside some block models solved by the new methods described in this chapter are compared with the corresponding induced fields obtained from the tensor integral equation method. This comparison is shown in Fig-2.6 to Fig-2.24 and is the subject matter of discussion in this section. In Fig-2.6, a single layer, homogeneous rectangular biological model with dimen- sions of 6 x 6 x 1 cm is illuminated by a plane wave with a vertically polarized field at end-on incidence. The total volume is divided into 36 subvolumes and each of them is 1 x l x 1 cm in dimensions. Fig-2.7 indicates the numerical solutions of the tensor integral equation due to symmetric part of the incident electric field, Ei = Ef, = x cos(koz) (2.5.1) with a frequency f = 100 MHz. Those numbers in the figure are the x— and 2- components of the total induced field at the centers of the cells. The y— component is very small compared with the other two components so that it can be disregarded. In this calculation, the conduc- tivity 0' is assumed to be 0.889 S/m and the permittivity e is assumed to be 71.780. Fig-2.6 A homogeneous rectangular biological model with dimensions 6 x 6 x 1 cm /// 1cm 29 § ‘< qm II It illuminated by an EM wave at end-on incidence. \\\ 71.7 0.889 S / m 30 Due to the symmetry of the geometry and hence the field distribution [11], [12], only the values of the induced fields in the first quadrant of the model are shown in Fig-2.7. Fig-2.8 shows the solutions of a model with the same dimension and material which was subdivided into 288 cells with each subvolume of dimensions 0.5 x 0.5 x 0.5 cm exposed to the same incident field as Fig-2.7. Each numerical value is obtained by taking average of the fields at the centers of eight neighboring subcells which are includeded in the original 1 cm3 cubic cell in order to compare with the numerical values in Fig-2.7. Fig-2.9 and Fig-2.10 are the similar results as those shown in Fig- 2.7 and Fig-2.8 except that the incident electric field is antisymmetrical: E’ = Ef, = x isin(koz) . (2.5.2) It is observed that the values in Fig-2.7 are quite close to those in Fig-2.8, but the values in Fig-2.9 are only about one third to one half of those in Fig-2.10 which are regarded as more accurate solutions. This is not surprising since we have already seen an example in section 2.2. Fig-2.11 shows the field distribution obtained by the iterative loop-EMF method in the same body with an antisymmetrical incident electric field of frequency 100 MHz. Here the body is subdivided into 36 cells of size 1 cm3. We find that the values in this figure are closer to those in Fig-2.10 than those in Fig-2.9. Fig-2.12 presents the results of the equivalent magnetic current method, with the same body and the same incident fields as the case of Fig-2.11. The body is again cut into 36 cells and we have added the effect due to the equivalent magnetic current between adjacent cells. The improvement in this case is not very significant since the electric field distribution is not changing rapidly and hence the effect due to the discontinuity assumption by using pulse-basis expansion is also not very noticeable. 31 IExl .0332 .0334 .0418 |E,| .0030 .0089 .0150 1cm 1 [ . , —> m (V/m) .0465 .0472 .0560 flora z .0014 .0041 .0063 y WET—633T "W .0005 .0013 .0021 4 Z Fig-2.7 Conventional EFIE solutions based on 36-cell division. E’ = x cos(koz) ‘ x f = 100 MI'IZ IEA .0353 .0363 .0430 IE,I .0036 .0107 .0170 .0511 .0524 .0597 .0017 .0051 .0076 .0575 .0590 .0664 .0005 .0016 .0024 D Fig-2.8 Conventional EFIE solutions ' based on 288-cell division. 32 4 x IEQI .0007 .0021 .0044 i i E' . 7 . 2 . I ,l 008 005 0021 1cm 4» nr(VWn) .0013 .0040 .0078 °"‘ 2 y .0037 .0029 .0011 "70018""70030“"700§5"' .0012 .0010 .0004 —I> Z Fig-2.9 Conventional EFIE solutions based on 36-cell division. E=xammm ‘ x f = 100 MHz IE; .0013 .0039 .0073 IEQI .0189 .0131 .0053 .0028 .0088 .0155 .0097 .0075 .0030 'TUU§5""70?09""TUT§5"‘ .0032 .0024 .0010 .443» Z Fig-2. 10 Conventional EFIE solutions based on 288-cell division. 33 153 .0025 .0082 .0166 lEg .0303 .0281 .0188 hitVWn) .0043 .0137 .0261 .0183 .0137 .0082 .0051 .0042 .0026 CD 2 Fig-2.11 Iterative loop-EMF solutions based on 36—cell division. Ei = x ism/:02) x ==l ‘ f 00 MHz IE5] .0007 .0022 .0045 (art .0070 .0054 .0022 .0014 .0041 .0081 .0039 .0030 .0012 .0013 .0010 .0004 D Fig-2. 12 Equivalent magnetic current method solutions based on 36—cell division. f“ 10 It ex; rap; crea metl antis; ment frequr MHz, discus. the res C856. this se( in this again. and pg anllSyn this ca- are of 34 Fig-2.13 to Fig-2.16 show the induced fields in the same body illuminated by an antisymmetrical incident electric field of frequency 300 MHz. Fig-2.13 shows the solution of tensor EFIE method with 36 cells, Fig-2.14 indicates the solution of same method with 288 cells, while Fig-2.15 and Fig-2.16 depict the solutions by iterative loop-EMF method and equivalent magnetic current method, respectively, with 36 cells. It is obvious that the improvements by the new methods are increased. The physical explanation is that as the frequency goes up, the electric field distribution varies more rapidly and hence those new methods will give more compensations to the inaccuracy created by the piece-wise constant assumption of fields in each cell in the conventional method. Fig-2.17 to Fig-2.24 illustrate the induced field in the same body, exposed to antisymmetric incident electric field of frequencies 500 and 700 MHz. The improve- ment of the iterative loop-EMF method is quite significant as we can observe when the frequency is 500 MHz. The effect is not so significant when the frequency is 700 MHz, as we observe in the figure, since there are certain limitations of this method as discussed at the end of section 2.3. The equivalent magnetic current method improves the results about 20 to 30 percents on the amplitude of the field distribution in each C356. An example of the induced field in a heterogeneous body is given at the end of this section. The iterative loop-EMF method is used to calculate the field distribution in this example. Fig-2.25 shows a heterogeneous body of dimensions 6 x 6 x 1 cm again. The shaded region indicates different material of conductivity 0 = 0.048 S/m and permittivity e = 7.4580, the remaining part is the same as the previous cases. An antisymmetrical incident electric field of frequency 500 MHz is used. The difficulty of this case is the determination of the impedance at the boundaries of those cells which are of different materials. The impedances which are related to the complex 35 A x 9 x /—1 15,] .0037 .0110 .0221 [5,] .0311 .0240 .0094 mm in (WM) .0087 .0204 .0385 fin" 1 .0175 .0134 .0053 y .0058 .0044 .0017 > Z Fig-2.13 Conventional EFIE solutions based on 36-cell division. E‘ = x ism/.02) x f = 300 MHz |E,| .0054 .0188 .0311 15,1 .0831 .0488 .0191 .0114 .0350 .0626 .0360 .0277 .0108 WWW‘ .0117 .0097 .0034 > Z Fig-2.14 Conventional EFIE solutions based on 288-cell division. 36 IE; .0088 0221 .0488 IE,I .0825 .0715 .0485 2 in (V/m) .0111 .0361 .0715 .0425 .0359 .0219 .0131 .0110 .0065 ’ Z Fig-2.15 Iterative loop-EMF solutions based on 36—cell division. 13‘ = x isin(koz) x f = 300 MHz A IE; .0039 .01 18 .0238 'IEQI .0350 .0272 .0109 .0072 .0217 .0420 .0193 .0149 .0060 .0064 .0050 .0020 25., Z Fig-2.16 Equivalent magnetic current method solutions based on 36-cell division. 37 x A» X IE} .0074 .0219 .0424 IBM .0814 .0470 .0184 . 1on1 In (Wu) .0138 .0408 .0742 {F10 m -z> .0354 .0270 .0107 y '70?87""70302""708§§"' .0118 .0090 .0038 4 . Z Fig-2. 17 Conventional EFIE solutions based on 36—cell division. E’ = x isin(koz) A x f = 500 MHz IEQI .0114 .0348 .0810 war .1288 .0981 .0378 .0245 .0738 .1250 . .0773 .0587 .0228 '70570""TU§2§""T?§5§"' .0258 .0914 .0075 D Z Fig-2.18 Conventional EFIE solutions based on 288-cell division. 38 x ll ‘ X IE; .0113 .0378 .0785 ‘3 IEU .1328 .1142 .0731 in Win) .0184 .0595 .1142 10m ( ‘K)HCNI I—:> .0646 .0540 .0323 . Y "W "68. 98 ' """.130"'_4 ‘ .0196 .0162 .0095 4 Z Fig-2.19 Iterative loop-EMF solutions based on 36-cell division. Ei = x isin(koz) x f = 500 MHz IEA .0080 .0238 .0487 IEQI .0715 .0552 .0223 .0151 .0446 .0830 .0407 .0312 .0127 .0137 .0105 .0043 D Fig-2.20 Equivalent magnetic current method solutions based on 36-cell division. 39 15; .0128 .0378 .0882 IEQI .1094 .0834 .0032 nr(VWn) .0244 .0717 .1232 .0881 .0502 .0204 '70363""7088§""77508"' .0224 .0171 .0894 4 Z Fig-2.21 Conventional EFIE solutions based on 36-cell division. E‘ = x isin(koz) x f = 700 MHz 18 lEal .0239 .0895 .1112 IE) .2898 .2025 .0775 .0540 .1559 .2395 .2025 .1557 .0504 '707UU"'"720T7""T§03§"' .0814 .0445 .0177 t» Z Fig-2.22 Conventional EFIE solutions based on 288-cell division. IE;| IEH hr(VWn) ‘x .0097 .0338 .0828 .1309 .1185 .0815 .0145 .0498 .1185 .0542 .0473 .0315 'TUT88""70383""712§§"‘ .0155 .0134 .0088 40 IE} IEU Fig-2.23 Iterative loop-EMF solutions based on 36-cell division. 1: it .0146 .0422 .0780 .1331 .1021 .0422 .0282 .0820 .1427 .0801 ..0812 .0255 .0352 .1022 .1 754 .0275 .0210 .0088 1cm -> [m 2 Y ’ Z Ei = x isin(koz) f= 700 NIHZ ' Fig-2.24 Equivalent magnetic current method solutions based on 36—cell division. 41 1cm 6cm 81= 71.7 so 0‘. 0.889 S/m 82:- 7.45 so 02:- 0.048 S/m Fig-2.25 A heterogeneous body of dimensions 6 x 6 x 1 cm with the parameters specified in the shaded and unshaded regions. 42 conductivities at boundaries of heterogeneous cells can be determined by two different approaches which do not yield the same result. The first approach simply uses the complex conductivity in each cell itself in the evaluation of the left hand side of Faraday’s law (2.3.2). The second approach takes the average of the two different complex conductivities of two adjacent heterogeneous cells for the impedance of the common side. Fig-2.26 and Fig-2.27 are the numerical results of these two different approaches. It is observed that the field distribution is about the same in the region away from the heterogeneity but is somewhat different in the area near the hetero- geneity. Further study is needed to determine which approach is more accurate. 43 15.1 .0163 .0018 .0212 .0212 .0018 .0163 152’ .0027 .0208 .0050 0004 , .0018 .0027 in (V/m) 3’. .0208 .0023 .0212 ’ '. . 0 .0018 . .0009 .0059 .0005 ,ij .0009 E‘=x isin(lcoz) .0221 .0023 .0225 .0019 15.500 MHz .0005 .0014 .0005 .0005 .0221 .0023 .0266 0266 .0023 .0221 .0005 .0014 .0005 .0046 .0189 .0005 .0208 .0023 . 257 .0257 .0023 .0208 .0009 .0059 .0005 .0005 .0059 .0009 .0163 .0018 .0212 .0212 .0018 0163 .0027 .0208 .0050 .0050 .0208 .0027 Fig-2.26 Iterative loop-EMF solutions in the heterogeneous body with impedances determined by the complex conductivity of each cell. the 44 15:1 .0178 .0001 0196 0148 .0027 0124 15,1 .0015 .0m .0008 in(V/m) .0236 .0023 0196 _ .0015 .0073 .0035 E‘=xisin(lcoz) .0251 .0022 0209 #SOOMI-Iz .0002 .0005 .0023 .0241 .0002 .0257 .0012 .0033 .0049 .0084 .0113 .0005 .0218 .0015 .0261 .0228 .0005 0192 .0018 .0075 .0030 .0032 .0007 .0009 .0166 .0014 .021-1 0203 .0013 0158 .0022 .0218 .0037 1:443 1.0192 .0037 Fig-2.27 Iterative loop-EMF solutions in the heterogeneous body with the impedances determined by the average of the complex conducnvities between two adjacent cells. CHAPT ER III INTERACTION OF ELECTROMAGNETIC FIELDS WITH F INITE, HETEROGENEOUS, DIELECTRIC, MAGNETIC AND LOSSY BODIES The induced EM fields in a finite, heterogeneous, dielectric, magnetic and lossy body irritated by an incident plane EM wave are investigated in this chapter. Field integral equations of various forms are used to solve this problem. The first formula- tion is based on the method of equivalent polarized currents that yield a set of coupled tensor integral equations [3], [14]. Another set of coupled integral equations can be derived using the free space scalar Green’s functions and the concepts of scalar and vector potentials maintained by equivalent currents and charges. This two sets of cou- pled integral equations are essentially equivalent and a proof will be given. The cou- pled tensor integral equations can be decoupled into a separated tensor electric field integral equation (Tensor EFIE) and a separated tensor magnetic field integral equation (Tensor MFIE) in different but equivalent forms [l3], [16]. The separated tensor field integral equations can also be expressed in terms of free space scalar Green’s functions instead of dyadic Green’s functions in forms of more conventional EFIE and MFIE. Derivations of the integral equations and relations of different formulations are presented. The numerical comparisons of various integral equations are also included. In section 3.1, we introduce the derivation of the coupled tensor integral equations. Section 3.2 illustrates the procedures of decoupling the set of equations derived in sec- tion 3.1 and the relations with another set of decoupled integral equations developed by Tai [13]. Those integral equations in terms of free space scalar Green’s function are presented in section 3.3. We will discuss and compare the numerical solutions of 45 46 these different equations in the last two sections. 3.1 Coupled Tensor Integral Equations A set of coupled tensor integral equation has been derived by Chen [3] in 1981 to relate the induced EM fields inside a finite, arbitrarily shaped, heterogeneous body with the EM fields of an incident time harmonic plane wave. A brief outline of derivation is given in this section. Fig-3.1 shows a finite heterogeneous system composed of dielectric, magnetic and lossy medium with arbitrary shape, being exposed to an incident plane EM wave (13’, H’) with an angular frequency 0). The conductivity, per- rrrittivity and permeability of medium consisting this system are all functions of loca- tion, i.e., o = o(r) (3.1.1) 8 = E(r) (3.1.2) 11 = 110') 1 (3.13) where u is a complex number since the system is assumed to possess some magnetic loss. The generalized Maxwell’s equations govern the electric and magnetic fields, E and H, maintained by a time harmonic source consisting of an electric current density J‘ and a magnetic current density J’” in a medium with permittivity e and permeability 11: V x E(r) = — J’"(r) + i0)u(r)H(r) (3.1.4) V x H(r) = J‘(r) —1‘0)£(r)E(r) (3.1.5) 47 I‘3(I').H(I') etr)'.11(r). on) Fig-3.1 An arbitrarily shaped magnetic lossy body in free space illuminated by an incident EM wave. 48 V°[u(r)H(r)l = p’"(r) (3.1.6) V-[e(r)E(r)] = p"(r) , (3.1.7) where the time harmonic factor e”"" is assumed and p‘, p’" are electric and magnetic volume charge density, respectively. The charge densities and current densities are related by the continuity equations as V-J‘(r) - imp‘(r) = 0 (3.1.8) V-J’"(r) — icop’"(r) = 0 . (3.1.9) Our derivation. is based on (3.1.4) to (3.1.9). In a source free region of free space, or in absence of the body, the incident electric and magnetic fields E’ and Hi must satisfy the following equations: V x E‘(r) = 165110 H’(r) (3.1.10) V x 11"(r) = —icoeo E’(r) . (3.1.11) The internal fields E and H inside the system which are induced by incident fields should also satisfy Maxwell’s equations: V x E(r) = im11(r) H(r) (3.1.12) V x H(r) = o(r)E(r) - itoe(r) E(r) , (3.1.13) or we can rewrite them as V x E(r) = - 3;,(r) +i03110H(r) (3.1.14) V x H(r) = 3,0) - 16120150) , (3.1.15) where .1240) = re(r)E(r) = {60) - iwl E(r) — so ]} E(r) (3.1.16) 49 E(r) = m(l’)H(l') = - i0)[ 90') - 110 ] H(I‘) (3.1.17) are defined as equivalent electric current density and equivalent magnetic current den- sity, 1:, and 1:", are complex electric and magnetic conductivities, respectively. The scattered fields E’ and H5 are defined as follows: E‘(r) = E(r) - E’(r) (3.1.18) H’(r) = H(r) — H’(r) . (3.1.19) Subtracting (3.1.10) and (3.1.11) from (3.1.14) and (3.1.15), we can obtain equa- tions in terms of E‘ and H‘: V x E’(r) = - Jz;(r) + icotrOH‘(r) (3.1.20) V x 113(1) = ngm — itonE‘(r) . (3.1.21) It is advantageous to express the scattering fields E‘ and H‘ as the sum of the electric mode fields, E: and Hi, which are excited by Jfiq, and the magnetic mode fields, Efn and Hf,” which are excited by J2: E‘ = E“; + Efn (3.1.22) H‘ = H: + Hf" . (3.1.23) Therefore (3.1.20) and (3.1.21) can be divided into two sets of differential equations of the electric mode and the magnetic mode: V x Eitr) =im110Hg(r) (3.1.24) V x H§(r) = 54(1) — 161801230 (3.1.25) and V x Efn(r) = — J39“) — impOanm (3.1.26) V x an(r) = - imeoEfnm . (3.1.27) 50 Now we can solve E‘ and HS in terms of .12., and J2. From (3.1.24) and (3.1.25), we have V x V x Ego) - k3 Ego) = 165110 .120) (3.1.28) V x V x Hg(r) - k3 H§(r) = V x ngm . (3.1.29) Similarly from (3.1.26) and (3.1.27), we have V x V x an(r) - k3 an(r) = £0380 Jg;(r) (3.1.30) V x V x Efn(r) - k3 Efn(r) = — V x .1310). (3.1.31) As mentioned in chapter H, the free space dyadic Green’s function 50 must satisfy the differential equation, V x V x Go(r,r’) - kg 80(r,r’) = 7’8(r-r’) , (3.1.32) and by Tai [4], L '6 where ¢(r,r’) is the free space scalar Green’s function. By the definition of the 500.0 =¢(r.r’) for m’. (3.1.33) Green’s function and the discussion of the singularity problem, that is, the correction term and the principal value integration, we obtain the expression of E: and H; [3] as Ego) = j , imnngq(r’)-8o(r,r') dv’ . e , , , J? (r) = P.V. ] , 1651101,,“ )-80(r,r) dv + 3in20 (3.1.34) Hm) = j v 1820130348001") dv’ I I I J’: (r) = P.V. j ,1 3:,(1- )-80(r,r) dv + 371E . (3.1.35) E; and H: can be determined by using (3.1.24) and (3.1.27): 51 s _ _ _i_ .- H.(r) — (”“0 V X Ee(r) = j v [V x 80(r,r')]-ng(r') dv’ S — _‘__ s Em(r) — 0380 V x Hm(r) = — I v [V x 80(r,r’)]-J’¢’;(r’) dv’ . The principle of superposition gives E(r) = E’(r) + E‘(r) = E’(r) + Eg(r) + Em) H(r) = H’(r) + H‘(r) = H‘(r) + Him + 11;,(r) , and this leads to the final results of the coupled tensor integral equations, i.e., [I + :32]: ]E(r) — P.V. I v impote(r’)E(r’)-Go(r,r') dv’ -1. Tm(r’)H(r’)'[V x 50(r,r’)] dv' = E’(r) [I + 23:2]HU) - P.V. I v imotm(r)H(r’)-80(r,r’) dv’ + I . r.(r’)E(r’)-1V x 80(r.r’)] w = H’(r) . Suppose we define 530.1") = @110800‘11") 9,141) = @8060031") 520.8) = - 53.033 = V X 500;) = V¢(r,r’) >< 7’, (3.1.36) (3.1.37) (3.1.38) (3.1.39) (3. 1.40) (3.1.41) (3.1.42) (3.1.43) (3.1.44) 52 then (3.1.40) and (3.1.41) can be written as [I + “‘(r)]E(r) — P.V. I , re(r’)E(r’)-5:(r,r’) dv’ 3018 0 +1 v Tm(r’)H(r’)'5Z(r.r’) dV’ = E’(r) (3-1-45) [1 + mm“) H(r) — P.V. I ,, tm(r’)H(r’)-G’,,';(r,r’) dv’ 3mllo +1 v T.(r’)E(r’)-5’."(r.r’) dv’ = H’tr) . (3. 1.46) Equations (3.1.45) and (3.1.46) are the so called coupled tensor integral equations and can be transformed into a system of linear algebraic equations via pulse-basis expansion and point-matching. Both the electric and magnetic field distributions, E(r) and H(r), can thus be solved simultaneously. More descriptions about the transforma- tion will be presented in section 3.4. 3.2 Decoupled Electric Field Integral Equation (EFIE) and Magnetic Field Integral Equation (MFIE) The coupled tensor integral equations (3.1.45) and (3.1.46) which we have derived in the preceding section can be decomposed into a separated EFIE and a separated MFIE. Let’s derive the tensor EFIE from (3.1.45) first. Rewriting (3.1.45): ite(r) , , [I + 30360 ]E(r) — P.V. I v t¢(r’)E(r’)°G:(r,r) dv + j , em(r')H(r')-5;(r,r') dv’ = E’tr) . (3.1.45) 53 Equation (3.1.45) has the unknown H in the second integral of the left hand side. To eliminate H, we can apply the Maxwell’s equation: _ - ___i_ , H(r’) — 03110") V x E(r’) . (3.2.1) The integrand of the integral can then be written as Mr) wutr’) tremor-5:180 = — [V x E(r’)l°5§.(r.r’) . (3.2.2) We need to make use of the following dyadic and vector identities [15] to remove the curl operation on E: V-(A x 5’) = (V x A)-6’ — A-(V x 8) (3.2.3) V x (6A) = (V6) x A + 6 (V x A) . (3.2.4) Equation (3.2.2) then becomes Tm(r')H(|")'5:(r.l") , 11,,(r’) . , itm(r’) , _ . _ {V [(01103] x E(r’)} 5:,(nr’) — {V x [WM]. )J} 5:0; ) , i‘cm(r’) , itm(r’) , , , _ {V [(01103] x E(r3}5:,(r,r) - C0110") E(r) [V x m(r,r )] _ V’- [:38 E(r’) x 6f,(r,r')] . (3.2.5) Since V’ x Gimr’) = V’ x V’ x80(r,r’) = kfifiour’) + 6(r—r’)r, (3.2.6) by integrating both sides of (3.2.5) over v and applying the divergence theorem to the last term of the right hand side, we have I , tm(r’)H(r’):G:,(r,r’) dv’ 54 . 11,,(r’) , , w ’) = I ”{V [(1)1100] x E(r )}'an(r,r’) dv — I v k2_’;)__ (0W ,’-E(r)) 800, r’) dv’ itm(r) itm(r’) ’ , I _ (0110') E(r) - I, 11 [(01100 E(r) x 5:0; )] ds , (3.2.7) Note that the second volume integral in the right hand side of (3.2.7) is valid in the sense of Cauchy principal value, a correction term is again needed to overcome the singularity problem, i.e., I v krzil—(Zfigmr’lgoflx’) dv’ = P.V.], 8° ——(”1—-;(,)') ———-’E(r) 830-, r’) dv Mr) E 3 2 8 + i30)rr(r) (r). ( . . ) Substituting (3.2.7) and (3.2.8) into (3.1.45) and arranging terms, the decoupled EFIE can be written [_3__:(r+ r+) 8 3‘0“) __]E(r) + P.V.] v ia)[e (r’) — -8%u%]E(r’ ) 5:0, r’) dv’ M[_—)(—l: )140] x E(r)-5:,(rx’) dv’ —j_,n[—:—:;:u 3E0» ) x (‘7‘ (r, r ’85)] --E’(r) , (3.2.9) where e (r) = e(r) +— -’——"('). (3.2.10) The decoupled MFIE can be derived through a similar procedure. Rewriting (3.1.46): it...“ ) [I + 3‘0110 ]H(r) - P. V. I tm(r ’)H(r') G’"(r, r’) dv’ 55 + I ., r.< H(r’)°5:’(r,r’) dv’ l‘ it, r’ , - I s n' #HU’) >< 5:”(rr’) ds’ = H‘(r) . (3.2.17) we (1") To summerize the results, we rewrite (3.2.9) and (3.2.17) as 2% £52 8") r’) [3u(r—)—+ 3 0 E(r)- P.V.I .A—Ic%[ £0 -—(-3-]E(r’) 80a ,3 d, _. J’ v E(r’) x V’ [%u3]-[V’ x 60(r,r’)] dv’ ‘1 [Lian M- “0]“! ><<‘5’o(r.r’)1ds’=E‘(r) (3.2.18) H(r’)-80(r,r’) dv’ +M “0.)" PHVIVI%[flfl-—t-EO_ 38 (1)3110 110 e (r’) 57 8*(r’) - so I.(nx [ €w> _ i [87") ' 80 s 8.0") Equations (3.2.18) and (3.2.19) are decoupled tensor EFIE and MFIE, and can be ]-[V’ x 0(r,r’)] dv’ [n x H(r’)]-[V’ x 0(r,r’)] ds’ = H’(r) (3.2.19) transformed into two systems of linear algebraic equations with a proper choice of basis functions. The electric field and magnetic field distributions are computed numerically in section 3.4. To check the validity of (3.2.18) and (3.2.19), they are compared with a set of decoupled integral equations for homogeneous body derived by Tai [13] here. Tai’s integral equations are: E(r) - (k2 - k5) j , 80(r,r’)-E(r') dv’ +[u-w 11 J I s 50(rfl'l" x V’ x E(r)] dr’ = E‘(r) (3.2.20) H(r) - (k2 - k%) I v 80(r,r’)-H(r’) dv’ + [’3 8:60] I 3 80(r,r’)-[n x V’ x H(r’)] ds’ = H‘(r) (3.2.21) where the complex permittivity 8' and the complex permeability p. are both constants inside the body and k = Vcozue' . (3.2.22) is the wave number in the homogeneous medium. He started from the fact that the EM fields both inside and outside the body must satisfy the Maxwell’s equations, then applied the vector-dyadic Green’s theorem, finally matched the boundary conditions to obtain the above equations. This set of 58 equations can be proved to be equivalent to the equations derived in this section when the body is homogeneous. The proof is given in the appendix A.1. More discussions about the decoupled tensor integral equations are presented in section 3.5. 3.3 Coupled and Decoupled Integral Equations in Terms of Free Space Scalar Green’s Function The problem of solving induced EM fields in a finite, heterogeneous, dielectric, magnetic and lossy body due to an incident plane EM wave as described in the begin- ning of this chapter can be solved through another approach. When such a body is exposed to an incident EM field, there must exist induced electric and magnetic volume current densities and induced electric and magnetic volume charge densities inside the body, in addition to induced electric and magnetic surface charge densities on the boundary as indicated in Fig-3.2. We can then treat these induced currents and charges as equivalent sources in free space, and the scattered fields maintained by these sources are determined via the concept of potentials. In Fig-3.2, ng, J27, pgq, p2}, represent volume equivalent electric, magnetic current densities and volume electric, magnetic charge densities, respectively, while pg and pf," stand for the electric and magnetic surface charge densities, respectively. The scalar and vector potentials, (be and A‘, due to the electric sources can be expressed as: ‘(r) = 21; j . pi,(r)¢(r.r'> dv’ + 8—10 I . p:(r’)¢(r,r’) ds' (3.3.1) A‘(r) = j , J§q(r’)¢(r,r’) dv’, (3.3.2) 59 manna (WQJZ) E(r).H(r) E(r).u(r) o(r) t“ A‘ / 1 - s ”0 F!" O Fig-3.2 Illustration of the equivalent sources induced by the incident EM wave in the volume region v and on the enclosing surface s. where v stands for the volume source region and s for the closed boundary surface enclosing region v, and Mar) = 5E (3.3.3) 4rtR is the free space scalar Green’s function. Similarly, the scalar and vector potentials, CD“ and A“, due to the magnetic sources are (17"(1') = :11: I v pZ'q(r’)¢(r,r’) dv’ + hi0- I s pg"(r’)q>(r,r’) ds’ (3.3.4) Am(r) = j , Jg;(r')¢(r,r’) dv’. (3.3.5) The equivalent volume current densities have the following relations with the induced EM fields: Jiqm = t.(r)E(r) (3.3.6) J20) = rm(r)H(r) (3.3.7) where re(r) = - ico(e‘(r) - 80) (3.3.8) r...<[J§4 (r’)¢(rr’) ] =¢(rr’)[ V >< V¢(r.r) (33-23) and from (3.3.12), (3.3.13), we can to express the scattering fields, E‘ and H3, in terms of the equivalent volume current densities: E’(r)= j .. 1— “(V’J§4(r))V¢(rr)+imqu§4 (r)¢(r.r') 1 dv' -J'swe;o(n°§4J (r’))V¢(rr’)ds’— —I V¢(r,’;"r)xJ (r”)dv (3.3.24) H‘(r>=J V¢1V¢(r.r') + k6r.(r’)E(r'>¢(r.-’)} dv' 63 + it] _. [ne(c.(r)E(r’»1V¢(r.-') ds’ + I . c...¢(r.r’>} dv’ + it] . 1n-(r...(r3HE(r3 x V¢(r.r') dv’ = H(r) , . (3.3.29) ‘- — 3.3.30 is the wave impedance in the free space. where Equation (3.3.28) and (3.3.29) are the coupled integral equations in terms of free space scalar Green’s function. They can also be transformed into a system of linear algebraic equations in a similar way as will be discussed in a latter section. The two different formulations of the coupled integral equations derived in sec- tion 3.1 and here are actually equivalent The proof is given in the appendix A2. The decoupled Tensor EFIE and MFIE (3.2.18) and (3.2.19) can also be expressed in terms of the free space scalar Green’s function instead of the free space dyadic Green’s functions. Since .1. 80(r,r,) = ( r‘i' k3 VV)¢(r,r’) , (3.1.33) (3.2.18) can be rewritten as __2“0 8 (1‘) e'(r’)_ _ll_o_ [3H(r) +_ 320 —-r—]E() Ivk<2)[ 80 ' Tit ) E(F'UWO'I‘) dv’ _ e'(r’) _ F10 P.V. I v [“80l1'(r_]E(r’)) VV¢(r, r’) dv’ - I v E(r’) x V’ [ELLIS—“0]{V’ x 80(r,r’)] dv’ _ u__(__r r)- “om x E(r’)]-[V’ x 0(r,r’)] ds' = E’(r) (3.3.31) “0' ) In appendix A.2, we have shown that I . [V'-A(r)1V¢(I',I‘) 0'3 t1(r ’) - Ho I " (r) X [ Mr) ——]E(r’ )¢(r, r’) dv' ] x V¢(r,r’) dv' — I 3 [Whn x E(r’)] x V¢(r,r’) ds’ = E’(r) (3.3.33) Similarly the decoupled MFIE (3.2.19) can be rewritten as 2’30 £9211- w- 80 [B—e—‘(o 3qu1”) I"k‘z’Lto e’m H(r’)¢(r,r’) dv’ —P.V. Jig—in -VV ,’d’ LL10 8.0.3] 0") MN) v -I H( rx') V’[—(e_:r:-)_] [V’ x 800, r’)] dv’ ‘l [E(r’)r-) 80 Suppose we make use of (A2.1) again, the second integral in the left hand side of [n x H(r’)]-[V’ x 80(r,r’)] ds’ = H’(r) (3.3.34) (3.3.34), denoted as [2, can be written: I I 80 I I I 12: JV{ . [2:12) _ 8‘(r’)]H(r)l}V¢(r,r) dv hand don. expn Spa. Can 66 H(r’) V¢(r,r’) ds’ S “0 8.0") +J_BU)_ E0 3 no 6'0) H(r) . (3.3.35) Let us apply (A2.12) once more, then the third and fourth integrals in the left hand side of (3.3.35), can be rewritten in terms of the free space scalar Green’s func- tion. Substituting (3.3.35) into (3.2.19) and using (A2.12), the decoupled MFIE can be expressed as follows: H(r’)¢(r,r’) dv’ g r *0 H(r)-Ivkg[flu:_)-—f-L e (r) 8 (1") 80 I _J' {V’ V-P—(lfl llo —8*(r,)]H(r’)]}V¢(r,r') dv °£&Q-—EL-H V7 ’df +Is H “0 8.03 (r) ¢ 01m) 0 -¢z(r 1") ¢,(r 1") H')x(r H(r’) [V x 5’0“, r ’)]- _ (3.4.2) Similarly in (3.1.46), the inner product H(r’)-80(r,r’) can be represented by a matrix product as H(r)-806;) f. 'l ¢(r,r’) + i¢n(r,r’) -1—¢,,(r,r’) i¢n(r,r') k3 k3 k% H 1 ’ ’ 1 ¢ ( ’) 1 «1 (r r') 1133 = --¢ (r,r) ¢(r.r) + — r,r — , 161(2) yx 1 k3 yy 1‘3 ’21 H:(r’) —¢zx(r’r’) —¢z (rar’) ¢(r:r’) + _¢zz(r9r’) L k% k3 ’ k?) 4 (3.4.3) 69 while the inner product E(r’)°[V x 80(r,r’)] can be represented by another matrix pro- duct as o -¢z(r,r’) ¢y(r,r’) Ex(r’) E(r’)-[V x 60(r,r’)] = ¢,(r,r') o —¢,(r,r’) Ey(r’) (3.4.4) -<|>y(r,r’) ¢x(r’r’) O E,(r’) where ¢x'x'(r,r’), ¢xp(r,r’) and x1, x2, x3 are the same as defined in section 2.4. After applying pulse-basis expansion and point-matching, the coupled tensor integral equations become a system of linear algebraic equations of order 6N: A B i [C D] [15] = [15] (3'45) where each of A, B, C and D is 31v x 3N matrix, E, H, 15" and H‘ are 3N x 1 column vectors. These matrices and vectors can be expressed as follows: A11 A12 A13 311 312 313 A = A21 A22 A23 B: 821 322 323 (3.46) A31 A32 A33 331 332 333 where Ape and BM are both N x N matrices for p,q=l,2,3, and the elements of these matrices are 1.180.!» (qu),,m = 5pq5mn [I + 30380 .1. + 1'qu P.V. I , te(r')[5pq¢(l'm"') + - k3 ¢.,.,(r,..r')1 dv', m,n=1,2,...,N (3.4.7) 3 (qu)mn = 21 €qu J. v tm(r’)¢x'(r,r’) dv’ . m,n=1,2,...,N (3.4.8) 70 Note that e is defined as Mr 1 p,q,r form a cyclic permutation em, = -1 p,q,r form a anti—cyclic permutation p,q,r=l,2,3 (3.4.9) 0 otherwise Matrices C and D are represented as C11 C12 C13 Du 012 013 C: C21 C22 C23 D: D21 D22 D23 (3.410) C31 C32 C33 D31 D32 D33 where CM and DM are also N x N matrices for p,$l,2,3, and the elements of these matrices are 3 (94),,“ = - )3 cm, I , te(r’)¢x'(r,r’) dv’, m,n=1,2,...,N (3.4.11) r=l itm(r,,) (qu),,,,, — Spqfim [1 + 30410 ] + £0160 P.V. I v" tm(r’)[8pq¢(r,,,r’) + 212_¢x,x,(rmr’)] dv’ . o m,n=l,2,...,N (3.4.12) The column vectors E, E’, H and H’ are E1 El E: E, E‘: 55 (3.4.13) E3 Egj H FH‘i. 1 o e H: H2 11': [1‘2 (3.4.14) ”3 ng where Ep, 15;, Hp and H}, are N x l vectors for p=1,2,3 with the n-th elements 71 (5,), = Exp(r,,) , n=1,2,...,N (3.4.15) (5;), = 520,) , n=l,2,...,N (3.4.16) (Hp),, = pr(r,,) , n=1,2,...,N (3.4.17) (11;), = Hip(r,,) , n=1,2,...,N (3.4.18) From the expressions above, it is noticed that the diagonal submatrices Bpp and CW of the matrices B and C are null matrices. The diagonals of the off-diagonal sub- matrices BM and CM are all zeros since the integration will be canceled out by the symmetry property. By solving the 6N x 6N matrix equation (3.4.5), we can obtain both electric and magnetic field distributions over the body simultaneously. The decoupled tensor integral equation (3.2.18) and (3.2.19) are investigated next. For a heterogeneous body, these equations contain two terms which involve the gra- dients of the complex permittivity and permeability functions. This will cause difficulty especially for heterogeneous body with jump discontinuities of parameters within the body. Therefore, we will concentrate on the cases of homogeneous bodies in the discussion of the decoupled tensor inteng equations. The decoupled tensor integral equations when applied to a homogeneous body can be written: 2110 e' 2' “o , , , . {-3—};- + 380]E(r) — P.V.! v kgL—O - TJE“ ) 800x ) dv - I s [u Ito Jln >< E(r’)]°[V’ X 50(r,r’)] ds’ = E(r) (3.4.19) H(r’)-(‘§"0(r,r') dv’ 2 £3 + .11_ [38 3140 H(r) —P.v.j , k3[-u% — :3 72 [n x H(r’)]-[V’ x 80(r,r’)] ds’ = H’(r) . (3.4.20) 8*- -Jls[ 8.80 It is sufficient to analyze the tensor EFIE only since the tensor MFIE can be han- dled similarly. In the tensor EFIE (3.4.19), the inner product E(r’)-80(r,r’) is exactly the same as that shown in (3.4.1) while the inner product [11 x E(r’)]-[V’ x 0(r,r’)] can be expressed in the following matrix product form: [n x E(r-311V x 800.0] 42450.1") + n3¢,(r,r’) -n1¢,(r.r’) -n1¢z(r.r’) Ex(r’) = " “n2¢x(l‘,|") "14’4".” + n3¢z(l‘,l") “"2¢z(l‘,l") 5,0") —n3¢x(r’r’) ’n3¢y(r’r’) nl¢x(r9r’) + n2¢y(rar’) £20") (3.4.21) where n1, n2 and n3 are the three components of the unit normal vector n as defined in section 2.4. Applying similar testing procedure as before, we can transform (3.4.19) into a 3N x 3N matrix equation: F .7 011 012 013 E1 E} 021 022 023 E2 = E‘z (3.422) G31 G32 033 E3 ES The elements of the N x N matrix GM for p,q=l,2,3 are: -1. (0,4),“ = Cl swam, + e2 P.V. j ,_ [5pq¢(r,,,r’) + k2 0 ¢x,.,(r,.,r’)] dV’ 3 + c3 j ,_ [8,, 2‘, nfibxi(r,,,r’) — (1 — Spq)np¢xq(rn,r’)] ds’, £44 i=1 m,n=1,2,...,N (3.4.23) 73 where c, = '23? 3:0 (3.4.24) 62 = - 1.4.2: .. 5”? (3.4.25) c3 = u :10 (3.4.26) and s”, stands for the surface of m-th cell which are common with the closed surface s enclosing v. We can now solve the matrix equation (3.4.22) to obtain the electric field distri- bution of the body. Similarly we can obtain the magnetic field distribution if we fol- low the same procedure to transform the MFIE into a matrix equation. The numerical solutions of the coupled tensor integral equations and decoupled tensor integral equations are compared in section 3.5. The coupled integral equations with scalar Green’s function (3.3.28) and (3.3.29) can be transformed similarly. Since these two equations contain the terms involving the divergences of E and H, they are in fact integro—differential equations instead of pure integral equations. To avoid the complexity of the equations, we again discuss this set of equations for the case of a homogeneous body where the volume charge density should vanish inside the body. Furthermore, the pulse-basis expansion of the induced field distributions will satisfy this charge-free condition inside each cell automatically except on the adjacent boundaries of the cells. Those boundaries of the adjacent cells are then assumed to contain some surface charge so that we should extend the domain of the surface integration when the pulse-basis expansion is applied. The set of integral equations when applied on a homogeneous body then becomes: 74 E(r) - £de . r.E(r’)¢(r.r) dv + 30-] .. [n'(t,,,H(r ))]V¢(r,r) d3 - I . I.E(r’) >< V¢(r.r’) dv’ = H‘(r) . (3.4.28) where I, and 1:", are both constants inside the body now. In (3.4.27) and (3.4.28), the inner products [n°E(r’)]V¢(r,r’) and [n-H(r’)]V¢(r,r’) are expressed in matrix forms as n.¢.(r,r’) can be represented by the same matrix product forms as those in (3.4.2) and (3.4.4). The system of linear algebraic equations of order 6N obtained after we applied pulse-basis expansion and point-matching is then 1841,5142: where the matrices B, C', and the column vectors E, H, E’ and H’ are the same as those (3.4.31) in (3.4.5). X and 13' are also 3N x 3N matrices: F— _ _ . ._ _ _ . _ 1:11 I112 1113 _ 1:11 £12 1:13 A: {21 1:22 11.23 D = 321 13.22 323 (3.432) LAM A23 A33 031 023 033 75 Km and 5M are both N x N matrices for p,q=l,2,3 with elements (,1;an = 5m [5m - iCkore j ,m ¢(rn,r’) dv’ ] iCr, ko + I§.nq¢xp(rmr’) d5, : m,n=1,2,...,N (3H433) _ ikotm , , mm)... = 6,, [6,... - T I ._ ¢ VII VI III y 11 Fig—3.3 A symmetric body partitioned into symmetrical octants, denoted by Roman numerals. The origin of the coordinate system is located at the center of the body. 78 (3.4.37) Eysm) = — we) = Ewe) = - mm = Ems) = - E,. = - H..(rs) = Hues) = - H..(r7) = Hum) (3.4.40) H,.(r1) = — H,. = - H,. = H,.(r7) = - H,.(r8) (3.4.41) st(|‘1) = H..(r2) = — Hum) = — Hum) = Hues) = Haas) = - Hum) = - z.(rg) (3.4.42) Similarly, let Ea(r) and Ha(r) denote the EM fields induced by the anti—symmetric mode of E‘(r) and H‘(r), then the three components of Ea(r) and Ha(r), denoted as EMU), Eya(r), Eza(r) and Hm(r), Hya(r), Hza(r), will obey the following relations: Exam) = Emu» = Emu.) = Emu.) = — ”(u-5) = - Emu.) = — mm) = - was) (3.4.43) 5 nyrl) = — £3.02) = Eyars) = - Emu.) = - Eya("5) = E,.(r6) = - E,.(r7) = 5,.(r8) (3.4.44) Emu.) = - 52.02) = - 52m) = Emu.) = Ewes) = - Emu.) = - Earn) = Enos) (3.4.45) Hxa(r1) = - 12102) = Hxa(r3) : _ xa(r4) = Hm(r5) = " Hxa(r6) = Hxa(r7) = _ xa(r8) 79 (3.4.46) Hya(|' 1) = Hya(r 2) = Hya(|' 3) = Hya(l‘4) = Hya(r5) = Hya(r6) = Hya(r 7) = Hya(r8) (3.4.47) Hza(l‘1) = Hza(l'2) = " Hm“ 3) = - Hza(|'4) = - Hza(|'5) = " Hza(|‘5) = ”2.107) = Hza(l'3) (3.4.48) By using the relations (3.4.37) to (3.4.48), we can reduce the number of unk- nowns in the system of linear algebraic equations by a factor of eight if a particular mode of incident field is given. To manipulate the reduced systems, we only match the points at the centers of those cells in the first octant, and then rearrange those linear algebraic equations by collecting the terms which possess symmetric properties. Once we find the induced field distributions in the first octant, the distributions in other octants can be easily obtained by employing the relations of (3.4.37) to (3.4.48). 3.5 Discussions Numerical examples of induced fields in a magnetically permeable body irritated by an incident plane wave are presented in this section. Solutions of various integral equations applied to different bodies are investigated and some comments about each set of equations are made. In Fig-3.4, a homogeneous cubic body of dimensions 6 x 6 x 6 cm is illuminated by an incident EM plane wave. If the incident EM wave is decomposed into sym- metric and antisymmetric modes, we can choose the origin of the coordinate system at the center of this cube so that we merely need to investigate the induced EM field 80 e:=5+i4 Ei p;=5+i3 f=200NIHZ i H +X F Z 6C!!! f L Fig-3.4 A homogeneous cubic body of dimensions 6 x 6 x 6 cm illuminated by an incident plane EM wave. 81 distributions of different modes in the first octant owing to the symmetry properties. It is noticed that when the body is non-magnetic, that is, 11" = no, the first equa- tion of the set of coupled integral equations is decoupled automatically and we will find that (3.1.45) becomes identical to (3.2.18). Similarly in the non-dielectric case 8" = 80, equation (3.1.46) will become identical to (3.2.19). We will study the numeri- cal solutions of the coupled tensor integral equations first and then investigate the effects of the magnetic material on the induced fields. The numbers presented in Fig- 3.5 to Fig-3.10 are all numerical solutions of (3.1.45) and (3.1.46). Fig-3.5 shows the induced electric field distribution in a homogeneous body of the dimensions specified in Fig-3.4 with permittivity e = 580, conductivity 0 = 0.0444 S/m and complex permeability u' = (5 + i3)uo, i.e., the relative complex permittivity e: = 5 + i4 and the relative complex permeability tr: = 5 + £3, excited by i C quency at 200 MHz. The body is divided into 216 cubic cells of size 1 x 1 x 1 cm, a symmetric mode of incident EM field (E‘, H‘) = (x cos(koz), y sin(koz)) of fre- but only the numerical solutions in the 27 cells of the first octant need to be presented We simply indicate the amplitude (in V/m) and phase angle (in degree) of the x— com- ponent of the electric field E in this figure since lEyl and [5,] are much smaller than IEXI in the symmetric mode solution. Fig-3.6 shows the distribution of EJr in a nonmagnetic cube with identical size and e: excited by the same incident fields. We observe that the numerical solutions in these two cases are almost the same, this tells us the mag- netic property of the material does not affect too much on the induced electric field when the incident electric field is symmetric and almost constant throughout the body. The reason for this phenomenon is that the incident magnetic field is zero at the center of the body, and also the permeability is not high enough to have a significant enhancement of the induced fields. 82 g: .-. 5 +14 T 11 u: = 5 + [3 ”I! 15,; in (Wm) .3734 .3819 .4344 ‘5’ m (a) .273 -27.3 -25.1 m 3 2 .4314 .4432 .4331 W 1 48.4 48.3 45.2 .4548 .4890 .4978 -25.9 -25.3 .252 l .3138 .3221 .3777 .3040 .3122 .3877 49.7 49.9 47.9 49.4 49.8 48.2 .3948 .4078 .4400 .3798 .3919 .4227 45.9 48.4 - 48.7 45.2 48.1 47.0 .4255 .4403 .4522 .4149 .4254 .4451 44.8 45.5 48.3 44.1 45.1 48.8 2 3 F' -3.5 Solutions for the symmetric mode of induced electric field obtained from the 1g coupled tensor integral eguations based on 216-cell drvrsron 1n the magnetic body. (x- component of field) H: = 851?! IE; in (Wm) .3597 .3791 .4345 A ‘5‘ m (a) -25.0 -27.7 -24.5 m 3 2 .4254 .4425 .4543 f 1 -25.5 -25.1 «24.3 / .4491 .4553 .4993 -25.9 -25.5 -24.1 1 .3077 .3175 .3771 .2959 .3059 .3555 -31.3 -31.1 -27.7 -31.5 -31.4 -25.1 .3545 .4005 .4400 .3577 .3534 .4224 -25.1 -27.9 -25.2 .254 .252 .257 .4159 .4321 .4523 .4009 .4155 .4447 -27.3 .270 -25.7 -27.5 -27.4 -25.1 2 3 a: = 5+ 14 83 Fig-3.6 Solutions for the symmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the non- magnetic body. (x- component of B field) 84 Now let us investigate the case for an incident field of the antisymmetric mode. Fig-3.7 and Fig-3.8 (indicate the induced electric field distributions in the magnetic body with the same parameters and dimensions as in Fig-3.5 excited by an incident EM field (E‘, H‘) = (x isin(koz), y €cos(koz)) of frequency at 200 MHz. The body is divided as before and only the cells of the first octant are of interests. In this case only y— component of the electric field is insignificant. The distribution of Ex is presented in Fig-3.7 and E, in Fig-3.8. The x- and z— components of the induced electric fields in the same non-magnetic body as described in Fig-3.6 excited by the same antisymmetric incident field are shown in Fig-3.9 and Fig-3.10. It is observed that the amplitude of the electric field E in the magnetic material increases by a factor of 2 compared with that in the non-magnetic material. This indicates that the magnetic material plays an important roll in the induced electric field excited by the incident EM field of antisymmetric mode which has a maximum incident magnetic field at the center of the body. Fig-3.11 to Fig-3.13 give the numerical solutions of the decoupled tensor integral equation (3.2.18) applied to the same magnetic body as described above. The distribu- tion of 5,, of symmetric mode is shown in Fig-3.11 while the distributions of E, and 15‘z of the antisymmetric mode are shown in Fig-3.12 and Fig-3.13. The frequency is 200 IVle as before. We find that not only is the induced field distribution of the sym- metric mode very close to that solved by the coupled tensor integral equations but also the distribution of the antisymmetric mode. From the examples shown above, it appears that the agreement of the numerical solutions between the two different sets of tensor integral equations is satisfactory. We will investigate the convergence of the numerical solutions next. e: = 5 + £4 T 11 u: = 5 + :3 85:11 15:1 in (V/m) .0130 .0407 .0835 A ‘5: i“ (a) 75.0 75.5 50.5 5.3,. 3 2 .0195 .0828 .1239 7 85.1 85.8 88.2 / .0229 .0731 .1409 87.8 88.3 90.7 1 .0135 .0428 .0895 .0141 .0448 .0930 80.3 80.5 83.3 81.8 81.9 84.2 .0224 .0722 .1418 .0237 .0784 .1488 89.9 89.9 91.1 91.8 91.4 92.1 .0288 .0854 .1832 .0284 .0909 .1721 92.3 92.4 93.4 1 93.9 93.7 94.4 2 3 85 Fig-3.7 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the mag- netic body. (x- component of E field) IE,I in (Wm) 15', in (’) Fig-3.8 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell divrsron 1n the mag- e,‘ = 5+ 1'4 11: = 5+ 13 .0975 .0751 .0339 .757 -77.2 -77.0 .0473 .0354 .0155 .755 -75.5 50.2 .0143 .0109 .0045 -7s.5 -75.0 -50.5 1 .1255 .1023 .0475 -75.4 .755 -77.1 .0629 .0497 .0228 .75.2 -75.5 -79.1 .0191 .0150 .0069 .74.1 .744 -75.7 2 86 § netic body. (2- component of E field) .1389 .1115 .0522 ~78.3 -78.8 ~77.4 .0698 .0553 .0258 -75.1 -75.5 -79.2 .0213 .0188 .0078 -73.9 -74.3 -78.8 3 5: = 5 + 1'4 T 1‘ “r. = Gem 15,; in (WM) .0095 .0292 .0559 4 E, in (°) 1 59.7 70.0 72.7 5a., 3 2 .0130 .0405 .0740 f 1 75.5 77.0 75.5 .0145 .0455 .0513 75.9 79.4 51.2 Io—— .. ——~1 1 .0053 .0257 .0511 .0052 .0253 .0505 55.3 55.5 71.5 55.3 55.5 71.5 .0123 .0355 .0709 .0121 .0379 .0599 75.3 75.7 75.5 75.5 75.9 75.5 .0141 .0441 .0791 .0139 .0434 .0752 75.5 79.1 50.9 75.5 79.2 51.0 2 3 87 Fig-3.9 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the non- magnetic body. (x- component of E field) 15:1 in (Win) 255m Fig-3. 10 5: = 5+ 14 n; .0353 .0255 .0052 -75.5 -74.7 .41.4 .0155 .0135 .0041 -74.5 -72.3 -57.9 .0057 .0042 .0013 .735 -71.2 -59.7 1 .0351 .0252 .0075 .7911 -7711 -51J .0198 .0148 .0050 .7511 -745 4220 .0051 .0045 .0015 .7594 .7391 -5533 2 88 § Ni .0392 .0291 .0082 -79.8 ~77.8 -53.3 .0203 .0150 .0052 -77.7 -75.7 -88.8 .0083 .0048 .0018 -78.3 -74.5 -87.7 3 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 216-cell division in the non—magnetic body. (x- component of E field) [5,] in (V/m) LE, in (’) Fig-3. 1 1 e: = 5 + 1'4 11: = 5 + 1'3 .3795 .3535 .4425 .277 .279 .245 .4394 .4495 .4918 -25.5 -25.1 -23.9 .4533 .4742 .5093 -2512 -2591 -2394 1 .3095 .3102 .3795 -32.1 32.5 25.0 .3557 .3929 .4429 -2931 -3013 -2511 .4175 .4219 .4554 -2511 -2911 -251' 2 89 § Ni .3075 .3051 .3744 -31.5 32.1 -25.0 .3531 .3550 .4305 -25.5 .295 -25.2 .4120 .4144 .4530 -27.5 -25.9 25.5 3 Solutions for the symmetric mode of induced electric field obtained from the tensor EFIE based on 216-cell division in the magnetic body. (x- com- ponent of E field) '90 fx 5fi=5+i4 A ll/ u:=5+1'3 em I / |E,| in (WM) .0125 .0357 .0799 ‘ Ex 1" (a) 73.1 72.5 75.5 5a,, '7 3 2 .0159 .0557 .1153 1 51.3 50.5 52.4 .0217 .0875 .1350 83.7 82.9 84.7 1 .0134 .0478 .0858 .0145 .0488 .0890 74.9 73.7 78.8 78.1 74.8 77.0 .0224 .0898 .1392 .0245 .0758 .1 457 88.7 82.2 83.5 84.8 83.2 84.0 .0284 .0828 .1831 .0292 .0904 .1718 88.2 84.8 85.8 87.2 85.8 88.2 2 3 Fig-3.12 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 216-cell division in the magnetic body. (x- component of E field) 15,! in (V/m) 48mm Fig-3. 13 5; = 5 + 1'4 Ll: = 5 + 13 .0853 .0871 .0280 284.1 ~84.3 -82.8 .0415 .0328 .0141 -82.4 -82.7 -88.2 .0130 .0103 .0045 ~80.7 81.0 -85.2 I .1 180 .0938 .0398 -85.7 -88.1 -85.5 .0831 .0501 .0227 -83.7 ~84.5 -88.1 .0200 .0159 .0072 -81.5 -82.3 -88.3 2 Solutions for the antisymmetric mode of induced elecnic field obtained from the tensor EFIE based on 216-cell division in the magnetic body. (2- component of E field) 91 i .1250 .1015 .0434 -55.1 .555 -55.4 '07‘2 .0555 .0257 '8” -55.1 59.0 .0230 .0153 .0537 52.2 53.1 .573 3 92 Since for the symmetric mode of incident field, regardless of the body being permeable or not, the induced electric field distribution does not change much in the previous examples. We also know from section 2.2 that the symmetric mode solution converges quite well in the non-magnetic body hence so does the symmetric mode solution in the magnetic body. In fact the numerical results will confirm the predic- tion. Thus, we omit the symmetric mode case and concentrate on the cases of antisymmetric mode. The convergence of the solutions of the coupled tensor integral equations are shown in Fig-3.14 to Fig-3.17. Fig-3.14 and Fig-3.15 show the distributions of E, and E, in the same magnetic body divided into 512 cubic cells excited by the same antisymmetric incident EM field. Each octant contains 64 cells and we depict the first octant distribution by dividing the octant into four l6-cell layers as shown in the figures. Each layer has a fixed y— coordinate. The numbers in Fig-3.14 are the ampli- tude and the phase angle of E, and in Fig-3.15 are of E,. We next divide the body into 1000 cells and present the first octant induced field distributions in Fig-3.16 and Fig-3.17. Now we divide the first octant into five 25-cell layers and the numbers in each cell again show the amplitude and phase of the electric field at the center of each cell. The distribution of E, is shown in Fig-3.16 and E, is in Fig-3.17. The agreement between these two sets of data is quite satisfactory. Fig-3.18 to Fig-3.21 indicate the induced electric field distribution of 512 and 1000 subdivisions of the same body solved by the decoupled tensor EFIE. The incident EM field is the same antisymmetric mode with frequency at 200 MHz. The distributions of E, and E, of 512 subdivision are shown in Fig-3.18 and Fig-3.19 and that of 1000 subdivision are shown in Fig-3.20 and Fig-3.21. Again the results of these two divisions agree well. 5fi=5+i4 u:==5-+13 IE;|01(VWn) LEAN") Fig-3.14 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 512-cell division in the 93 .0091 7315 .0278 7413 .0487 7515 .0829 7915 .0129 831) .0402 8357 .0729 841) .1210 8715 .0157 8711 .0488 8715 .0880 8815 .1415 901) .0171 8857 .0532 8912 .0953 9012 .1506 9213 .0093 7917 .0290 80A. .0521 801! .0903 831 .0153 911) .0479 91.1 .0877 9191 .1438 9113 .0193 9413 .0602 9411 .1090 9414 .1724 941) .0212 9515 .0661 9515 .1188 9515 .1847 9612 3 magnetic body. (711- component of E field) .0089 7715 .0277 7812 .0496 791) .0866 821) .0143 8811 8817 .0821 8912 .1360 9015 .0178 921) .0558 9212 .1013 9215 .1618 9317 .0196 9313 .0611 9315 .1101 941) .1730 951) .0096 8015 .0297 801) .0532 8115 .0919 8315 .0158 921) .0494 921) .0902 9117 .1474 9211 .0199 9512 .0623 9512 .1125 9551 .1773 9515 .0220 9611 .0685 9611 .1228 9613 .1903 9617 4 94 +1: 80m 5, = 5 + £4 ‘ 2 85111 11: = 5+ 13 f 1 2 y / L |E,I in (Vim) .1052 .0950 .0701 .0273 .1341 .1213 .0922 .0399 4 E, in (a) -74.7 -75.1 -75.7 -74.4 -74.4 -74.8 -75.4 -74.8 .0832 .0552 .0388 .0155 .0815 .0723 .0530 .0233 -74.7 -74.9 -75.5 -79.1 -73.4 -73.8 -74.2 -77.8 .0338 .0292 .0202 .0084 .0437 .0385 .0278 .0124 ~74.3 -74.4 -75.2 -81 .3 -72.5 -72.8 -73.2 -78.4 .0108 .0091 .0083 .0027 .0138 .0121 .0087 .0039 -74.1 -74.3 -75.2 -52.5 -72.0 -72.1 -72.7 -78.8 1 2 .1487 .1348 .1029 .0452 .1555 .1410 .1077 .0475 ~74.3 -74.7 ~75.3 ~75.2 -74.2 ~74.8 -75.2 -75.3 .0921 .0821 .0807 .0272 .0970 .0888 .0842 .0289 -73.2 -73.4 -74.0 -78.0 -73.1 -73.3 -73.9 -77.9 .0499 .0441 .0322 .0148 .0528 .0488 .0342 .0158 -72.1 -72.2 -72.9 -75.3 -72.0 -72.1 -72.7 -75.3 .0158 .0139 .0101 .0048 .0187 .0148 .0107 .0049 -71.5 -71 .8 -72.3 -78.5 -71.4 -71 .5 -72.2 -78.4 3 4 Fig-3. 15 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on 512-cell division in the magnetic body. (2- component of E field) 5:=5+1‘4 fi=5+fi |E§|ht(th) LE, in (’) 95 .0070 7115 .0212 721) .0362 73J. .0532 7411 .0823 7811 .0066 7417 .0201 7512 .0349 7612 .0527 7711 .0841 801i .0093 8012 .0287 8011 .0504 821) .0772 8311 .1174 8611 .0100 8517 .0309 861. .0549 8612 .0851 8717 .1298 8911 .0114 8511 .0350 8515 .0616 8615 .0938 871) .1382 9012 .0126 9012 .0390 9015 .0691 9115 .1062 9157 .1559 931) , .0125 8737 .0394 8812 .0691 891) .1043 9012 .1502 9212 .0144 9211 9215 .0782 93A. .1188 9357 .1704 9413 .0136 881) .0417 8913 .0728 90J. .1093 9112 .1557 93A. .0153 9313 .0471 9315 .0826 941) .1247 9415 .1769 9515 Fig-3.16 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations based on lOOGcell division in the magnetic body. (x- component of E field) IE,| in (V/m) 45: in (a) 96 Fig-3.168 .0068 .020 .0364 .0551 .0873 .0070 .0215 .0374 .0565 .0892 77.1 77.5 78.3 79.2 81.8 78.2 78.6 79.3 80.1 82.3 .0106 .0329F0584 .0906 .1369 .0111 .0342 .0607 09391411 88.7 89.0 89.5 89.9 90.8 90.3 90.5 90.8 91.0 91.6 .0136 .0422 .0745 .1142.1660 .0143 .0441 .0779 .1190.1721 93.0 93.2 93.4 93.7 94.4 94.5 94.6 94.7 94.7 95.1 .0156 .04810846 12821822 .0164 0505.0887 .1340.1894 94.9 95.0 95.3 95.5 96.1 96.3 96.3 96.4 96.5 96.8 .0166 .0511 .0895 .1348.1894 .0174 .0 .0938 .1410 .197 95.7 95.8 96.0 96.3 96.8 97.0 97.1 97.1 97.3 97. 4 .0071 .021 .0379 .0573 .0901 78.7 79.0 79.7 80.5 82.6 .0113 03480618 0955.1431 90.9 91.1 91.3 91.4 91.9 .0146 .0450 .0795 .1213.1750 95.1 95.2 95.2 95.2 95.5 .0168 .051 .0906 .1367.1929 96.9 96.9 96.9 96.9 97.1 .0178 .054 .0959 .1439 .2009 97.6 97.6 97.6 97.7 97.9 s d=5+m ¢=5+a 97 \ IE,I in (Vim) 1E, in (0) .1105 ~73.6 .103 ~73.9 .0882 ~74.5 .0621 ~75.1 .0216 . ~71.9 ~ 1 368 73.3 .m9. 436- 1120 74.1 082010332 ~74.7 ~72.8 .0735 ~73.7 .067 ~73. 9 .0557 ~74. 4 .0373 ~75.1 .0140 ~78.7 .0923 ~72.6 .085 ~72.8 .0726 ~73.1 .0511 ~73.7 .0218 ~77.2 .0470 ~73.6 .042 ~73. 7 .0347 74. 2 .02281 ~75.4 .0091 ~82.4 .0590 ~72.0 .0545 ~72. 1 .0453 ~72.3 .0314 ~73.1 .0137 ~78.8 .0261 ~73.5 .023 ~73. 7 .0190 ~74. 3 .0133 ~76.1 ~85. 1 .0052 .0328 ~71.5 .0301 ~71.6 .0249 ~71.8 .01 71 ~73.0 .0076 ~80. 1 .0084 ~73.5 .007 ~73.7 .0060 ~74. 3 .0039 ~76.6 001 . 7 .0105 ~86.5 ~71.2 .009 ~71.4 .0079 ~71.7 .0054 ~73.0 .0025 ~80.8 2 Fig-3.17 Solutions for the antisymmetric mode of induced electric field obtained from the 6013 led tensor integral equations based on 1000-cell division in the magnetic 1 body. (z- component of E field) 98 15.1 in (Wmfi 524 25mm .1 .1254 09240384 .1620 .1531.1333 09830413 -73.3 -73.5 -74.1 ~74.6 -73.4 -73.2 -73.4 .730 -74.5 ~73.6 .1046 .097 .0831 05930259 .1122 .104 .0894 064010283 -72.4 -72.5 -72.9 -73.5 -77.3 -72.2 -72.4 .727 .733 -77.3 .0675 .062 .0525 036910165 .0728 .067 .0569 .0402 .0182 -71.5 .710 «71.8 .727 ~78.5 -71.3 -71.4 ~71.6 -72.5 ~78.5 .0377 034810290 .0202 .0092 .0409 037810316 .0221 .0102 -70.9 -70.9 -71.2 -72.3 "-79.4 ~70.6 -70.7 -71.0 -72.0 -79.2 .0121 01120093 00640030 .0132 0121.0101 0071.0033 -70.5 ~70.6 -70.8 -72.1 -79.8 -70.3 -70.3 ~70.6 -71.8 -79.5 3 4 .1166 .157 .1370 .1010[0426 -73.1 -73.4 -73.9 -74.5 -74.7 .1159 .1083I .0923 .0662 .0293 -72.2 -72.3 ~72.6 -73.3 -77.3 .0754 .0701 .0590 041810189 :n27m-N5mu-n5 .0424 03920328 .0231 .0166 -70.5 ~70.6 ~70.8 -71.9 -79.1 .0137 .012 .0105 00740034 -70.2 -70.2 -70.5 -71.7 .-79.5 Fig-3. l7a 99 5"“ / O . g“ E, = 5 + 14 2 “cm 3 ‘— Gcm __..I |E,l in (Vim) .0092 .0279 .0475 ..0804 .0091 .0279 .0481 .0841 L Ex in (a) 72.7 72.9 72.9 76.2 75.3 75.3 74.8 77.5 .0129 .0398 .0701 .1174 .0147 .0454 .0809 .1355 81.4 81.5 81.4 83.6 85.7 85.3 84.4 85.7 .0153 .0471 .0833 .1383 .0180 .0557 .0999 .1650 84.6 84.8 84.7 86.9 89.4 88.9 87.9 89.1 .0164 .0506 .0895 .1477 .0195 .0605 .1088 .1785 85.8 86.0 86.0 88.3 90.7 90.3 89.3 90.6 .0099 .0303 .0519 .0877 .0102 .0313 .0533 .0891 77.5 77.3 76.5 78.2 78.4 78.1 77.1 78.5 .0164 .0505 .0888 .1430 .0172 .0528 .0920 .1464 88.2 87.6 86.2 86.5 89.1 88.5 86.9 86.8 .0202 .0626 .1106 .1756 .0213 .0657 .1151 .1803 91.9 91.2 89.7 89.9 92.8 92.1 90.4 90.2 .0221 .0682 .1208 .1906 .0233 .0717 .1259 .1960 93.3 92.7 91.2 91.3 94.3 93.5 91.8 91.7 3 4 Fig-3.18 Solutions for- the antisymmetric mode of induced electric field 'obtained from the tensor EFIE based on 512-cell division in the magnetic body. (x- component of E field) 100 f x 6"" / 6: =5 +14 3 4 z 1'; =5+ i3 2 / ’7 J. ‘— IE,I in (Win) .0973 .0861 .0622 .0213 .1328 .1184 .0873 .0344 421:: (a, -80.0 -80.3 ~80.5 -76.6 -80.3 ~80.8 81.3 .79.4 .0561 .0497 .0361 .0147 .0828 .0739 .0549 .0240 ~79.4 ~79.5 ~79.6 ~83.0 ~78.0 ~78.5 ~79.4 ~83.2 .0306 .0271 .0197 .0084 .0453 .0405 .0302 .0135 ~78.2 ~78.l ~78.l ~83.2 ~75.6 ~76.0 ~77.0 ~81.8 .0097 .0086 .0063 .0028 .0144 .0129 .0096 .0044 ~77.7 ~77.5 ~77.5 ~83.5 ~74.4 ~74.8 ~75.7 ~81.2 1 2 .1473 .1312 .0970 .0389 .1539 .1372 .1014 .0409 ~80.6 ~81.1 ~81.8 ~80.7 ~80.7 ~81.2 ~81.9 ~81.0 .0962 .0860 .0639 .0283 .1020 .0911 .0677 .0030 ~78.0 ~78.6 ~79.8 ~83.9 ~78.0 ~78.6 ~79.9 ~84.2 .0541 .0484 .0361 .0163 .0580 .0519 .0388 .0176 ~75.3 ~75.9 ~77.2 ~82.4 ~75.3 ~75.9 ~77.3 ~82.6 .0174 .0156 .0117 .0053 .1879 .0168 .0126 .0058 ~74.0 ~74.6 ~75.9 ~81.6 ~73.9 ~74.6 ~76.0 ~81.8 3 4 Fig-3.19 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 512-cell division in the magnetic body. (2- component of B field) - 101 d=5+m Lc=5+B 1 L l« ..._.| EQmOWOQNI .LE;in(?) 7111 .0215 721) .0365 7211 .0523 7215 .0803 751) .0067 741) .0204 741) .0351 7415 .0512 7415 .0818 7712 .0095 8012 .0290 8011 .0503 8013 .0750 81J .1145 8315 .0103 851) .0317 851) .0555 8411 .0840 8411 .1292 851) .0113 841) .0346 8412 8415 .0901 841) .1358 8711 .0127 89J. .0392 891) .0695 881) .1050 8815 .1590 891) .0125 8511 .0383 861) 8615 .0995 861) .1484 8911 .0143 911) 901) .0773 9011 .1179 9015 .1764 92J .0131 8611 8611 .0697 8713 .1040 8711 .1542 9011 .0150 9117 .0462 9117 .0812 9115 .1240 9115 .1845 931. Fig-3.20 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 1000~ce11 division in the magnetic body. (x- component of E field) IE;|h1(VWn) LE, in (0) 102 .0072 761) .0221 7611 .0379 7617 .0549 7613 f .0076 78.4 .0851 771) .0230 7812 .0394 771) .0567 7712 .0868 7811 .0114 8813 .0351 881) .0613 8715 .0917 8615 .1362 8611 .0122 9011 .0373 8917 .0647 881) .0959 8715 .1402 8711 .0143 9215 .0439 9212 .0769 9115 .1158 9015 .1693 9011 .0153 9412 .0469 9311 .0817 931) .1218 9115 .1752 9111 .0160 9415 .0494 9412 .0865 9315 .1305 9217 1.1888 931) .0172 9612 .0529 9511 .0921 951) .1375 9317 .1960 9315 .0169 9513 .0519 951) .0910 9415 .1374 9315 .1978 941) .0181 971) .0556 9615 .0970 9511 .1449 9415 .2057 9415 .0077 791) .0235 7811 .0402 7811 .0576 7715 .0876 7515 .0125 9011 .0383 9011 .0663 8915 .0978 881) .1420 8717 .0158 951) .0484 9415 .0839 9315 .1244 921. .1779 9115 .0178 971) .0545 9615 .0947 9515 .1407 9481 .1994 9317 .0187 9711 .0574 9713 .0998 9611 .1483 951) .2095 9417 Fig-3.208 103 l\ 1 €=5+M ¢=5+B 1521 in (VIM) LQmm \ .1040 ~77.5 .0966 ~77.8 .0811 ~78.3 .0555 ~78.4 .0168 ~71.5 .1390 ~77.3 .u% -WJ .1103 ~78.4 .0781 ~78.8 .0287 ~75. l .0662 ~77.8 .0613 ~77.9 .05 12 ~77.9 .0353 ~77.9 .0136 ~80.9 .0934 ~75.4 .0872 ~75.7 .0743 ~76.3 .0532 ~77 .2 .0227 ~80.8 .0427 ~77.2 .0393 ~77.1 .0327 ~76.9 .0226 ~76.9 .0093 ~82.4 .0596 ~73.7 .0557 ~73.8 .0474 ~74.2 .0341 ~75 . l .0150 80.2 .0239 ~76.9 .0220 -MJ .0182 ~76.4 .0126 ~76.5 .0055 ~83.4 .0333 ~72.5 .031 1 ~72.6 .0264 ~72.8 .0190 ~73.6 .0086 ~79.8 .0077 ~76.7 .(X)7 l ~76.5 .0058 ~76. 1 .0040 ~76.3 .0018 ~83.8 .0107 ~72.0 .0100 ~72.0 .0085 ~72. 1 .0061 ~72.9 .0028 ~79.5 Fig-3.21 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE based on 1000-cell division in the magnetic body. (2- component of E field) 104 15,: in (Wmf 1550 23mm ~7715 .1449 ~771) .1233 ~7817 .0876 ~7913 .0331 ~7611 .1647 ~7715 .1539 ~781) .1309 ~7811 .0931 ~7915 .0357 ~7713 .1090 ~751) .1020 ~7511 .0871 ~76.2 .0626 ~7711 .0272 ~8115 .1177 ~741) .1101 ~7513 .0941 ~76J. mm -n4 .0297 ~8111 .0713 ~7211 .0667 ~73.1 .0571 ~7317 .0413 ~751) .0185 ~8015 .0782 ~7211 .0732 ~7211 .0627 ~7315 .0454 ~751) .0206 ~8015 .0402 ~7112 .0376 ~7115 .0322 -721) .0233 -7313 .0107 57911 .0445 ~7011 .0417 ~71.1 .0357 ~7111 .0260 ~7312 .0121 ~7911 .0130 ~7015 .0121 ~7017 .0104 ~7112 .0075 ~7211 .0035 ~7913 .0144 ~691) .0135 ~7012 .0116 ~701) bu ins .1693 ~7715 .1582 ~781) .1345 ~7811 .0957 ~7915 .0369 ~7715 .1217 ~7411 .1139 ~7513 .0973 ~76J. .0702 ~7711 .0309 ~81.9 .0814 ~7213 .0762 ~7217 .0653 ~7315 mu .M9 .0215 ~80.7 ~7015 .0436 ~701) .0374 ~7117 .0272 ~7312 .0127 ~7911 .0151 4%17 .0142 ~701) .0122 ~7017 .0089 ~72.2 .0042 ~7913 50mm 105 We observe that in this particular body excited by an a antisymmetric mode of incident EM field of frequency at 200 MHz, the numerical solutions of both coupled tensor integral equation and decoupled tensor EFIE achieve a good convergence as evi- denced from the numbers shown in Fig-3.14 to Fig-3.21. For practical purposes, the materials used in industry, for example, the coating materials of the airplane, are of the parameters around 8 = 2080 and it ‘~" 10110. Work- ers who are interested in the backward scattering EM fields of this kind of materials can find the induced EM fields inside the body and then calculate the EM fields out- side the body from the equivalent sources which are related to the induced EM fields in the body. An example is given next. Fig-3.22 shows a larger rectangular model of permeable material of dimensions 12 x 6 x 12 cm with e: = 20 + £5 and p: = 5 +13, irritated by an incident plane EM wave polarized in x— direction and propagating toward z— direction of frequency at 200 MHz. ' Fig-3.23 and Fig-3.24 present the symmetric mode solutions of E, of coupled ten- sor integral equations and tensor EFIE respectively. The body is divided into 256 cubic cells of dimensions 1.5 x 1.5 x 1.5 cm. Again owing to symmetry property only the distribution of the first octant is shown. It is observed that the two sets of solu- tions are very close to each other and in fact they are close to the induced fields of the non-magnetic case which are not presented when all the conditions remain the same except if = 1.10. The antisymmetric mode solutions of EJ: and E2 of two different equations are indicated in Fig-3.25 to Fig-3.28. The non-magnetic case is shown in Fig-3.29 and Fig-3.30 for reference. We observe the agreement of the numerical solutions between the two sets of integral equations from the comparison of the numbers shown in those 106 E d=M+w u:=5+i3 H . f=200MHz Tx —> 2 12cm y 0 le 12cm » 6““ Fig-3.22 A homogeneous cubic body of dimensions 12 x 12 x 6 cm illuminated by an incident plane EM wave. e; =20+15 u:=5+13 12cm 107 lExl in (Vlm)[ (E: in (a) .1691 .1692 .1688 .1932 .1512 .1497 .1451 .1617 -9.8 .103 .1r.3 -125 -3.9 -5.1 -7.7 -12.4 .2127 .2108 .2069 .2090 .2242 .2184 .2055 .1917 .9.1 .100 .121 .156 -0.6 -2.4 6.5 -145 .2369 .2332 . .2251 .2150 .2650 .2559 .2355 .2051 -8.7 .9.9 .120 .175 . 0.8 -12 -6.0 .150 2483 .2437 .2334 .2174 .2845 .2737 .2496 .2112 -8.5 -9.9 -12.8 -18.5 1.4 0.7 .5.7 -16.4 Fig-3.23 Solutions for the symmetric mode of induced electric field obtained from the coupled tensor integral equations in the rectangular magnetic body. (x- component of E field) 108 6cm 6; =20+15 T. 12cm 2 11::54-1‘3 y 19— .1 —»1 1831111141») .1686 .1688 .1675 .2000 .1351 .1341 .1299 .1623 42,1111") -lO.6 -107 -111 -10.6 -112 .115 .123 -ll.6 .2125 .2113 .2079 .2240 .1923 .1884 .1794 .1961 -9.9 -102 -10.9 -10.7 .104 -109 -121 ‘-11.7 .2361 .2337 .2277 .2353 .2221 .2162 .2030 .2119 -9.6 -10.0 -107 -10.8 .101 -107 -121 -11.8 .2470 .2440 .2367 .2399 .2365 .2296 .2144 .2190 -95 -9.8 -10.7 -10.8 -9.9 -10.6 12.1 -ll.8 2 Fig-3.24 Solutions for the symmetric mode of induced electric field obtained from the tensor EFIE in the rectangular magnetic body. (311- component of B field) 109 66m ———> e: =20+i5 2 2 12¢!“ ‘ u: = 5 + 33 y k— 12cm —O' 15,: in (Wm)! .0112 .0341 .0587 .0987 .0122 .0371 .0632 .1037 ‘53 “‘ (a) 93.2 92.8 91.6 89.9 102.1 100.8 98.2 932 .0210 .0642 .1110 .1734 .0252 .0764 .1300 .1949 98.2 97.5 95.9 93.6 106.2 104.8 104.8 96.9 .0277 .0843 .1446 .2188 .0339 .1024 .1726 .2499 100.4 99.6 97.8 95.4 . 108.1 106.7 103.6 98.8 .0313 .0942 .1611 .2404 .0383 .1153 .1934 .2760 101:3 1005 9&7 963 10911 10:75 194:5 991 Fig-3.25 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations 1n the rectangular magnetic body. (x- component of B field) . 110 6cm 8: = 20 + £5 —" 12cm 2 u; = 5 + 13 f ‘ Y <-—— 12cm ——u IE,I in (V/m) .1975 .1739 .1251 .0486 W .2372 _.2094 .1521 .0622 ‘5‘ i” (a) -79.2 -79.6 -80.2 ~78.3 -76.3 -76.8 -77.7 -76.7 .1298 .1130 .0793 .0307 .1626 .1419 .1006 .4048 -76.1 -76.5 -77.1 -76.2 .71.0 -71.4 .72.1 .71.5 .0748 .0649 .0452 .0175 .0954 .0829 .0583 .0234 -73.8 .74.2 ~74.8 -74.1 - -67.4 -67.9 -68.4 457.6 .0244 .0211 .0147 .0057 .0314 .0272 .0191 .0077 JZJ Jll ~7317 ~731L ~65] - _-_fi_6-& -65-8 Fig-3.26 Solutions for the antisymmetric mode of induced electric field obtained from the coupled tensor integral equations in the rectangular magnetic body. (z- component of E field) 111 6cm 6: = 20 + 15 -——> 1261» 2 z p; = 5 + 13 ‘7 ‘ y k—_ 12:17! _.| “5,. in (Wm)! .0104 .0314 .0527 .0889 .0103 .0313 .0522 .0894 LE; i" (’) 86.3 86.1 85.7 85.7 88.7 88.5 87.6 86.9 .0188 .0571 .0968 .1534 .0208 .0629 .1058 .1668 89.3 89.1 88.4 88.2 91.7 91.3 90.2 89.4 .0242 .0734 .1243 .1933 .0275 .0831 .1398 .2153 90.4 90.1 89.4 89.3 92.8 92.3 91.3 90.5 .0268 .0813 .1377 .2125 .0308 .0931 .1565 .2388 9‘18 905 89.9 891 933 92.8 91-7 90.9 1 2 Fig-3.27 Solutions for the antisymmetric mode of induced electric field obtained fé'om the tensor EFIE in the rectangular magnetic body. (x- component of field) 112 Goat a: = 20 + 15 ' —> 12cm 2 z p; 2 5+ 13 f ‘ y |¢_._ 12cm —.' 15,: in (V/m) .1613 .1415 .1010 .0366 .1922 .1688 .1612 .0468 “‘3 i“ (a) -87.1 -87.2 -87.0 -84.2 -86.3 -86.4 -86.4 «84.8 .1059 .0927 .0660 .0253 .1309 .1146 .0818 .0325 -86.4 -86.4 -86.3 -85.3 -84.8 -85.0 -85.1 -84.7 .0617 .0540 .0384 .0149 .0771 .0675 .0481 .0192 -85.4 -85.5 485.4 -84.5 - -83.5 -83.6 ~83.8 -83.3 .0203 .0177 .0126 .0050 .0254 .0222 .0159 .0064 -35 n .850 .84_9___.84_3_ .82 9 -830 .8311 -828 Fig-3.28 Solutions for the antisymmetric mode. of induced electric field obtained from the tensor EFIE in the rectangular magnetic body. (z- component of B field) 113 m / . , ——> g, = 20 + 15 2 12cm 1 u; = 1 f y F.— 12cm —.| 15,; in (w...) .0078 .0238 .0407 .0672 .0069 .0210 .0360 .0611 “5* m (a) 83.2 83.2 83.1 83.3 83.6 83.5 83.4 83.5 .0133 .0405 .0698 .1076 .0129 .0392 .0676 .1044 85.9 85.8 85.6 85.6 86.2 86.0 85.8 85.7 .0168 .0511 .0878 .1318 .0166 .0506 .0868 .1303 86.8 86.7 86.5 86.5 , 87.0 86.9 86.6 86.6 .0185 .0562 .0964 .1432 .0185 .0562 .0961 .1426 871 87-1 MJL, 87 3 871 871) 8711 1 2 Fig-3.29 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular non-magnetic body. (x- component of E field) 114 Gem 6: = 20 +15 —_' 120"! j 2 z e 1 u, =1 Y I*— ..... _.. |E,| in (Vlm)l .0953 .0828 .0569 .0161 .0990 .0863 0599 0187 ‘5' m (a) -89.2 -89.1 -88.5 -82.1 -89.3 -89.2 -88.8 -84.2 .0635 .0550 .0377 .0125 .0660 .0573 .0395 .0138 -88.1 ~88.4 -87.9 -84.9 -88.6 -88.5 -88.1 -86.3 .0364 .0314 .0215 .0072 .0379 .0328 .0226 .0079 -88.0 -87.9 -87.3 -84.7 -88.0 -87.9 -87.5 -85.8 .0119 .0102 .0070 .0024 .0123 .0107 .0073 .0026 -817 -87fi £111 -84L, -8717 -816 -87-L .8515 1 2 Fig-3.30 Solutions for the antisymmetric mode of induced electric field obtained from the tensor EFIE in the rectangular non-magnetic body. (z— component of E field) 115 figures and the significant differences in the induced fields between magnetic and non- magnetic cases. The coupled tensor integral equations will become a 6N x 6N matrix equation after transformation via pulse-basis expansion and point-matching if the body is divided into N cells while the decoupled EFIE will become a 3N x 3N matrix equa- tion. Obviously, the former needs four times of the computer memory storage as that of the latter thus requires more computation time to solve the matrix equation by exist- ing numerical methods. However, in the tensor EFIE, we have an extra surface integration term to evaluate when the transformation is performed in contrast to the coupled tensor integral equations. The other advantage of the coupled tensor integral equations is that the heterogeneous problem can be handled easilier by applying this set of equations since we only need to vary the parameters 1:, and 1:", in the coefficients of the elements in the transformed matrix equation. An example of heterogeneous case is given as here. Fig-3.31 shows a hetero- geneous body of dimensions 6 x 6 x 6 cm. The shaded region in the center consists of permeable material with e: = 20 + £5 and u: = 5 + £3 and the rest of the body is non-magnetic material with e: = 20 + £5 and 11:: 1. Again we concentrate on the antisymmetric mode of incident field and omit the case of symmetric mode due to the same reason as previous examples. The frequency is the same 200 MHz. Fig-3.32 shows the distribution of 5,, E2 and Hy of the first octant of the hetero- geneous body described in Fig-3.31. The reason we indicate y- component of induced magnetic field only is simply because this component is the only significant one since H‘ is polarized in y- direction and almost uniformly distributed. The body is divided into 64 cubic cells with dimensions of 1.5 x 1.5 x 1.5 cm. Fig-3.33 presents 5,, E, and I-ly of a homogeneous non-magnetic cube with e: = 20 + £5 and of the same EI=120 + (5)120 ui=uo e; = (20 4 1511:.) Fig-3.31 115 :15 + 13100 116 A heterogeneous cubic body of dimensions 6 x 6 x 6 cm with different parameters specified in the shaded and unshaded regions. MT V 15,1111 (V/m) .0087 .0304 .0314 45,111 (0’ 83.2 83.2 35.0 .0146 .0470 .0530 84.9 84.8 86.9 . 1 . 165 IE,I in (Wm) . .0291 .0124 0 Us”, in (°) -91.2 -89.6 -88.3 0097 .0044 - .0074 -90.8 -90.1 -86.4 ‘ 1 My in W 1.1 0.2 -3.8 -19 .0038 .0028 .0021 4.5 1.1 -3.9 1 2 Fig-3.32 Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations in the heterogeneous body. T p Z 60m 2 1 l y / 4—— 115,1 in (V/m) .0071 .0271 .0062 .0246 LE; i“ (0) 80.4 81.3 80.1 81.1 .0115 .0401 .0107 .0379 81.8 82.5 81.5 82.3 1 2 IE I in (VIII!) .0221 .0090 .0226 .0095 2 25,111 (") -94.2 -924 -947 -93.3 .0065 .0028 .0065 .0028 -94.1 -93.4 -94.7 -944 1 2 1111,1111 (A/m) .0027 .0026 .0027 .0027 M, in W 0.1 0.0 0.1 0.0 .0027 .0027 .0027 .0027 0.2 0.1 0.3 0.1 1 2 Fig-3.33 Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations in the non-magnetic homogeneous body. 119 dimensions and incident field for comparison. We observe that the induced electric field in the heterogeneous case is larger than that in the homogeneous case, especially in the cell consisting of magnetic material, but the induced magnetic field becomes smaller in that cell. We may use a simple picture to explain this result. Since the B field has to be continuous inside the body, the magnetic cell which has larger value of u then has a smaller strength of the H field. By the definition of the magnetization M M=£—H, (3.5.1) 110 the smaller H implies a larger magnetization M and hence a larger circulatory electric ‘ fields. Fig-3.34 to Fig-3.36 show the distributions of 5,, E2 and Hy of the above hetero- geneous example with 512 subdivisions. A fair convergence is indicated by the com- parison of the numbers in Fig-3.33 and Fig-3.34 to Fig-3.36. The convergence and agreement in the numerical solutions of the coupled tensor integral equations and the tensor EFIE shown in the above examples, however, deteriorate when the relative permittivity 8, goes up to the value about 50. The unsa- tisfactory results also occur if we increase the frequency of incident EM fields. It appears that there are certain limitations for the numerical solutions of the integral equations. A series of numerical results of the two set of integral equations have been compared to investigate the limitations of the convergence and agreement properties. Table-3.1 and Table-3.1a summarize these properties under various circumstances such as different frequencies of the incident EM fields or parameters of the materials. It is noted that in the cases of 1.1: = l, the agreement between two sets of equations is always satisfactory since they are identical equations. 120 E 'n V/m) .0041 .0124 02100344 .0038 .0118 .0196 .osfil L 21111507) 84.4 84.4 84.2 84.1 85.6 86.1 85.8 84.8 x . 1 .0215 .0368 .0561 .0075 .0231 .0385 .0563 0:775 87.4 87.0 86.8 88.7 89.0 88.7 87.7 . .0279 .0473 .0701 .0101 .0312 .0515 .0724 (831972 88.6 88.3 88.0 89.7 90.2 89.9 89.1 .0103 .0311 .0525 .0769 .0114 .0353 .0582 .0805 89.2 89.1 88.8 88.6 90.0 90.6 90.4 89.6 1 2 .0042 .0136 .0214 .0323 .0047 .0150 .0226 .0327 88.0 89.3 87.9 85.5 90.5 91.4 89.1 85.9 .0088 .0285 .0440 .0585 .0102 .0324 .0475 .0601 91.5 92.5 91.1 88.7 89.3 . ” ” .0800 90.9 .0903 91.6 3 4 Fig-3.34 Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations based on 512-cell division in the heterogeneous body. (x- component of E field) IEAintV/m) .0552 .0477 .0326 .0098 .0618 .0533 .0365 .0120 45,1116) -87.6 -87.6 -87.3 -82.0 -87.0 -87.0 -86.9 -83.4 .0383 .0329 .0224 .0077 .0459 .0393 02660097 -87.0 -87.0 -86.7 -84.4 -85.8 -85.8 -85.8 -84.6 .0227 .0195 .0132 .0047 .0281 .0240 .0162 .0059 -86.6 -86.6 -86.3 -84.3 -85.7 -85.7 -85.5 -84.6 .0075 .0064 .0044 .0015 .0090 .0077 .0052 .0019 -86.4 -86.3 -86.1 -84.4 -86.2 -86.2 -85.9 -85.4 1 2 .0687 ,0589 .0393 .0132 .0733 .0625 04190139 -86.0 -86.1 -86.2 ~83.2 -85.3 -85.5 -85.8 -83.0 .0589 .0499 .0328 .0120 .0671 .0565 03670134 -83.8 ~83.8 -83.9 -82.9 -82.8 -82.0 ' .0220 .0081 .0262 .0096 -82.7 -82.0 -81.0 -80.2 .0067 .0024 u _" .0081 .0029 -83.7 -83.5 : -8l.4 -81.2 3 Fig-3.35 Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations based on 512-cell division in the heterogeneous body. (z- component of E field) lHyl in (A/m) .0028 .0028 .0027 .0026 .0027 .0027 .0026 .0026 AH, in (0) 1.0 0.9 0.5 0.1 O. 3 0. 2 -0.1 -0.4 .0031 .0030 .0029 .0027 .0031 .0031 0028 .0026 2.6 2.3 1.5 0.5 2. 9 2.8 1.2 -0.1 .0035 .0034 .0030 .0028 .0043 .0042 .0031 .0026 3.8 3.5 2. 3 0.9 5.9 6.6 2. 8 O. 2 .0037 .0035 .0031 .0028 .0043 .0043 .0031 .0026 4.1 3.8 2. 6 1.0 5.1 5.9 2.9 0.2 l 2 .0024 .0024 .0024 .0025 -2.1 -l.9 -l.6 -1.2 .0024 -l.6 .0024 -2.0 .0023 -2.2 Fig-3.36 Solutions for the antisymmetric mode of induced EM field obtained from the coupled tensor integral equations based on 512-cell division in the heterogeneous body. (y- component of H field) 123 Table-3. l Summary of Agreement and Convergence Properties for the Coupled tensor IE and tensor EFIE (I) s;=50+110 e;=5+i4 Type of Incident u; = u; = u; = ' = u; = u: = Electric Field 50+1'10 5+1'3 l+i0 50+i10 5+i3 1+1'0 E C E C E C E C E C E C conv. X X X X X X F X F O O O f = _ srn agr. X X 0 X F 0 750MHz conv. X X X X X X X X 0 O 0 cos agr. X X 0 X X 0 conv. X X X X X X 0 O 0 0 O O f = . srn agr. X X 0 O O O 200MHz conv. O X F O O O O O O O O 0 cos agr. X 0 O F O O E: Tensor EFIE C: Coupled Tensor IE 0: Satisfactory F: Fair X: Poor 124 Table-3. la Summary of Agreement and Convergence Properties for the Coupled tensor IE and tensor EFIE (II) e:=50+i10 g;=5+i4 Type of Incident . = . = . = u; = u; = u; = Electric Field 50+i10 5+1”3 1+i0 504-110 5+i3 l+i0 E C E C E C E C E C E C conv. X X X X F O O O O O O f = . srn agr. X F O O O O SOMHz conv. O O O O O O O O O O O 0 cos agr. O O O O O O conv. X X X X F O O O O O O f = . sin ._... agr. X F O O O 0 lOMHz conv. O O O O O O O O O O O 0 cos agr. O O O O O O E: Tensor EFIE C: Coupled Tensor IE 0: Satisfactory F: Fair X: Poor 125 We now gain some insights about the numerical solution behaviors for the cou- pled tensor integral equations and the tensor EFIE by comparing the numerical results of the induced electric field distributions presented above. However, to investigate the behavior of the magnetic field distribution, the solutions of tensor MFIE and the cou- pled equations should be compared. Since both sets of equations are dual in E and H, it is logical for us to expect the numerical solution behavior of the induced magnetic field to be similar to that of the electric field if we interchange 8* and 11*, and also symmetric and antisymmetric modes of incident fields. In fact, the numerical results which are not presented here will confirm our expectation. There are some advantages in the decoupled tensor EFIE and MFIE which we derived in section 3.2 over the formulations derived by Tai [13] although they are equivalent in the homogeneous case. One thing is that there exist the terms of V x E and V x H in Tai’s equations. This will cause difficulty when one attempts to solve the equations numerically. The other advantage of our equations is that they are derived for a heterogeneous system so that they can be used to handle more compli- cated cases, although some numerical skill is needed to calculate the terms involving the gradient of the permittivity or permeability functions. The set of coupled integral equations in terms of the free space scalar Green’s functions (3.3.28) and (3.3.29) has also been investigated when it is applied to a homo- geneous body. As we mentioned before, there are no induced volume charge inside the homogeneous body therefore the terms of VE and V-H should vanish in the equa- tions. However, if we simply solve the reduced equations (3.4.27) and (3.4.28) via pulse-basis expansion and point-matching, we will find that the numerical results are completely different from what we obtained by the coupled tensor integral equations and tensor EFIE. The reason is that the pulse-basis expansion assumption satisfies the charge free condition inside each cell, but the jump discontinuities created by the 126 pulse-basis at the boundaries of adjacent cells will imply some charge distributions on the cell surfaces which need to be taken into account in the evaluation of the induced EM fields. Therefore we need to extend the domain of the surface integration from the enclosing surface s to all the surrounding surfaces of each cell in (3.4.27) and (3.4.28). After this extension process, we are able to get numerical results very close to what we have obtained by the tensor integral equations. The numerical results are also omitted. To compare the advantages and disadvantages of the two sets of coupled integral equations, a few things need to be pointed out. Since these two sets of equations are both transformed into matrix equations of order 6N, they cost equally in computer memory storage. In the tensor equations, we need to handle the singularity problem with a correction term before we get into computer but the formulations turn out to be easier to program since there are only volume integrations involved. On the other hand, we need to calculate both the volume and surface integrations in the other set of coupled integral equations. The arguments of the two different formulations of decoupled integral equations are similar to that of the two sets of coupled integral equations except that both sets of decoupled equations contain the surface integrations. We should also notice that only the domains of the surface integrations involving the n-E and n-H terms which imply surface charge distributions need to be extended to all the surrounding surfaces of each cell. Let us summarize the relations among the four sets of integral equations briefly. The two sets of coupled integral equations will give similar numerical results provided the surface integration terms are properly treated when we apply the pulse-basis expan- sion and point-matching to solve the induced field distributions in a homogeneous body, and so will the two sets of decoupled integral equations. The coupled integral 127 equations can determine both E and H simultaneously but will require four times of memory storage in the computer and need a longer CPU time to solve matrix equa- tions of larger size. The agreement of the numerical results between the coupled integral equations and the decoupled ones is satisfactory for certain ranges. However, in the heterogeneous cases, it is easy to program for the coupled tensor integral equa- tions. CHAPTER IV PLANE WAVE SPECTRUM ANALYSIS OF A NONUNIFORM PLANE WAVE PASSING THROUGH A LAYER OF LOSSY MATERIAL This chapter presents an application of electromagnetic wave for detecting a small movement of a biological body behind a barrier. A microwave life detection system Operating at a frequency of 10 GHz [20], [21] has been constructed to measure the breathing and heart beats of a human subject located at a distance of 100 feet. The principle of the system is to illuminate a human subject with a low intensity microwave beam, and then extract the breathing and heart signals from the modulated back-scattered wave with a detecting system. In order to detect small body vibrations behind a barrier, a system constructed on the same principle with an operating fre- quency at 2 GHz is developed because of the better penetration ability of EM waves with a lower frequency. A series of experiments have been conducted to measure the breathing and heart signals of a human subject behind a thick layer of bricks with the microwave life detection systems. Plane wave spectrum theory [17], [18] is used to analyze the nonuniform plane wave radiated from the antenna and passing through a layer of lossy material in order to predict the field distribution on the other side of the barrier. The predicted electric field distributions and experimental results on the detec: tion of breathing and heart signals behind brick wall of various thicknesses are presented. In section 4.1, the plane wave spectrum analysis is introduced. Some predictions of the elecuic field distributions of the nonuniform plane wave passing through a layer of lossy material are shown in section 4.2. A brief description of the microwave detection systems is included in section 4.3. Section 4.4 presents experimental results 128 129 of detecting the small vibrations due to breathing and heart beats of a human body behind a brick wall. 4.1 Plane Wave Spectrum Analysis The plane wave spectrum approach has been summarized by Paris [17]. Simply speaking, any arbitrary electric field distribution of a particular frequency can be represented as a superposition of plane waves traveling in different directions with different amplitudes and the same frequency. The resultant expansion is known as a model expansion of the arbitrary monochromatic wave. The object of this expansion is to determine the unknown amplitudes and directions of propagation of the plane waves in the superposition. A plane wave in free space must satisfy the Maxwell’s equations which lead to the vector Helmholtz equation: V2E(r) + 1% E(r) % 0 (4.1.1) where k3 = (0211020 . (4.1.2) The trial solution of (4.1.1) can be written as E = A(k)e-J‘” (4.1.3) where the time harmonic factor of aim is assumed, j = ‘1——1 is used in this chapter and k = xkx + yky + zk, (4.1.4) denotes the direction of propagation of the plane wave represented by (4.1.3). The 130 three Cartesian components of k should satisfy the following relations: k§+k§+k§=kfi=m2uoeo. (4.1.5) Thus at a frequency, only two of the components of k can be independently specified. Suppose k, and ky are independent components, then \lké-ki-kfi, k§+kzskfi y k = 1 (4.1.6) The negative radical is chosen specifically in order to ensure that the wave is bounded at infinity. Since the region is charge free, we have V-E = O . (4.1-7) Substitute (4.1.4) into (4.1.7) gives k-A(k) = 0 (4.1.8) or 12,4,(1‘) + 194,01) + k,4,(k) = 0 . (4.1.9) This implies only two components of A(k) are independent for all k. Let these be Ax and Ay, then A,(k) = "kl—[154414) + k,4,(k)] . (4.1.10) The general solution for E(r) can be constructed as linear combinations over kx and ky if A(k) is known: E(r) = 1:1: A(k)e"fi"" dk,dky (4.1.11) 131 where A(k) is the so called plane wave spectrum. The expression of (4.1.11) is essen- tially a 2-dimensional inverse Fourier transform when the z coordinate is specified. Suppose we aim to predict a electric field distribution of a nonuniform plane wave passing through a layer of lossy barrier with thickness d as shown in Fig-4.1. The complex permittivity of the barrier is 8' and the permeability is 110. An antenna is placed right next to the barrier and the aperture is on the z=0 plane. The aperture field of the antenna is then denoted as E(x,y,0). Now we can find the plane wave spectrum of the aperture field on the 2:0 plane by the twoodimensional Fourier transform: 4.0) = fngl" I" 5.05.018” * ’9” dxdy (4.1.12) A,(k) = 21:11" Ey(x,y,0)e""-" + ’9” dxdy (4.1.13) It is noticed that only tangential components of E field on the z=O plane are needed to determine the spectrum A(k) since A, can be derived from (4.1.10). We treat a nonuniform plane wave as a linear combination of uniform plane waves coming from all directions. In order to analyze a single uniform plane wave passing through the one-layer barrier, a very simple approach [19] by means of wave matrix and transmission-line theory is used. This analysis is valid for both cases that the barrier consists of loseless (0' = 0) or lossy (0' at 0) material. The lossy layer is replaced by an equivalent transmission-line circuit as shown in Fig-4.2. The transmis- sion coefficients of the barrier for an obliquely incident plane wave can be determined as follows: The reflection and transmission coefficients at the two interfaces are z-1 1 Z+1 ( ) R2- 1‘2 (4.1.15) 132 Fig-4.1 An incident plane EM wave on a lossy layer of thinkness d. 1""'——- 1 Fig-4.2 Equivalent circuit for a lossy dielectric layer. 133 T1=1+R1= (4.1.16) 1 Z + 2 1+Z T2=1+R2= (4.1.17) where Z is the normalized impedance of the section of line representing the lossy layer. Using the wave matrix theory [19] gives the over-all transmission coefficient 7172 l 2. -1 T: . . = cos¢+j srnq) , (4.1.18) where 9 = k0 K — sinzei (4.1.19) is the electrical length of the lossy layer of thickness d, K = e: is the relative complex permittivity and 9,- is the angle of incidence measured from the interface normal in Fig-4.1. For the cases of perpendicularly polarized and parallelly polarized incident waves, we have different expressions for the normalized impedances: Z = K - 311120,. (4 1 20) P Kcosei ' ' cosO; ZN = . 2 VIC - srn 9" where the subscripts N and P stand for perpendicular and parallel polarization, respec- (4.1.21) tively. Similarly for the two different over-all transmission coefficients: .1 + Z}, . ‘1 T p = cos¢ + 1 51nd) (4.1.22) 22? 1 + Z}, _ ‘1 TN = 0056 + j srn¢ . (4.1.23) 221v 134 Notice that in a particular direction of incidence k, k'Z 1‘2 cost) = = — 4.1.24 ‘ Ikl Ill k0 ( ) sinOi = -iko—k—3 . (4.1.25) The normal vector aN of the incidence plane for a uniform plane wave traveling in the direction of k can be expressed as aN = z x k = -xky + ykx. (4.1.26) Thus the perpendicularly polarized component of A, denoted as AN, is the projection of A on aN: _ (A-aN)aN ~- Ia I2 (4.1.27) N and the parallelly polarized component of A, denoted as Ap, can be obtained by sub- tracting AN from A: AP = A — AN . (4.1.28) The direction of a single uniform plane wave remains the same after passing through the barrier by the Snell’s law. So we simply multiply the two different over- all transmission coefficients with AN and Ap to obtain AN’ and Ap’, the two different polarized components of the plane wave spectrum, behind the barrier: AN' = TNAN (4.1.29) Ap’ = TPAP . (4.1.30) The total plane wave spectrum behind the barrier A’ is then A, = A’(kX’ky) = AN, '1' AP, = TNAN + TPAP . (4.1.31) 135 Therefore we can determine the electric field distribution behind the barrier at any par- ticular 2 through the inverse two-dimensional Fourier transform: E(x,y,z) = fl A’(kx,k,,kz)e'j[k"‘ + ’9’ + ““01 dkxdky (4.1.32) where k2 is related to k, and k, by (4.1.6). Now we can predict the E field distribution at any point behind the barrier for a given antenna aperture field distribution, or in other words, a nonuniform plane wave, passing through a infinite one-layer barrier. Some examples simulating the practical situations are presented in next section. 4.2 Predictions of Field distributions for a Nonuniform Plane Wave Passing through a Layer of Lossy Material Some examples of the predicted electric field distributions for a nonuniform plane wave passing through a layer of lossy material are presented in this section. Suppose some antenna aperture field distribution are given, we can use the formulas derived in section 4.1 to calculate the desired field distributions behind the layer. We use the aperture field of an open ended rectangular wave guide with dimen- sion a and b (b < a) as the nonuniform plane wave at r=0 plane in our example as shown in Fig-4.3. The origin of the coordinate system is set at the center of the aper- ture. The E field of T1310 mode in a rectangular wave guide is E(x,y) = x Bows? e'jB’ (4.2.1) where N 11: [5 = k% .. 23 (4.2.2) e n m a m m. a n m _ mm. 3 A. g 137 is the propagation constant of the z— direction. On the 2:0 plane, we have E,(x,y,0) = P(x,y) Bows? (4.2.3) Ey(x,y,0) = 0 . (4.2.4) where P(x,y) is defined as < 6:1,; 4:13.344 By using (4.1.12), we obtain the plane wave spectrum of this aperture field: 1 0° 0° . 4.00 = 2.721 [ 5.05.0140“ " *1” My Eon =._... 1,: fl .k’y 41:2 me, dx cos a e’ dy 1 . kg) 1 kya 1 E0 SlnT COST = ‘1:— ’9. J _—i - (4.2.6) ' _ (:2 A,(k) = 0 . (4.2.7) From (4.1.10), we will have kx A,(k) = -k—A,(k) . (4.2.8) 2 The perpendicularly polarized and parallelly polarized components of the plane wave spectrum A are _ (A'aN)aN _ Axky AN - - lale 1‘3 + k3 (—xky + ykx) (4.2.9) 138 k3 1,1. k A =A—A =4 x——+ ’ —z—" . (4.2.10) P N x é+§ yé+6 @ Then the total plane wave spectrum A’ behind the barrier is A’(kx,ky) = TNAN '1' TpAp x 71ng + Tpkg + kzky(Tp -' TN) _ Tpkx 4.2.11 x ki-i-k; y [cg-Pk; z [(2 ( ) Suppose we like to know the electric field distribution on the plane at a distance 5 from the barrier, this field distribution can be obtained from (4.1.32) as E(x,y,d+s) = f' [°° A'(k,,k,,k,)e"'"‘-’ f "r’ " ’9" dkxdky . . (4.2.12) The two-dimensional discrete Fourier transform (DF'I') [23], [24] is used to evalu- ate (4.2.12). The spectrum on the plane at a distance 5 from the wall can be obtained by multiplying a phase factor e’jk" to A’, then we can calculate the value of spectrum at each discrete grid point on the kx—ky plane. The electric fields at corresponding grid points on this plane 2 = d+s can be obtained by taking two-dimensional inverse DFI‘. The fast Fourier transform (FFI‘) algorithm was used to perform the DFT in our calcu- lation because for efficiency computation. Fig-4.4 shows the amplitude of the x— com- ponent of plane wave spectrum le’l on the plane of s = 0.2 m. In this example, the dimensions of the aperture are a = 32 cm, b = 15 cm and the value of E0 is assumed to be 1 Wm. The relative complex permittivity a: of the barrier is 5 - j0.3. Since the field decays very fast when k0 S W (evanescent waves), we can truncate the spectrum at k, = il.lko and ky = il.lko and assume that the function values of the spectrum are all zero outside the square. The frequency of this nonuniform wave is 2 GHz, so the boundaries shown on the figure are kx = i1.1k0 = i46.08 m‘1 and (810—4) 139 M"! 3.92919 ' 2.6l9‘46‘ 1.30973 ' .000000 -/' // Fig-4.4 The magnitude of the x— component of plane wave spectrum behind the bar- rier (d = 0.2 m). 140 ky = i1.lk0 = $46.08 m‘l. We sample 100 points in k, direction and 30 points in ky direction. The numbers of sampling points are decided by the Nyquist rate which prevents the aliasing of the result of the inverse DFI‘. After taking two-dimensional inverse FFI', we obtain the x— component of electric field distribution E, on the same plane, where IExl is shown in Fig-4.5. The bounds for the truncated domain of IExl, x S $3.4 m and y S $1.02 m, are determined by the sampling rates and so are the bounds for the truncated domain, k, = 5:46.08 m-1 and k, = 21:46.08 m-l, of 14,1. It is noticed that since Ax’ is an even function of k, and k), EJr is also even function of x and y by the Fourier transform theory [22]. Fig-4.6 indicates the amplitude of the 2- component of plane wave spectrum |A,’| and Fig—4.7 the presents 2- component of electric field distribution lEzl on the same plane. Now, since A,’ is odd in k, but even in Icy, E, is odd in x and even in y. There- fore lAz’l is zero when k, = 0 and lEzl is zero when x = O as observed in the figures. The y— component is much smaller than the other two components so that it is neglected in this example. If we increase the distance 3 to 0.4 m, we expect that the field distribution spreads out and the peak value decreases. The numerical simulations in Figs-4.8 and 4.9 confirm our predictions. Only the electric field distributions are shown in the figures since the amplitude of spectrum remains the same in spite of the variation of s which only introduces a phase change to the spectrum functions. Another example for the barrier with a different thickness of d = 0.5 m is shown in Figs-4.10 to 4.15. We observe similar results except that the amplitudes of both spectrum and electric field are smaller than those in the previous case because of more attenuation caused by the thicker barrier. 141 IE}! (V/m) ,_s m“ 6 9 2. .19?99? - .098998 ‘ Fig-4.5 The magnitude of the x— component of electric field distribution behind the 0.2 m) at a distance of 0.2 m. barrier (d (X10 142 lAz'l l . L18269 '1 .988‘160 “ .‘19‘1830 ‘ :x/ ///’ , .000000 - Fig-4.6 The magnitude of the z- com nent f l ' rier (d = 0.2 m). po 0 p ane wave spectrum behind the bar- 143 1521 (V/m) - 9 w «I. 9 0. .061072 - .030536 - Fig-4.7 The magnitude of the 2- component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.2 m. 144 IE}! (V/m) q 1. ”a 0 9 1.. 12710? - .063553 ‘ 0.2 m) at a distance of 0.4 m. Fig-4.8 The magnitude of the x— component of electric field distribution behind the barrier (d 145 lEzl ( V/m) . 057696 “ .038‘1’6‘1 ‘ .019232 “ /9.v W *4/71! ’ /; ,/ Fig-4.9 The magnitude of the z— component of electric field distribution behind the barrier (d = 0.2 m) at a distance of 0.4 m. 146 1A,] 1 . 62%?1 " 1.083L‘i ‘ a3o..o_E 2.. do 88?... .325 2.4.3.. 23338.... _ 3...: 1 .83: _ _ _ m ESSEME 32.133 emu-.30.. — 3...... _ ..eee.e 32.2.... 1 JL #5:... 16.....2381— ewweu...“ isosgfieemq 12.3.8.6... «3:98.... . 3. s 333.48é .3333 /_ 11 L .32....- eeugcuaafl 1 .39.... #2.“: fi 3...... «32...... _ 12.2586... 32.3.6864 : L5:58 A V 13.233238... .39.... a. c a. e __ £33.... 1 , 1...... 1. 1. _ assessflr _ 1 2.. 8.6. 1 e... 2 A VWV 153.. 1 1 . «36...? 1 3:29... - .3233... 233.8... 41 _ _|..|._ «339.3... vexed—toms... boo: 3.26-.3 .8— :8 155 portion of the amplifier signal for monitoring its intensity. This monitoring is mainly for checking how well the clutter is canceled. The mixing of the amplified, body scat- tered signal and a reference signal (7 to 10 mW) in the double-balance mixer produces a low-frequency breathing and heart signals which modulate the scattered microwave from the body. This output from the mixer is amplified by an operational amplifier and then it passes through a low-pass filter (4 Hz cut-off) before reaching a recorder. The two systems of different bands have the same circuit diagram, except that various components for different frequency bands are used. 4.4 Experimental Results of Detecting Movements of a Human Body across a Wall by Using Microwave Detection Systems The experimental results of detecting movements of a human body at a distance away from the system can be found in [21]. In this section we will present the test results on the detection of the heart and breathing signals of a human subject who is located behind brick walls of various thickness, using both the X-band and the L-band system. We have found that with the X-band system, it was possible to detect the heart and breathing signals of a human subject through a brick barrier of up to about 15 inches thickness. However, if the wall was thicker than that, the detection become difficult with the X-band system. The L-band (2 GHz) system was then design for the purpose of detecting the heart and breathing signals of human subjects who were behind a very thick wall or buried under a thick layer of rubble. 156 As mentioned in section 4.3, the L-band system has the essentially same circuit arrangement as that of the X-band system, but the radiation beam of the former can penetrate a thicker wall than the latter due to its lower Operating frequency. With the L-band system, it is possible to penetrate the brick barrier of up to about 34 inches thickness. Fig-4.17 depicts the experimental setup of the first series of experiments. A brick wall (3 feet wide and 4.5 feet high) of various thicknesses was lined with microwave absorbers along the edges. New Orlean’s homestead bricks were used. A human sub- ject sat behind the brick wall within a distance of l to 2 feet- The antenna of the life- detection system was placed close to the other side of the brick wall- Fig-4.18 shows the heart and breathing signals of a human subject measured by the 10 GHz system through one layer (3 3/8") and two layers (6 3/ ") of dry bricks. In each recorded graph, the breathing signal, the heart signal (the subject holding breath) and the background noise were included. Both the heart and breathing signals were clearly detected. Fig-4.19 shows the heart and breathing signals of a human sub- ject measured by the 10 GHz system through three layers (10 1/8"), four layers (13 1/2") and five layers (16 7/8") of dry bricks. It is observed that both the heart and breathing signals were detected through four layers of bricks, but only the breathing signal was measured through five layers of brick. It is noted that as the thickness of the brick wall was increased, the amplifier gain of the system was increased accord- ingly. We estimate that with the 10 GHz system, it is possible to penetrate a dry brick wall of about 15 inches thickness- Fig-4.20 shows the heart and breathing signals of a human subject measured by the 2 GHz system through five layers (16 7/8") and six layers (20 1/4") of dry bricks- In each case, both the heart and breathing signals were easily detected. It is noted that for a thinner brick wall, it was even easier to measure these signals. Fig-4.21 shows 157 ABSORBER BRICK WALL HUMAN OBJECT MICROWAVE LIFE-DETECTION SYSTEM SIDE-VIEW TOP-VIEW Fig-4.17 Experimental set up for the measurement of heart and breathing signals of .a human object through brick wall, using microwave life detection system. 158 - breathing heart beat breathing background [ m_1‘ noise FL—fiz l lay er 3‘. fl (3 3/8")° :———=:——=—_=E—-‘__;‘—:== " —-— —_—=—_:_ dry bricksf _:.._'_____._.._ ___==-E.=-_-;._ _ __ _‘ .,,___,,__._ breathing heart beat background noise 2 layers (6 3/4") of dry bricks Fig—4. 18 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection system. 159 lO-GHz Life-Detection System it breathingT— heart beat f background ‘1 - ._. .. noise 3 layers E j; 5E”— __— (10 1/8") of =————E=—-E:=I= M“ dry bricks 1 sec. I—breathi T— heart beat background .i noise 'ng 4 layers (13 l/Z") of dry bricks r.— breathing —cr heart beat background -+ noise 5 layers (l6 7/8") of dry bricks Fig-4.19 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection system. Z-GHz Life-Detection System #— breathing 1+r- heart beat background - noise 5 layers (16 7/8") of dry bricks '— breathing heart beat background e' noise - - 6 layers (20 l/4") of dry bricks Fig-4.20 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. 161 Z-GHz Li fe-Oetection System l.— breathing ar— heart beat —T backgrOund' _ _ __ nmse 7 layers ff (23 5/8") of dry bricks 8 layers (27") of dry bricks Fig-4.21 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. 162 the results for the cases of seven layers (23 5/8") and eight layers (27") of dry bricks. Clear heart and breathing signals of a human subject were measured in each case. We were able to measure the heart and breathing signals of a human subject through ten layers of dry bricks with the 2 GHz system. This leads to an estimation that with the 2 GHz system, it is possible to penetrate a dry brick wall of about 34 inches thickness. We also tested the effect of moisture on the performance of the life-detection sys- tems. For this purpose one layer of the brick wall was constructed with wet bricks. When we used wet bricks which were soaked in water for several hours, the perfor- mance of the system was affected insignificantly. However, a significant effect was observed when we used very wet bricks .which were soaked in water for three days. Fig-4.22 shows the heart and breathing signals of a human subject measured through three layers (10 1/8") of very wet bricks, using the 10 GHz system. It is observed that with one layer of very wet bricks present, the penetration of 10 GHz microwave beam was severely hampered. It is then conjectured that the 10 GHz system may not be effective in detecting the heart and breathing signals of human subjects through a very wet barrier. This difficulty may be overcome by using the 2 GHz system. Fig-4.23 shows the results of the same experiment of Fig-4.22 except using the 2 GB: system. It is observed that one layer of very wet bricks reduced the magnitudes of the meas- ured heart and breathing signals but we were still able to detect these signals clearly without much difficulty. Therefore, it is reasonable to conclude that for the purpose of detecting vital signs of human subjects through a thick layer of wet rubble, the life- detection system should be designed to operate at low frequencies, such as lower than 1 GHz. The second series of experiments was conducted with a setup depicted in Fig- 4.24. Varies layers of bricks were laid on a wooden frame which formed a cavity for a human subject to lie down in it. Microwave absorbers were used to line the sides of 163 lO-GHz Life-Detection System +breathing+_ heart beat _____,'.._ background noise | 3 layers (101/8")0 i-_-__ —' — - dry bricks ’ ____:_ e-" ' 3::— — <- .._'-,,g.. I'm»: £2 fir-T: {i aaagéfigz. i _. .:... 31.24%" '-" ‘ " :21 fiéfiéfia§§= -~ —=~———-._* - LE. _ __ ‘ —-l l— 1 sec. l. breathingT— heart beat ————+— background noise “l 2 layers (6 3/4") of_ dry bricks ‘ and 1 layer (3 3/8") of wet bricks Fig-4.22 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 10 GHz life detection system. One layer of wet bricks severely hampered the penetration of the microwave beam. Z-GHz Life-Detection System - -o- q l.— breathing +— heart beat + backgr0und —1 noise _ M_ 3 layers (lo l/8") of w dry bricks r— breathing ar— heart beat background "' noise 2 layers (6 3/4") of dry bricks and 1 layer (3 3/8") of wet brick Fig—4.23 Heart and breathing signals of a human subject measured through brick walls of various thicknesses using the 2 GHz life detection system. One layer of wet bricks had a considerable effect on the penetration of the nucrovvave beanL 165 BRICK LAYERS ' /(\' ABSORBE . MI CROHAVE L I FE- DETECTION SYSTEM O SIDE-VIEW FRONT- VI EN Fig-4.24 Experimental set up for the measurement of heart and breathing signals of a human object under layers of bricks, using microwave life detection system. 166 this brick structure. This setup simulated a human subject trapped under a thick layer of rubble. The antenna of the life-detection system was placed on the top of the brick structure aiming at the human subject. Fig-4.25 shows the heart and breathing signals of a human subject lying with face-up or face-down position under three layers (10 1/8") of dry bricks, measured by the 10 GHz system. Both the heart and breathing signals were clearly detected for each position. Fig-4.26 shows the similar results for the case of four layers (13 1/2") of dry bricks, with the 10 GHz system. When the thickness of the brick structure exceeded more than five layers (16 7/8") of bricks, the performance of the 10 GHz system became marginal. Fig-4.27 to Fig—4.29 show the heart and breathing signals of a human subject lying with face-up or face-down position under five layers (16 7/8"), six layers (20 1/4") and seven layers (23 5/8") of dry bricks, respectively, measured by the 2 GHz system. In each of these figures, the heart and breathing signals were both clearly recorded. As the thickness of the brick structure was increased, the amplifier gain of the system was increased accordingly. It was found that it is easy to penetrate a pile of dry rubble up to about three feet thickness with the 2 GHz life-detection system. 3 layers (10 l/8") of dry bricks. subject face-up 3 layers (10 l/8") of dry bricks. subject face-down 167 lO-GHz Life-Detection System breath- +—- heart beat -+¢ breathing-1-background -1 ing _ t::.&___:__’_ " r _:_:'_;-:' ‘ 2H: - ----- ,3? in - '- T LT - -_ ____ __ g " -O-o-. 1-. —. - - —.-O - — . u >-- - —. . .. . o ‘ --. I .c- ———-. - - --. g o -—‘ ~00 - o —-_ I. breathing +— heart beat +— background noise e—n g. .1 s; ' "—"rt'is- '-"l W — "_ ’ .. -t.; | -_..__ l'l I I .1 I Pa I I . r r r --r—-.—f f —- ——-—-— -—-—-- j :- L E T47 i 2 i L ' k - ——— ?’ Era . : P '. _ ' —" .. L53.E ' [7' “ b b — Fig-4.25 Heart and breathing signals of a human subject, lying with face-up or face- down position under 3 layers of bricks, measured by the 10 GHz life detec- tion system. 168 lO-GHz Life-Detection System ‘0- breathing +— heart beat ——41¢ backgrouno‘Y noise _fi_ . . -_._. - -. -. - - 4 layers --__F ---- - -———— ——---—~——-— ----- (l3 1/2") of _ ,_. .. -_- - _-- _ __ -- ~ _ E l dry bricks. ,_ .. - ...- .. subject ; ___.__ ,_ _ ,__,_ ‘___ face-up ; T W ~ . r WW——m ,_ S ‘ —- #— 1 sec. breathing heart beat -'+background~' noise _"-f_'i;__-.- _ __ __-_~___ 4 layers ,__ _,_ __ - 9M (l3 l/Z") of a; .. .___“ ____. dry bricks. § § __ __ _ _ _ _ ,__________ _. subject T"? 7 "T. -- __ ____ ._--‘__- -— face-down “" ‘lTTTW: ‘T Tfl— - _--____-_ Fig-4.26 Heart and breathing signals of a human subject, lying with face-up or face- down position under 4 layers of bricks, measured by the 10 GHz life detec- tion system. 5 layers (16 7/8") of dry bricks. subject face-uD 5 layers (16 7/8") of dry bricks subject face-down 169 Z-GHz Life-Detection System ’_ breathing _+_ heart beat bzgkground‘i ' ise ‘ ‘f— ‘- -..-__.__.. 3;. [5.3. "f ... - :’ ‘ TV: —_:~_§:T:_:__-—:: i"?- f - E- ' ~ - -E-‘.=‘_=. '3 E g..__F"- 2" ii V F -__...- ' F L- , . — l" i- L 4: ? LA _L L _' - lirt- ;§ 2. b 3 background T.— breathing .t.__ heart beat _+noise+ ’- P --—-- - - — ——=-.._._ ‘ l- i.:.§§gj.:_;;;,.ysg'. rs: -. -?::“EEE= ? ‘ 1? E3'“'5¥T€T:L"7 T"' "‘ EL'-;:J-t -; t ‘ ' ' __ 3‘53:- _;r ..' Fig-4.27 Heart and breathing signals of a human subject, lying with face-up or face- down position under 5 layers of bricks, measured by the 2 GHz life detec- tion system. 6 layers (20 l/4") of dry bricks. subject face-up 6 layers (20 l/4") of dry bricks. subject face-down 170 Z-GHz Life-Detection System ’w-breathing +__ heart beat . background ——‘i. noise *1 ‘fi '~ _ ____ _L .ggu . .L-_“ _-__ . Fifi-3:91.556”. .‘LI: E? :- i'xfliild . . f A; T -_— C” ,. g- __ ‘2? 3 __,_._._._..—_.. '2:- breathing 11.__ heart beat .c-L‘f .‘_: 1 sec. background I noise T1 “ 'UWflgfilyinTT"i: F... _._i ' ‘ _‘i 3, L ‘- T-‘ Gama-neuiqfutf ____ 1 i tr; flu? §4V1~____._ -_ __ l 'il ;g L1 u i I .-- _-_ ; 4 . 7f l . LT}? . .:' r L" . L ._. i i 4‘” if---____ . - L _ __ _ _ __ - _ — _ Fig-4.28 Heart and breathing signals of a human subject, lying with face-up or face- down position under 6 layers of bricks, measured by the 2 GHz life detec- tion system. 7 layers (23 5/8“) of dry bricks. subject face-up 7 layers (23 5/8") of dry bricks. subject face-down 171 Z-GHz Lif e-Detection System background 'i' noise "l I'— breathing +_ heart beat - . z‘ r: f“ - .é ~- 4 ~-‘ e—— — — g- 2 .- 2. W r- z . ' . . E "I 3¥£.:. . 4- % 3.. 3' . ‘.-."’, f " .2 .- _ I .' L- I" "" "' " “'1‘ Z: - - — 0.- — A i.- 1 sec. rbreathing 4... heart beat background 'f"noise T1 .E‘ l. =:_ ._ - ___. _ [l 5' T 'Eri‘; ’ - E- 5;: f. 3:.- _ __ ____-____ Fig-4.29 Heart and breathing signals of a human subject, lying with face-up or face- down position under 7 layers of bricks, measured by the 2 GHz life detec- tion system. CHAPTER V SUMMARY This thesis presents of the study on the quantification of interaction of elec- tromagnetic fields with finite heterogeneous bodies and an application of electromag- netic waves for detecting a small movement of a biological body behind a barrier. In the study on the quantification of interaction of electromagnetic fields with finite non-magnetic lossy bodies, an iterative Imp-EMF method and an equivalent magnetic current compensation method have been developed to improve the efficiency and accuracy of the existing tensor integral equation method. The induced electric field of electric mode which is excited by symmetrical part of incident electric field can be solved accurately from the tensor EFIE with the method of moment and pulse- basis expansion. However, the induced electric field of the magnetic mode which is excited by the antisymmetrical part of incident electric field can not be determined accurately by the same method. A series of numerical examples at various frequencies for the induced field distributions in a rectangular biological body are presented. It is observed that for an antisymmetric incident electric field in the range of several hun- dred MHz, the latter method can improve the accuracy of the numerical solution of induced electric field while the former method achieves more improvement. It is felt that the latter method can be further refined if a proper magnetic current can be found to compensate the discontinuity of the tangential electric field at the air-body interface, or the outrnost boundary of the body. One can also use the iterative loop-EMF method to calculate the induced fields in a heterogeneous body, but the determination of the impedance at the boundaries of the cells which are of different materials needs 172 173 further investigation. The interaction of EM fields with finite lossy magnetic bodies has also been stu- died in this thesis. A set of coupled tensor integral equations has been derived to solve for the induced EM field in a finite heterogeneous body with arbitrary electric parameters which is exposed to an incident EM field. This set of equations can be decoupled into a separate tensor EFIE and a separate tensor MFIE. If the coupled ten- sor integral equations are used, we need to solve for the unknown fields E and H simultaneously while we can solve E and H separately if the decoupled equations are used. This means twice the number of unknown variables when the coupled equations are solved numerically. Both coupled and decoupled integral equations can be expressed in terms of the free space scalar Green’s functions instead of the dyadic Green’s functions. With pulse-basis expansion and point-matching, either solving the tensor integral equations or the integral equations in terms of the scalar Green’s func- tions will give similar numerical results. One advantage of the coupled tensor integral equations is that they are easy to be formulated in the heterogeneous case. The agree- ment of the numerical solutions of the coupled and decoupled equations was observed for the cases of low permittivity (e, < 20) and frequency below a few hundred MHz. The phenomenon that the induced electric field is enhanced by the magnetic material in a finite body has also been investigated through the comparison of the numerical solutions for a magnetic body and a non-magnetic body with the same dimensions and complex permittivity. In the study of the application of electromagnetic wave for detecting a small movement of a biological body behind a barrier, a nonuniform plane wave passing through a layer of lossy barrier has been analyzed by using the plane wave spectrum analysis. Also a series of experiments were conducted to measure the breathing and heart signals of a human subject behind a thick layer of bricks with microwave life 174 detection systems. The predicted electric field behind the barrier distributions calcu- lated by the FFI‘ algorithm indicate some perturbations of the original electric field distributions due to the barrier. Varying the thickness of the barriers causes different attenuations and perturbations to the electric field distributions behind the barrier. Experimental results on the detection of small vibrations of a biological body using the microwave life detection systems clearly indicate the breathing and heart signals of a human subject located behind a brick wall of up to 3 feet thick. The comparison of the results obtained from the X-band and L-band systems confirms that the EM waves of lower frequencies achieve better penetration through a barrier. APPENDICES APPENDIX I We will prove that the decoupled tensor EFIE and MFIE derived in section 3.2 are equivalent to the set of decoupled integral equations derived by Tai [13] when they are applied to a homogeneous body. When (3.2.18) and (3.2.19) are applied to a homogeneous body, i.e., re, 1",, 6* and it are not functions of location, then the equations becomes: 2 I :12. ., e [Bu 380 ‘1. [ll — “'0 ’ u 2 6? +-l‘— H(r)-P.V.],kfi J’— ~52- 38 3110 “o e T I s a e The decoupled integral equations by Tai are ]E(r) - P.V.j v k436- - -:£]E(r’)-80(r,r’) dv’ ][n x E(r’)l°[V’ >< o(r,r’)l ds’ = 13’0“) H(r’)°80(r,r’) dv’ [n X H(r’)l'[V' >< o(r.r’)] ds’ = H’(r) . E(r) - (k2 — k3) j , 80(r,r’)-E(r’) dv’ + [u :40] is 50(r,r’)'[n x V’ X E(r)] 613' = E‘l") H(r) - (k2 — 1:3) I v 80(r,r’)-H(r’) dv’ . + [e e- 80] l . 800m" x V’ x H(r)] ds' = H‘< x on “r (r) ( > which is essentially equivalent to (3.2.20). Therefore we can conclude that (Al.l) is equivalent to (3.2.20) since (A13) is simply a short form of (A1.1). The remaining thing is to prove (Al.7). Proof: Tai denoted the total electric and magnetic field distributions outside the body as E2(r) and H2(r), the induced field distributions inside the body as Em and H1(r) and the incident fields as E") and H“), while we use E(r), H(r), E’(r) and H’(r) instead of E1(r), H1(r), E") and H") as shown in Fig-A.1. Note that the outward nor- mal vector of region I (inside of the body) in the figure of [13] is denoted as n1, and that of region 11 (outside of the body) as n2. We use the notation r1 = n1 = — n2 to denote the normal vector when the boundaries conditions are applied on the closed sur- face 3 enclosing region 1. Equations (15) to (17) in Tai’s paper are needed to prove this equality. On s the boundary conditions are 178 5.. Fig-AA Different noradons of the parameters and variables in a permeable body used by Tai andthis thesis whenitisilluminawd by a plane wave. 179 nX(E—E2)=O, (15) VxE nx VXE- 2 =0, (16) l.1 “0 hence nxVxE=-“EO-nxVxE2=[l+ u-HOJnXVxEZ Thus, we can have - I . n-1E2< 80(r,r’)] ‘15 u- l;l = E‘(r) — I No] I _, n°[(V’ x E(r’)) x 50(r,r')] ds’ . (17) From equation (17) of [13], we have u- 11 E’(r) - I no] I s n.[(V’ x E(r’)) x 50(r,r’)] ds’ = - I .. n'lE2(|") x (V’ x 0(l',|")) + (V' x E2(r’)) >< 50(r,r’>1 ds’ - I“ l-l-ouoI I , n°[(V’ x E2(r’) x 80(r,r’)] ds’ = - I . [(n x E2(r’»- dv’ - I . 1n-A(r')1V<1>(r.r’) ds’ = P.V. I , A(r’)-VV1V¢(r.r) ds’ + I .,[n~A(r’)1V¢(r.r’) ds’ + I WVEA(r’)-VV¢(r,r’) dv’ (A25) 183 where v is the volume source region, s is the closed boundary surface enclosing region v, Vs is a small sphere centered at r with radius 8 and Se is the closed boundary surface enclosing region v8, as shown in Fig-A.2. Note that the unit normal vector u of the surfaces 3+3e enclosing v—vs points outward to the the surfaces, i.e., it is in the oppo- site direction of the outward normal of the small sphere vE when we consider the sur- face integration on se in (A25). We use the following expression by the definition of Cauchy principal value, P.V. I v A(r’)-VV¢(r,r’) dv’ = limo I H,‘ A(r’)-VV¢(r,r’) dv’ . (A26) 8 Let us take limits for both side of (A25) when 8 goes to zero, we have I ,. [V’-A(r’)lV¢(r.r’) dv’ - I , [n'A(r’)lV¢(r.r’) ds’ = lim I ,¢[n'A(r’)]V¢(r,r’) ds’ + P.V. I v A(r’)-VV¢(r,r’) dv’ e->0 = P.V. I v A(r’)-VV¢(r,r’) dv’ - A? , (A2.7) since lim I s [n-A(r’)]V¢(r,r’) ds’ = — 315)- (A28) e—rO ‘ 3 for a continuous vector function A(r). The result of (A28) can be carried out by fun- damental calculus as shown in [2] and [3]. Now let us compare (3.1.40) and (3.3.28) term by term: I v impote(r’)E(r')q>(r,r’) dv’ = is- I v kgre(r’)E(r’)¢(r,r’) dv’ ; (A29) 0 by (A2.7), he“) E(r) — P.V. I v rmtto te(r’)E(r’)-VV¢(r,r’) dv’ 30380 kg 184 Fig-A2 Illustration of the exclusion of a small region which contains the singularity point when evaluating the principal integration. 185 = _ ICEI v [V'.1e(r’)E(r’)]V¢(r,r’) dv' + it I s [n-te(r’)E(r')]V¢(r,r’) ds' ; (A210) and I . tm(r’)H(r’)'[V >< 806x31 dv’ = I . «ammo x V¢(r.r’) dv’ (A2.11) since A(r’)-[V >< 80(r.r’)] = A(r’)-[V¢(r.r3 x 71 = A(r’) >< V¢(r,r’) (A2.12) for any vector function A(r’) and hence in particular, (A2.12) holds when A0") = H(r’). From (A2.9), (A2.10) and (A2.11), we can conclude that (3.1.40) and (3.3.28) are equivalent. It is noted that the limiting processes and the evaluation of the Cauchy principal values are all based on the assumption that the excluded volume vs is a small sphere. The different infinitesimal geometries will end up with different results as mentioned by Van Bladal [2] and Chen [9]. Similarly, we can prove that (3.1.41) and (3.3.29) are also equivalent. COMPUTER PROGRAMS Fortran-77 Program Guide for the Coupled Tensor Integral Equations This program is used to calculate the induced EM field distribution in a finite, hetero- geneous, dielectric, magnetic and lossy body exposed to an incident EM field. COMPUTER PROGRAM I Description of key variables and arrays for using the program: Variable or Description Array Name NT no. of cells in first octant of a symmetric body, integer variable DL length of a side of each cubic cell, in (m), real variable X(L,I) L coordinate of the center of the I-th cell, in (m), L=1,2,3=x,y,z coordinate, real array SIG(I) conductivity of the I-th cell, in (Sim), real array EPR(I) relative permittivity of the I-th cell, real array MUR(I) relative permeability of the I-th cell, complex array symmetry of incident EM field, INCI is l or 2. INCI 1 stands for antisymmetric and 2 for symmetric, integer variable f uenc of the incident EM field, in , FREQ rec1 y (MHZ) real variable 186 187 Input data file structure:(free formats are used for all the variables) Line No. Variable Name 1 NT,DL 2 (X(L,1),L=l,3), SIG(I), EPR(1), MUR(I) 3 (X(L,2),L=l,3), SIG(2), EPR(Z), MUR(2) NT+1 (X(L,NT),L=1,3), SIG(NT), EPR(N'I'), MUR(NT) NT+2 INCI, FREQ The following data are for the symmetry of geometry and EM field distributions and should remain at the end of this data file. NT+3 l 1 l NT+4 -1 1 1 NT+5 -l -1 l NT+6 l-l 1 NT+7 1 1-1 NT+8 -1 1-1 NT+9 -l -l -1 NT+10 l -1 -l NT+11 llll-l-l-l-l NT+12 1-1 l-l-l 1-1 1 NT+13 1-1-1 1 l-l-ll NT+14 1 1 l l 1 l l l NT+15 1-11-1 l-ll-l NT+16 1-1-11-11 1-1 NT+17 1-11-1 1-11-1 NT+18 l 1 l l l l l l NT+19' ll-l-l-l-lll NT+20 l-l l-l-l l-ll NT+21 l l l l-l-l-l-l NT+22 1 1-1-1 1 1-1-1 188 Output data file structure: There will be 5 numbers in each line. The first one is an integer and indicates the numbering of the cell. The 2nd and 3rd numbers stand for the phasor (real and imaginary parts) of EM field of each cell. The 4th and 5th numbers stand for the amplitude (in V/m or A/m) and phase angle (in degree) of the EM field. Three com- ponents of EM field are all printed out. The program listing is attached from the next page. 189 ctifirttti*fiitriittttititttttiiitiiiittfiiflitittitiittttttttttitttitittflttttittttt c Coupled Tensor Integral Equations C*****t*tti***i*ttiitifltttititttitfiflfiitifittifi*t************t***ii****fittitttttit c This program is used to calculate the induced EM field distribution in c a finite, heterogeneous, dielectric, magnetic and lossy body exposed to c an incident EM field Cfit*tiiittttiiittttit*tiitfititttiitititittiiit********t**tt***ttititittititrttit PROGRAM COUPLEIE PARAMETER (NTMAX-64,NTMAX3-l92,NTMAXG-384,NTMAX61-385 5'NTMAXB'SlleTMAX83-1536) REAL X(3,NTMAX8),SIG(NTMAX),EPR(NTMAX) COMPLEX A(2,3.NTMAX83).D(NTMAX6,NTMAX61) $.MUR(NTMAX),TAUE(NTMAX),TAUM(NTMAX) $.CON1(NTMAX).CONZ(NTMAX),CON3(NTMAX),CON4(NTMAX) INTEGER NS(NTMAX.6) Cttittifii*fi*fl*'**ittiftiiflifit***ifiiiiffliitttifiiifi*fl*****fii********i*********titt c Input data: cttttit!****i***tttittiit******fi****fiitfitti*tittttiiiiiitflttiiitittflittitittiiti c NT-No. of cells in first octant of a symmetric body, integer DL-length of a side of each cubic cell, in (m) X(L,I)-L coordinate of the center of the i-th cell, in (m), L-l,2,3-x,y,z coordinate SIG(I)-conductivity of the i-th cell, in (S/m) EPR(I)-relative permittivity of the i-th cell, a real number MUR(Il-relative permeability of the i-th cell, a complex number ***t*iiifltitflttiiiiitttfltflfi*****t******t*i**t***********tt***titti*iiititititit open (l,file-’cplin.dat’) read (l,*) NT,DL do 391 i-l,nt read (1,*) X(l,I),X(2,I),X(2,I),SIGII),EPR(I),MUR(I) cit*tttttiittitfiiitiitiiittttfltttfitttttttttittitittii*tititiiit*iititflitttttttti nt3-3*nt nt6-6*nt nt8-8*nt nt83-24*nt nt61-nt6+l call main(x,sig,epr,a,d,mur,taue,taum,conl,con2 $,con3,con4,ns,NT,NT3,NT6,NT61,NT8,NT83) stop end ()0 OCTCIO O subroutine main(x,sig,epr,a,d,,mur,taue,taum,conl,con2 S,con3,con4,ns,NT,NT3,NT6,NT61,NT8,NT83) REAL MUO,K0,XJV(8,3),X(3,NT8),XC(3),x1(3),SIG(NT),EPR(NT) COMPLEX A(2,3,NT83),D(NT6,NT61) $.JC,MUR(NT),MU,mur2,TAUE(NT),TAUM(NT) $.SUMV,DT1,DT2,CON1(NT),CON2(NT),CON3(NT),CON4(NT) INTEGER P,Q,NS(NT,6),ISE(2,3,8),ISM(2,3,8),IQ(3,8) COMMON /A/JC,K0,PI/F/DL,HDL,QDL,DS,Dv OPEN (4,?ILE-'cplout.dat') nt2-2'nt nt4-4'nt ntS-S'nt ct**t**i**tfi**iiifliitittttititttttiittitiitittttttttflttiiitit*fifittitflttittittttt c Input data: cttttfltflittttttitit*tttiflifitt*ttt*ttttitt*ttttttiittttttttiit*ifltitittittfltitttt c INCI-smetricity of incident EM field, INCI is l or 2 c 1 stands for antisymmetric and 2 for symmetric c FREQ-frequency of the incident EM field, in (MHz) ciiiifliifitfliflflflfififiiflitiii*ti**t***ii*fit**i**t*fl**fl*******t************fi§*tiititi READ (1¢*) INCI,?REQ c*tii**i***t***fl*i*******fl***fl*****ii******tfli**************i***t*i**fl*i******tt c Output Format: Citflirtfittitittftflitttitittttifi*ttttfltttflitittititttttiittttt*titttttittittfiitit WRITE (4,*) 'HETRO COUPLED IE WITH 8 QUA SYMMETRY' IF (INCI.EO.1) WRITE (4,*) ’Ei - x j*sinkz, Hi - y coskz/ZO' 190 IF (INCI.EQ.2) WRITE (4,*) 'Ei ' X COSKZ' Hi ' y j'sinkz/ZO’ DO 73 I'1,NT Cttitititittititiittitttfiitittttittiittttttiittttttttitititttitt**tfitttit*ttttrr 73 WRITE(4,50) IISIG(I)IEPR(I)IMUR(I) 50 FORMAT (IS,F15.8,3X,F15.8,3X,2E15.8) Ci********fi******it*****iiifi********************t****t**************i*i**twtttit c IQ(L,J)-sign of L cooridnate of center of each cell in J-th octant, c IQ is 1 or -1 c ISEtINCI,L,J)-sign of L component of E field distribution in J-th octant, c ISE is l or -1, INCI is l or 2 c ISM(INCI,L,J)-sign of L component of H field distribution in J-th octant, c ISM is 1 or -1, INCI is l or 2 ctr*********fli**tttfititi*t*t*ti**iti*******************tittiitttiiifii*ttttittit DO 31 9.7-1! 8 31 READ (l,*) (IQ(L,J),L-l,3) D0 32 I-1'3 32 READ (1,*) (I$E(1'L,J),J-l,8) DO 33 1.1: 3 33 READ (l,*) (ISE(2,L,J),J-l,8) DO 52 I'l,3 52 READ (1,*) (ISM(1,L,J),J-l,8) DO 53 I‘ll 3 53 READ (1,*) (ISM(2,L,J),J‘1,8) c*fi***************************i***'*********fi*************fi***********fi***tttft JC'(0.,1.) PI-DATAN(1.D0)*4. WG-2.D6*PI*EREQ K0-WG/3.D8 EPO'l.E-9/36./PI MUG-4.E-7*PI DO 74 I'1,NT MU-MUR(I)*MUO EP‘EPR(I)*EPO TAUE(I)'SIG(I)‘JC*WG*(EP-EPO) TAUM(I)--JC*WG'(MU-MUOI CON1(II'-K0*TAUE(I)*JC’WG*MUO/4./PI CON2(I)-+KO**2*TAUM(I)/4./PI CON3(I)'-KO**2*TAUE(I)/4./PI 74 CON4(I)"KO*TAUM(I)*JC*WG*EPO/4./PI Ds-DL*DL DV‘DS*DL HDL-DL/Z. QDL-DL/4. Cit****fl*fi**fi*i****tfi*fl*fl***fi*****fiflt*i*fifl*fi*t*tfl1ft.t#*****fi*ttiffiifittitflt**if! c Calculate the corresponding coordinates from the 2nd to 8th octants ctitttittflitttit?*itflfliififitttitfiitiflititt*itittit'iwitiiittittttttttittttflittttt DO 15 I-l,NT DO 15 L-l,3 15 X(L,I+(M-l)*NT)-X(L,I)*IQ(L,M) Cifltfl****.'**************fl*iiQ*******i*tiitfiiftitittit!tit!tittittiiifiiftfiffifii c Filling matrix elements c**********iflt**fi***t***fifiii******t****ttiiittiflttititfiitfittiititifltttttttfititifl DO 18 I-l.NT PRINT *,I DO 79 L-l,3 79 XCiL)-X(L.I) DO 18 MQ-1,2 DO 17 J-l,NT8 JMODNT-J-(J-1)/NT*NT DO 78 L-1,3 78 Xl(L)-X(L,J) DO 17 Q-1,3 LL-J+(Q-ll*NT8 DO 17 P-l,3 CALL VPT(X1,XJV,QDL) 191 CALL VINTE(I,J,XC,XJV,MQ,P,Q,SUMV) IF (MQ.EQ.1) THEN A(1,P,LL)-CON1(JMODNT)*SUMV A(2,P,LL)'CON4(JMODNT)*SUMV IF ((I.EQ.J).AND.(P.EQ.Q)) $A(1,P,LL)-A(1,P,LL)+1.+JC*TAUE(JMODNT)/3./WG/EPO IF ((I.EQ.J).AND.(P.EQ.Q)) $A(2,P,LL)-A(2.P,LL)+1.+JC*TAUM(JMODNT)/3./WG/MUO ELSEIF (MQ.EQ.2) THEN A(l,P,LL)-CON2(JMODNT)*SUMV A(Z.P.LL)‘CON3(JMODNT)*SUMV ENDIF l7 CONTINUE Citiitti*******t**fii*fi**t*******ittt******flifii***it*fi******tfifi**tiitifittiiflirt? c Using symmetric properties to reduce matrix size ctittit********t*tt*ttiifiitiiiiitittittititfittiittttttttitttittttittttttt*tttttt DO 41 K-1,NT DO 41 9-103 DO 41 Q-l,3 DTl-(0.,0.) DT2-(0..0.) KT-K+(Q-1)*NT IF (MQ.EQ.1) THEN DO 42 M-1,8 DT1-A(1.P.(Q-l)*NT8+K+(M-l)*NT)*ISE(INCI,Q,M)+DT1 42 DT2-A(2,P.(Q-1)*NT8+K+(M-1)*NT)*I$M(INCI,Q.M)+DT2 ELSEIF (MQ.EQ.2) THEN DO 43 H-118 DTl-A(1,P.(Q-l)*NT8+K+(M-1)*NT)'ISM(INCI,Q,M)+DT1 43 DT2-A(2,P,(Q-l)*NT8+K+(M-1)*NT)*ISE(INCI,Q,M)+DT2 ENDIF D(I+(P-1)*NT,KT+(MQ-1)*NT3)-DT1 41 D(I+(P-1)*NT+NT3,KT+(2-MQ)*NT3)-DT2 l8 CONTINUE c*ifii*fiit**ti*tfiitiififiiitfififlfiittittitiiitififlititi*fltififltiiflfitiflflffiifittfiittiiit c Filling the vector of incident EM field in the center of each cell Ctit!*tfiitttttififlti*ttiiitttfititi**flitfitttt*tfifltittifltflttti*ttitittflitiflttt*ttifi DO 35 I-1,NT B-K0*X(3.I) IF (INCI.EQ.1) THEN C-SIN(B) E-COS(B) D(I,NT61)-CMPLX(0.,C) D(I+NT4,NT61)-CMPLX(E/120/PI,0.) ELSEIF (INCI.EQ.2) THEN C-COS(B) E-SIN(B) D(I.NT61)-CMPLX(C,O.) D(I+NT4,NT6l)-CMPLX(O.,E/lZO/PI) ENDIF D(I+NT,NT61)-(O..O.) D(I+NT2,NT61)-(O.,O.) D(I+NT3,NT61)-(0.,O.) 35 D(I+NTS,NT61)-(0.,O.) cttifiif*Qiitii'itfiflifl'itiififi*t****t*itflfl.*fifliiifliiflttiififi*tiittiiiiiiflflfliiifittit c SolVing the matrix equation ctttttttitflittttit!it*iii*tfiiflitiitti*itttfititittttitfltit*ttttttitittitttttttttt CALL CMATPA(-1,D,NT6,1,DET,1.3-38,NT61) ciiii***i**9***fi*i***fi*fl*********fi******fl*i********ii*fl******t*fiifii**flfifl****tit! c Print out the induced EM fields at the center of each cell cittittttttttittttttt*ttititttiiitttt*tttttifittttittfi*ttttttiititiiiittttttit??? c FABS-Magnitute of the EM fields, in (V/m) or (A/m) c PHASE-Phase angle of the EM fields, in degree ctfl*t***t*ti**t*ifi*ttt*i**tttittfittttttt*****t*****fii*t*ttfitiitflttttttttttttittt DO 36 K-l,6 IF (K.EQ.1) WRITE (4,*) 'EX' 192 IF (K.EQ.2) WRITE (4,*) ’EY' IF (K.EQ.3) WRITE (4,*) ’EZ’ IF (K.EQ.4) WRITE (4,*) 'HX' IF (K.EQ.5) WRITE (4.*) 'HY' IF (K.EQ.6) WRITE (4,*) 'HZ' DO 36 I'1,NT Il-I+NT*(K-l) FABS-CABS(D(I1,NT61)) PHASE-ATANZ(AIMAG(D(II,NT61)),REAL(D(I1,NT61)))*l80/PI 36 WRITE (4,150) I,D(I1,NT61),FABS,PHASE 150 FORMAT (1X,I4,2E20.11,3X,E15.8,F10.4) RETURN END Ctiii*fi*ifl*******i**fi***i**i*******t***iftififlift*ttiffl***i**********tfitiitiittt c This subroutine is used to calculate each entry of the matrix except the c diagonal element via numerical integration, the integration is approximated c simply by cutting a cell into 8 subcells and summing up the function value c in each subcell ctttiiii.fittttittttttittiifiittfittit*ttttttttittttttttitt§tffittitt§ttittittttitit SUBROUTINE VINTE(M,N.X,XJ,MQ,P.Q.ST) REAL X(3),X1(3),XJ(8,3) COMPLEX GPQ,GP,ST,GB INTEGER P,Q COMMON /F/DL,HDL,QDL,DS,DV ST-(0.,0.) IF (MQ.EQ.1) THEN IF (M.EQ.N) GO TO 69 DO 30 J-1.8 DO 46 L-1.3 46 X1(L)-XJ(J.L) 3O ST-ST+GPQ(P,Q,X,X1) ST-ST*DV/8. GO TO 99 69 IF (P.NE.Q) GO TO 99 ST-GB(DL) ELSEIF (MQ.EQ.2) THEN IF (P.EQ.Q) GO TO 99 K-6-P-Q DO 40 J-1,8 DO 56 L-l,3 S6 x1(L)-XJ(J,L) 40 ST-ST+GP(KIX'XI) KD-(P-Q)/2 IF (KD.EQ.0) ST-ST*(P-Q) IF (KD.NE.0) ST--ST'KD ST-ST*DV/8. ENDIF 99 RETURN END cfiiflfi*fiiffifltflfiifiititfitfiiftif*fi*tfiitit*ifl*tffi'itit*flf**********iit*ii*iititfiit c This function subroutine is used to evaluate the 2nd derivative of the c free space Green's function at center of each cell Cittitttitttttitt*ttittttttttttiitfittttflttttiwitfitttttttfi*itittflifltttitiiitttttt FUNCTION GPQ(P.Q.X,X1) REAL K0,X(3),X1(3),XD(3) COMPLEX JC.GPQ,8,C,D INTEGER P,Q COMMON /A/JC.KO.PI DO 75 L-1,3 XD(L)-X(L)-X1(L) IF (ABS(XD(L)).LT.1.D-8) XD(L)-0. 7S CONTINUE 193 R-SQRT(XD(1)*XD(1)+XD(2)*XD(2)+XD(3)*XD(3)) A-KO'R D-CEXP(JC*A) C-(3.-A*A-3.*JC*A)*XD(P)*XD(Q)/R/R IF (P.NE.Q) THEN GPQ'C*D/A/A/A ELSEIF (P.EQ.Q) THEN B-A*A-l.+JC*A GPQ-(B+C)*D/A/A/A ENDIF RETURN END cit************fit*ffl*********ii**fi************tfifl*i*******t***********i**t*i**tt c This function subroutine is used to evaluate the lst derivative of the c free space Green's function at center of each cell Ctttwflfittti*ttiiiitiitiittitittitit*flitiittitfliiitt*ti*fl*t*t*tt*tititititit*itit FUNCTION GP(P.X,X1) REAL K0,X(3),X1(3),XD(3) COMPLEX JC,GP,C,D INTEGER P COMMON /A/JC,KO.PI DO 75 L-l,3 XD(L)-X(L)-Xl(L) IF (ABS(XD(L)).LT.1.D-8) XD(L)-O. 7S CONTINUE R-SQRT(XD(1)*XD(1)+XD(2)*XD(2)+XD(3)*XD(3)) A-K0*R D-CEXP(JC*A) c-(JC*A-l.)*XD(P)/R GP-C*D/A/A RETURN END cfli*ii*'**fi********fl**fiti***********ffiifliflitflflfiiflfltfliii**fl****fl************fiit! c This function subroutine is used to calculate the value of the diagonal c element of the matrix, this value includes a pricinpal integration plus c a correction term C**t**********titttittiifiittiitit******t****itttt************i**t*****ttiitttivw FUNCTION GB(DL) REAL K0 COMPLEX GB,JC COMMON /A/JC,K0,PI DV-DL**3 AN-(3.*DV/4./PI)**(1./3.) GB-4.*PI*2.*(CEXP(JC*K0*AN)*(l.-JC*K0*AN)-l.)/(3.*K0**3) RETURN END citit****************§***********************i******f*ttflttifittttii*t****§*i***t c This subroutine is used to calculate the coordinates of 8 subcells c of each cell in order to perform the numerical integration cit*tttttitii*tttitiittitttttii*ttttttiittittitttttt************t****ttittttttit SUBROUTINE VPT(B,BN,QRDL) REAL QRDL,B(3),BN(8,3) BN(l,l)-B(l)+QRDL BN(l,2)-B(2)+QRDL BN(1,3)-B(3)+QRDL BN(2,l)-B(1)+QRDL BN(2,2)-B(2)+QRDL BN(2,3)-B(3)-QRDL BN(3.1)-B(1)+QRDL BN(3,2)-B(2)-QRDL 194 BN(3,3)-B(3)+QRDL 5N(4,l)-B(l)+QRDL BN(4,2)-B(2)-QRDL BN(4,3)-B(3)-QRDL BN(5,1)'B(l)-QRDL BN(S,2)-B(2)+QRDL BN(S,3)-B(3)+QRDL BN(6,l)-B(l)-QRDL BN(6,2)-B(2)+QRDL BN(6,3)-B(3)-QRDL BN(7,1)'B(l)-QRDL BN(7,2)-B(2)-QRDL BN(7,3)-B(3)+QRDL BN(8,1)-B(l)-QRDL BN(8,2)-B(2)-QRDL BN(8,3)-B(3)-QRDL RETURN END cii****fi***fi***§***itiifliiflittiitiifififliifiit*i**tfi*******fl*ti***itttttittiifittfiii c This subroutine is used to solve the matrix equation via Gaussion c ellimination Citit*ttttiiittiiitti*itttittfltitttttttititii*itittttfitfitttttttittttiitttiwititr SUBROUTINE CMATPA(IJOB.A,N,M,DET,EP,N031) COMPLEX A.B.DET,CONST,S DIMENSION A(N,N031) 30 FORMAT(1X,4ZHTHE DETERMINANT OF THE SYSTEM EQUALS ZERO./ llX,36HTHE PROGRAM CANNOT HANDLE THIS CASE.//) DET-l. NPl-N+l NPM-N+M NMl-N-l IF(IJOB) 2,1,2 1 DO 3 I-1,N NPI-N+I A(I.NPI)-1. IP1-I+1 DO 3 J-IP1,N NPJ-N+J A(I,NPJ)-O. A(J,NPI)-O. DO 4 J-l,NM1 C-CABS(A(J.J)) JPl-J+l DO 5 I-JPl'N D-CABS(A(I.J)) IF(C‘D) 615,5 6 DET--DET DO 7 K-J,NPM B-A(IIK) A(I.K)-A(J,K) 7 A(J.K)-B C-D 5 CONTINUE IF(CABS(A(J.J))-EP) 14,15,15 15 DO 4 I-JP1,N CONST-A(I.J)/A(J,J) DO 4 K-JP1,NPM 4 A(I,K)-A(I,K)-CONST*A(J,K) IF(CABS(A(N,N))-EP) 14,18,18 l4 DET-O. ' 'IF(IJOB) 16,16,17 16 PRINT 30 17 RETURN C 18 DO 11 I-1.N NU) C 18 195 11 DET-DET*A(I,I) 10 19 12 IF(IJOB) 10,10,17 DO 12 I-1,N K-N-I+l KPl-K+1 DO 12 L-NP1,NPM S-O. IF(N-KP1) 12,19,19 DO 13 J-KPle 5-S+A(K,J)*A(J,L) A(K'L)'(A(K'L)-S)/A(K,K) RETURN END COMPUTER PROGRAM II Fortran-77 Program Guide for the Decoupled Tensor EFIE This program is used to calculate the induced electric field distribution in a finite, dielectric, magnetic and lossy body exposed to an incident electric field. Description of key variables and arrays for using the program: Variable or Description Array Name NT no. of cells in first octant of a symmetric body, integer variable DL length of a side of each cubic cell, in (m), real variable X(L I) L coordinate of the center of the I-th cell, in (m), ’ L=1,2,3=x,y,z coordinate, real array surface index, which is actually the sign of the normal vector of the K-th face of the I-th cell, NS is l, or -1 NS(I K) if the face is on the outmost surface, otherwise NS is 0. ’ K=1,2,...,6 where K=1,2,3 stand for the outword normal vectors pointing to +x,+y,+z directions, and K=4,5,6 for -x,-y,-z directions, integer array SIG conductivity, in (S/m), real variable EPR relative permittivity, real variable MUR relative permeability, complex variable symmetry of incident electric field, INCI is 1 or 2. INCI 1 stands for antisymmetric and 2 for symmetric, integer variable f none of the incident electric field, in z , FREQ rcq .Y (MH ) real vanable 196 197 Input data file structure:(free formats are used for all the variables) Line No. Variable Name 1 NT,DL,SIG,EPR,MUR 2 (X(L,1),L=l,3'). (NS(1.K).K=1.5) 3 (X(L,2),L=1,3), (NS(2,K),K=1,6) N'I‘+l (X(LNT),L=1,3), (NS(NT,K),K=1,6) NT+2 INCI, FREQ The following data are for the symmetry of geometry and electric field distributions and should remain at the end of this data file. NT+3 l l 1 NT+4 -l 1 l NT+5 -l -1 1 NT+6 l -1 l NT+7 1 1-1 NT+8 -1 1-1 NT+9 -l-1-1 NT+10 1-1-1 NT+ll 1 1 l 1 -l -1 -1-1 NT+12 1-1 1-1 -1 1-1 1 NT+13 1-1-1 1 l-l-ll NT+14 l l 1 l 1 l l l NT+15 1-1 1-1 1 -1 1-1 NT+16 1-1-11-111-1 198 Output data file structure: There will be 5 numbers in each line. The first one is an integer and indicates the numbering of the cell. The 2nd and 3rd numbers stand for the phasor (real and imaginary parts) of electric field of each cell. The 4th and 5th numbers stand for the amplitude (in V/m) and phase angle (in degree) of electric field. Three components of electric field are all printed out. The program listing is attached from the next page. 199 Ct!ititit*tttflitfiitffifiittiiiittttttittiflti*ttttiflttitiiiritttttiiiittttttvttttw c Decoupled Tensor EFIE Cttttflrifltwitti*iiittttititittitttttt***t***t*itttitittttittittttttttitttitkttrr c This program is used to calculate the induced E field distribution in a c finite, dielectric, magnetic and lossy body exposed to an incident field Ctttwiiti******t*ifittii*fittti'tiittfltiitflfltttifiiittttttttflititiiiitittttitwtti’ttt PROGRAM DECOPULEEFIE PARAMETER (NTMAX-343,NTMAX3-1029,NTMAX31-1030 $,NTMAX8-2744,NTMAX83-8232) REAL X(3,NTMAX8) COMPLEX A(2,3,NTMAX83),D(NTMAX3,NTMAX31) INTEGER NS(NTMAX8,6) cit*fifliiiffiiitiiiifliifltt*********************i********************i*****fitittit c Input data: Citit*iittttiff*t*flttttfiiiititttiittttttfitttt*fittfiitttttttttfltitttttfiiitttttwit. c NT-No. of cells in first octant of a symmetric body, integer DL-length of a side of each cubic cell, in (m) X(L,I)-L coordinate of the center of the i-th cell, in (m), L-l,2,3-x,y,z coordinate SIG-conductivity, in (S/m) EPR-relative permittivity, a real number MGR-relative permeability, a complex number NS(I,K)-surface index, which is actually the sign of the normal vector of the K-th face of the I-th cell, N5 is 1 or -1 if the face is on the outmost surface, otherwise N5 is 0. K-1,2,...,6 where K-1,2,3 stand for the outword normal point to +x,+y,+z directions, and K-4,S,6 for the -x,-y,-z directions *tiiiifliifittitititflttit.ifliiitflirttiiii?itiiifliittiiflt'kfi*titfifltitiikitiittit!it? open (l,file-'efiein.dat') read (l,*) NT,DL,SIG,EPR,MUR do 391 i-1,nt read (1,*) X(1,I).X(2,I),X(2,I). $NS(I,1),NS(I,2),NS(I,3),NS(I,4),NS(I,S),NS(I,6) citiiti*ttttifittiitttitttitittiititittittfitiittflttittiiiti’!tiiiifiiitiitiitfiittit NT3-3*NT NT8-8*NT NT83-24*NT NT31-NT3+1 CALL EFIE(SIG,EPR,MUR,X,A,D,NS,NT,NT3,NT31,NT8,NT83) STOP END 0(3()O 0(7()0 0(3()0 O SUBROUTINE EFIE(SIG,EPR,MUR,X,A,D,NS,NT,NT3,NT31 $.NT8,NT83) REAL MUO,K0,XJV(8,3),XJS(4,3,6),X(3,NT8),XC(3),X1(3) COMPLEX A(3,NT83),D(NT3,NT31),GD(3,3) S,JC,EPS,MUR,MU,CON1,CON2,SUMV,SUMS,DT INTEGER P,Q,NS(NT8,6),IS(2,3,8),IQ(3,8) COMMON /A/JC,K0,PI,EPO,EP,EPS,MUO,MU,CON1,CON2 $/F/DL,HDL,QDL,DS,DV OPEN (4,FILE-’efieout.dat’) NT2-2*NT c*iiiifififiitt*****ttflfltfi*t*fliittiflifi******i**ii****f****t**fiiififiififit**fifi*ttiiii* c Input data: C****t**********t***t*tttttttttttittiflittiitttttttfltittitttii*ttttttittittitttti c INCI-smetricity of incident E field, INCI is 1 or 2 c 1 stands for antisymmetric and 2 for symmetric c FREQ-frequency of the incident EM field, in (MHz) cfiiiflfliififltfliififlt*fliiiitifititflfi*****t*i********fi******i*****iitfltiflfiffi****t*iii READ (1,*) INCI,FREQ ct**********i***i**i******itfiiittiiitt***ti***************t****fi*******i******tt c Output Format: Cti*tttititttttitttittttt*tititfltttittttitt*tttttitfittttttiflt*ttttttttttflttttitw WRITE (4,*) 'DECOUPLED EFIE' IF (INCI.EQ.1) WRITE (4,*) 'INCI - x j*sinkz' 200 IF (INCI.EQ.2) WRITE (4,*) 'INCI - x coskz' WRITE(4,*) 'DL -’ ,DL WRITE(4,*) ’FREQ -’ ,FREQ WRITE(4,*) 'SIG -' ,SIG WRITE(4,*) ’EPR -' ,EPR WRITE(4,*) 'MUR -' ,MUR Ct*itiitt9*titittifiitttifittttitfitittttttttttttttittiwtitttttittttfititttiittrtw't c IQ(L,J)-sign of L coordinate of center of each cell in J-th octant, c IQ is 1 or -1 c IS(INCI,L,J)-sign of L component of E field distribution in J-th octant, c ISE is 1 or -1, INCI is 1 or 2 Citittifittitittiiti*ttifliit*tttttiititt******it**t******t**tt**tittttttttttitttt DO 31 J-1,8 31 READ (l,*) (IQ(I,J),I-1,3) DO 32 I'lr3 32 READ (1,*) (IS(1,I,J),J-1,8) DO 33 I-1,3 33 READ (l,*) (IS(2,I,J),J-1,8) Ctittttittitffiiiitfitifitiititttiiiitt***t***titttifittitttt'itititttiiiiiittttittt JC-(O.,1.) PI-ATAN(1.)*4. WG-2.E6*PI*FREQ K0-WG/3.E8 EPO-l.E-9/36./PI MUO-4.E-7*PI MU-MUR*MUO EP-EPR*EPO EPs-EP+JC*SIG/WG CONl--KO**3*(EPS/EPO-MUO/MU)/4./PI CON2-K0**2*(MU-MUO)/MU/4./PI PRINT *,JC,PI,WG,K0,EPO,MUO,EP,MU,EPS,CONl,CON2 DS-DL*DL DV-DS*DL HDL-DL/Z. QDL-DL/4. Citttttti*tiiiitttittitfltiitttttiitittttiitffitittfit****it*ttittflttiittitflitttttt c Calculate the corresponding coordinates and the surface indecies c from the 2nd to 8th octants Cttttttfltittfltttitfittttttit***ittttttfltittflttitit*ttttitttiitittttttititttiflirt! DO 15 I-1,NT DO 15 M-2,8 DO 15 L-l,3 15 X(L,I+(M-1)*NT)-X(L,I)*IQ(L,M) DO 16 I-1,NT DO 16 M-2,8 DO 16 L-1,3 IF (IQ(L,M).GE.0) THEN NS(I+(M-1)*NT,L)-NS(I,L)*IQ(L,M) NS(I+(M-l)*NT,L+3)-NS(I,L+3)*IQ(L,M) ELSEIF (IQ(L,M).LT.0) THEN N$(I+(M-1)*NT,L)-NS(I,L+3)*IQ(L,M) NS(I+(M-1)*NT,L+3)-NS(I,L)*IQ(L,M) ENDIF 16 CONTINUE CALL SELF(GD) cfl*********fl****tt**it*it*ittifitt*********ttttt****fiti*ti*t**********tti*ttitit c Filling matrix elements citfiittti*flttfit*tttfittittittittflttiiiitttittittttiittittttt*tttiiflittttt.**tt*tt DO 18 I-1,NT PRINT *,I DO 79 L-1,3 79 XCtLl-XtL,I) DO 17 J-1,NT8 DO 78 L-l,3 78 X1(L)-X(L,J) DO 17 Q-1,3 201 LL-J+(Q-l)*NT8 DO 17 P-1,3 IF (I.EQ.J) THEN CALL SPT(X1,XJS,HDL,QDL) CALL SINTE(XC,J,XJS,P,Q,SUMS,NT8,NS) A(P,LL)-GD(P,Q)+CON2*SUMS ELSEIF (I.NE.J) THEN CALL VPT(X1,XJV,QDL) CALL VINTE(XC,XJV,PIQ,SUMV) CALL SPT(X1,XJS,HDL,QDL) CALL SINTE(XC,J,XJ$,P,Q,SUMS,NT8,NS) A(P,LL)-CON1*SUMV+CON2*SUMS ENDIF l7 CONTINUE citt****i********it******t*i*fli********fi*fittt************fi********i*********titi c Using symmetric properties to reduce matrix size cw*itit***iifli*ii*t***t****t**ittttitfiitiittiitii****fi***t*i******i*titttititiwt DO 41 K-l,NT DO 41 P-1,3 DO 41 Q-l,3 DT-(0.,0.) DO 42 M-1,8 42 DT-A(P,(Q-l)*NT8+K+(M-1)*NT)*IS(INCI,Q,M)+DT 41 D(I+(P-1)*NT,K+(Q-1)*NT)-DT 18 CONTINUE C**it*iitiiiiii*********t*fi****ti*iii*******tttiifiiitfi****fl****i***fifltfifitittrtit c Filling the vector of incident E field in the center of each cell Ctrttiiititflttititfi*tttiiittitttttttitittittttttt*tttttttittttiiiittt'iittiflirt! DO 35 I-1,NT B-KO*X(3,I) IF (INCI.EQ.1) THEN C-SIN(B) D(I,NT31)-DCMPLX(0.,C) ELSEIF (INCI.EQ.2) THEN c-COS(B) D(I,NT31)-DCMPLX(C,0.) ENDIF D(I+NT,NT31)-(0.,0.) 35 D(I+NT2,NT31)-(0.,0.) cfitti*fi****fifi*****i**fli*fl****fi***tfiitfli'tif*ttiiifit*flti*fl*fl*t*******i***ifi*iiiii c Solving the matrix equation Cttttittttiiittti**i***t*t*i**t**tt*******t********t*i*titt*iiiitttitt*****i*ttt CALL CMATPA(-1,D,NT3,1,DET,1.E-38,NT31) cittfiiifliiiiiit********fi*fi*****f*****tifl*****fi**#*flitt************i**********i*i c Print out the induced EM fields at the center of each cell cittt*ifltfitttt*ittiiiiiiittiitti*titit*tiiittititi*tiittittwtiittitwttittttitti c FABS-Magnitute of the E fields, in (V/m) c PHASE-Phase angle of the E fields, in degree Citittitittfliit*fiiitiitttttititti*tiifiittttttttiitt***tti*****fl****ii*fltttitttit DO 36 K-1,3 IF (K.EO.1) WRITE (4,*) 'EX' IF (K.EQ.2) WRITE (4,*) ’EY' IF (K.EQ.3) WRITE (4,*) ’E2' DO 36 I-1,NT Il-I+NT*(K-1) EABS-CABS(D(I1,NT31)) PHASE-ATANZ(AIMAG(D(11,NT31)).REAL(D(Il,NT31)))*180/PI 36 WRITE (4,150) I,D(I1,NT31),EABS,PHASE 150 FORMAT (1X,I4,2E20.11,3X,E15.8,F10.4) STOP END cit******i*************************************************************i******tt c This subroutine is used to calculate the volume integration, the c integration is approximated simply by cutting a cell into 8 subcells 202 c and summing up the function value in each subcell cifitt*t'ktiwitt*'i*t**i***tti*tttiflittiiitttiiit'ttttifrivttt'ittttitrtrtittwtctr SUBROUTINE VINTE(X,XJ,P,Q,ST) REAL X(3),X1(3).XJ(8,3) COMPLEX GPQ,ST INTEGER PIQ COMMON /F/DL,HDL,QDL,DS,DV ST-(O.,0.) DO 30 J-1,8 DO 46 L-1,3 46 X1(L)-XJ(J,L) 3O ST-ST+GPQ(P,Q,X,X1) ST-ST'DV/8. RETURN END ctt*ttiifiiiifi'fit‘kiitflitffiflfiifit‘tti*tttii’it‘ti’********fi*******it*titfiiiittitttttr c This subroutine is used to calculate the surface integration, the c integration is approximated simply by cutting a surface into 4 pieces c and summing up the function value in each piece Cttttit*tt**i*itittttttttittittiitiiitiitttti*titttittttiiitiiiiititiititttttitt SUBROUTINE SINTEtX,M,XJ,P,Q,STT,NT8,NS) REAL X(3),Xl(3),XJ(4,3,6) COMPLEX GP,ST,STT INTEGER P,Q,NS(NT8,6) COMMON /F/DL,HDL,QDL,DS,DV STT-(0.,0.) IF (P.EQ.Q) GO TO 39 DO 31 K-P,P+3,3 IF (NS(M,K).EQ.0) GO TO 31 ST-(0.,0.) DO 30 J-1,4 46 X1(L)-XJ(J,L,K) 30 ST-ST+GP(Q,X,X1) STT--ST*NS(M,K)+STT 31 CONTINUE GO TO 99 39 IF (P.EQ.1) THEN IP1-2 IP2-3 ELSEIF (P.EQ.2) THEN IPl-l IP2-3 ELSEIF (P.EQ.3) THEN IPl-l IP2-2 ENDIF DO 41 K-IP1,IP1+3,3 IF (NS(M,K).EQ.0) GO TO 41 ST-(O.,0.) DO 40 J-1,4 DO 56 L-l,3 56 X1(L)-XJ(J,L,K) 4O ST-ST+GP(IP1,X,X1) STT-ST'NS(M,K)+STT 41 CONTINUE DO 51 K-IP2,IP2+3,3 IF (NS(M,K).EQ.0) GO TO 51 ST-(0.,O.) DO 50 J-1,4 DO 66 L-l,3 50 ST-ST+GP(IP2,X,X1) STT-ST*NS(M,K)+STT 203 51 CONTINUE 99 STT-STT*DS/4. RETURN END ct*****§***i***fi**********fi**i*****i*******fi*****iifltt****i****itfit.*tiiitiittt c This subroutine is used to calculate the volume integrations involved c with the singularity problem cfi********titiiittititiittttt*******t*ittiitiittitflt****ttii*ti**ii*tti**ittttit SUBROUTINE SELF(GD) COMPLEX DT,GNN,GD(3,3) INTEGER P,Q COMMON /F/DL,HDL,QDL,DS,DV DO 81 9-113 DO 81 Q-l,3 IF (P .EQ . Q) THEN DT-GNNtDL) ELSEIF (P.NE.Q) THEN DT-(O.,0.) ENDIF GD(P,Q)-DT 81 CONTINUE RETURN END ctti***ifl***fi*fi***ifli*i**fltfi***fi******i********fiftii**fi***t****flit*fi*****'****ii c This function subroutine is used to evaluate the 2nd derivative of the c free space Green's function at center of each cell cittttitttitiiittititittitittttttttfl*****ttt**tttiittttttttifiittittititttrtttitt FUNCTION GPQ(P,pr,Xl) REAL KO,MUO,X(3),X1(3),XD(3) COMPLEX JC,EPS,MU,CON1,CON2,GPQ,B,C,D INTEGER P,O COMMON /A/JC,K0,PI,EPO,EP,EPS,MUO,MU,CON1,CON2 DO 75 L-1,3 XDtL)-X(L)-X1(L) IF (ABS(XD(L)).LT.1.E-8) XD(L)-0. 75 CONTINUE R-SQRT(XD(1)*XD(1)+XD(2)*XD(2)+XD(3)*XD(3)) A-K0*R D-CEXPtJC*A) c-(3.-A*A-3.*JC*A)*XDtP)*XD(Q)/R/R IF (P.NE.Q) THEN GPQ-C*D/A/A/A ELSEIF (P.EQ.Q) THEN B-A*A-l.+JC*A GPQ-(B+C)*D/A/A/A ENDIF RETURN END c***fl****i****fi**fi*flflflt*f**#***************tflflifiiftttfl*i**§*******i*ttfiffititttt c This function subroutine is used to evaluate the lst derivative of the c free space Green’s function at center of each cell ctittfiiititiflii*ttttfliitittitttittit****titi*t*ttititititttt**tt**t*titittiittwt FUNCTION GP(P,X,X1) REAL KO,MUO,X(3),X1(3),XD(3) . COMPLEX JC.EPS,MU,CON1,CON2,GP,C,D INTEGER P COMMON /A/JC,K0,PI,EPO,EP,EPS,MUO,MU,CON1,CON2 DO 75 L-1,3 XDtL)-X(L)-X1(L) IF (ABS(XD(L)).LT.1.E-8) XD(L)-0. 204 75 CONTINUE R-SQRT(XD(1)*XD(1)+XD(2)*XD(2)+XD(3)*XD(3)) A-K0*R D-CEXP(JC*A) C-(JC'A-l.)*XD(P)/R GP-C*D/A/A RETURN END cit*fiii'iiiiiii*fitiiiii**fli*fl**iiti*tittfl*fit**fit****fitt*fii*t*itfltflfiittittttt1*? c This function subroutine is used to calculate the value of the diagonal c element of the matrix, this value includes a pricinpal integration plus c a correction term ctifit!tittttiflfliiiittflitiflit*tfli*ttt*tfltiittitttitititttttt*tttfiitiitiititxiiit FUNCTION GNN(DL) REAL KO,MUO COMPLEX JC,EPS,MU,CON1,CON2,GNN,GC,GB COMMON /A/JC,K0,PI,EPO,EP,EPS,MUO,MU,CON1,CON2 Dv-DL**3 AN-(3.*DV/4./PI)**(1./3.) GC-2.*MUO/3./MU+EPS/3./EPO GB-CON1*4.*PI*2.*(CEXP(JC*KO*AN)*(1.-JC*K0*AN)-1.)/(3.*K0**3) GNN-GC+GB RETURN END ctififl*iiitit.flitt******fi*****§tifitifft***fi****i..i***fi**i****fl*i*fii*iiiitititiit c This subroutine is used to calculate the coordinates of 8 subcells c of each cell in order to perform the numerical integration Ctitttit,tittiiiiititttttii*itttittttittifftit.*ttttfiitttt'ittiittttt*ittt*t*t'* ' SUBROUTINE VPT(B,BN,QRDL) REAL QRDL,B(3),BN(8,3) BN(1,1)-B(1)+QRDL BN(1,2)-B(2)+QRDL BN(1,3)-B(3)+ORDL BN(2,l)-B(1)+QRDL BN(2,2)-B(2)+QRDL BN(2,3)-8(3)-QRDL BN(3,1)-B(l)+QRDL BN(3,2)-B(2)-QRDL BN(3,3)-B(3)+QRDL BN(4,l)-B(1)+QRDL BN(4,2)-B(2)-QRDL BN(4,3)-B(3)-QRDL BN(5,l)-B(1)-QRDL BN(S,2)-8(2)+QRDL BN(5,3)-8(3)+QRDL BN(6,1)-8(1)-QRDL BN(6,2)-B(2)+QRDL BN(6,3)-B(3)-ORDL BN(7,1)-B(l)-QRDL BN(7,2)-B(2)-QRDL BN(7,3)-B(3)+QRDL BN(8,1)-8(l)-QRDL BN(8,2)-B(2)-QRDL BN(8,3)-B(3)-QRDL RETURN END Cit*iif****t*ifi***fi********iiiti***fi**ttitiflt****fliifiii*i*#***flt****t***i***fitti c This subroutine is used to calculate the coordinates of 4 pieces of c 6 surfaces on a cell in order to perform the numerical integration ctt*itiit*tiitt*itttifititittfitflitfittitttttttttfitiittititiflttitttttttttitttttitit 205 SUBROUTINE SPT(B,BS,HFDL,QRDL) REAL QRDL,HFDL,B(3),BS(4, 3, 6),X(3,5) DO 20 L-l,3 X(L,l)-B(L)-HFDL X1Lyz)‘B(L)‘QRDL X(Ly3)‘B(L) X(L,4)'B(L)+QRDL X(L,5)'B(L)+HFDL BS(1,1,1)'X(1,5) BS(2,1,l)-X(1,5) 88(39 1' 1)-X(1'S) BS(4,1,1)‘X(1,5) BS(l,2'1)-X(2'2) BS(2,2,1)'X(2,2) BS(3r211)-X(204) 35(4r211)-X(214) BS(1,3,1)-X(3,2) BS(2,3,1)-X(3,4) BS(3,3,1)-X(3,4) 85(4r3tl)-X(3o2) 35(1, 1,4)'X(l,l) 35(2, 1,4)'X(1,1) BS(4,1,4)-X(1,1) BS(1,2,4)‘X(2'2) BS(2,2,4)-X(2,2) 85(3, 2, 4)-X(2, 4) BS(4,2,4)-X(2,4) 85(1, 3, 4)-X(3, 2) BS(2,3,4)'X(3,4) 35(3, 3, 4)-X(3, 4) 85(41314)-X(312) 35(lr 1'2)-X(1I4) BS(2,1,2)'X(1,4) BS(3,1,2)‘X(1,2) 85(4, 1, 2)-X(1, 2) BS(1,2,2)'X(2,5) BS(2,2,2)'X(2,5) 33(312r2)-X(215) 83(41 2! 2)-X(2I 5) BS(1I3'2)-X(3I2) 85(21 3' 2)-X(3I 4) 35(3, 3, 2)-X(3, 4) 85(4, 3, 2)-X(3,2) 85(1, 1' S)-X(1I4) 35(2, 1, 5)'X(l,4) 35(3111 S)-X(112) 35(4, 1,5)'X(1,2) 38(112r 5)-X(2rl) BS(2,2, 5)-X(2,l) 85(312r S)-X(211) BS(4,2, 5)-X(2,l) 85(113, 5)-X(312) BS(2,3, 5)‘X(3,4) BS(3,3,5)-X(3,4) 35(4, 3, 5)'X(3,2) 85(10113)-X(114) BS(2,1,3)-X(1,4) BS(3( 1,3)-X(1,2) 35(4, 1, 3)-X(1, 2) BS(1p2,3)-X(2,4) BS(2,2,3)-X(2,2) 206 BS(3.2,3)-X(2,2) 85(4,2,3)-X(2,4) 85(1’3I3)-X(3l5) BS(2.3,3)-X(3,5) BSt3,3,3)-X(3,S) BS(4,3,3)'X(3,S) 88(1’1'6)-X(1l4) BS(2,1,6)'X(1,4) BS(3,1,6)-X(l,2) BS(4,1,6)-X(1,2) 85(112I6)-X(204) BS(2,2, 6)'X(2.2) BS(3,2,6)'X(2,2) BS(4,2,6)'X(2,4) 85(1, 3, 6)-X(3,l) 88(293r6)-X(3r1) BS(3I3'6)-X(3I1) 85(4, 3, 6)-X(3, l) RETURN END cittifi***fi*fli**fiifiitiifiifflfiifiiflfifitii*ifflfii*flifl*fififiiiflfl'iit**i*tifl*fiiii*iiiittit c This subroutine is used to solve the matrix equation via Gaussion c ellimination Cit***************i********itit*************fl***fi********ifi********************* SUBROUTINE CMATPA(IJOB,A,N,M,DET,EP,N031) COMPLEX A,B,DET,CONST,S DIMENSION A(N,N031) 3O FORMATth,42HTHE DETERMINANT OF THE SYSTEM EQUALS ZERO./ 11X,36HTHE PROGRAM CANNOT HANDLE THIS CASE.//) DET-l. NPl-N+l NPM-N+M NMl-N-l IF(IJOB) 2,1,2 1 DO 3 I-1,N NPI-N+I A(I,NPI)-1. IP1-I+1 DO 3 J-IP1,N NPJ-N+J A(I,NPJ)-O. A(J,NPI)-0. DO 4 J-1,NM1 C-CABS(A(J,J)) JPl-J+1 DO 5 I-JP1,N D-CABS(A(1,J)) IF(C‘D) 61515 6 DET--DET DO 7 K-J,NPM B-A(I,K) A(I,K)-A(J,K) 7 A(JtK)-B C-D 5 CONTINUE IF(CABS(A(J,J))-EP) 14,15,15 15 DO 4 I-JP1,N CONST‘A‘IIJ)/A(JIJ) DO 4 K-JP1,NPM 4 A(I,K)-A(I,K)-CONST*A(J,K) IF(CAES(A(N,N))-EP) 14,18,18 14 DET-O. IF(IJOB) 16,16,17 h)U H (‘) (‘1 207 16 PRINT 30 17 RETURN 18 DO 11 I-1,N 11 DET=DET*A(I,I) IF(IJOB) 10,10,17 10 DO 12 I-1,N K=N-I+1 KPl-K+l DO 12 L-NP1,NPM S-O. IF(N-KP1) 12,19,19 19 DO 13 J-KP1,N 13 $-S+A(K,J)*A(J,L) 12 A(K,L)-(A(K,L)-S)/A(K,K) RETURN END BIBLIOGRAPHY 10. BIBLIOGRAPHY D. E. Livesay and K. M. Chen, "Electromagnetic fields induced inside arbitrarily shaped biological bodies," IEEE Trans. on Microwave Theory and Techiques, vol. MTT-22, pp. 1273-1280, December 1974. J. 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Livesay, "Electromagnetic Fields Induced in and Scattered by Biological Systems Exposed to Nonionizing Electromagnetic Radiation," Ph. D. Dissertation, Michigan State University, East Lansing, MI 1975. C. T. Tai, "A note on the integral equations for the scattering of a plane wave by an electromagnetically permeable body," Electromagnetics 5:79-88, 1985. R. F. Harrington, Field Computation by Moment Method, McMillan, NY 1968. J. Van Bladel, Dyadic analysis, Appendix 3, Electromagnetic Fields, McGraw- Hill, NY 1964. H. Wang, Y.Q. Liang and KM. Chen, "Quantification of induced EM fields in finite heterogeneous bodies," 1985 North American Radio Science Meeting and International IEEE/APS Symposium, Univ. of British Columbia, Vancouver, Canada, June 1985. D. T. Paris, W. M. Leach, Jr. and E. B. Joy, "Basic theory of probe-compensated near-field measurements," IEEE Trans. on Antennas and Propagation, vol. AP-26, pp. 373-379, May 1978. I. Chatterjee, O. P. Gandhi, M. J. Hagmann and A. 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Misra, "Microwave Life-Detection Systems," Finial Report to Naval Medical Research and Development Command, Contract N00014-82-C- 0355, Division of Engineering Research, Michigan State University, East Lansing, MI, November 1986. D. H. Schaubert, D. R. Wilton, and A. W. Glisson, "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric boies," IEEE Trans. on Antennas and Propagation, vol. AP-32, pp. 77- 85, January 1984. C. T. Tsai, H. Massoudi, C. H. Dumey, and M. F. Iskander, "A procedure for cal- culation fields inside arbitrarily shaped, inhomogeneous dielectric bodies using linear basis functions with the moment method", IEEE Trans. on Microwave Theory and Techniques, vol. M'IT-34, November 1986.