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'.‘.,' .Wg} '- " £4. -'1-' .v r 'ny I'Sq‘m'l .vv, II”! .fg’éizflf '5' I i’t'J‘: 91-7": ’ I afisssfifi If}? 101w," ~51?” , M3,: 7}; - jgfh galf 16:} NW “'ch . :31.) ‘* * ’1 ' 1111‘ " "I" ”Eat“. é-\". 4,0,6“ 10765707 mm 31' M WWI!!! lflllllll lull Ill!!! llllllllllflljll 1293 mooosa 80 LIBRARY Michigan State University This is to certify that the dissertation entitled INTERACTION OF ELF-LF ELECTROMAGNETIC FIELDS WITH BIOLOGICAL SYSTEMS AND CONDUCTING OBJECTS presented by Huey-Ru Chuang has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engr. I/Zfléék Major professor Date I O” 4- / af/L "(Iliumur 1' - 1'1 In '1' - . 0-1 1 . MSU RETURNING MATERIALS: Place in book drop to Die/572‘? Z 153 LJBRARJES remove this checkout from .—:—. your record. FINES will be charged if book is returned after the date stamped below. ‘ 12 r ‘. INT! s) INTERACTION OF ELF-LF ELECTROMAGNETIC FIELDS WITH ? BIOLOGICAL SYSTEMS AND CONDUCTING OBJECTS By Huey-Ru Chuang 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 Copyright° by Huey-Ru Chung 1987 To my beloved parents, brothers, and my future wife iii En wil hcl thci pani Phi ACKNOWLEDGEMENTS The longer in the journey of a man’s life, the more he realizes how much of his achievements have come from the help of his family, friends, and fellow travelers. I owe a great intellectual debt to my academic advisor, Dr. Kun-Mu Chen, for his enthusiastic guidance and advice, without which this research could not have been accom- plished. I am deeply grateful to Dr. Dennis P. Nyquist for the valuable assistance he gave me in the course of this research. Grateful thanks extend to committee members, Dr. Byron C. Drachman, Professor of the Department of Mathematics, and Dr. Edward Rothwell for many helpful comments and criticisms to this dissertation. A special thank is due to Dr. Chun-Ju Lin, Professor of the Department of Electrical Engineering at the University of Detroit. Most part of this research was in joint effort with him. This dissertation was word-processed by PC-WRITE. Mr. Steven S. Leung’s help in generating mathematical formulas and Greek symbols is highly appreciated. Finally, I wish to express the most sincere gratitude to my parents and brothers for their love and encouragement. Their spoken and unspoken words of caring have accom- panied me from the very beginning of my education to the final accomplishment of this PhD. degree. iv IN bod clcc ant: indr and Situ: ABSTRACT INTERACTION OF ELF-LF ELECTROMAGNETIC FIELDS WITH BIOLOGICAL SYSTEMS AND CONDUCTING OBJECTS By Huey—Ru Chuang The knowledge of the induced electric field and current in a biological body is very important for investigating the potential health hazard due to the electromagnetic field of an extremely high voltage (El-IV) power line and ELF antenna systems. This thesis presents a new numerical technique for quantifying the induced electric field at the body surface and inside the body as well as the steady and the transient current induced in a biological body that has a realistic shape and situated in a realistic environment, when exposed to ELF-LF electromagnetic fields. As an introduction, the interaction of an ELF-LP electric field with a biological system is considered. An integral equation for the induced surface charge density (surface charge integral equation) is developed. By using the Moment Method, the body surface is partitioned into a number of patches for numerical calculation. After the induced surface charge density is determined, the electric field and the current density induced inside the body can be computed on the basis of the conservation of electric charge, Ohm’s law, and Maxwell’s equations. lVarious numerical results have been generated to compare with closed form solutions and existing experimental results for establishing the accuracy of the present method. The agreement is excellent. The surface charge integral equation (SCIE) method has also been applied to realistic models of the human body standing on the ground with various grounding impedances at various frequencies of the incident electric as 3] mm Huey-Ru Chuang field. Various sizes and postures of the human body are considered. The induced surface charge, the short-circuit current, and the induced electric field as well as the induced current density inside the body are quantified. The frequency range limitation of the SCIE method is also investigated and is found to be valid up to the HF range. In order to compare the surface charge integral equation method with other theoretical methods, a tensor electric field integral equation (EFIE) has been applied to calculate the induced electric field inside the biological body. By subdividing the body into many small cubical cells, the method of moments is used to transform the integral equation to a large matrix equation. The Conjugate Gradient Method (CGM) and the Guass-Seidel Method (GSM) are employed to solve the matrix equation iteratively. Numerical results and convergence rates of both methods are compared and discussed. The coupled surface charge integral equations have also been applied to determine floating potentials and short-circuit currents of human body and metallic object at proximity immersed in a 60-Hz electric field. Equivalent circuits for computing induced currents of a human body and a vehicle are constructed. The shock current created between the vehicle and the nearby human body under the exposure of a high voltage 60-Hz electric field is analyzed by utilizing the equivalent circuit model. To treat heterogeneous-body problem, the surface charge integral equation (SCIE) method is combined with an impedance network method by which the body is modeled as an equivalent impedance network and the induced surface charge is viewed as an equivalent current source. The current flowing at any impedance can be determined on the basis of the Kirchhoff’s current law and from which the induced current density and electric field inside the body can be mapped. A conducting concentric- sphere is chosen to test the validity and accuracy of the present method. The method can be applied to any heterogeneous biological body when exposed to ELF-LP electric field. CH2U?EER.J. CHEAPTTE! 2 CTUKPTTH! 3 CEBU?IEFII4 TABLE OF CONTENTS INTRODUCTION ......................................... INTERACTION OF ELF ELECTRIC FIELDS WITH BIOLOGICAL BODIES ................................... 2.1 Introduction ....................................... 2.2 Theory ............................................. 2.2.1 Surface Charge Integral Equation ............ 2.2.2 Transformation of Integral Equation - Moment Method ............................. 2.2.3 Induced Current Inside the Body ............. APPLICATIONS OF SURFACE CHARGE INTEGRAL EQUATION METHOD TO SPHERICAL AND SPHEROIDAL MODELS OF BIOLOGICAL BODIES ................................... Introduction ....................................... Closed Form Solution for a Sphere and Comparison with Present Numerical Results .......... 3.3 Sphere above the Ground Plane ...................... 3.4 Applications to Spheroidal Models of Biological Bodies .................................. 3.5 Spheroid above the Ground Plane .................... NH 3. 3. APPLICATIONS OF SURFACE CHARGE INTEGRAL EQUATION METHOD TO ANIMAL AND HUMAN BODIES ................ 1 Introduction ....................................... 2 Numerical Results for Animal Bodies and Comparison with Existing Experimental Results ............................... 4.3 Numerical Results for Human Model and Comparison with Existing Experimental and Theoretical Results ............................ 4.4 Numerical Results for the Model of Adult Man with Arbitrary Posture ......................... 4.5 Frequency Range Limitation of Surface Charge Integral Equation Method ............................................. 4. 4. \Im 15 21 29 29 29 36 53 58 69 69 69 75 82 CRUKPTEEI 5 CHAJUHNR 6 CHEAPTTHI 7 CHAIUHHR 8 TENSOR ELECTRIC FIELD INTEGRAL EQUATION METHOD FOR INDUCED ELECTRIC FIELDS INSIDE BIOLOGICAL BODIES ................................... 5.1 Introduction ....................................... 5.2 Theoretical Development ............................ 5.3 Moment Method Solution of Tensor ' Integral Equation .................................. 5.4 Iterative Methods for Solving the Matrix Equation .................................... 5.4.1 Conjugate Gradient Method (CGM) ............. 5.4.2 Guass-Seidel Method (GSM) ................... 5.4.3 Initial Guess for the Iterative Method ...... 5.4.4 Numerical Example ........................... 5.5 Numerical Results for the Human Model and Comparison with the SCIE Method ................ ANALYSIS OF SHOCK CURRENT BETWEEN HUMAN BODY AND METALLIC OBJECTS UNDER THE EXPOSURE OF ELF ELECTRIC FIELDS ............................. 6.1 Introduction ....................................... 6.2 Theoretical Development of the Shock Current Analysis ............................. 6.2.1 Equivalent-circuit Models ................... 6.2.2 Numerical Examples .......................... 6.2.3 Equivalent Circuits for a Man and a Vehicle at Proximity .................. 6.3 Transient Analysis of the Shock Current ............ CALCULATION OF ELECTRIC CURRENTS INSIDE HETEROGENEOUS BODIES INDUCED BY ELF-LE ELECTRIC FIELDS ............................. 7.1 Introduction ....................................... 7.2 Theoretical Development of the Impedance Network Method ........................... 7.3 Numerical Examples and Comparison with Analytic Solutions ................................. CONCLUSIONS ........................................... vi APPL'DIX l ‘EPEIIDIX 2 APP;sz 3 APPEDIX 4 39:? Var"! \ He's! go“ DU. U 1:3 APPENDIX 1 . . . ............................. 203 APPENDIX 2 . . . ...................................................... 205 APPENDIX 3 . . ....................................................... 209 APPENDIX 4 . . ....................................................... 217 BIBLIOGRAPHY ....................................................... 228 vii fig. 2.1. Fig. 2.2. Fig. 2.3. $512. Fig. 2.1. Fig. 2.2. Fig. 2.3. Fig. 2.4. Fig. 2.5. Fig. 12 6 Fig. 2.7. Fig. 3.1. Fig. 3.2. Fig. 3.3. Fig. 3-4- Fig. 3.5. Fig. 3.6- LIST OF FIGURES A human standing on the ground is exposed to an ELF electric field E exp(jwt) maintained by an BIN power line. Induced current inside the human body is J 1exp(j¢nt) . A man standing on the ground is exposed to an electric field of ELF range. The contacts between the feet as well as the left hand and the ground are represented by zLi (i - 1 ~ 3). iInduced surface charges at the interface of different media . Conductivity and dielectric constant of various tissues in the ELF range. Geometries for integration of Eq. (2.8) when m - n. ..... 17 Geometry for calculating the induced current in the body. Cylindrical geometry for calculating the induced current density inside the body. The induced surface electric field and the induced surface charge density on a perfectly conducting sphere immersed in an uniform electric field. The surface of a perfectly conducting sphere is divided into 2N rings and each ring occupies a radian angle of A0 (- n/ZN). Geometries of two sample rings on the spherical surface. Comparison of numerical results by the present mrthod with the exact solution on the surface charge density of a perfectly conducting sphere induced by an uniform electric field. A perfectly conducting sphere located above the ground with a grounding impedance ZL is illuminated by an uniform electric field. Geometries of three sample rings on the surfaces of a sphere and its image sphere. viii c.' n. 0"- (F) >.' 4- w 0-, an LA, LAJ LL, on] (1"! \‘ LAD “\ ..b. .31 0‘0“ r’l ' (W '1 U ’ ‘I ‘.Q- 7.1 ', 0‘ "9'; ("p 0' () ,1 '4 k.) :2. ‘- t) GE 5:? Q .17 A ‘ 6,; 0?. liig. I?ig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 3. 3. 3. 7. 8. 9. .10. .11. .12. .13. .14. .15. .16. .17. The distribution of surface charge density on a floating perfectly conducting sphere located at an infinite distance from the ground induced by an uniform electric field. ................................ 44 Surface charge distributions on perfectly conducting spheres located at various distances from the ground and exposed to an uniform electric field. .............. 45 Comparison of theoretical results by the present method with empirical results on the short-circuit current for a biological sphere as a function of the radius when exposed to a l kV/m, 60-Hz electric field. ........................................ 47 Theoretical results on the short-circuit current for a biological sphere as a function of the grounding resistance when exposed to a l kV/m, 60-Hz electric field. .................................. 48 Theoretical results on the short-circuit current for a biological sphere as a function of the grounding capacitance when exposed to a l kV/m, 60-Hz electric field. .................................. 49 Theoretical results on the short-circuit current for a biological sphere as a function of the grounding inductance when exposed to a l kV/m, 60-Hz electric field. .................................. 50 A displacement current, I, is coupled from the overhead electrode through a capacitance C to a surface area element A of the subject body. ............ 52 A perfectly conducting prolate spheroid with major axis a and minor axis b immersed in an uniform electric field. The surface of the spheroid is divided into N rings for numerical calculation. ........ 54 Geometries of two sample rings on the surface of a spheroid. .............................................. 55 Distributions of surface charge density on spheroids with two different axis-ratios (a/b - l & a/b - 2) induced by an uniform electric field. ....... 59 A spheroid located above the ground with a grounding impedance Z exposed to an uniform electric field. Geometries of three sample rings on the surfaces of the spheroid and the image spheroid are also shown in the figure. ................. 60 ix Fig. p D an : 3.4;4. r841. :51 ('1 n O O In .0. r1 '2 0'3 Fig. Fig. Fig. Fig. Fig. . 3.18. . 3.19. 3.20. The distribution of surface charge density on a floating perfectly conducting spheroid located at an infinite distance from the ground induced by an uniform electric field. ................................ 65 Comparison of theoretical results by the present method with empirical results on the short-circuit current for spheroidal models of man with various axis ratios when exposed to a l kV/m, 60-Hz electric field. ........................................ 67 Floating potentials of spheroids with the same axis-ratio (a/b - 2.2) and various sizes when exposed to a l kV/m, 60-Hz electric field. ............. 68 Experimental guinea pig and the theoretical model for the SCIE numerical computation. .................... 71 Experimental results of Kaune and Miller on the electric field enhancement factor, the sectional current and the short-circuit current for a grounded guinea pig exposed to a 60-Hz, 10 kV/m electric field. ........................................ 72 Theoretical results by the present method on the electric field enhancemect factor, the sectional current and the short-circuit current for a grounded guinea pig exposed to a 60-Hz, 10 kV/m electric field. ........................................ 73 Experimental model of man (height - 45 cm) and its theoretical model for the SCIE numerical computation. ........................................... 76 Theoretical results on the electric field enhancement factor and the short-circuit current for the experimental model of man (used by Kaune & Forsythe) standing on the ground and exposed to a 60-Hz, 1kV/m electric field. ........................... 79 Theoretical results on the electric field enhancement factor and the short-circuit current for the experimental model of man (used by Kaune & Forsythe) standing on the ground and exposed to a 60-Hz, 1kV/m electric field. ........................... 80 Comparison of theoretical results by the present method with experimental results of Kaune and forsythe on vertical and horizontal current densities for a grounded human model exposed to a 60-Hz, lO kV/m electric field. Inducgd current densities are given in units of nA/cm . ................ 81 0" A. "l o - (F0 0 (Fl ‘5- 4.6. ‘C p _4 '0 -Ct Hmtn H10 H nzrrhovi U) Eé SE ms 'fi Fig. Figp Fig” Fig. Fig. Fig. Fig. Fig. Fig. Fig. .10. .11. .12. .13. .14. .15. .16. Theoretical results on the electric field enhancement factor and the short-circuit current for a realistic model of man standing on the ground and exposed to a 60-Hz, 1 kV/m electric field. ................................................. 83 Theoretical results on the electric field enhancement factor and the short-circuit current for a realistic model of man with hands stretching horizontally standing on the ground being exposed to a 60-Hz, 1 kV/m electric field. ..................... 84 Short-circuit current as a function of grounding resistance. ............................................ 86 Short-circuit current as a function of grounding capacitance ............................................. 87 Short-circuit current as a function of grounding inductance. ............................................ 88 Human model and experimental set-up by Chiba et a1 [15]. .................................................. 90 Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional currents for a grounded human model exposed to a 60-Hz, 1 kV/m electric field. ............. 91 Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional current densities for a grounded human model exposed to a 60-Hz, 1 kV/m electric field. Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional currents for a grounded human model with hands stretching horizontally exposed to a 60-Hz, 1 kV/m electric field. ................................. 93 Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional currents densities for a grounded human model with hands stretching horizontally exposed to a 60-Hz, 1 kV/m electric field. ..................... 94 Comparison of theoretical results by the SCIE method with experimental results of Candi et a1 [23] on the short-circuit current for a grounded human model (~1.75 m in height) at frequencies from 20 MHz to 50 MHz. ................................. 96 xi 0 (Fl 0 I”. 1.5.5. to! CD Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. .10. .11. .12. An arbitrarily-shaped biological body in free space, illuminated by an EM plane wave. ................ 100 The biological body is partitioned into many small cubic cells for numerical calculation of the induced electric field. ........................................ 105 A biological tissue-layer exposed to a 60on EM plane wave. ............................................ 117 Convergences of the CGM and the GSM when solving the induced current in the tissue-layer of Fig. 5.3 by the EFIE method. .................................... 118 Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3, when illuminated by a 60-Hz, 1 kV/m EM plane wave. .......... 119 Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3, when illuminated by a 60-Hz, l kV/m EM plane wave. .......... 120 Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3, when illuminated by a 60-Hz, l kV/m EM plane wave. ............................................ 121 Comparison of the CGM and the GSM with the GEM on induced current densities for the tissue-layer of Fig. 5.3, illuminated by a 60-Hz, 1 kV/m EM plane wave. Induced current densities are given in units of A/cm . ........................................ 122 Geometries of the theoretical human model for the numerical calculation of the EFIE method. ........................................... 123 Comparison of experimental human model with theoretical models for the SCIE method and The EFIE method which approximate the experimental model. ................................................. 124 The theoretical human model of Fig. 5.9 standing on the ground is exposed to a 60-Hz EM plane wave. ............................................ 125 Comparison of theoretical results by the EFIE method with that by the SCIE method and experimental results of Kaune and Forsythe on vertical current densities for a grounded human model exposed to a 60-Hz, 10 kV/m EM field. Inducgd current densities are given in units of nA/cm . ................................................ 126 xii a.. e- v 17" .-. a. r (7" I (n: , .. (I‘D J I," ‘_. ~ (.7‘ 0‘ 813.Convefi for th 5.11 be Large : nearby discha: elect: A bioL ground elect: are Th this 5' A biol standi Z.‘ ant E2? e1 figure whole COZpar flethod on the curzen the co: ICs: Vi 0n the A nan I the grc IFig. 5.13. Convergences of the CGM and the GSM when solving for the induced current in the body model of Fig. 5.11 based on the EFIE method. ......................... 130 IFig. 6.1. Large shock currents flow between a vehicle and a nearby human body through direct contact or spark discharge, when both are exposed to the ELF electric field of an EHV power line. ................... 132 Fig. 6.2. A biological body standing on the ground with a grounding impedance Z is exposed to an ELF electric field. The circuits shown in the figure are Thévenin and Norton equivalent circuit for this system. ........................................... 134 Fig. 6.3. A biological body b and a nearby metallic object c standing on the ground with grounding impedances ZEE'and Z , respectively, are both exposed to an E electr c field. The circuit shown in the figure is the Thévenin equivalent circuit for the whole system. .......................................... 136 Fig. 6.4. Comparison of numerical results by the present method with experimental measurements by Reilly on the floating potential and the short-circuit current of a Plymouth automobile. Also shown are the convergences of numerical values for d and 1c": with respect to the partitioned patch ciiirmber on the vehicle's metallic surface. ..................... 141 Fig. 6.5. A man and a vehicle (Plymouth) standing on the ground are both exposed to an ELF electric field. ........................................ 142 Fig. 6.6. The floating potential of a man and the vehicle (a) (Plymouth) potential (with tires) as functions of their separation distance. .......................... 144 Fig. 6.6. The short-circuit current of a man and the current (b) flowing between a vehicle (Plymouth) and the ground as functions of their separation distance. ............. 145 Fig. 6.7. Scattered electric fields maintained by induced surface charges on the surface of a man and a vehicle. ............................................... 146 Fig. 6.8 The floating potential of a vehicle (Plymouth) (a) as functions of the separation distance between the vehicle and a grounded man. ........................ 147 Fig. 6.8 The short-circuit currents of a vehicle (Plymouth) (b) as functions of the separation distance between the vehicle and a grounded man. ........................ 148 xiii 9.. ... .., .., ,.. O ‘1‘ a a‘ r. .—-O t.6.13 .5 fl; 0 III. 0‘ ; .4 L." 1"! m ;. 0‘ Ecuivi height under fi 1d. . Relatf . Inter: . Reill} transi which elect: . An eq: grount Of a l . Cozpa: lethoc 0f Rei betvee $6515: elect: . I ' 3“Liner: the EC ShOck and a eXPOSL The cc ' Naefi 5h0ck Fig. Fig. Fig” Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 6. 9. .10. .11. .12. .13. .14. .15. .16. Equivalent circuits of a grounded man (180 cm in height) and a vehicle (Plymouth) at proximity, under the exposure of a lkV/m, 60-Hz electric field. ................................................. 150 Relation between body impedance and voltage [33]. ....... 151 Internal resistance of the human body [33]. ............ 152 Reilly's [32] experimental set-up and measured transient current for 8 Plymouth automobile which was exposed to a 4.5 kV/m, 60-Hz electric field. ........................................ 154 An equivalent circuit for a vehicle and a grounding resistance R under the exposure of a 4.5 kV/m, 60-Hz eIectric field. ................... 155 Comparison of numerical results by the present method (dash lines) with experimental results of Reilly [32] on the transient shock current between a vehicle (Plymouth) and a grounding resistance, both exposed to a 4.5 kV/m, 60-Hz electric field. ........................................ 157 Numerical results by the present method based on the equivalent circuit of Fig. 6.9 on the transient shock current between a man (180 cm in height) and a vehicle (Plymouth) at proximity, under the exposure of a 4.5 kV/m, 60-Hz electric field. The contact resistance is assumed to be 1.5 km. ........ 163 Numerical results by the present method based on the equivalent circuit of Fig. 6.9 on the transient shock current between a man (180 cm in height) and a vehicle (Plymouth) at proximity, under the exposure of a 4.5 kV/m, 60-Hz electric field. The contact resistance is assumed to be 1.0 k0. ........ 164 A heterogeneous biological body standing on the ground is exposed to an ELF-LF electric field. ......... 166 A volume cell is represented by 3 orthogonal impedances (zcu’ Zc , law) for the current to flow along (u, v, wY directions. ....................... 168 Impedances connected to the nth node of a inner volume cell and the KCIaof the current flow. ........... 170 Impedances connected to the nth node of an outer cell. There is a net current terminating at the outer cell surface. ................................ 173 xiv ... I I!" '~ n.‘. 7" .g‘. A cor hoaog a sit The t densi insic expos The d (pna "'C COECE 1 in": A con V It"; needs due : An ei secti . A sec r°1 On .311 when 1 Fig” 7.5. A.conducting concentric-sphere located in a homogeneous conducting medium is immersed in a slowly time-varying electric field. .................. 176 Fig. 7.6. The uniform dist ibutions of the induced current density J (nA/cm ) and electric field E (10 kV/m) inside a homogeneous fat and a muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. ............. 180 Fig. 7.7. The di tributions of the induced current density J (nA/cm ) and electric field E (10 kV/m) inside a concentric fat-muscle sphere when exposed to a 60-Hz, l kV/m electric field. ................................. 181 Fig. 7.8. A concentric sphere is immersed in a slowly time- varying electric field. Only an eighth-sphere needs to be considered for numerical computation due to the symmetry. ................................... 183 Fig. 7.9. An eighth-sphere is cut into M sections and each section is divided into (N x N) sector-cells. .......... 184 Fig. 7.10. A sector-cell is modeled as 3 impedances (2:, 2:, Zw) for the current flowing along (r, 0, o) directions. w1 ~ W6 are listed in Fig. 7.9. ........................ 185 Fig. 7.11. An equivalent impedance network for an eighth-sphere when exposed to an ELF-LF electric field. .............. 187 Fig. 7.12. Comparison of the analytical and numerical solutions on the induced current density J inside a fat sphere when exposed to a 60-Hz, 1 kV/mzelectric field. J is given in the unit of nA/cm . ...................... 191 Fig. 7.13. Comparison of the analytical and numerical solutions on the induced current density J inside a muscle sphere when exposed to a 60-Hz,21 kV/m electric field. J is given in the unit of nA/cm . ...................... 192 Fig. 7.14. Comparison of the analytical and numerical solutions on the induced current density J inside a concentric fat—muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. J is given in the unit of nA/cmz. ..... 193 Fig. 7.15. Comparison of the analytical and numerical solutions on the induced current density J inside a concentric fat-muscle sphere when exposed to a 60-Hz, l kV/m electric field. J is given in the unit of nA/cmz. ..... 194 Fig. 7.16. Comparison of the analytical and numerical solutions on the induced current density J inside a concentric fat-muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. J is given in the unit of nA/cmz. ..... 195 'l ,g 717 ,E 1.18 .i 119 Lg. All is. m. (a) ~irill. (5) i 54.1. s. m 0.. Conpar on the muscle elect: . Corpar on the muscle electr . Compar on the muscle ele t: . Equiva respor Equiva COECaC Equiw'va respor (the s A bio: t ime . V'I - A con homege A Slow Fig. Fig. Fig. Fig. Fig. Fig. Fig. . 7.17 7.18. 7.19. A2.1. A3.1. (a) A3.1. (b) A4.1. A4.2. Comparison of the analytical and numerical solutions on the induced current J density inside a concentric muscle-fat sphere when exposed to a 60-Hz, 1 kV/m electric field. J is given in the unit of nA/cmz. ..... 196 Comparison of the analytical and numerical solutions on the induced current density J inside a concentric muscle-fat sphere when exposed to a 60-Hz, 1 kV/m electric field. J is given in the unit of nA/cmz. ..... 197 Comparison of the analytical and numerical solutions on the induced current density J inside a concentric muscle-fat sphere when exposed to a 60-Hz, 1 kV/m electric field. J is given in the unit of nA/cmz. ..... 198 Equivalent circuit for computing the steady-state response of the equivalent circuit of Fig. 6.9. ........ 206 Equivalent circuit of Fig. 6.9 before the spark contact (the switch in the circuit is open). Equivalent circuit for computing the transient response of the equivalent circuit of Fig. 6.9 (the switch in the circuit is closed). A biological body is immersed in a slowly time-varying electric field. ........................... 218 A conducting concentric-sphere located in a homogeneous conducting medium is immersed in a slowly time-varying electric field. Table 4.1 Table 6.1 Co: on re: Bet Table 4.1 Table 6 . 1 LIST OF TABLES Comparison of experimental and empirical results on the short-circuit current with theoretical results by the present method. Body resistance of adult and child in ohms [33] . xvii electromagne izportant su fields of 81‘. 5.753915 beco Resea: QC 593131 c CEHUPTEE! l INHHNJDUCHKUUN The interaction of extremely low frequency (ELF; 0-100 Hz) electromagnetic fields with human body has become an increasingly important subject, since a potential health hazard due to the EM fields of an extremely high voltage (El-IV) power line and ELF antenna systems become a public concern [1-4]. Research in the area of the relation between ELF electricity and health can be traced to 1960s. Researchers in U.S.S.R. published several reports of nonspecific physiological complaints among switch- yard workers when line voltages were raised to levels of 400-500 kV [4]. Since then, the biological effects and the interaction of ELF electric fields with human body have been studied extensively. Many workers have investigated this subject experimentally or empirically [5-10]. However, all the experiments were conducted on animals or scale model of man, and it is necessary to extrapolate these experimental data to provide information for human risk analysis. This is not an easy task if a reliable theoretical method for predicting the interaction of ELF EM fields with human body is not available. Theoretical studies on this subject have been conducted by a number of researchers, but they invariably used over simplified body geometries or inaccurate methods. Spiegle [11] used spheres as human models and Shiau et al. [12] used spheroidal models of man, and their results from these idealized models mutate bloc) ratified, but hi an. . : ”90qu n’ '- naaabae \- F 'I;“ p 7 ”Mia CL 3.. . I . a :y.‘.ndr‘.cal - ._.' Q ‘ frag-lca‘ x'a '9 ‘u finite charge “$5.3: env' 12:6“ may have little practical values. Spiegle [13-14] has also used a more accurate block model and an electric field integral equation (EFIE) method, but his results disagree with experimental results mainly due to insufficient partition of the body model in the numerical calculation. Chiba et al. [15] used a finite-element method (FEM) and a body model of revolution geometry. Their results are still not accurate for a realistic human body. Kaune and McCreary [16] developed a numerical method on a cylindrical model of man. Since this model is over simplified, the practical values of their results are questionable. This thesis presents a recently developed numerical technique, surface charge integral equation (SCIE) method, which is based on a realistic model of man with arbitrary shape and posture, and under a realistic environmental condition such as assuming arbitrary grounding impedances between some parts of the body and the ground. The method is developed on the basis of an integral equation for the induced surface charge density, the Ohm's law, and the conservation of electric charge. This integral equation method can be used to quantify the induced electric field at the body surface and inside the body, the induced body current, as well as effects of the grounding impedance by an incident ELF electric field. And it is found that this method is possible to predict experimental results on the interaction of human body with LF-HF electromagnetic fields. This seems to extend the validity of the surface charge integral equation method to LP or even up to HF range. It is also noted that this surface charge integral equation method is numerically quite efficient. In chapter 2, the theoretical development for calculating the induced surface charge and the induced body current of an arbitrarily shaped biolo; Ascalar intl fte method or a aatrix eqm Chapte: surface char; effects when are compared €39 integral and analybiCC' (.2 C ‘ a STOL‘ndec 7‘31““ tt agreement is :“qc;; ‘ .4..L..y the ”hues. “be " the inc it; {I 133“... _ “5‘ :3 “‘ Eke‘ ‘ Shaped biological body, when exposed to an ELF electric fields, is given. .A.sca1ar integral equation for the induced surface charge is derived. The method of moments is used to transform the integral equation into a matrix equation for numerical solution. Chapter 3 contains calculated numerical results of the induced surface charge density of spherical and spheroidal models of biological objects when exposed to ELF electric fields. These numerical results are compared with existing analytical solutions to check the validity of the integral equation method. An excellent agreement between numerical and analytical solutions is obtained. Also the short-circuit currents of a grounded biological sphere and a grounded biological spheroid are calculated to compare with the existing empirical formulas. A very good agreement is observed. In chapter 4 the surface charge integral equation is applied to quantify the interaction of 60-Hz electric fields with animal and human bodies. The induced electric fields at the body's surface and inside the body, the induced body current, and effects of the grounding impedances are quantified and compared with existing experimental results. An excellent agreement is obtained. Also it is found that, for an inductive grounding impedance, there is a possible resonance phenomenon when the impedance has a value in the order of 100 MD for a man with a height of 180 cm. For this case, there may be a very large current induced in the body causing possible health hazard. The SCIE method is also employed to compute the short-circuit currents of the human body (1.8 m in height) induced by electric fields at frequencies over the ELF and HF range (60 Hz ~ 50 MHz) and they are compared with experimental results measured by .a..- us worker a I .k 81:33 15 sole ] ‘T'e frequency is up to the I field and car: electric fie.t 53:3 he :hod, free space dye :Eée- The p; 35:210d far the :5 13599-35. 12‘ 3:3“ 5',“ ~GLe 5“ pro ‘ I :15 - “9516:. 3‘}; ‘L N “'9 Guess . . 7“ . ‘*C:“v. ‘ 6"; 79;? ‘5 ‘- n a ' 1“ V .L‘ssEd ChaPtEr ”idem a 5101: €va .. L‘sure 0f *k 5H" 2:4“ '8’; ~‘ ‘I - ‘ " be H “aCI be “Ua:::‘ 5 Pete“: ltxic‘e at Pro}. la ‘neSe various workers. A satisfactory agreement is obtained up to about 40 MHz which is the resonant frequency for the human body. This indicates that the frequency range applicability of the SCIE method for the human body is up to the first resonant frequency in the HF range. Chapter 5 is devoted to the calculation of the induced electric field and current in a model of man as used in chapter 4 by the tensor electric field integral equation (EFIE) method, when exposed to ELF EM fields. Instead of using the low-frequency approximations as the SCIE method, this tensor integral equation method is derived by using the free space dyadic Green’s function, and is valid for the whole frequency range. The purpose of using this method is to compare with the SCIE method for the numerical accuracy and efficiency. By using the method of moments, in order to yield accurate numerical results from EFIE method, the body needs to be partitioned into many small cubical cells. This will create the problem of solving a very large matrix equation. To overcome this problem, two iterative methods - the Conjugate Gradient Method (CGM) and the Guass-Seidel Method (GSM) are employed to solve the matrix equation iteratively. The convergence rate of the CGM and GSM are compared and discussed. Chapter 6 covers the study of transient response of the interaction between a biological body and a nearby metallic object both under the exposure of the ELF electric field of a high-voltage power line. The coupled surface charge integral equation method is applied to determine floating potentials and short-circuit currents of a human body and a vehicle at proximity both immersed in a 60-Hz high-voltage electric field. Based on these data, the equivalent circuits for a human body and a nearby vehicle are constructed. The shock current flowing between a human body 33;: a ‘I'Ehic &e In the; 25-2! electri ISCIE) aet..oc heterogeneous hinted body-s :urtents flow: Txrerzt law an it 5‘ aawed. :teck the ‘-’al 1'. fat ”mentric fie and a vehicle through direct or spark contact is then analyzed using the equivalent-circuit model. Numerical results of the present method and experimental results by Reilly [32] are compared and discussed. In chapter 7, the problem of heterogeneous bodies exposed to ELF-LP electric fields is treated. The surface charge integral equation (SCIE) method is combined with an impedance network method by which a heterogeneous body is modeled as an equivalent impedance network and the induced body-surface charge is viewed as an equivalent current source. The currents flowing in the network can be determined based on the Kirchhoff '5 current law and from which the induced current densities inside the body can be mapped. A conducting concentric-sphere is chosen as a test case to check the validity of the present method. Induced currents inside muscle- fat concentric-spheres with various thickness immersed in 60-Hz electric fields are computed and compared with analytical solutions. Chapter 8 is the conclusion. Y 99" 21 “ :eruy K 33 electric o . P imtztles o; “I The indu factor 0' ground, . Lire shot Wrens}! e‘ect‘ic it) The indu (d) the Elec elEC‘viC . .ze validity CEHKPTTHE 2 INTERACTION OF ELF ELECTRIC FIELDS WITH BIOLOGICAL BODIES 2-1 W In this chapter, the theoretical study on the interaction of ELF electric fields with biological bodies will be presented. The quantities of interest to be discussed are : (a) The induced surface charge and the electric field enhancement factor on an arbitrarily shaped biological body standing on the Iground, when exposed to an ELF electric field. (b) The short-circuit current from the biological body to the ground through grounding impedances Z '3 under the exposure of an ELF L electric field. (c) The induced body currents and current densities as well as (d) the electric field inside the biological body due to the ELF electric field. The validity of this theoretical method will be verified by applying it to idealized geometries of a sphere and a spheroid for which closed form solutions exist. The accuracy of the method will also be checked by comparing with existing experimental results for a guinea pig and a phantom model of man. '.'r.en a bu “rd and hen gr... ""“ current l'sc at. electr. M .Le de‘v' 51;: the ground be 1: 317-113! frquer In order a 5.5-re ~,een [32‘0"9 In .re C£‘--'asi. A‘aenSlOn ELF elem:- :2) 1“is: ’J\ “ ll- . J duke L“Cuce SF-‘7 Ma‘ c038 Sin 2-2 Tum When a biological system, such as a human body standing on the ground and being exposed to an ELF electric field (Fig. 2.1), a short- circuit current will be maintained between the human body and the ground. Also an electric current is induced inside the body and surface charge on the body surface. Consider a geometry of a human body standing on the ground being exposed to an ELF electric field, as shown in Fig. 2.2. Contact impedances between the ground and the human body are represented by ZL1 (i - 1 ~ k). The impressed ELF electric field at the location of the body is assumed to be spatially uniform and oscillates with an angular frequency of w. In order to simplify the problem, following approximations which have been proved valid [5] will be used: (1) The quasi-static approximation will be used, since the body dimension is small compared with the wavelength of the impressed ELF electric field. (2) The body will be assumed to be equipotential with an unknown potential ¢b' (3) The ground effect will be taken into account by the method of image. (4) The induced electric charge inside the body is assumed to be small compared with the induced charge on the body surface. Since under the quasistatic approximation, the induced surface charge a q - (2(c and e tr surface in one I the nedi higher t t are gr 0f diffe assumpci, 0) Inside cl Breater 1 0n the f; not be Conside CClpared with charge at the interface between media 1 and 2 is given by, j n - e2(91/92 - 61/62)(Q-E1), where 9 is the complex conductivity I and e the permittivity (shown in the Fig. 2.3). Thus, the induced surface charge is proportional to the normal component of E field in one region and also depends on the discontinuity of 9 and e of the media. (a El) at the body's surface is many orders of magnitude higher than that inside the body, and the discontinuity of 9 and c are greater at the body-air interface than at the interface of different body tissues (as shown in Fig. 2.4). Thus, this assumption is valid. (5) Inside the body, the conducting current is assumed to be much greater than the capacitive current. This assumption is based on the fact that a >> we inside the body in the ELF range. The effect due to the magnetic field of ELF electromagnetic fields will not be considered here, since this effect is believed to be insignificant compared with that due to the electric field. 2.2-1 W The first step in this approach is to determine the electric charge induced on the body surface by the impressed electric field. The induced surface charge density n(f) and the body potential ¢b can be expressed as unknowns in a pair of integral equations as follows. EHV POWER LINE / l-‘l'lll'll!lllllu'l' 'l'I'l'l "l'l'lll ' I |"Il"i!l'|‘:l'lll' I'll ll‘ll'll'!ll.l|‘|l‘nll llllll "I I ll, .l'|'l"'I'-‘l"|"l'l'll!l’ I I, I I n.I'I'll'|"I"l|ll"ll'Il I I, / I, 1.! ll|""'l'lll"ll'l llllllll I, l/ I / I / / Induced p(jwt) maintained by an EHV power line. A human standing on the ground is exposed to an ELF electric inside the human body is Jiexp(jwt). field E ex Fig. 2.1. 0 current Anna “3- 2.2 Fig. 2.2. 10 H' r .3 ;, IIL'i. 1.27 J. L 214 ‘r.".___1vvrg___.._ 33:. '1" IMAGE 11. 'E 3% ‘3 erupt fl. th:t1; ‘3 “E. 11 is. 3 j 1:1 EFF :91 ‘~k[ KL +1"? 4, entrd - \I A man standing on the ground is exposed to an electric field of ELF range. The contacts between the feet as well as the left hand and the ground are represented by zL1 (1 - 1 ~ 3). cog com dis; fre4 m F15 2.3. ll E0 (to. 0) auscui (6‘2) 3 i 3 ":5 5. m- (9) . - L .' , .' .. . -Jz 1 11 '.‘ "" "" O ‘ .—M-f—L-—-r FAT 9 - ', A K .' «5".'..' ( 1) f «’1 32 >01 W§CLF'.(?2)/' 'Jz ' 92 >31 complex conductivity: 9 - a + jut conducting current density : 3c - cE * Ag , J - aE - (a + jw¢)E displacement current density: 3d - jug - mm A A Jln " ”'2n " "131.1 " ”232:: zit ' E2t - W A free charge density : "t - «232“ - £131“ - «2(01/92 - cl/czn-I1n polarization charge density: "p - (s2 - (1)32n - total charge density: n: - at + qp Fig. 2.3. Induced surface charges at the interface of different media. liln .. —ep .-. --h D , i!_ P»: .1. rnu A J ‘I- in uz<_nz:c .-._ ..-Jllu--. C—v: ,9... -2 -.1iL .s- “v 9 I. n \ ‘. I- N..- 12 . LIVER ..' MUSCLE ”s. HEART MUSCLE g 105 __ . BRAIN B ; . LUNG 5 - U U - E D p... 0 L” d - .— e FAT CD 105 1 l lillllll l 1 l I Illnxl 10‘2 ‘0‘] 1 CONDUCTIV ITY 0( S/m) Fig. 2.4. Conductivity and dielectric constant of various tissues in the ELF range [15], hysic. over the body is maintained .* e I 00.1) wtzch 1: l3 Physically, the body potential ob, which is spatitally constant over the body, can be considered as sum of the potential ¢s(¥) which is maintained by the induced surface charge n(?), and the potential ¢°(;) which is maintained by the impressed electric field, ¢s - ¢b (2.1) By using the quasistatic approximation and considering the ground image effect, ¢s(;) can be expressed as l 1 ¢ (¥) - -————-— n<¥'> -———-—-——— as: (2.2) 5 4x60 [8+8i I? - f'l where S is the body surface and Si is the surface of the body image, f is a field point on the body surface and ¥' represents a source point on the body surface and the image surface. The potential ¢o(;) can be expressed in terms of the impressed electric field, ¢o(;) - -Eoz, for the geometry of Fig. 2.2. The body potential db is an unknown quantity and its value depends on the body geometry, the impressed electric field and the grounding impedances Z Li' Equation (2.1) can now be rewritten as J 1 n<¥') _—___—:—T_ ds' + ¢o(¥> - ¢b (2.3) S+S Eq. (2.3) 1 second equa conserva: io Loving be t . . cur: nts f. r4 r4 14 Eq. (2.3) is a scalar integral equation for the induced charge density n(;) and the unknown body potential ¢b. To determine n(;) and db, another equation is needed. This second equation is obtained on the basis of Ohm's law and the conservation of electric charge. Let IL be the total current flowing between the body and the ground. It is the sum of the currents flowing through the grounding impedances 2L1: IL - db-[(1/ZL1) + (l/ZLZ) + ..... + (l/ZLk)] k - ¢b-[ 2 (l/ZLi)] (2.4) i-l On other hand, IL can be expressed in terms of the total surface charge Q, based on the conservation of electric charge, as IL - ij.- jw J n(?'> ds' (2.5) S Combining Eqs. (2.4) and (2.5), db can be written as ¢b- k J n(1:')ds' (2.6) ) s Eqs. (2.6) is the desired second equation which has n(;) and.¢b as the unknowns. 2.2.2 Iransi The Hs‘ at certain d Hethod, the as (n - 1 ~ :1 e}. 1.. -at mt) ar . '0 pomt I on I erpressed as N1 2\ 3'1 he , "Eere 15 2 2.2 Iranai2rnati2n_2f_Intssral_Essesisn_;_n9aent_§sthed The Moment Method is applied to solve the integral equation at certain discrete points in the region of interest. In the Moment Method, the total surface of the body is partitioned into N patches ASn (n - 1 ~ N). These patches are assumed to be electrically small so that n(;) are assumed to be constant. Mathematically, at a observation point fm on the body surface, the integral equation (2.3) can be expressed as N 1 "n "n a 2 ——-ds'- ———ds' +¢ + m ASn Irm - r'I by the charge of ASn patch (-nn) g ds' - potential at r maintained 1 -> +, i m AS r - r I n m i by the charge of as: patch and f“ is the central point of the ASIn patch and fn is that of ASn patch. Since the patches are assumed to be small, for m i n, the above integrals can be approximated as, 'a‘ten I - n, 1 "O u it" v r - 1 .s a Emmi. this 5“. (2.8), 1 Circular dis} Sives the fol l6 "n , "nAsn + + d8 - + + AS Ir — r'I Ir - r | m m n (2.8) "n , "nASn 1 + + ds - + +1 AS Ir - r'l Ir — r I n m i m n When m - n, fm is at the center of ASn patch and f’ is within ASn, thus, I}?m — ¥'| will vanish in the first integral of Eq. (2.8). However, this singularity is removable through the integral of Eq. (2.8). First the ASm patch is approximated by a equivalent circular disk Dm with the same area, as shown in Fig. 2.5. This gives the following relationship: am--———- (2.9) where ASm is the surface area of the mth patch and am is the radius of the disk Dm' Now, for m - n, the first integral in the Eq. (2.8) can be written as ds 2« a Rdeo Jh R J0 J0 R m - 21ram - 2] « ASm (2.10) l7 R P Q as," "Fm 7h 0 9 y x As". = in?“ p m J Dm : equivalent disk of the patch AS,“ Fig. 2.5. Geometries for integration of Eq. (2 8) when m - n. Also Eq. (3 59- (2.7) a: 4" J rd‘ '4 ll J N ‘ he {e 0n 18 Also Eq. (2.6) can be approximated as jw N ¢ - z (7) A8) (2.11) b k n-l n n [ 2: (1/ )1 1_1 71.1 Eq. (2.7) and (2.11) are then transformed into a matrix equation as P - I - -I P 1 “11 M12 °°°°°° ”m 1 "l 4‘01 M21 M22 ........ MZN -1 "2 ¢02 M111 M112 °°°°°° MNN "'1 "N ¢oN R .1351 A52 ........ ASN -[1§1(l/ZL1)]/ij b¢bd _ o J where (2.12) 1 1 l M - AS + .+ - + +1 m i n mn 4t: Ir - r I Ir - r I o m n m n l 2/ a l M - AS - m - n m lure “‘ (as )1/2 l? -?1I m m m ¢on - ¢o(rn) - - Eoz In this ”tri? (5+1) x (5' I I potential at Equipotent ia; the nzrix as r 1 The conventiO‘ the latrix 9C. éass~5eidel ‘ if the matrix There (I) The case 11 21 19 In this matrix equation, [M][n]-[¢], the matrix [M] represents a (N + l) x (N + 1) matrix and [n] as well as [d] are (N + 1) column matrices. [an] and [don] are charge density and impressed field potential at the central point of the nth patch. db is the unknown equipotential of the body. [an] and db can be determined by inverting the matrix as follows : [n] - [MJ'1[¢] (2.13) The conventional Guass Elimination Method (GEM) can be used to solve the matrix equation, and the Conjugate Gradient Method (CGM) or the Guass-Seidel Method (GSM) can also be applied to solve it iteratively if the matrix size becomes excessively large. There are two special cases of interest: (1) The case when the body is shorted to the ground, 2 - O for any Li i. The body potential db will be zero, therefore, Eq. (2.12) is reduced to a (N x N) matrix equation, with the last column and the last row of the matrix in Eq. (2.12) removed, as q n q n - IN "1 ¢ol 21 22 """""" 2N "2 45oz - - - (2.14) 'M'Nl M'NZ ........ MN'N -14 “"N‘ -¢oN-I The unknown surface charge density [an] can then be determined accordi The cas all 1. zero, t 11 21 20 according. (2) The case when the body is isolated from the ground, ZLi - w for all i. Hence the last element of the matrix of Eq. (2.12) becomes zero, the matrix equation is ’ _-Po r1 M11 M12 ........ M l ¢ol 21 22 """""" 2N "2 %2 -- . (2.15) MN1 MN2 ........ MNN -l "N ¢oN AS AS This implies that N 2 "n ASn - O (2.16) n91 that is, the total net charge on the body is zero. For this case [an] and db are determined from Eq. (2.15). After the induced surface charge density is determined, the induced electric field at the body surface is simply obtained from Es - n/eo (2.17) assuming th. the body 51:: The 1 3f the indm field, E ..'E s < st: :ched a: ixT—‘osed to '. field a: so: already a .€ 21 assuming that the induced electric field is totally perpendicular to the body surface. The electric field enhancement factor is defined as the ratio of the induced electric field at the surface to the impressed electric field, Es/Eo. This value can exceed 10 at the head or the tip of a stretched arm and hand as calculated in chapter 4. Thus, when a man is exposed to the electric field of an.EHFV power line, the induced electric field at some points of the body can be extremely high since E0 is already a very high value in this case. 2.2.3 e e B d After the induced surface charge density "n at any point on the body surface is determined, the induced current density inside the body can be determined on the basis of the conservation of the electric charge and employing Maxwell's equations. The first quantity to be determined is the total sectional current at any cross section of the body. Referring to Fig. 2.6, it is assumed that the positive sectional current at any cross section of the body is directed toward the head. As for example, three sectional currents will be considered: the sectional current at the chest 11’ the sectional current at the lower abdomen 12 , and the sectional current at the right arm 13 . If the equation of the conservation of electric charge, V-J + jwp - O, is integrated over the volume V1 which encloses the . s . a 22 A n ---------- -------------‘------------1 P------------- Geometry for calculating the induced current in the body. Fig. 2.6. upper had;- obtained. A . where 1'11 1 L sectional cross sec: P°rtion of cutting th 01’ 23 upper body above the chest cross section (see Fig. 2.6), it can be obtained, S A .. ' I1 - - I (nl-J1)ds' - jw I n ds' (2.18) S c1 1 where 31 is the unit vector pointing outward from V1, Sc1 is the cross sectional area at the chest, 81 is the body surface above the chest cross section of Scl’ and 31 is the current density at Scl' A similar integral over the volume V which includes the 2’ portion of the body above the lower abdomen and has a boundary surface cutting through the lower abdomen section 802 and a grounding impedance 2L3 connecting the left hand to the ground, will lead to A -. I _ _ _' I I (nZ-J2)ds IL3 Jw I n ds S S c2 2 or (2.19) 12 + 1L3 - jw J n ds' 82 where 32 is the unit vector pointing outward from V2, J2 is the current density at Sc2’ $2 is the body surface enclosed by V2, and IL3 is the current flowing from the ground to the left hand through the grounding impedance 2L3' 1L3 can be easily determined from ¢b/ZL3 . I Ano . the right a; 1 I there :1 is 3 sectional a: ’2 Is :ne 5: J the phenoae: ~0 that flow. One the 53.01' : Sc 24 Another similar integral over the volume V3 which contains the right arm and hand leads to 1 - (3 -J )ds' - —jw 0 ds' (2.20) 3 3 3 s 5 c3 3 where n3 is the unit vector pointed outward from V3, Sc3 is the cross sectional area at the right arm, 33 is the current density at Sc3 and $3 is the surface of the right arm and hand enclosed in V3. It is noted that 33 and 33 are in the same direction. Therefore, the expression of I3 has a negative sign in Eq. (2.20). This negative sign will lead to the phenomenon that the current in the arm flows in the opposite direction to that flowing in the other parts of the body. One of the most important quantities concerning the body current is the short-circuit current Isc' ISC is defined as the current flowing between the feet and the ground when the grounding impedances between the feet and the ground, 2 and Z are zero and other grounding L1 L2 ’ impedances between other parts of the body and the ground, such as ZL3’ are infinity (open circuit). Isc can be easily obtained by the same approach as above and is given by Isc - jw Is 0 ds (2.21) where hargl poten‘ ‘ I ”~01 Q. U so A. ,. ’vr. Q eev' 5 01' ‘ I”. ‘1 av“ ;‘_ 25 where S includes the total body surface, and n is the induced surface charge density at the body surface under the condition that the body potential db is zero. After the determination of the sectional current, the volume density of the induced current inside the body 3 can be determined from the Maxwell's equation, -e 4 V x H - (a + jwe)E or (2.22) V-[(a + jwe)E] - o For the ELF-LF range, a >> we inside the human body, therefore v (0%) - v.3 - o (2.23) Eq. (2.23) can be used to predict the distribution of 3 inside the body with the prior knowledge of the sectional current I at any cross section of the body. Assume that 3 at any cross section of the body has only two components: the longitudinal component J and the radial component Jr. I This approximation assumes a cylindrical geometry for the body cross section as shown in Fig. 2.7, and also ignores the circumferential component Jw. due. __> *§) ”'-I_~\‘ /’ \\ 4 1’: ‘x I \4/_'r ‘ ”F <-—-<-—76-{ be I i \ I V \ ¢ Eb Fig. 2.7. Cylindrical geometry for calculating the induced current density inside the body. LIA ‘—4 here I is the cross 5 is applied. but it may based on :1‘ Equation (a i3d€pendent <1 . .g ‘ ‘s n. 27 The longitudinal component J1 can be approximately obtained as J3 - I/Sc (2.24) where I is the already determined total sectional current and Sc is the cross sectional area of the body at the position where Eq. (2.24) is applied. The calculation of J! is valid for a homogeneous body, but it may also be a fair approximation for a heterogeneous body, based on the finding by Spiegel [14] using the electric field integral equation (EFIE) method that the induced current density 3 is rather independent of the electric parameters of the body at the ELF range. Now that J1 is determined at any cross section of the body, the radial component J? can be derived from Eq. (2.23), using a cylindrical geometry (shown in Fig. 2.7), as follows: From (assuming J¢ component is zero) # l a 6 V-J - -- (rJr) + J2 - O (2.25) r ar 62 it leads to 3 6J (rJr) - —r( ) (2.26) ar 82 After where '4. use b( 51‘. #15:! 28 After integrating both sides, it gives ) (2.27) where r is the radial distance from the center of the cross section, and (aJl/ae) is the rate of change of J! in the longitudinal direction. Since J! is known at any cross section, the value of (6J1/61) can be estimated easily. Eq. (2.27) indicates that Jr is zero at the center of the body and linearly increases toward the body surface. The direction of Jr is dictated by the sign of (8J£/8£). After J is dertermined, the electric field induced inside the body is determined from E - 37.: (2.28) and the SAR (specific absorption rate) value is calculated from 1 SAR (W/Kg) - |J|2(p (2.29) 20 where p is the volune density of mass in Kg/m3. APPLI( 1.1 1; I: eiuatic the Sc; isolate Also t} h to I": 5“ en N If? 7’ Co @535 Q 503:; G: Pk.‘~ enq‘se < CHEUEPERLZB APPLICATIONS OF SURFACE CHARGE INTEGRAL EQUATION METHOD TO SPHERICAL AND SPHEROIDAL MODELS OF BIOLOGICAL BODIES 3-1 Introduction In order to compare the numerical solutions of the integral equation developed in chapter 2 with existing analytical solutions, the SCIE method is applied to an isolated conducting sphere and an isolated conducting spheroid under the exposure of ELF electric fields. Also the case of a sphere locating above the ground plane and exposed to ELF electric fields as well as the case of a spheroid under the same condition are studied, for simulating a human or an animal body standing on the ground. 3.2 u o a r nd om ar ion with W Consider a perfectly conducting sphere with radius a which is immersed in an uniform electric field as shown in Fig. 3.1. Analytical solutions of the induced surface electric field E(a,0) and the surface charge density n(a,0) can be written as [17] 29 30 Eo surface electric field-EKG) =’?3Eocos(0) surface charge density n(0) = £OE(0) = Bthocos(9) Fig. 3.1. The induced surface electric field and the induced surface charge density on a perfectly conducting sphere immersed in an uniform electric field. ZN ri shown $23161 '30 s; Enda the a: , ' I '1‘ . veins tug 9f the a. .“e re 31 E(a,9) - 3Eocosaf (3.1) n(a,0) - eoEr(a,0) - 3: E c030 (3.2) o o For numerical calculation, the sphere surface is divided into 2N rings and each ring occupies a radian angle of A0 (- r/2N), as shown in Fig. 3.2(a). Due to the rotational symmetry, the induced surface charge is assumed to be uniform within each ring. Consider two sample rings as shown in Fig. 3.2(b): the nth ring with radius pn and a reference source point Q(;n), f - (r , 0 , an), as well as n n n the mth ring with radius pm and a reference field point P(;m)’ fm - (rm, on, em). The distance between the field point P and the source point Q is Rmn‘ In order to calculate Rmn’ a more detailed geometries of these two rings are depicted in Fig. 3.3. It can be found that R - l? - F | mn m n 2 2 1/2 - [Zmn + d l (3.3) where 2 2 2 zmn - (Zm - Zn) - (acosflm - acoson) din - (asinom)2 + (asinon)2 - 2(asin0m)(asin0n)cos(¢m - ¢n) 32 2N rings , A9= 7r/2N radius = a mth ring nth ring Fig. 3.2. The surface of a perfectly conducting sphere is divided into gN rings and each ring occupies a radian angle of A9 - n/ N). 33 rm = rn = a 1m = rmCOSOm , 2n = rncos n Zmn= Zm ' Zn i . l . dmn I I l I I I l I ' I 1 . i “N i : ' I \ . i I l i I 1 x v" Pm = rmsinom Pn = rnsinon dgn= 9% + 9% - thphcos(Qm-yh) y Fig. 3.3. Geometries of two sample rings on the spherical surface. and he: 34 and hence R - [2a2(l - coso coso - sine sine cos(¢ - ¢ )]1/2 (3 4) mn m n m n m n ' If the surface charge density at the nth ring is denoted as "n’ the scalar potential at the field point P of mth ring due to the surface charge at the nth ring can be expressed as n 1 ¢ — n J ds' (3-5) S n where Sn - surface area of the nth ring "n - surface charge density of the nth ring The rotational symmetry gives (for simplicity, let um - 0) "n « 0n+A0/2 azsino'dO’dw' ¢ - mn 2x6 0 On-Afl/Z [2a2(l - cosomcoso' - sinéimsinO'cosyfi')]1/2 - anfln (3.6) The total scalar potential at the field P maintained by the surface charge on the entire sphere surface is then obtained as can be 35 2XN ¢m- z (ann) (3.7) npl From the symmetry of the sphere, it can be seen that surface charges at two symmetrical rings on the upper and lower hemispheres have the same magnitude but different polarity. ¢m can then be written N O 8 es“, - “31 (Human - Hmnn) (3.8) where 0 indicates the upper hemisphere and s the lower hemisphere. By applying Eq. (3.8) to the equation (2.14), a matrix equation can be derived ll 12 21 22 2N "2 ¢oz - - - (3.9) “MNl MN2 ........ MNN -l‘ -"N‘ h¢oN4 where M _ Mo _ 3 mn mn ¢on - -Eozn Oh n. (l 36 and 1 x 0 +A0/2 asinfl'do'dp’ o n an - 1/2 2t: 0 0 -A0/2 [2(1 - c050 c030' — sin0 sin0'cos¢')] o n m m l x 0 ,+A0/2 asinfl'dfi'dw' s n M - 1/2 mu 2te O 0 ,-A0/2 [2(1 - cosfl c030' — sine sin0'cos¢')] o n m m 0 5 0 5 «/2, 9 , - x - 0 n n n The numerical results for the case which the sphere is divided into 20 rings are shown in Fig. 3.4. Numerical results of the closed form solution are also listed in the figure for comparison. It is apparent that the agreement between the numerical solution and the closed form solution is excellent. The validity of the surface charge integral equation method is confirmed. 3.3 v an To simulate a human or an animal body standing on the ground, a biological sphere located above the ground plane with a grounding impedance Z and exposed to an ELF electric field is studied in this L section. Consider a biological sphere with radius a located at distance d from the ground plane and immersed in an uniform ELF 37 "(0)=¢.E(0) EXACT SOLUTION "(0) (x 10'1°) exact sol. numerical sol. 4.5 0.2644 0.2578 . 13.5 0.2579 0.2561 'E; 22.5 0.2450 0.2452 31.5 0.2262 0.2267 40.5 0.2017 0.2021 49.5 0.1723 0.1724 58.5 0.1386 0.1385 67.5 0.1015 0.1014 * ‘_ ¢ 76.5 0.0620 0.0618 A 85.5 0.0221 0.0211 255 '+ -+ s s A“ 4- n- -e \R A O c. ¢ “ & ¢ «9 4 T‘ A 4- .6 .5 h [z_ ¢' *' e 4:1 4' '2 L - ¢ 1? -b + F - - - - - NUMERICAL RESULT [EL - -' - - I (4: - - 1— -I \u:, .. -' - -‘7 \L - - - -] \e' - - - ~37 : c27' Fig. 3.4. Comparison of numerical results by the present method with the exact solution on the surface charge density of a perfectly conducting sphere induced by an uniform electric field. o. t 38 electric field as shown in Fig. 3.5. To take into account of the ground effect, an image sphere is considered. The problem becomes the case of two spheres with a seperation of 2d immersed in an uniform electric field . Again, for numerical calculation each spherical surface is divided into N rings, as shown in Fig. 3.6. The induced surface charge is assumed to be uniform within each ring, due to the rotational symmetry. Also induced surface charges on each ring and that on its image ring have the same magnitude but opposite polarity. Now consider the mth ring with radius pm and a reference field point P(;m), ¥m - (rm, 0m, ¢m), the nth ring with radius pm and a source 4 -+ point Q(rn), rn - (rn, 0n, 0n) as well as the image n th ring with radius pn, and a source point Qi(;n,), ; - (r ,, 0n,, $n')’ as shown n' n in Fig. 3.6. The distance Rmn (-|?m - ¥n|) is the same as Eqs. (3.4). '14 To calculate the distance Rmn' (-I - ¥n,|) between the field point P m and the image source point Q1, it can be found that R . - I? - ? .l - [22 . + d2 .11/2 (3.11) mn m n mn mn where 22 ' - (z - Z ,)2 - (2D + acoso + acosfl ,)2 mn m n m n d2 - (asinfi )2 + (asinfl )2 - 2(asin0 )(asin6 )cos(¢ - 0 ) mn' m n’ m n' m n' and hence 95 e‘ I Ia 39 OBJECT SPHERE E0 radius = a T I, GROUND j“ 5 ZL ’ a " L‘- I ’ “ I I \\‘ I ‘\ I \ I \ I \ I \ I l I . i 1 ‘ I \ I \ I \ ll IMAGE SPHERE \\ ’ \ ll \\ ’1' Fig. 3.5. A perfectly conducting sphere located above the ground is illuminated by an uniform with a grounding impedance ZL electric field. 40 mth ring I‘ Rmn =I_".m _ T,” nth ring oml 0(7) V " — 0 9n 3 D u _/ Rm”: = (2%”. + dEmM/Z 7+///////////// ////////7/ "f“ *zmn. = 2D + rmcosern + rn'cosonn /’ ‘N IMAGE SPHERE Fig. 3.6. Geometries of three sample rings on the surfaces of a sphere and its image sphere. 11”.: 41 2 Rmn' - [2a (1 + cosomcoson, - sinomsinon,cos(¢m — ¢n') + 4aD(cosom + cosl9n,)]]'/2 (3.12) If the surface charge density at the nth ring is denoted as "n’ then it is (-qn) at the n'th ring. The scalar potential at the field point P of mth ring due to the surface charges at the nth and the n'th ring can be expressed as d» -¢° +01 (313) mn . where mn and Sn - surface area of the nth ring Sn,- surface area of the n'th ring The rotational symmetry gives (for simplicity, let pm - O) 42 n t 0 +A0/2 azsino'dO'dw' ¢o _ n n mn 4st ° 9 -A9/2 [2a2(l - c030 cosfl' - sin9 sinfi'cosnlz'nl/2 o n m m o - annn 2 1 ‘0 fl 0 ,+A0/2 a sino'do'dw' ¢-“ “ mn 4st 0 0 ,-A0/2 [2a2(1 + coso cos6' - sine sino'cosp') + o n m m 4aD(cosom + cosa')]1/2 i - an(-nn) (3.14) The total scalar potential at the field point P maintained by the total surface charge of the sphere and the image sphere is then obtained as N o i ¢m - nil (an - an)nn (3. 15) By applying Eq. (3.15) to the equation (2.12), a matrix equation is obtained D - q P 1 M11 M12 .......... MIN -1 f "1 4501 M21 M22 .......... M2N -1 n2 ¢02 - - ' (3.16) “NI MNZ .......... MNN -1 "N ¢oN _ s1 32 .......... SN —1/(jsz)d _¢b‘ _ 0 . 43 where M -M° -M1 mn mn mn ¢on - _Eozn and l x 0 +A0/2 asino'do'dw' o n 4W6 0 0 -A0/2 [2(1 - cosfl c050' — sine sin0'cos¢’)] o n m m . l w 0 ,+A0/2 asinfl'do'dp' Ml -— n mn 4,6 0 o ,-A9/2 [2(1 + coso cosfl' - sine sin0'c05¢') + o n m m 4aD(cosOIn + coso')]1/2 Since there is no closed form solution for the problem of two spheres, the way to check the validity of the numerical solution is to let the seperation between the two spheres approaching infinity and then compute the induced surface charge. Numerical results of this case should be the same as that of an isolated sphere. Fig. 3.7 shows results on induced surface charges of both cases. A good agreement is obtained. The validity of this integral equation method for a conducting object above the ground is then confirmed. Also surface charge distributions on a sphere located at various distances from the ground plane are shown in Fig. 3.8. It can be seen that the 44 0"“ GROUND d-—-ao ' induced surface charge density "(9) ,L 10 "(9) (x 10' ) present case isolated sphere 4. 13. 22. 31. 40. 49. 58. 67. 76. 85. U'IUIUIUIUUUIUIUUIUI 0000000000 * "(180° — .2578 .2560 .2451 .2265 .2021 .1724 .1386 .1012 .0620 .0211 9) - -n(0) .2578 .2561 .2452 .2267 .2021 .1724 .1385 .1014 .0618 .0211 0000000000 0° 5 o 5 90° Fig. 3.7. The distribution of surface charge density on a floating perfectly conducting sphere located at an infinite distance from the ground induced by an uniform electric field. _.~__‘_A.-.-.- _ a, Fig. 3.8. 45 Surface charge distributions on perfectly conducting spheres located at various distances from the ground and exposed to an uniform electric field. (D ‘1 n. M" 46 closer the sphere to the ground, the surface charge distribution will be more asymmetric along the vertical direction due to a stronger coupling effect. The short-circuit current Isc of a grounded sphere (ZL - 0) can be expressed as (from Eq. (2.5)) N 15c - jw E (nnSn) (3.17) n-l The empirical formula of the short-circuit current of a grounded biological body can be written as [8][9] 15c - 4.2 x 10‘5 x onw2/3 (3.18) where ISC is in pA/(kV/m), f in Hz, E0 in kV/m, and w(weight) in g. To compare empirical results with numerically evaluated results, short- circuit currents of various size (weight) spheres are calculated and shown in Fig. 3.9. It is observed that the agreement between numerical solutions and existing empirical results is very good. It is of interest to find the effect of grounding impedances on the short-circuit current of a biological body exposed to a 60 Hz electric field. The grounding impedance is assumed to be resistive, capacitive or inductive. The short-circuit current as the function of grounding impedance is shown in Fig. 3.10 ~ 3.12. For a sphere with a radius of 0.25m (volume - 6.5 x 103 cm3, corresponding to a 65 Kg sphere), 47 N(WEIGHT)=(4/3)wa3 1 E0 .r 0:2 CI“ ' —e—I5RESENT THEORY --o-'- EMPIRICAL FORMULA ISC Isc=4.5x10'5xonw2/3 (#A/(kV/m))__ 'ISC(HA) f (Hz) 6 — ‘50 (kV/m) 1H (9) 5 — 4 L 3 - 2 I- '1 1- 0 l l A L E L l L 4 8 (cm) 5 10 15 20 25 30 35 c .1. .L. .1_. .1. .J_. ..1_ ._1_ ,__ 0.524 4.189 14.14 33.51 55,45 113,10 179,50 W (k9) Fig. 3.9. Comparison of theoretical results by the present method with empirical results on the short—circuit current for a biological sphere as a function of the radius when exposed to a l kV/m, 60-Hz electric field. a. ‘ee 48 T d A L f/////////////////////////// I SC a=25 cm (F'A/(kV/fl”). d: 2 cm 8 . 7 4 6 d 5 .1 4 .1 3 a! 2 .1 1 d 0 l 10 102 103 104 105 106 107 103 107(0) 2L Fig. 3.10. Theoretical results on the short-circuit current for a biological sphere as a function of the grounding resistance when exposed to a l kV/m, 60-Hz electric field. 49 1 m 0 T c d J. . /7////////////////////////// ZL=l1/ijI ISC (HA/(kV/M) a=25 cm - d=2cm 7 q .6 _, 5 a 4 q 3 d 2 .. 1 v-1 0 L l 4 L 1 L 1 1 _L 10 102 103 104 105 105 107 108 109(0) zL 10"3 10'4 10'5 10'5 10'7 10‘8 10'9 10"0 (F) c ' (x 0.265) Fig. 3.11. Theoretical results on the short-circuit current for a biological sphere as a function of the grounding capacitance when exposed to a l kV/m, 60—Hz electric field. 50 EB ; L d I I ZL=|3°’L| sc (HA/(kV/m)) a=25 cm 11 7 d= 2 cm 8 -1 I 7 d 6 q 5 d 4 4 3 q 2 - 1 - 0 #1_ 12 13 l_4 45 16 17 £8 E 10 10 10 10 10 10 10 10 10 (a) ZL 10" 1 10 102 103 10Z“ 105 105 107 (x 0.265) M L Fig. 3.12. Theoretical results on the short-circuit current for a biological sphere as a function of the grounding inductance when exposed to a l kV/m. 60-Hz electric field. 51 the short-circuit current is found to remain practically unchanged, maintaining a value of about 4.5 pA/(kV/m), when the grounding impedance is varied from zero to 5.0 MB. Only after the grounding impedance exceeds the value of 5.0 M0, the short-circuit current starts to fall for a resistive or capacitive grounding impedance. For an inductive grounding impedance, there is a possible resonance phenomenon when the impedance has a value in the order of 20 M0, corresponding to an inductance of 0.53 x 105 henries. This implies that in a very unlikely case when the grounding impedance is an extremely large inductance of about 53000 henries, there may be a very large current induced in the body. The physical explanation of this phenomenon can be discussed from a circuit point of view. Let C denoted as the capacitance between a small area element A on the sphere surface and the energized electrode (such as the power line), as shown in Fig. 3.13, a displacement current, I, will pass from the overhead electrode through C, then through A, and to the ground through the grounding impedance 2 Since the sphere is assumed as L' perfectly conducting, the current I will not flow through inside of the sphere. The total capacitance Ct between the electrode and the whole sphere surface can be derived by the equivalent circuit method (Chapter 7). The impedance of this capacitance Ct is found to be very large. Hence for a resistive or capacitive grounding impedance, the current I starts to fall only after the grounding impedance becomes as large as the impedance of Ct’ While for an inductive grounding impedance, the resonance phenomenon is observed, when the inductance 52 HIGH VOLTAGE ELECTRODE __<:> surface area e1ement A BODY GROUND . O . O . 0 ‘ . e 9 ' e e 0 Fig. 3.13. A displacement current. I, is coupled from the overhead electrode through a capacitance C to a surface area element A of the subject body. l‘. '1 r0 53 has a certain value L in series with the capacitance Ct for which (ij + l/ijt) - 0. 3-4 An21isasi2ns_s2_§2hsr2ida1_n9dsls_si_£ielesisal_§edis§ Since geometric shapes of many living objects such as human or animal bodies are far from spherical, a spheroid is considered to be a more realistic model for a biological object than a sphere. Consider a perfectly conducting prolate spheroid with major axis a and minor axis b immersed in an uniform electric field, as shown in Fig. 3.14. For numerical calaulation, the spheroidal surface is divided into N rings, as shown in Fig. 3.14. The induced surface charge is assumed to be uniform within each ring due to the rotational symmetry. Consider two sample rings as shown in Fig. 3.15; the nth ring with radius pn and a reference source point Q(rn), E - (r , Rn’ p“), and the mth ring with n n -9 -v radius pm and a reference field point P(rm), rm - (rm, 0m, ¢m). Rmn 15 the distance between the field point P and the source point Q. It can be found that (Fig. 3.15) [18] _ [22 + d2 11/2 mn mn mn (3'19) where 2 2 2 zmn - (Zm - Zn) - (rmcosom rncoson) N dInn - (rmsinom)2 + (rnsinon)2 - 2(rmsin0m)(rnsin9n)cos(1bm — wn) 54 N rings ’ 493 ”/N I ’ I” 111., Fig. 3.14. A perfectly conducting prolate spheroid with major axis a and minor axis b immersed in an uniform electric field. The surface of the spheroid is divided into N rings for numerical calculation. mth ring nth ring 55 0 :11 f PFm) 1/2 Rmn 0(7‘71) (21%11 + drfin) Pm = rmsinom X dmn= ( sz + Dnz - Znhphcosvh)]/2 rnsmon Fig. 3.15. Geometries of two sample rings on the surface of a spheroid. 56 and ab ab r - r m [(asinfim)2 + (bcosem)2]l/2 , n [(asinon)2 + (bcosan)]l/ 2 The scalar potential ¢mn at the field point P due to the surface charge at the nth ring can be expressed as 1 "n ¢ ' ——————’ J ————————- ds' (3.20) S where Sn - surface area of the nth ring "n - surface charge density of the nth ring It can be found that (Appendix 1) d8 - r sinfl d¢ d1 (3.21) n n n n n where ab d0 [(a2 - b2)sin9 cosfl ]2 11/2 n n n dA _ 1 + n [(asinon)2 + (bcoson)2]l/2 { [(asinon)2 + (bcoson)2]2 By Using Eq. (3.20) - (3.21) and rotational symmetry we have (let wm - O) 57 ¢mn ' annn (3.22) where 1 In 10 +A0/2 abr sinfl' M - n n x mm 2 _ , 2 , 2 1/2 «60 0 0n A8/2 Rmn[(asin0 ) + (bcosfi ) ] 2 2 [(a2 - b2)sin0'cosfi']2 1/2 1 + 2 d6'dw" [(asinO') + (bcosfi') ] The total scalar potential at the field point P maintained by the total surface charge on the entire spheroid is 0 - E (annn). (3.23) By applying Eq. (3.23) to the equation (2.14), a matrix equation is given 1’ - " P " I- 1 M11 M12 .......... MIN 1 "1 ¢01 M21 M22 .......... M2N —l 02 ¢02 - - - (3.24) _ MN1 MNZ .......... MNN -l .. _ "N J L ¢oN . 58 where ¢on - -Eozn "n - surface charge density of the nth ring ¢b - equipotential of the spheroid Fig. 3.16 depicts two numerical examples of spheroids with the same volume but different axis ratio. The surface of each spheroid is divided into 20 rings. Numerical results of the induced surface charge for the spheroid with axis ratio equal to 1 should be the same as that of a sphere given in Fig. 3.4. It can be seen that the agreement is excellent. Also it is observed that the electric field enhancement factor at the top of the spheroid increases as the axis ratio increases, since the spheroid becomes thinner and sharper. 3.5 Spheroid above thevground Plggg To simulate an animal or human body standing on the ground, a biological prolate spheroid located above the ground plane with a grounding impedance Z and exposed to an ELF electric field, as shown L in Fig. 3.17, is studied in this section. Again, an image spheroid is considered for the ground effect. Each spheroidal surface is divided into N rings. The induced surface charge within each ring is assumed to be uniform due to the rotational symmetry. Consider S9 a : major axis 9° "(9) (X10-1O) b : minor axis 4,5 0,2577 13.5 0.2562 _ 22.5 0.2450 - 31.5 0.2267 a/b - 1 (sphere) 40.5 0.2020 49.5 0.1726 58.5 0.1388 67.5 0.1014 76.5 0.0620 85.5 0.0223 4 ‘1 ti 0 90 , 0(0) (x10’13) 4.5 0.4750 13.5 0.3754 22.5 0.2667 31.5 0.2023 40.5 0.1475 _ 49.5 0.1022 58.5 0.0753 a/b = 2 67.5 0.0521 76.5 0.0316 85.5 0.0085 1 1*. Fig. 3.16. Distributions of surface charge density on spheroids with two different axis-ratios (a/b - l & a/b - 2) induced by an uniform electric field. 60 mth ring nth ring ' IT! 0 v ZL l , ///f///// ////////7/ D //8 "’Znn' = 20 + rmcosem + rn'coson- / / I ,’ 1 / IMAGE SPHERDID l / 1 ' . 1 0 1 1 90' 1 I, -. n'th ring 15:::;:: dmn' \\ ‘TQU‘nW \ / \ / \ z \ Fig. 3.17. A spheroid located above the ground with a grounding impedance Z exposed to an uniform electric field. Geometries of three sample rings on the surfaces of the spheroid and the image spheroid are also shown in the figure. 61 mth ring : radius pm & a reference field point P(;m), rm - (r , 6 , w ) nth ring : radius pn & a reference source point Q(fn), f - (r , an, un) n'th ring : radius pn, & a reference source point Qi(fn,), -. r - (r ,, 0 n' n n" 4“,), (n th ring - image ring of nth ring) as shown in Fig. 3.17. The distances between the field point P and source points Q & Q' are found to be Rmn - [Zin + dinll/z (3.25) R .- [22 . + d2 .11/2 mn mn mn where R , Z and d are in Eq. (3.19) and Z , & d , are mn mn mn mn mm 22 ,- (Z - Z ,)2 - (2D + r c050 + r ,c050 ,)2 mn m n m m n n 2 . 2 . 2 . . dmn'- (rm31n9m) + (rn,51n0n,) - 2(rms1n9m)(rn51n6n,)cos(¢»m - ¢ ) n! and ab ab rm ' 2 2 1/2 rn" 2 2 [(asinom) + (bcosom) ] , [(a51n0n,) + (bcosfin,) ] 1/2 Let the surface charge density of the nth ring denoted as "n and (-nn) is that of the n'th ring. Now the scalar potential at the 62 field point P due to the surface charge on the nth & n'th ring can be expressed as + ¢ (3.26) where n 1 0° - ————E——— -——————— ds' m“ 416 s R n ‘0 1 ,1 _ ____2___ 1 d5. mn S ' n and Sn - surface area of the nth ring Sn,- surface area of the n'th ring Using Eq. (3.25) ~ (3.26) and rotational symmetry, it leads to (let ¢m =0) (3.27) 1 ¢ - an(-nn) where 63 l n 0 +A9/2 abr sinfi' o n n an - 2 1/2 x 2neo 0 0n-A0/2 Rmn[(asin0 ) + (bcosB )] [(a2 - b2)sin9’coso']2 1/2 1 + da'dw' [(asinfl')2 + (bcosfl')2]2 . l n 0 ,+A0/2 abr ,sinfl' 1 n n an ' 2 1/2 x 21110 0 0n,-A9/2 Rmn,[(asin0 ) + (bcosfi )] [(a2 - b2)sin9'coso']2 1/2 1 + da'du' [(asinfl')2 + (bcosfl')2]2 The total scalar potential at P maintained by the total surface charge on the object spheroid and the image spheroid is then obtained as N o i ¢m - n51 (an - an)nn (3.28) By applying Eq. (3.28) to the matrix equation (2.15), a matrix equation is obtained as M11 M12 .......... MIN -1 "1 F ¢ol M21 M22 .......... MZN -1 02 402 - - - (3.29) MN1 MN2 .......... MNN -l "N ¢ON _ s1 s2 .......... SN -1/(ijL)j _¢bj _ o 1 0‘ Ya. 64 where é - — £02 on n To check the validity of the numerical solution. let (d ~ 0) & (ZL ~ 0) and then compare the numerical result of this case with that of an isolated spheroid in the free space as shown in Fig. 3.18. The agreement is very good. The short-circuit current of a grounded spheroid (ZL - 0) is N Isc - jw “fl (nnSn) (3.30) It is known that the empirical formula of the short-circuit current, Eq. (3.18), is good for a biological object with a body shape close to a sphere. Since a human body is far from spherical, the empirical formula for the short-circuit current given by Deno [5] and used by Chiba et al. [15], 1 - s 4 x 10'9 x (f/60)E H2 <3 31) ac ' o . ' where Isc in pA/(kV/m), f in Hz, so in kV/m, H (height) in m, was found to be much better for a human body. It is interesting to find out the 65 0 a/b = 2 "(0) : induced surface charge density an. E0 1. d-—-ao GROUND gJ_ /77//////////////////7//7/7E 0° n(o) (x 10‘1°) present case isolated spheroid 4.5 0.4751 0.4750 13.5 0.3753 0.3754 22.5 0.2667 0.2667 31.5 0.2023 0.2023 40.5 0.1475 0.1475 49.5 0.1032 0.1022 58.5 0.0753 0.0753 67.5 0.0521 0.0521 76.5 0.0316 0.0316 85.5 0.0085 0.0085 * "(180o - 0) - -n(0) 00 s o 5 90° Fig. 3.18. The distribution of surface charge density on a floating perfectly conducting spheroid located at an infinite distance from the ground induced by an uniform electric field. ‘ 66 relation of the short-circuit current and the floating potential of a human body with respect to height and weight, when exposed to ELF electric fields. A spheroid is a good model for a human body for this aspect, since the major axis and the minor axis of the spheroid can be adjusted to simulate the physical shape of human body. First, short-circuit currents of various spheroids with the same volume (weight) but different axis ratio are calculated and shown in Fig. 3.19. It can be observed that the short-circuit current will increase as the spheroid becomes longer and thinner. As shown in Fig. 3.19, it seems that a spheroid with the axis ratio of 2.2 is the closest model for a man to calculate the short-circuit current. Now, floating potentials of various spheroids with the same axis ratio (- 2.2) but different volume (weight) are calculated and shown in Fig. 3.20. It can be seen that the floating potential will increase as the spheroid becomes longer (higher and larger), and the function of the floating potential with respect to the height (H) of the spheroid is approximately linear as ¢b z aEOH , a z 4.5 ~ 5.0 . (3.32) where ¢b in kV, E0 in kV/m and H in m. 67 EMPIRICAL FORMULA 1 ‘ - .- , 1 1‘ \ , , , , 0 I$C - 5.4-10 9-H2-Eo-f/60 \ \ I . ‘\ , ‘ " I for H = 1.70 m (HEIGHT) Xx \, 1 , , w . 65 kg (HEIGHT) . , 15c =15.6flA/kV/m _. \ / E° \ A ” I ( ) ‘ 7 ' f = 60 Hz ~. 13'; / 1 .1 1 t a : major axis b : minor axis a/b 1.2 1.8 2.2 3 a 28.23 cm 37.0 cm 42.3 cm 52.0 cm 0 23.53 cm 20.6 cm 19.2 cm 17.3 cm H(=2a) 56.46 cm 74.0 cm 84.6 cm 104.0 cm H 65.0 kg 65.0 kg 65.0 kg 65.0 kg Isc 7.5 flA/kV/m 11 flA/kV/m 16 flA/kV/m 25 uA/kV/m Fig. 3.19. Comparison of theoretical results by the present method with empirical results on the shortocircuit current for spheroidal models of man with various axis ratios when exposed to a l kV/m, 60-82 electric field. ¢b a 240 V 4’5 = 364 v H = 80 cm 68 a: major axis 0: minor axis HEIGHT: H=Za Eg= 1 kVém 4’6 = 281 v t"O : : ...’ .Oee H 60 cm 405 V J H = 90 cm (a/b=2.2) 411, = 323 v H = 70 cm 4’b = 487 V H = 110 cm Fig. 3.20. Floating potentials of spheroids with the same axis-ratio (a/b - 2.2) and various sizes when exposed to a l kV/m, 60-Hz electric field. w R1 51 '3. st ‘3 re, ‘A.') CHAPTER 4 APPLICATIONS OF SURFACE CHARGE INTEGRAL EQUATION METHOD TO ANIMAL AND HUMAN BODIES 4-1 1113224223128 The validity of the surface charge integral equation (SCIE) method developed in chapter 2 was confirmed in chapter 3. In this chapter, the SCIE method is applied to quantify the interaction between 60-Hz electric fields and animals as well as human bodies. Theoretical results will be compared with existing experimental results to check the accuracy of this method. Also in order to investigate the frequency range limitation of the SCIE method, the short-circuit current of a human body induced by LF-HF electric fields is computed by the surface charge integral equation method and compared with experimental results by other workers. 4.2 i a Bod e om a ison with W In this section, the SCIE method is applied to quantify the short-circuit current, the induced surface charge and body current of a grounded guinea pig with a length of 22 cm and a height of 8 cm. An incident ELF electric field of 60-Hz and 10 kV/m has been considered in order to compare with the experimental results of Kaune and Miller [9]. In the experiment by Kaune & Miller, two guinea pigs, weighting 69 W a.— 1.- - 70 794 and 844 g, were adopted. Surface areas of two subjects were measured to be 0.05 m2 and 0.053 m2. A simplistic model of a guinea pig which simulates the actual animal used by Kaune & Miller is shown in Fig. 4.1. For the numerical calculation, the body surface of the guinea pig is partitioned into 228 patches. Due to the symmetry of the body along the x-axis, only a half of the body surface (114 patches) will be computed. Let the central point of the mth patch be P(?m), ; - (xm, ym, 2m), and that of the nth patch I . - m be Q(rn), rn - (xn, yn, zn), then the matrix equation (2.14) of the SCIE method can be written as, 11 12 21 22 ¢oz - - - (4.1) MNl MNZ .......... MNN -1 "N ¢oN AS AS where n - surface charge density of the nth patch ASn - surface area of the nth patch' ASn l l l l “mu - + + + + -> - + + _ -> -> 416 Ir - r Ir - r [r - r Ir - r I m 4n for m a n (1) C111 71 u/T 2.1/1 ANTémon POSTERIOR RUMP NECK CHEST ANTERIOR POSTERlOR CHEST aeoomew EXPERIMENTAL GUINEA PIG (Kaune & Mi11er) 'T' Fig. 4.1. 22 cm THEORETICAL MODEL FOR GUINEA PIG Experimental guinea pig and the theoretical model for the SCIE numerical computation. 72 for m - n and N4 rln - (xnv ynv Zn), 2n - (x 2 -y 2 Zn) 5 5 r---"1 ;3n - (xn’ yn’ -zn)’ ;4n - (x ’ -y ’ -2 ) After surface charge densities are computed, the short-circuit current can be calculated by using Eq. (2.5). The experimental results of Kaune & Miller and numerically evaluated results on the surface electric field enhancement factor, the short-circuit current and the axial sectional current are shown in Fig. 4.2(a) & (b) for comparison. Fig. 4.2(a) shows the electric field enhancement factors measured at various locations on the surface and the sectional currents at five cross sections of a guinea pig. Fig. 4.2(b) shows the electric field enhancement factors and five sectional currents calculated for the theoretical model of the guinea pig. Good agreement is obvious when the corresponding experimental and numerical values of these quantities shown in these two figures are compared. The tables at the bottom of Fig. 4.2(a) & (b) indicate the comparison of the measured short-circuit currents and the calculated values. The measured short-circuits for two guinea pigs are 0.219 and 0.225 pA/(kV/m) and the calculate result is 0.212 pA/(kV/m). The results shown in Fig. 4.2 demonstrate the accuracy of the SCIE method. 73 (ELECTRIC FIELD ENHANCEMENT FACTOR) 2-5 2.4 1 1 12.6 1 1 1.8 . , 1 300 nA :1. 4 nA :_, 420 nA 1,380 n 1 ' ' 1 1 | 1 0.8 1 1 1. 1 c | Q ' 2 l I ' 1 I l I ' 1 ' I 1 7" 1— ANTERIOR _ POSTERIOR 1 RUMP NECK . CHEST ANTERIOR POSTERIOR CHEST ABDOMEN :******************************* EXP (Kaune & Miller) ; :******************************* Surf 8“ 41’“ £218.22 .ILQEALLE‘L/JELL (1) 0.050 m2 794 g 0.219 (2) 0.053 m2 844 g 0.225 Fig. 4.2(a). Experimental results of Kaune and Miller on the electric field enhancement factor, the sectional current and the short-circuit current for a grounded guinea pig exposed to a 60-Hz, 10 kV/m electric field. A 74 (ELECTRIC FIELD ENHANCEMENT FACTOR) oor—v—NNNu 0.5 0.2 0.022 0.0021 0.0076 0.4 0.12 0.02 0.0015 0.0041 0.8 0.4 0.12 0.02 .0013 0034 ‘ .1.0 0.5 0.2 0.03 0.0003 0.0034 0.4 0.12 0.02 0.0013 0.0037 0.02 .0018 .0051 0 0. 4mJnA -1.-- 320 M 1‘- 3.0 nA I 1-- 1"350 M 1 (SECTIONAL CURRENT) ****************** ; PRESENT THEORY * ****************** Surface Area JEEEQBE Ig£_(flA/(kV/m)) T90 8 0.212 0.048 m2 Fig. 4.2(b). Theoretical results by the present method on the electric field enhancement factor, the sectional current and the short-circuit current for a grounded guinea pig exposed to a 60-Hz, 10 kV/m electric field. 75 4.3 flgmgriggl Results for Human Model and Comparison with Ezigging Experimental and Theoretical Results In th experiment by Kaune and Forsythe [10], induced current densities were measured in a homogeneous grounded phantom model of man (45 cm in height, as shown in Fig. 4.3) exposed to a vertical, 60-Hz electric field. To compare the present method with their experimental results, the SCIE method is employed to quantify the interaction of a 60-Hz, 10 kV/m electric field with a similar theoretical model of man (Fig. 4.3). In the numerical calculation, in order to reduce the size of the matrix equation, the theoretical human model is constructed to have 4-quarter symmetries along the x-axis and the y-axis. The body surface is partitioned into 472 patches and that leads to 118 unknowns with a quarter-body symmetry. Let the central point of the mth patch -9 4 be P(rm), rm - (xm, ym, 2m), and that of the nth patch be Q(rn), rn - (xn, yn, Zn)' The matrix equation of the surface charge integral equation can then be written as 11 12 21 22 2N "2 ¢02 MNl MN2 .......... MNN —l "N ¢oN AS AS .......... AS 0 ¢b O where 76 .cofiumuaaeoo fimUwumEDc mHUm ecu wow Hones ~mofiuwuoo£u mug 6cm AEu no I uzmfiosv :mE we Hobos AnacoEMuoaxm .m.¢ .wwh 2.55m: Emmott 52 no doc: ._ -> --+ + __, _, - + + mfin Irm - rSnl Irm - r6nl Irm - r7nI lrm - r8nI AS { 2/fl I 1 l M --——-—— '—————————— + -————————— + —————————— + -————-———— m 1/2 -> -> + -> -> + 4er (AS ) Irm - rzml Irm - r3m| Irm — ram] 1 l 1 l -+ + --> -> _—> -> --+ .,. m-n Irm - rSml Irm - r6mI Irm - r7ml 'rm - r8ml and rln - (xn, yn. 2n). an - (-xn, ya. 2 ) r3n - (xn. -yn. 2n). rtm - (-xn. -yn. 2n) r5n - (xn. yn. -zn). r6n - (-xn. yn. -zn) ; - (x p _y 9 -2 )v ; - (‘X 0 _y ’ -2 ) 7n n n n 8n n 78 After surface charge densities have been computed, the short-circuit current and induced body currents can be calculated accordingly. Fig. 4.4(a) & (b) shows numerical results of the surface electric field enhancement factor and the short-circuit current of the theoretical human model. Fig. 4.5 depicts the comparison of the experimental and numerical results of the induced current density inside the experimental model and the corresponding theoretical model of man. The right figure shows the measured current densities at various points inside the body. The induced current is mainly longitudinal (or vertical) with a small radial (or horizontal) component as shown in the figure. Notice that the direction of the radial component is outward in the chest region but it is inward at the neck and the abdomen region. The most interesting observation is that the induced current in the arm is directed downward, or in the opposite direction to the current flowing in other parts of the body. The numerical results on the induced current density in the theoretical model which approximates the experimental model are shown in the left figure. It can be seen that the computed current densities, in amplitude and direction, at various points inside the body agree very well with the measured values. It is also noted that the present theory correctly predicted the reversed direction of the induced current in the arm, and the directions of the radial components of the currents at different parts of the body. The results of Fig. 4.5 again give a positive verification on the accuracy of the SCIE technique. FRONT VIEW 1 .1 0.8 0.5 0.1 Isc- 12.75 M (PRESENT meow) Isc- 10.95 “A (BASED ON Isc-5.4-‘|0‘9-H2-E-f/60 see Chiba et al.) Fig. 4.4(a). Theoretical results on the electric field enhancement factor and the short-circuit current for the experimental model of man (used by Kaune 6: Forsythe) standing on the ground and exposed to a 60-Hz, lkV/m electric field. 80 l 2,3 Ln.mn ---“ 0.7 .3I.13l I l I .L .. O 45 cm 1'6 lfloaI -4 -4--4 I 2.0 9:9.9: SIDE VIEN 0.9 0.7 " 0.4 * - L 9.9 7 7 7 / 7 Fig. 4.4(b). Theoretical results on the electric field enhancement factor and the short-circuit current for the experimental model of men (used by Kaune & Forsythe) standing on the ground and exposed to a 60-Hz, lkV/m electric field. I III}; *4) -—e E§t-'"’ —) —. —§ -—-e 5+»; ~43 8‘ ' r—§ -e u H 0 _.§ —’ p—OO at. _s -—eSt——e N gar. .__. 1293 1293 Present Method Fig. 4.5. 81 180 v ...-V.'.v :-. '\t".n:"y ryv—qv-y - -. "r 3;}; . ;.. _ . - ..o-.v.""r - * .9 , 3 ‘3... .33.”; '-..~-‘- ‘. [J 4 . . {no}: I if; nib-5%" ' x‘ N .I' ' ' s. . 9 n . ~; 9 ;,J;; i '- 2'3": £7 TTITT 296 277 B6 . ., 311 297r ‘3 'F' L L011 .138 «173 Kaune & Forsythe Comparison of theoretical results by the present method with experimental results of Kaune and forsythe on vertical and horizontal current densities for a grounded human model exposed to a 60-Hz, 10 kV/m electric field. Induced current densities are given in units of nA/cm . ' {fi'fii‘t'fi‘ .‘~ - f. 5' I 91. T3201 ,1: lil ' ig. ,3" - r. ‘ w. .r.-.- 5,499,. 9 ~‘-‘ ' 23' 1"- ' T' ‘ ' a. '. 7M. 3'- ' .1440 few 3~.. 40 «30 (cm) 82 4.4 e c Re 3 fo the Model of Adult Man with In this section, the surface charge integral equation method is applied to a model of man with a height of 180 cm and a weight of 68.2 kg standing on the ground plane with grounding impedances, and is exposed to a 60-Hz electric field of l kV/m. For numerical calculation, the body surface is partitioned into 424 patches leading to 106 unknowns with a quarter body symmetry. Fig. 4.6 shows the calculated electric field enhancement factors on the body surface and the short-circuit current of a man standing upright and in direct contact (short-circuit case) with the ground. The enhancement factor on the head is about 18 which is almost the same as the measurement reported by Deno [S]. The calculated short-circuit current is 18.0 pA/(kV/m). This value is very close to 17.5 pA/(kV/m) which is calculated with the empirical formula of I - 5.4 x 10'9 x (f/60)E H2 (4.3) sc 0 used by Deno [5] and Chiba et al. [15]. Fig. 4.7 depicts the electric field enhancement factors and the short-circuit current in the same man with stretched arms indeced by the same electric field as the case of Fig. 4.6. It is observed that the surface electric field can be very high at the tip of the hand due to its sharp geometry. Also it is noted that when the arms are stretched, the short-circuit current is increased 23.3 pA. This value is quite different from the value of 17.5 pA W H-180 cm I\ l [7 Fig. 4.6. #1 <—1 N 4:. n 3 4:4» 3 +> _. 1..» n 3 4+; _. u n 3 .11. .4 N ’"I—" O n 3 Isc = 18.0uA (Present Theory) 1“ . 17.5 101 (Based on 15:: 5.4-10‘9-H2-E-f/60 see Chiba et a1.) Theoretical results on the electric field enhancement factor and the short-circuit current for a realistic model of man standing on the ground and exposed to a 60-Hz, l kV/m electric field. 17 14 13 9 1 8 3. .5 6.7 29 13 0 11 1217 4 v 3'3 4'3 6. 6.6 3'210 13 11 .1 - 23.3 PA (911555111 111501111) 15:: 17.5 #A (BASED ON Isc=5.4-1O'9-HZ-E-f/60) ISC Fig. 4.7. Theoretical results on the electric field enhancement factor and the short-circuit current for a realistic model of man with hands stretching horizontally standing on the ground being exposed to a 60-Hz. l kV/m electric field. 85 if the same formula used by Deno [5] and Chiba et al. [15] is used. It is because stretching arms will increase more induced surface charge and hence short-circuit current. In general, we found that the induced electric field at the body's surface, the short-circuit current, and consequently the induced current inside the body, are strongly dependent on the body geometry and posture. Even though the phenomenon involed is rather complicated, the present method is capable of predicting them. The effect of grounding impedances on the short-circuit current of a man exposed to a 60-Hz electric field of l kV/m is given in Fig. 4.8 to 4.10. Grounding impedances are assumed to be resistive, capacitive or inductive. As the same phenomena observed in Sec. 3.3, for the resistive or capacitive impedance, the short-circuit current is found to remain practically unchanged, maintaining a value of about 18 uA/(kV/m) when the grounding impedance is varied from zero to about 10 MG. Only after the grounding impedance exceeds the value of 10 MD, the short-circuit current starts to fall down. For an inductive grounding impedance, there is a_possible resonance phenomenon when the impedance has a value of 100 M9, corresponding to an inductance of 3 x 105 H (henries). This implies that in a in a very unlikely case when the grounding impedance is an extremely large inductance of about 3 x 105 H, there may be a very large current induced in the body causing health hazard. Chiba et al. [15] applied the Finite Element Method (FEM) to analyze induced current densities inside an axially symmetrical human model of revolutional geometry, when exposed to a 60-Hz electric field as shown .mosmumfimmm wzupczouw uo comuussw a mm uzmuuao amzouuo-uuozm .m.q .wum as s so. we. a: as me .2 m2 ~2 2 @— .N— .m— @— aw Um 2:32.: \ Al.\)¥~\<\\» 87 moszMQQOo wcwtcsomw uo :cmuoczu m as scosmso uwsomwo-uuosm .¢.q .wwm u A: :-c_xmm.~ mbzmofi Nuc-xmod m-c_xmm.~ c 3 1 a d d . 1 q 1 Cd— :oa_<:.:~ E. as as a2 so. me a: 1: ~2 c A N l v o . a 33.3: n 4N @— oma «\s _ _ omH N\H . ~— 0 U . v— . @— 2 8 32.3.3. Um \ AA=\\\1‘\\a.\ \ 180 cm fij V V V 'T V HEIGHT (cm) ’ l. a L 4L 1 l l. l 1—— . e . e g s0 0e...'.'.‘ .‘o. a... e Method 0 50 100 150 200 "m“ nA/anZ/(kV/m) Fig. 4.13. Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional current densities for a grounded human model exposed to a 60- Hz, 1 kV/m electric field. 93 0 Theoretical I ’ ‘1 I _ \ + Experimental 15:,- X 'O-Data from [5] e ‘ ' 100- U 2 A r . 9 E I E 50- ‘ L 0 5 Chiba [15] I ; ' 1504 =1; 3:- I 100 ‘ S .3; 50 q Lav. . I ' T 0 s 10 1'0 20 25 uA/(kV/m) Present. method” Fig. 4.14. Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional currents for a grounded human model with hands stretching horizontally exposed to a éO-Hz, 1 kV/m electric field. 94 71"" 1 50 - I . .::2. i -e— Theoretical \ -N- Experimental 1CK3 5 x 2 .5. E 50- 1 l I L 7' 7 . 7 7 0 50 100 150 200 Chiba [15] nA/cmz/(kV/m) 180 'fIf—V jvvvv f1 HEIGHT (cm) IJ Q I 180 cm —--—— —" r V 1 r " ' 0 50 100 150‘ 200 nA/cmz/ own» A _ l 4 Present'hethod Fig. 4 15. Comparison of theoretical results by the present method with results of Chiba et al on vertical sectional currents densities for a grounded human model with hands streching horizontally exposed to a 60-Hz, l kV/m electric field. 4.1 and the 51101 at freq numeric agreeme 40 MHz. data 51 which stand: resona human Polar Chen methc for i 95 4.1 and Fig. 4.16 show the comparison of experimental results on the short-circuit current, measured by various workers [19] ~ [23] at frequencies over the ELF and HF range, and the corresponding numerical results generated by the present method. A satisfactory agreement is obtained between theory and experiment up to about 40 MHz. It is interesting to note that in Fig. 4.16 the experimental data show a maximum current being measured at the frequency of 40 MHz,. which is the resonant frequency for the human body (1.7m ~ 1.8m) standing on the ground. This is consistent with the fact that the resonant frequency of the human body in free space, at which the human body absorbs a maximum RF energy in response to a vertically polarized EM wave, is 80 MHz. This resonant frequency was found by Chen et al. [27] using the electric field integral equation (EFIE) method and it has been verified experimentally by many workers [28], for a human model in free space. It is well known and as depicted in Fig. 4.16 that the induced current in the body starts to decrease once the frequency exceeds the resonant frequency. However, the SCIE method predicts a linear increase of the induced current with the frequency. This indicates that the SCIE method may be applicable up to the first resonant frequency in the HF range when the method is applied to the human body. Isc (mA/(v/m)) 96 e EXPERIMENT [23] -- SCIE 4' TT’ 1' F ’1 20 30 4O 50 60 FREQUENCY (MHz) Fig. 4.16. Comparison of theoretical results by the SCIE method with experimental results of Candi et al.[23] on the short-circuit current for a grounded human model (~l.75 m in height) at frequencies from 20 MHz to 50 MHz. 97 Table 4.1 Comparison of experimental and empirical results on the short-circuit current with theoretical results by the present method. Isc (mA/(V/m)) H - 1.75 m H - 1.75 m H - 1.80 m f (MHz) Eq. (*) MEASURED PRESENT METHOD subject barefoot 0.630 0.208 0.210 0.189 0.700 0.232 0.280 0.212. 1.510 0.499 0.391 0.453 27.405 9.060 9.330 8.230 subject barefoot & arms raised 0.700 0.384 0.272 1.510 0.555 0.586 27.405 9.850 10.600 comparison data 0.146 [21] 0.048 0.035 0.044 0.720 [20] 0.238 0.277 0.216 0.920 0.304 0.316 0.276 1.145 0.379 0.366 0.344 1.350 0.447 0.405 0.405 1.470 0.486 0.560 0.441 27.000 [22] 8.930 8.400 8.130 2 (*) Isc - 0.100 x sou fMHz [19] 4‘ CEUUHEER 5 TENSOR ELECTRIC FIELD INTEGRAL EQUATION METHOD FOR INDUCED ELECTRIC FIELDS INSIDE BIOLOGICAL BODIES 591 1.113121111331211 In this chapter, in order to compare the surface charge integral equation (SCIE) method with other existing theoretical methods, the tensor electric field integral equation (EFIE) method is applied to quantify the interaction between ELF electromagnetic fields and biological bodies. The induced electric field and current inside an arbitrarily- shaped biological body can be quantified by this tensor integral equation method. Instead of using low-frequency approximations as the SCIE method, the EFIE method is an exact formulation for quantifying the interaction of electromagnetic fields with biological bodies. Hence, it is appropriate to compare the accuracy and efficiency of the SCIE method with the EFIE method. Also, iterative methods will be employed to solve a large matrix equation transformed from the tensor integral equation. The convergence rate of various iterative algorithms will be compared. 592 W The tensor integral equation method used in this chapter was developed by Chen and Liversay [26]. When a biological body is 98 99 illuminated by an electromagnetic wave, an electromagnetic field is induced inside the body and an electromagnetic wave is scattered by the body in the region exterior to the body. In general, the biological body is an irregularly-shaped heterogeneous system with frequency dependent permittivity and conductivity. The induced electromagnetic field inside the body and the scattered electromagnetic wave will depend on the body's physiological parameters, geometry as well as frequency and polarization of the incident electromagnetic field. The electric field induced inside the body is the key quantity which determines the induced current inside the body. Consider a finite irregularly-shaped biological body, as shown in Fig. 5.1, with electrical parameters expressed as: conductivity : a - a(w, * 11 V permitivity : e - ¢(w, ¥) permeability : p - no where the biological body is assumed to be a non-magnetic medium. When it is illuminated by an incident electromagnetic wave with an electric field 31(2) and a magnetic field 81(2), the incident EM fields in the free space satisfy the following Maxwell's equations : v x §1(¥) - -jwpofii(;) (5.1a) v x 81(2) - jee°§i(?) (5.1b) ~01 4 V - E (r) - 0 (5.16) v - fi (?) - 0 (5.1a) 100 free space biological body Fig. 5.1. An arbitrarily-shaped biological body in free space, illuminated by an EM plane wave. where e o 101 and #0 are the permitivity and permeability of the free space. The induced charge and current in the body produce scattered fields ‘E’S (f) and R (2). field and the scattered field : Thus the total field is the sum of the incident (5.2a) (5.2b) where E(;) and R(f) are the total electric field and magnetic field. The total EM field existing at any point in space including the biological system should also satisfy the Maxwell's By using derived : equations : E<¥> - -jwpofi(¥) (5. fi(?) - a(E)E(¥) + jwe(¥)fi(?) (s. [ + jwe<¥)>E<¥)1 - 0 <5. - fi(¥) - 0 (5. Eqs. (5.1) through (5.3), the following equations can be ‘E’S PU» ( > - -quofis(¥) (E) 514 (3) - [a(?) + jc(e(¥) - eo)]E(¥) + jweOE 3a) 3b) 3c) 3d) (5.4a) (5.4b) 102 An equivalent volume current density 3eq(;) is defined as 4 -e 4-0-9 Jeq(r) - r(r)E(r) (5.5) where ¥(t) - a(¥) + Jw(e(¥) - Co) is the equivalent complex conductivity. Eq. (5.4b) can then be rewritten as v x fi‘(¥) - 3 (2) + jwe E‘(¥) ’ (5.6) sq o 3eq(;) has two components; a(;)E(;) the conduction current component and jw(¢(;) - ¢O)E(;) the polarization current component. jeq(f) exists only inside the biological system. From the equation of continuity for 3eq(f), an equivalent volume charge density peq(;) can be defined as v . 3eq(t) + jwpeq(r) - 0 (5.7) Taking divergence of Eq. (5.6) and using Eq. (5.7), it leads to v - 33(2) - pmd’veo (5.8) Now, Maxwell's equations for Es(f) and fis(f) can be expressed in -e -e terms of J.q(r) and peq as 103 v x Es“) - -3cpofis(¥) (5.9a) v x 713(7) - 'J'eqd?) + jweoEs(-f) (5.91:) v - 353d?) - peq(;)/¢o (5.9c) v - 11%?) - 0 (5.9.1) 19:90:) - P.V. I 3 (r')-G(r,r')dv' - —9L— (5.10) as V ijeo where 4 4 W 4 4 8(t,t') - -jwe [ I + ]w(r,t') o k2 0 e-Jkol'f-i" I P.V. symbol means the principle value of the integral and 8(33') is the free space dyadic Green's function. By substituting Eq. (5.10) into (5.2a) and rearranging terms, a tensor electric field integral equation (EFIE) can be obtained 104 9.29 . .... ..... . [1 + -————-—]§(t) - P.V. r(r')E(r')-G(r,r')dv' - E 3jw¢o V 1(I) (5.11) i where 7(f) and E (f) are known quantities and E(?) is the unknown total field inside the body. 5.3 nsaant_nsth2d_ssisti9ne2f_Isnaer_lntssrsl_£snaticn In the actual calculation, it is very difficult to solve the tensor integral equation by performing the integral which involves unknown E(;) inside the integral. The Moment Method provides some simplification and approximation to solve the integral equation numerically. In the Moment Method, the total volume of the biological body is partitioned into N subvolumes or cells as shown in Fig. 5.2. E(;) and r(¥) are assumed to be constant within each cell. Therefore, the accuracy of this method depends upon the dimensions of the cell. The integral equation (5.11) can be transformed into 3N simultaneous equations for (Ex’ E , Ez) unknown quantities at central points of Y N cells by point matching method. These simultaneous equations can be written into a matrix form as i 1°99.) [an] [an] 13991 [Ex] 1 [cu] ‘ny] 16,21 [2y] - - [2,1,1 (5.121 16231 [Gzyl [0,21 12,1 12,1 Fig. 5.2. 105 free space Biological Body 0 (partitioned into N cells) n'th cell : 10(73): ((7%), #0) 71771) = 017;) i' 341(65‘71) - ‘0) The biological body is partitioned into many small cubic cells for numerical calculation of the induced electric field. 4’Z ,.j 1.5"“ . 1 106 where the matrix [C] represents a (3N x 3N) matrix, while [E] and [E1] are 3N column matrices expressing E and E1 at central points of N cells. The explicit expressions for each (N x N) submatrix [Gx x ] elements, P q (P99 - 1,2,3), are given in [26]. Let (x1 - x, x2 - y, x3 - 2) Then the (m,n)th off-diagonal element of the [Gx x ] matrix is P 9 ~ -a -jwu k r(r )AV e an 0 - ° ° “ n x (5.13) x x 4na3 P q “I [(a2 -l - jo )6 + cosommcosomm(3 - a2 +-31a )] m # n a: n: pq X x a: u: ’ P q where 1: - k , R - I; - f u: mn mn m n m n mn x - x coso _ _P__P__ x R p mm m n x - x coso _ __S____S__ x R q mn ; - (xm xm xm) ' f - (xn xn x“) m 1’ 2' 3 ’ n 1’ 2’ 3 AVn - volume of the nth cell 107 The (m,n)th diagonal element of [Gx x ] matrix is P q Zpr r(; ) T(; ) cm --6 {—-°———“—[e'3koan(1+jke)-1]+[1+ In1 q Pq 3ko o m 3jweo where 3AV m 1/3 (5.14). 4: After all elements of the [G] matrix are determined, the total induced electric field E(;) inside the body can be obtained by inverting the matrix as 1 -1 i [E] - [G] {-E 1 4 (5 15) The induced body current density 3(f) can then be determined as 3(13) - r(¥)E(?) (5.16) 5.4 W In order to yield accurate numerical results from the tensor electric field integral equation method, sometimes the body needs to be partitioned into a large number of small cells. This will produce a large matrix equation transformed from the integral equation. 108 Conventional methods for solving the matrix equation, such as Guassian Elimination Method (GEM) or LU decomposition, require the computer memory storage of the whole matrix. When the matrix equation becomes excessively large, it often exceeds the storage capacity of the computer. Iterative techniques offter the possibility of solving large matrix equations without requiring the storage of any matrix. In a survey of numerical techniques for solving large system of linear equations, Sarkar et al. [36] compare the relative advantages and disadvantages of various iterative methods. Here the Conjugate Gradient Method (CGM) and the Guass-Seidel Method (GSM) are adopted to solve the matrix equation iteratively. 59491 W The basic principles of the Conjugate Gradient Method are described in an original paper by Mestenes and Stiefel [29]. For the solution of any complex matrix equation, Ax - b, the iterative process of the CGM starting with an initial guess x is defined as : 0 r1.'- Aug: -113 H pl - -A ro H 2 2 t,, - IIA rnll / IIAPnII . n = 1.2.3 rn+1 - rn + tnApn H 2 H 2 109 and xn+1 - xn + tnpn p a -AHr + q p n+1 n n+1 1'1 where AH : the complex conjugate transpose matrix of the matrix A rn_: the residual vector of iteration pn : the direction vector of iteration It has been proven that [31], if A.is a positive-definite matrix, the CGM converges to the exact solution of the matrix equation in at most N-step iterations (N - the order of the matrix A), assuming no round-off errors occurs. This finite-step convergence is an important advantage of the CGM over other iteration methods. However practically, due to round-off error in the computation, the CGM needs more than N steps iteration for the desired solution. But the iteration can be terminated at an earlier step, if the degree of accuracy is within the acceptable range. Also, in order to guarantee the positive-definite nature of the matrix A, instead of solving the matrix equation, Ax - b, we will solve the matrix equation, [AHAJX - A310 (5.17) since [AHA] must be positive-definite and the solutions of both matrix llO equations are the same. In order to check the convergence of the iterative process, the following measure of accurracy is defined : nun-w ERROR RATE '- (5.18) llbll A value of- 10-4 may be a reasonably good choice for this measure. 594.2 W The description of the Guess-Seidel method can be found in [31]. For the complex matrix equation, Ax - b, the matrix can be written as, A-L+D+U (5.19) where D,L,U are the diagonal, lower, uper triangular matrices of the matrix A. Thus the matrix equation is rewritten as [D + L]x + Ux :- b (5.20) Starting with an initial guess x0, the iterative process of the GSM is defined as [p + 1.1): - -an + b (5.21) n+1 or 111 x - -[D + L]-lan + [p + L]’1 n+1 1" l - M xn + [p + L)” b (5.22) G The matrix M I -[D + L] -117 is called the Guass-Seidel matrix. It can G be shown [31] that the GSM iteration process will converge for an arbitrary initial guess if and only if p(M ) - max |.\ (M )| < 1 (5.23) G 151511 1 G where p(MG) : spectrum radius of the matrix MG i G) : ith eigenvalue of the matrix MG ‘ It has also been shown that [31] , if the matrix A is symmetric and positive-definite, then the spectrum radius p(MG) is smaller than 1. Hence, to guarantee the convergence of the GSM, we will solve the following matrix equation, [AHA1x - AHb (5.24) The convergence rate of the GSM can be improved by so-called Successive Over-Relaxation (S.O.R) process (or accelerated Guass- Seidel Method). The S.O.R method is a modified scheme of the GSM. Instead of using X of Eq. (5.22) in the iteration process, the n+1 following equation is used, 112 i :x '— wx n+1 + (1 - mxn xrrl - w(—[D + L]’1[an + b]) + (1 - 99)::n «125) where the parameter w is a constant that can be determined so as to yield the fastest rate of the convergence. The Guass-Seidel Method is the special case w - l of Eq. (5.25). It has been proven that [31], any fixed constant w, l < w < 2, will accelerate the rate of convergence in the iteration process. 5 4-3 Initisl_§usss_Isr_ths_Itsratirs_Msthsd Iterative methods as discussed in previous sections, the CGM and the GSM, will converge for any initial guess as long as the matrix is positive-definite. However, a good initial guess (close to the exact solution) can reduce iterative times to yield a solution in a desired accuracy. A good choice for the initial guess xo of the iterative process for solving a matrix equation Ax - b is o I- “11 b1/3111 O 2 x b2/a22 x0 _ . _ , (5.26) ° b -xN- - N/aNN- where [aii] are diagonal elements of the matrix A and [bi] are column 113 elements of b. This is equivalent to solve a matrix equation on - b (5.27) where D is the diagonal matrix of the matrix A. If the matrix A is diagonally dominant, xo of Eq. (5.26) is a fairly good choice for the iteration. A better choice for x0 is to solve a tridiagonal matrix equation which is T20 - b (5.28) where T is the tridiagonal matrix of the matrix A, F q 811 312 321 822 823 832 833 834 Obviously the solution of this matrix equation is closer to the exact solution of the original matrix equation Ax - b than that of the diagonal matrix equation (5.27). The tridiagonal matrix equation of Eq. (5.28) can be solved by a simple algorithm - Thomas Method [34] as follows, an: 114 q r - 2,3 ........ N a a r rr r,r-i r-i qr-i - ar-i,r/wr-1 and 81 - bl/wl gr - (hr — 3r r-lgr-1)/Wr r - 2,3 ........ N and these transform (5.28) to er. I A? o o xr - gr - qrxr+1 r - 1,2 ........ N-l (5.29) If the w, q, and g are calculated in order of increasing r, it follows that (5.29) can be used to calculate the x0 in order of decreasing r, o o o o that is, xN, xN-1, ...... x2, XI. The algorithm is easily described in matrix notation. The T matrix is decomposed into two triangular matrices r - wQ ' (5.30) where 115 Pwl q a21 w2 0 a w W _ 32 3 0 1. 171.; .1 and F l w1 7 l w2 0 w 3 Q - O wN-l 1 1 . Thus wQ!o ' b (5.31) By defining the column vector 3 8 ' Q30 (5.32) Eq. (5.31) can be written as Wg - b (5-33) The algorithm (5.29) can be derived from solutions of matrix equations Eq. (5.32) and (5.33). 116 5-4-4 Wants In order to compare the convergence rate of the CGM and the GSM when employed to solve the matrix equation transformed from the EFIE Eq. (5.12) at the ELF range, a numerical example on a biological tissue- layer illuminated by a plane EM wave of 60-Hz (Fig. 5.3) is given. The dielectric constant at and the conductivity 0 of the tissue-layer are assumed to be 7 x 104 and 0.04 (am).1 at this frequency. The tissue-layer is illuminated by the incident wave at normal incidence, and the incident electric field is directed in the x-direction, as shown in Fig. 5.3. The tissue-layer is partitioned into 144 cubic cells. This leads to 108 (- 3 x 36) unknowns in the matrix equation (5.12) when a quarter-body symmetry is applied. Convergences of the CGM and the GSM when solving the matrix equation for the induced current in the tissue-layer are shown in Fig. 5.4. Comparisons of the x-component induced current densities ‘computed by using the GEM (Guassian Elimination Hethod), the CGM and the GSM (300 iterations) are shown in Figs. 5.5 ~ 5.8. It can be observed that the GSM has a faster convergence rate than the CGM for solving the matrix equation of EFIE at the ELF range. More numerical examples will be shown in the following section. In order to compare the scalar SCIE method with the tensor EFIE method, the electric field integral equation is applied to quantify the induced current inside the phantom model of man used in the experiment by Kaune and Forsythe [10] , when illuminated by a 60-Hz plane EM wave. 117 i HY INCIDENT WAVE * FREQUENCY 60 Hz 0.04 ZS/m 7x104 6 o * TISSUE: d 6 Fig. 5.3. A biological tissue-layer exposed to a 60-Hz EM plane wave ERROR RATE 118 1 1t 4» 4 1 - ---- CGM .1 L 10-11'> 4 4 D J 10-21 ‘* GSM + . 4 10‘3 1 4 4 T l 4 4 L - l L 4 L 1 1 k L 50 100 150 200 250 300 ITERATION NUMBER Fig. 5.4. Convergences of the CGM and the GSM when solving the induced current in the tissue-layer of Fig. 5.3 by the EFIE method. 119 4'01 x = 0.5 cm -o- GEM '0‘ CGM 3.5-1 a GSM D 3.0- Jx (x 10'5 A/cmz) 2.5-4 2.04 1.5-4 1.0 g , . , . e e . , . j o 2 4 6 a 10 12 Fig. 5.5. Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3. when illuminated by a 60-Hz. l kV/m EM plane wave. 120 4.0“. A x = 1.5 cm -o— GEM NE 3 3.5-4 ¢: “5’ <3 3; 3.0+ X 3 2.54 2.0d 1.5-4 1.0 r f t I t T t r r I f 1 0 2 4 6 8 10 12 y (cm) Fig. 5.6. Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3, when illuminated by a 60-Hz. 1 kV/m EM plane wave. 121 4.0— x=2.5 cm + GEM A —a— CGM N— E 3.5.. a GSM < “‘3 9- 3.04 :1 >< —3 2.54 2.0-1 1.5-4 1.0 . Fig. 5.7. Distribution of the x-component induced current densities in the tissue-layer of Fig. 5.3. when illuminated by a 60-Hz, l kV/m EM plane wave. CGM (300 iterations) GSM (300 iterations) GEM. F211;. 5 .EB. illuminated by a 60-Hz, l kV/m EM plane wave current densities are given in units of A/cm2 layer of Fig. 5.3, J11 1.01 1.79 1.20 1 '1 1.71 1.09 1.01 1.99 1.00 1.7011 01 1.10 1.91 2.99 2.00 2.17 1.27 1 19 1.10 2.11 2.10 2.10 2.00 1.01 0 , a (31°? 2.91 1.97 2.91 1.71 1.90 1-99 2-3- 3-3311.10 1..9 2.1913-191 fl. v 6 Jv .109 .001 .092 .907 701 .900 .170 .101 .170 .191 .910 .-70 191 .10 .007 .999 1.01 1.11 1.10 1.10 1.11 1.01 .911 .090 ., (010 ) .009 111 .197 .110 191 .110 .099 .111 .110 .190 .199 .279 Jz ' 7.20 9.79 9.90 9.70 10.. 9.90 9.91 0.99 9.07 11 . 7.19 9.19 1.19 1.20 1.90 1.71 1.00 0.19 1.01 0.19 0.01 0.00 0.91 0 11 , , , 7 (0 10‘31) il.- 11.7 11..11.. 100010.. 1.90 0.59 9.03 7.! 9.101;.) x. J11 1.01 1.0911.00 1.09 1.07 1.0011.71 1.71 1 7071.79 1.90 1.97 1.10 1.10 1.10 1.10 1.21 1.19 1.29 1.90 1.99 1.09 1.91 1.10 (- ,g,,) 1.99 1.10 1.10 1.01 1.09 1.00 2.91 1.99 1.09 1.09 1.79 9.00 v 6 J17 .000 .019 .011 .000 .009 .119 .107 .917 .0711 017 .701 .979 .000 .019 .010 .001 .000 .090 .119 .109 .107 .100 .100 .010 (‘ 10-51 .000 .011 .011 .019 .000 .091 .009 .000 .097 .110 .119 .109 6 J2 0.0010.11 0.00 0.00 0.90 9.02 0.91l0.17 0.00 0.97 7.09 10.7 . .7. .6 .76. .9. 6.7 . . .7 1 9 1 0 1 19 9 1 9 1 10 1 9 1 99 1 1 0 9 19 1 1 (0 10-119 7.0 7.00 7.00 7.99 7.01 0.11 0.11 7.99 0.70 0.1 9.00 9.97 91 J1: 1. 1.00 1.09 1.09 1.0951.70{1.70 1.7 1.71 1.70 1.79 1.91 1.10 1.10 1.10 1.10 1.27 2.17 2.20lz.19 1.11 1.19 1.19 9.00 (. 10") . . . . . . . . . . .0 .91 290290190191191192191190719011 1 99 - )7 3v .091 .191 .199 .190 .0971.990 .0901.709 .019{.911 .907 1.07 .011 .007 .119 .190 .199 .201 .100f.1101.900i.000 .091 .009 (' 10") .007 .011 .097 .091 .000 .000 .090[.100].111 .199 .101 .109 A .1 0.91 0.90 0.70 7.11 7.00 0.09]0.00 0.0910.71 0.71 0.91 10.9] 0.10 9.00 9.09 9.79 9.01 0.91 2.01 1.11 0.70 1.1119.17 9.00] (0 10“*1 0.00[0.11 0.01 0.91 0.09 0.91 0.19 7. 0.99 7.07 9.10 9.19 Comparison of the COM and the GSM with the GEM on induced current densities for the tissue- Induced 123 These results are then compared with the numerical results by the SCIE method for the same model (sec. 4.2). Fig. 5.9 depicts the theoretical model of man which approximates the experimental model. Also, the experimental model and theoretical models for the SCIE method as well as the EFIE method are shown in Fig. 5.10 for comparison. The model used for the EFIE method is subdivided into several hundred cubical cells and a half-body symmetry is applied to reduce the number of unknowns. The electric properties of the body are assumed to be er - 7 x 104 and a - 0.04 (am)-1 at the frequency of 60-Hz. The model is illuminated by the incident wave at normal incidence, and the incident electric field is directed in the x-direction, as shown in Fig. 5.11. Fig. 5.12 depicts the comparison of numerical results by the EFIE method and corresponding numerical results by the SCIE method as well as experimental results of the induced current density inside the body. In the numerical calculation of the EFIE method, two cases have been studied : (1) the body is subdivided into 536 cubic cells which leads to a (804 x 804) matrix equation with a half-body symmetry (268 cells) and (2) the body is subdivided into 674 cubic cells which leads to a (1011 x 1011) matrix equation with a half-body symmetry (337 cells). Also in the calculation, the incident electric field is assumed to be 10 kV/m. Due to the symmetry, only one half of the body is shown for both cases. In the Fig. 5.12 it can be observed that, numerical results of the first case (268 cells) by the EFIE method disagree with those by the SCIE method and experimental results. Specifically, the direction of the induced current in the arm is upward which is opposite to the downward direction predicted by the SCIE method and experimental 124 45 cm Fig. 5.9. Geometries of the theoretical human model for the numerical calculation of the EFIE method. 125 EXPERIMENT (Kaune & Forsythe) SCIE 003900 0’ L g . 7"‘ [FLI I F1 F1 r~ F EFIE Fig. 5.10. Comparison of experimental human model with theoretical -models for the SCIE method and The EFIE method which approximate the experimental model. 126 _ 4 ‘r - 7 x 10 i a = 0.04 Ex 1 GROUND H _____. z y INCIDENT WAVE f = 60-Hz :‘ '1 g L, IMAGE 300v : : g 1 .‘ fl ,: i I I l I-.. r_l fl '1 i 3 n‘L - -/ Fig. 5.11. The theoretical human model of Fig. 5.9 standing on the ground is exposed to a 60-Hz EM plane wave. SCIE 268 cells (804 x 804) matrix equatiom Fig. 5.12. EXPERIMENT (Kaune & Forsythe) 337 cells (1011 x 1011) matrix equation EFIE Comparison of theoretical results by the EFIE method with that by the SCIE method and experimental results of Kaune and Forsythe on vertical current densities for a grounded human model exposed to a 60-Hz, 10 kV/m EM fifild. Induced current densities are given in units of nA/cm . 128 measurement. This disagreement is mainly due to insufficient partitioning of the body model in the numerical calculation. The accuracy of numerical results by the EFIE method can be improved by subdividing the body into more cells. Fig. 5.12 shows that the second case (337 cells) yields more agreeable numerical results than the first case, when compared with corresponding results by the SCIE method and experimental results. As shown in the figure, the induced current density in the leg is 1500 nA/cm2 which is close to 1293 nA/cm2 calculated by the SCIE method and 1440 nA/cm2 measured in the experiment. Moreover, the induced current in the arm is directed downward now, which is consistent with the result given by the SCIE method and the experiment. The accuracy of the EFIE method can be further improved by more partitioning of the body at the expense of excessive computer memory and CPU time. Also, the comparison of the convergence rate for solving the matrix equation by the CGM and the GSM is shown in FIg. 5.13. It is noticed that the GSM again has a faster convergence rate than the CGM at the ELF range. From the above comparison we note that, for quantifying the interaction of ELF electromagnetic fields with a human body, the EFIE method needs to solve a ~ (1000 x 1000) matrix equation while the SCIE method only requires ~ (100 x 100) matrix equation, also the EFIE method takes much more computational time to calculate matrix elements than the SCIE method. Furthermore, the SCIE method can be used to calculate the body's surface electric field and the short-circuit current through grounding impedances of the body which cannot be achieved by the EFIE method. To conclude, the SCIE method has much better numerical accuracy and computational efficiency than the EFIE method 129 in quantifting the interaction between biological bodies (or conducting bodies) and low frequency EM fields. ERROR RATE 130 CGM f fi 0 200 T r T r 400 600 1 f 800 ITERATION NUMBER ' 1000 Fig. 5.13. Convergences of the CGM and the GSM when solving for the induced current in the body model of Fig. 5.11 based on the EFIE method. CHUU?TE5{ 6 ANALYSIS OF SHOCK CURRENT BETWEEN HUMAN BODY AND METALLIC OBJECTS UNDER THE EXPOSURE OF ELF ELECTRIC FIELDS 61 11139111051211 In previous chapters, the SCIE method was applied to quantify the steady-state induced current and field of the biological body under the exposure of the ELF-LF electric field. In this chapter, transient response of the interaction between a human body and a metallic object both exposed to the ELF electric field of a high—voltage transmission line is studied. The SCIE method and the equivalent-circuit model will be adopted to analyze the shock current flowing between a human body and a conducting object. Numerical results on the transient shock current predicted by the present method will be compared with the experimental results by Reilly [32]. 6-2 Warrant—000L111; When a man and a nearby metallic object, such as a vehicle, are both illuminated by the ELF electric field emitted from an EHV power line, both the human body and the metallic object will acquire certain induced body potentials. If they are located very close, a large shock current may be generated between the man and the vehicle through direct contact or spark discharges, as shown in Fig. 6.1. This shock current is 131 132 T (large transient current) .- .0 1: 4’: ¢b ‘ 0 9971 96+ '901# ' 991 I 9 i 411 1 *91 11 0,, I F‘. :.(: * d“. (a) Human body short-circuited to the ground 1—-—> 3* 0.1;. 9119 4| 1 ‘0- n r u H '9- tr O l+¢¢¢6 Of’ ‘ 1“"? 1#*+P 9906+96¢++ +9100..- 000-. .I.. ‘ C e I (b) Human body with grounding impedances Fig. 6.1. Large shock currents flow between a vehicle and a nearby human body through direct contact or spark discharge, when both are exposed to the ELF electric field of an EHV power line. 133 due to a large potential difference between two objects (~ kV, if Eo - l kV/m), and it may cause serious hazards to the human body. To analyze the shock current, we will develop equivalent-circuit models for the human body and the vehicle based on their floating potentials and short-circuit currents calculated by the SCIE method. The time-dependent behavior of the shock current is then determined from a transient analysis of a system composed of the equivalent circuits. 6.2.1 Eguivglgnt-Circuit Models Consider a biological body standing on the ground plane with a grounding impedance ZL and exposed to an ELF electric field, as shown in Fig. 6.2. This system can be modeled by two kinds of equivalent circuits [3l] as shown in Fig. 6.2: (a) Thevenin equivalent circuit a Zeq - ¢bo/Isc (b) Norton equivalent circuit a qu - Isc/¢bo (qu - 1/zeq) where ¢bo is the floating potential of the body (ZL - w), ISC is the short-circuit current of the body (ZL - 0) and Zeq is the equivalent impedance of the body with respect to the system in the equivalent circuit. ¢ob and Isc can be determined by using the SCIE method as discussed in Sec. 2.2. When a biological body and a nearby metallic object are both exposed to an ELF electric field, the same equivalent circuit models 134 E0 11 I GROUND ZEQ ¢b ’b ‘00 ZL Isc Zeq Z1, _’ 1 '1 Theyerin equivalent circuit Norton equivalent circuit " *bo ' Open-circuit body potential ' Isc - short-circuit current a‘b (2L 8‘) a I (ZL a O) 200 ' ¢bo / Isc Fig. 6.2. A biological body standing on the ground with a grounding impedance Z is exposed to an ELF electric field. The circuits shown in the figure are the Thevenin and Norton equivalent circuit for this system. 135 can be constructed to represent the whole system. Fig. 6.3 depicts the the Thevenin equivalent circuit of a biological body and a metallic object at proximity. It is important to note that, due to the coupling effect between the biological and metallic objects, the floating potential and the short-circuit current of each object will be different from those when each object is isolated. To determine the coupled floating potential and the coupled short-circuit current of each object, a set of coupled surface-charge integral equation are developed as follows: 1 J n(;’) , ds' + ¢ (r ) - d (6.1a) i i + +, bo b b 0000 sb+sb+sc+sc Irb - r | 1 n(?') a * ds' + 0 (r ) - ¢ (6.1b) 4x0 3 +si+s +51 I? - r'I °° ° c o b b c c c Ib - ¢b/ZLb - Jwa - Jw J "(r')d8' (6 1C) s b Ic ' ¢c/ZLC ' Jch ' Jw J3 n(r')dS' (6.1d) C where Sb - surface area of the biological body b 5: - surface area of the image of the biological body b Fig. 6.3. 0"“ 136 .17 ‘0 U A biological body b and a nearby metallic object c standing on the ground with grounding impedances ZLb and Z , respectively, are both exposed to an ELF electr 9 field. The circuit shown in the figure is the Thevenin equivalent circuit for the whole system. 137 to l surface area of the metallic object c m I surface area of the image of the metallic object c body potential of the biological body b 6~ 0' I body potential of the metallic object c 9- I H I current flowing through the impedance ZLb H I current flowing through the impedance ZLc ¢bo(;b) - impressed potential on the bilogical body b maintained by the incident electric field - -Eozb ¢co(;c) - impressed potential on the metallic object c maintained by the incident electric field - -E z o c and rb - (xb. yb. 2b) rc - (xc’ yc’ zc) By using the Moment Method, the body surface S and the object surface Sc b are partitioned into N and K patches, respectively. Let Pb(;bn) be the central point of the A8: patch on the body b and Pc(;cn) be that of the A8: patch on the object c, the induced'surface charge density on each patch is assumed to be a constant and represented by n: (for the patch ASb n) and 138 n: (for the patch ASS). Eqs. (6.1a) ~ (6.ld) can then be transformed into a (N + K + 2) x (N + K + 2) matrix equation rbb bb bb _ be bc bc 1-b- 0b- M11 M12 “111 1 M11 M12 "1x 0 ’71 ¢01 bb bb bb be be be b b M21 M22 M2N '1 M21 M22 "2K 0 "2 4’02 bb bb bb be be be b b "N1 M192“"‘1111 ’1 FLn1 M112""“1111 ° "N 45011 b b b 051 052 . . esN -1/(jszb) o o o 0 0b - _ 0 ch cb cb cc cc cc c c M11 M12 M111 0 M11 M12 "1K ‘1 "1 ‘501 ch ch ch cc cc cc c c M21 M22 M211 0 M21 M22 sz '1 ”2 4‘02 cb cb cb cc cc cc c c ”K1 “K2 0 ”x1 “x2 ' ”xx ‘1 "x ¢ox C C C 1 o o o 0 A31 2152215K -1/(302Lc)d A“ _ 0 . where (6.2) 1 1 1 ":2 "' AS: { .1 .0 ’ .. -1 } 4'60 Irbm - rbn| Irbm - rbnl -: _- ._. . ' ._" -_ 'IZTL' ' ,. bc mn cb CC mn bb bc cb CC 4x0 4x0 4x0 4r: 410 410 430 o 139 c l ASn + + Irbm - rcnl l ..;{ .. .. Ircm - rbnl c 1 ASn + + Ir -r l cm cn Ash { 2/ w m b 1/2 (ASm) c l ASm + + Irbm - rcml l ..;{ .. . Ircm - rbml Ase 217 m (05;)1/2 for m # n for m - n 140 and ¢ - _Eozbn ’ rbn (xbn’ ybn’ zbn) ¢ - -Eozcn ’ ;cn - (xcn’ ycn’ zcn) The (N +'K +62) unknowns, (ab ... nb, ¢ ) and (ac ... "C, d ), can be' 1 N b l K c determined from this matrix equation. 6.2-2 We: To analyze the shock current between a human body and a nearby vehicle both exposed to the electric field of a 60-Hz EHFV power line, a Plymouth automobile (L x W x H - 5.43 x 1.98 x 1.32 m) has been chosen for study. Fig. 6.4 shows the numerical results of the floating potential and the short-circuit current of the vehicle as functions of the total partitioned patch numbers of the vehicle's metallic surface. The figure also shows the comparison of numerical results on the floating potential and the short-circuit current of the vehicle with the corresponding experimental results [32]. A satisfactory agreement between theory and experiment was obtained. Next, to observe the coupling effect between a human body and a vehicle at proximity, a man (180 cm in height) and a Plymouth automobile on the same ground and both exposed to a l kV/m, 60-Hz electric field as shown in Fig. 6.5, are studied. The capacitance of the vehicle tires is assumed to have a value of 500 pF (z -j5.3 M0 at the frequency of 60-Hz), and the grounding impedance of the man is assumed to be infinity (floating). The floating potential ¢bo of the body and the 141 } 5 43m —{ F" 2m —‘l +——2m 1 OTZZm T [4‘ i ‘1- 0.65m L__ | 1.32m 0 T l I 0.250 i PLYMOUTH (L x W x H) - 5.43 x 1.98 x 1.32 (m) MEASUREMENT PRESENT 1051:1100 (19o patches) ¢co 367 V/ (kV/m) 428 V/ (kV/m) 1c“ .093 mA/kV/m .108 mA/kV/m IQK ¢Eo o 2 I ‘50 if ¢c(V/kV/m) ' 3 020 1 l 330 1 .. . 3 ‘ Whit; —x IcselmA/kV/m) 001 ‘ d9 4 0 I I « 9 “ ~11 f 1' * ‘ ‘ r 4— ‘ 9 a» h b 00 00 90 120 190 200 320 Pa“ mm “3 Fig. 6.4. Comparison of numerical results by the present method with experimental measurements by Reilly on the floating potential and the short-circuit current of a Plymouth automobile. Also shown are the convergences of numerical values for 0 and I ‘with respect to the partitioned patch number on.the iiehicle'cs”e metallic surface. 142 I ‘5 . . - . . . ’ \ ' - . ‘. " *-- ------------- -- - . O :' - ' u . 2 , ' ‘- ——' :"l:: r-1--.--,--v---'--.--~-;--.~-'--'--r '3'”. P‘4"P'I'-*--'-<--l---Och—Jr-L--‘ . H‘J‘. b.‘--&. L A A" ..- a '1 l“..- L ......... a "1.3.! S u._u..._.__. :L-Jj b P--. IMAGE "“4 s'v-t‘." "E. ‘P. IMAGE Fig. 6.5. A man and a vehicle (Plymouth) standing on the ground are both exposed to an ELF electric field. 143 car potential ¢c are calculated as functions of the separation D between the body and the vehicle, and the results are shown in Fig. 6.6(a). The short-circuit current 1b“:°f the body and the current Ic of the vehicle flowing through the tires to the ground, used in the previous example, are calculated as functions of the separation D with results shown in Fig. 6.6(b). It is noted to calculate Ibsc’ the grounding impedance of the man is set to zero (short-circuited to the ground). It can be observed that, as the distance between the man and the vehicle is deduced the body and vehicle potentials and the currents of both objects flowing to the ground decrease, which indicate an increasing coupling effect. The physical explanation for the decrease of the potentials and currents due to the coupling effect is the following. The scattered electric field from one object tends to cancel the original electric field which illuminates the other object, as depicted in Fig. 6.7. This effect reduces the strength of the net incident field to the body and hence reduces the body and vehicle potentials and the currents of both objects. It is also observed that the coupling has less effect on the vehicle than on the man since the physical dimension of the vehicle is much larger than that of the man. The coupling effect between a car and a grounded man can be calculated the same way. Fig. 6.8(a) depicts the floating potential ¢co of the car (the car is floating) and Fig. 6.8(b) shows the short- circuit currents Icu:°f the car (the car is short circuited to the ground), all as functions of the separation distance D between the vehicle and the grounded man. The coupling effect which slightly reduces the floating potential and the short-circuit current of the vehicle is again observed. 144 0”} —-> ‘1" m 18 cm 132 cm ' ' I t n l -L M t— D -t I floating. grounding impedance of the vehicle i-j5.3 M0 5 500 p? 300! POTENTIAL E0 = 1 W/m 4» (W) 9) nan (-01.6 kV) 1 5 y . . I 2 )- // 0.9 - 0'5 b vehicle (+0.32 kV) __.___ site 0.3 (r- o—e—f O L L l A'Mcm) 5 10 ' SO lOO Fig. 6.6.(a) The floating potential of a man and the vehicle (Plymouth) potential (with tires) as functions of their separation distance. T 132 cm 1+4 .. ..Vf all i 145 short-circuited to ground grounding impedance of the vehicle - 35.3 M9 30 26 22 l8 14 10 I A "1 r] I 'ffT 71 TTfi 5 500 pF OUA) VEHICLE ("t 30%) _+_ i HANG-oldflA) s 10 . so ' 100 ’1’ (cm) Fig. 6.6.(b) The short-circuit current of a man and the current flowing between a vehicle (Plymouth) and the ground as functions of their separation distance. 146 "'§ "Jr-.\‘ ‘ \ an“ I /? I "‘ v \ l ’ ","\ R ‘ 4.1 ‘ e 4: 1+ T1 1+ .+ + + 1+ + i |_.| __*tg * I E? Ea' \ t i F+ I T ¥ : :+ I I ’ I .. , O I; ~ , v Y V IL I 1 I \ I E ‘I ' h '~) -> -) . L \ o I “ . .“" - --k ...... - - - - - ‘ I.' I +. +, +. +1 + I? T'L; .? I? + 71" 'Jld F‘Y".'u“u’2"".".“,-".“.".‘:"."l l‘.‘,g. rt‘:" -.--0-- "a"‘“.‘:f':?:'r firs-0 fiJ‘ bg-Q .. t z -I - ' -‘..- I 'I ' t: L ......... a "-7! L._._- DTL‘J p$peJ .‘Qdc‘ at? G" b ---- SCATTERED ELECTRIC FIELD Fig. 6.7. Scattered electric fields maintained by induced surface charges on the surface of a man and a vehicle. 147 132 cm A |-— D -4 i snort-circuited to ground floating E0 3 1 kV/m ¢eo(Ith) ‘bo ! open-circuit voltage of the human body (1.2 kV) ”lbs; a short-circuit current of the human body (15.5 HA) Rg - contact resistance or spark resistance between the human body a the car Cb - capacitance of the human body with respect to the system (35 PF) Rb a human body resistance Fig. 6.9. Equivalent circuits of a grounded man (180 cm in height) and a vehicle (Plymouth) at proximity. under the exposure of a lkV/m, 60-Hz electric field. 151 s / ' --- use-m . :: ...... 3 Elli/'1’? mm: w: ‘ \\ ‘8‘ “ \\ a “ “ \‘ \\ \‘ \ \ \\ \ \ ‘\ \ \ s. \ \ \ \ Body Impedance (kn) I I ’I I a "N’IIII’IIII’IIIII / “ ‘4’ I ’I I l ‘ .d‘qll’lllllilllil’l’ I // $a I I] I - I r “JUCQIIJIUIIJ‘I. 0 IN no as a see Voltage (V) Fig. 6.10. Relation between body impedance and voltage [33] . amen-firm iguana-wan} ""1: _9.«ue.c.__'1—_u ‘W’r‘c .02 W? A“! (IF-40) MW“ 13.800 1.3“ 1.”. malle- 1.500 010 I” Aw 4,038 I“ ILA.” 1,181 CHM (MM?) Elma. $0.“. 6.8.. 38.4“ 5,". Minimum {.81. I“ 1.000 1.370 Amen 0.0“ 1.“! MI. Mil 'n I me. e! meets tested “$5.! manhole Table 6.1. Body resistance of adult and child in ohms [33] . ' usumulul values any 152 Augrosntnme Vatue ol Human Body esoslance at 230 vons. 50 Hz Skull and bony pans oi caution: . ‘x. (.‘Mng am The vaiues oi new 3000 N near oruocos) lance uuutuu how «(0 r u—Sonsmvo mesons oi and shunt] nu! U0 used to: u: mnen- 29cm nets and showers tauon or sa ety en- interns! home gmeenng purposes :50: ussue wuumm other sup- pon m 19 unlamauui. 750“ \z’mmdm \ }« Lowereideolann 11! 30mm Palmendlingers 1" ....Sole Fig. 6.11. Internal resistance of the human body [33] . 153 6.3 Transigng Analysis of the Shock Current The experiment by Reilly [32] for measuring the spark discharge current is illustrated in Fig. 6.12. The vehicle was a Plymouth and the human body was replaced by a load resistor R The experimental set-up d' was exposed to the electric field of a 60-Hz, 500 kV transmission line. The electric field was approximately 4.5 kV/m. In order to compare theoretical results of the present method with experimental results, the equivalent circuit for the human body in Fig. 6.9 is replaced by a resistor Rd as shown in Fig. 6.13. Since the open-circuit voltage is 0.428 kV for the incident field E0 - 1 kV/m, it is 1.926 kV when Eo - 4.5 kV/m. Hence, ¢c0(t) in the Fig. 6.13 can be expressed as ¢CO(c) - acmejwt, w - 2m x 60 (6.4) ‘where ¢cm - 1.926 kV. when the switch in the equivalent circuit is open (t < 0), the voltage across the capacitor Ct is Z ¢1(c) - ¢cm 0‘ e3“ (6.5) 2 + Z Cc Ct where ZCc - l/ijc ; th - l/ijt After the switch is closed (t > 0), the time response of the current Ish(t) flowing through the resistor Rd can be obtained from the circuit 154 9'06"!!!" man OI. Chm lull I0 rah-do U wow Current and voltage meaSurement system Spark instrumentation [ I l - I . Vuhuclu un (null. ”d ' LU Ml. VU I I485 V l Least-mucus ll! 1.1.1-. LJ. L.__ ._1_1 L_- ' l I 1 v I U J.0 4.0 5.0 Tune (u!) Measured transient current for a Plymouth automobile Fig. 6.12. Reilly's [32] experimental set-up and measured transient current for a Plymouth automobile which was exposed to a 4.5 kV/m, 60-Hz electric field. 155 mb———-.-— 0 b-s P- rah-fins. H ,—...,... r— 7.1 p... :0 (l l ¢co Ct ‘3 3 Rd + ‘1 vii-r an ( Rt is neglected since Rt >> th ) Fig. 6.13. An equivalent circuit for a vehicle and a grounding resistance Rd under the exposure of a 4.5 kV/m, 60-Hz electric field. 156 analysis as Ish(t) - koexp(-t/(cc + Ct)Rd] + Re(1me3”t) (6.6) where k0 - a constant to be determined Im - [dam x (Zcfl Rd>/(zcc + 0.5 ps. To investigate the realistic situation of the shock current between a vehicle and a grounded human body, the equivalent circuits in Fig. 6.9 should be adopted and analyzed. Since ¢co - 0.407 kV and ¢bo - 1.2 kV for E -- 1 kV, when E - 4.5 kV d and d can be expressed as o o co bo ¢co(t) - ¢cmejwt’ ¢cm - 1.832 kV ”them " ¢bmejwt' ¢bm - 5.1.00 kV “-3) where w - 2n x 60 Before the spark contact between the vehicle and the human body (the switch in the equivalent circuits of Fig: 6.9 is open for t < 0), the voltages across the capacitor Ct (¢1(t)), the resistor Rb (¢2(t)), and the capacitor Cb (¢3(t)) are 159 _ jwt ¢l(t) ¢CmIZCt/(th + ch)]e jwt ¢21e where ZCc - l/JwCC , th - l/Jth , ZCb - l/ijb When the spark contact happens (the switch in the equivalent circuits is closed for t > 0) the total time response of voltages ¢1(t) and ¢2(t) across the capacitor CC and the resistor Rb are (APPENDICES 2 & 3) Re[¢l(t)] - k3exp(alt) + kaexp(a2t) + Re(¢lmejwt) (6.10) Re[¢2(t)] - klexp(alt) + kzexp(a2t) + Re(¢2mejwt) where the first two terms of RHS in these equations are transient responses due to discharges of capacitors, and third terms are steady- state responses due to voltage sources ¢c0(t) and ¢b0(t). In APPENDIX 2, ¢1m and ¢2m are found as 160 Ich2 + ¢bm(Jwa/Rg) ¢ 1m 2 Y1Y2 - (l/Rg) (6.11) d ¢bm(Jwa)Yl + (Icm/Rgl_ 2m 2 YIYZ - (l/Rg) where Y1 - jw(Cc + ct) + l/Rg Y2 - chb + 1/Rb + 1/Rg Icm - chc¢cm In APPENDIX 3, the constants k - k and a ~ a in Eq. (6.10) are found 1 4 l 2 to be - 1 - R C C C 1 01 Zch [g/Rb+ b/( C+ C) + ]+ 8 { 1 2 1 }l/2 [ (R /Rb + C /(C + C ) + 1)] - g b c t 2CbR8 Cb(Cc + Ct)RgRb - 1 a2 - 20 R [Kg/Rb + Cb/(Cc + Ct) + l] - be { 1 2 1 }1/2 [ (R/Rb+cb/(C+C)+1)1- g c t 2CbRg Cb(Cc + Ct)RgRb 161 and (6.12) 7? I 1 + Re[¢2m](l + Rg/Rb + chgaz)/(cbagal - chgaz) w I 2 "k1 ‘ Re[¢2m] w I 3 (1 + Rg/Rb + CbRga1)kl W I 4 (1 + Rg/Rb + CbRga2)k2 where ¢1(0) - dim X [CC/(CC + Ct)] and ¢1m as well as ¢2m can be found in Eq. (6.11). After ¢1(t) and ¢2(t) are determined, the shock current Ish(t) can then be calculated by Re1¢1/ + (v - v )/(Z” )1/2 Jo - 11.1 1-11:] 1-11.1 14.111 13‘! 11:1 1.1 . [(A,,J + ‘A913’/21 (7.19) And the z-component of current density in (i, j)th cell is z r I . Ji,j - Ji’jcoso - Ji jsino (7.20) To compare with the analytical solution, the induced currents in several cases of homogeneous spheres and concentric-spheres exposed to an incident electric field of 60-Hz frequency have been computed. Figs. 12 and 13 show comparisons between analytical and numerical solutions on the induced currents for a homogeneous fat and a homogeneous muscle sphere. A (10 x 10) subdivision was made for a section of an eighth of a sphere which was cut into 10 sections (N - M - 10, see Figs. 7.9 and 7.10). It is observed that, in numerical solutions uniform distributions of the induced current are achieved which agrees with the analytical solutions. Fig. 14 to Fig. 19 depict comparisons between analytical solutions and numerical solutions (M - 10, N - 12) on induced currents for concentric fat-muscle and muscle-fat spheres with various thicknesses. Again, excellent agreements between analytic and numerical solutions are achieved. The validity of the impedance network method is thus confirmed. 190 The method can be applied to any heterogeneous biological body as long as a proper impedance network is constructed to model the body. 191 10 FAT a 0.028 (Qm)-1 e, 2.5 x 105 analytical solution (uniform distribution) numerical solution Fig. 7.12. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a fat sphere when exposed to a 60-Hz, 1 kV/m electric field. 192 MUSCLE a 0.11 (0m)' ‘r 1.9 x 106 1 analytical solution (uniform distribution) lllll numerical solution Fig. 7.13. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. MUSCLE l7 analytical solution (uniform distribution in the inner core) numerical solution Fig. 7.14. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric fat- muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. 194 (Sill 9 9.7 $9.8 analytical solution (uniform distribution in the inner core) 6 92 96 517 numerical solution Fig. 7.15. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric fat- muscle sphere when exposed to a 60-Hz, 1 kV/m electric field. analytical solution (uniform distribution in the inner core) MUSCLE numerical solution Fig. 7.16. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric fat- muscle sphere when exposed to a 60-Hz, l kV/m electric field. 196 analytical solution MUSCLE (uniform distribution in the inner core) b/a 2 MUSCLE numerical solution Fig. 7.17 Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric muscle- fat sphere when exposed to a 60-Hz, l kV/m electric field. 197 analytical solution MUSCLE (uniform distribution in the inner core) numerical solution Fig. 7.18. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric muscle- fat sphere when exposed to a 60-Hz, l kV/m electric field. analytical solution (uniform distribution in the inner core) b/a = l.25 numerical solution Fig. 7.19. Comparison of the analytical andznumerical solutions on the induced current density J (nA/cm ) inside a concentric muscle- fat sphere when exposed to a 60-Hz, l kV/m electric field. CHUU?TEF113 CONCLUSIONS This thesis presents a new numerical technique, the surface charge integral equation (SCIE) method, for quantifying the interaction of ELF-LF electric fields with biological bodies and conducting objects. The body-model is realistic and can have arbitrary shape, posture, composition, and is under a realistic environmental condition such as connecting with arbitrary grounding impedances. The induced electric field at the body surface, the induced electric field and current density inside the body, effects of grounding impedances, as well as the transient shock current can be analyzed when bodies are exposed to ELF-LF electric fields. The present method is numerically efficient and accurate, and is found to be applicable up to HF range. First, based on the quasi-static approximation, a scalar integral equation for the induced body-surface charge density was derived when the body is illuminated by an ELF-LF electric field. By using the Moment Method, the body surface is partitioned into a number of patches and the integral equation is transformed into a matrix equation for numerical calculation. From the calculated induced surface charge and using the Ohm's law and the conservation law of electric charge, the induced current density inside the body can be determined. Since this integral equation is for the scalar surface charge, not a vector field quantity, the matrix equation transformed from the integral equation has a much smaller size 199 200 than that from the conventional tensor electric field integral equation (EFIE). This renders the SCIE method especially efficient for numerical computation. A variety of numerical examples for spherical and spheroidal models have been calculated and those numerical results were compared with analytical solutions and existing experimental results for establishing the validity and accuracy of the present method. The SCIE method was further employed to quantify the induced surface charge, the induced body current density and electric field, and the short-circuit current for animal and human body models with various sizes (heights) and postures when the models are exposed to 60-Hz electric fields. An excellent agreement between the numerical results and existing experimental results was obtained. The effects of grounding impedances have also been calculated. It was found that, for an inductive grounding impedance, it is possible to cause a resonance and excite a large current inside the body when the inductance has a value in the order of 3 x 105 H. This may cause a serious health hazard for human body. To investigate the frequency range limitation of the SCIE method, the present method was employed to compute the short-circuit current of a human body (180 cm in height) induced by incident electric field at frequencies over the ELF and HF range (60 Hz ~ 50 MHz). These numerical data were compared with existing measured results, and it was found that a satisfactory agreement was obtained up to the first body resonant frequency of about 40 MHz. This seems to extend the frequency range applicability of the SCIE method to the HF range. To compare the SCIE method with the conventional EFIE method, the EFIE method was applied to calculate the induced current excited by a 201 60-Hz EM fields in a model of man used in the SCIE method. The EFIE method is valid for the whole frequency range without making any low frequency approximation as implicit in the SCIE method. Hence, it is appropriate to compare the accuracy and efficiency of these two methods. The comparison shows that the SCIE method has much better numerical efficiency and accuracy than the EFIE method at the low frequency range. In the course of application of the EFIE method, various iterative methods such as the Conjugate Gradient Method (CGM) and the Guass-Seidel Method (GSM) have been used to solve the large matrix equation transformed from the EFIE method. The convergent rate for these iterative methods were compared and discussed. When a human body and a nearby metallic object, such as a vehicle, are both exposed to a strong electric field of a power line, a large shock current may flow between them through direct or spark contact; this will cause serious health hazards to the human body. To analyze this transient shock-current phenomenon, the coupled surface charge integral equations were applied to determine floating potentials and short-circuit currents of a human body and a nearby vehicle both under the exposure of a 60-Hz strong electric field. The equivalent circuits of a human body and a vehicle at proximity were then constructed based on these data. From the equivalent-circuit model, the transient shock current phenomenon was analyzed. Numerical results were compared with existing experimental results and discussed. Lastly, a more challenging and difficult problem of a heterogeneous-body was studied. The problem was successfully solved by combining the SCIE method with an impedance network method. The body 202 is modeled as an equivalent impedance network and the induced surface charges at the body surface are viewed as equivalent current sources which are connected at the outer boundary of the impedance network. The currents flowing in the impedance network can be determined on the basis of Kirchhoff's current law CKCIQ, and from which the induced current density and electric field inside the body can be mapped. A muscle-fat concentric-sphere was chosen as a test case because there exists an analytical solution at the low frequency range for this geometry. Induced currents inside concentric-spheres with various thicknesses immersed in 60-Hz electric fields were computed and compared with corresponding analytical solutions. An excellent agreement was obtained, and thus the validity of the method was confirmed. This method can be applied to any heterogeneous biological body when it is exposed to ELF-LP electric fields, provided an impedance network is appropriately constructed to model the body. An interesting future topic for this research is to quantify the interaction of low-frequency magnetic fields with biological bodies. The findings from such a study may be very useful for some medical therapies. APPENDIX APPENDIX 1 surface integration = Jr dsn dsn = rnsinondvn - dAn 203 204 z dA= (c122 + deWZ ) x z = rcoso , x = rsino dz = -rsin0 + cosOdr dx = rcoso + sinOdr and ab r = 2 (a sinzo + b2c0529)l/2 -absin0coso(a2 - b2) dr = (azsinza + bzcoszo)3/2 hence dA = (dz2 + (1:63)”2 l/2 (dr2 + rzdez) 2 ab sin29c0529(a = 1 + azsinze + bzcoszo[ (azsinzo + b c0529 )2 APPENDIX 2 STEADY-STATE RESPONSE OF THE EQUIVALENT CIRCUIT The equivalent circuit in Fig. 7.9 can be replaced by the other equivalent circuit as shown in the Fig. A2.l. The voltages across the, s s . capacitor Ct and resistor Rb are ¢1 and ¢2' The voltage source Vco ls replaced by a current source Ico which is 1co(c) - ijc¢co(t) (A2 1) Let ¢:(t), ¢;(t), and Ico(t) be represented as s jwt ¢1(t) - ¢1me (A2.2a) s _ jwt ¢2(t) ¢2me (A2.2b) _ jwt Ico(t) Icme (A2.2c) where Icm - ijC¢Cm Then, from KCIntwo equations can be obtained as follows, 205 206 Fig. A2.l. Equivalent circuit for computing the steady-state response of the equivalent circuit of Fig. 6 .9. 207 Icm - i1m/(ch//Zc.) + <¢lm - ¢2m>/Rg (A2.3a) <¢1m - amp/Rg - ¢2m/Rd + <¢2m — ubmvzCb (A2.3b) where ch//ZCt ' 0), for transient analysis the circuit in Fig. A3.1(b) is used. Let the voltages across 209 210 ¢co C? Ct Q: g Rb ¢bo Fig. A3.1(a). Equivalent circuit of Fig. 6.9 before the spark contact ' (the switch in the circuit is open). CC ¢1(t) Rg ¢2(t) Cb J,|-—-—-e aft—Wo— VP 11(t) ‘5 0t 3: 3 Rb 4—‘F-Al ‘f"":77 17!? vivr Fig. A3.1(b). Equivalent circuit for computing the transient response of the equivalent circuit of Fig. 6.9 (the switch in the circuit is closed). 211 the capacitor Ct and Rb be d: and d5. Two equations can be obtained from this circuit by using KCL and KVL, ¢§ d¢§ -—————— + cb———————— - 11(t) (A3.2a) Rb dt 1 t t . Il(t)R + -——— 11(t)dt - -¢2(t) (A3.2b) ' 5 (cc + cc) 0 Substituting (A3.2a) into (A3.2b) leads to t R d¢§ 1 c ¢§ d¢§ ¢2(c)[1 + —5] - - chg - -———————— [ + 0b ]dt (A3.3) Rb dt (cc + ct) 0 Rb dt Eq. (A3.3) can be differentiated to become 2 t t t d d (t) 1 R C dd (t) d (t) ___2__2_.+ ( g + b + l) 2 + 2 - O (A3.4) dt CbRg Rb ct dt cb(cc + Ct)RgRb The solution ¢2(t) of Eq. (A3.4) is , C ¢2(t) - klexp(a1t) + kzexp(azt) (A3.5) 212 where k1 & k2 - constants to be determined a1 & a2 - roots of the characteristic equation of (A3.4) and - 1 a1 - 2C R [Rg/Rb + Cb/(Cc + CC) + 1] + bs { 1 2 1 }1/2 [ (R /Rb + C /(C + C ) + 1)] - g b c t 2CbRg Cb(Cc + Ct)RgRb - 1 a2 - 2C R [Rg/Rb + Cb/(Cc + CC) + 1] — b g { 1 2 1 }1/2 [ (R /Rb + C /(C + C ) + 1)] — g b c t 2CbRg Cb(Cc + Ct)RgRb By using the same procedures as above, ¢:(t) can be obtained as ¢§(c) - k3exp(a1t) + kaexp(a2t) (A3.6) where 213 k3 & k4 - constants to be determined To determine k1 ~ k4, two equations are employed which can be derived from the circuit in Fig. A3.1(b) by using KCL and KVL, t: c ¢l(t) - Il(t)Rg + ¢2(t) (A3.7a) t C . ¢2 — cbd¢2 - [¢§/dc)1Rg + ¢§ t t - [1 + Rg/Rb]¢2(t) + [CbRg]d¢2(t)/dt (A3.8) . t t Since ¢2(t) - klexp(alt) + kzexp(azt) and ¢1(t) - k3exp(a1t) + kaexp(a2t) Eq. (A3.8) can be rearranged to be (I ¢1(t) - [1 + Rg/Rb + CbRga1]klexp(alt) + [1 + Rg + CbRgaz]k2exp(a2t) - k3exp(alt) + kaexp(azt) (A3.9) Hence, the relationsbetween k1 ~ k2 and k3 ~ k4 are obtained as 214 k3 - (1 + Rg/Rb + ch a )k g 1 1 (A3.10) k4 - (1 + Rg/Rb + CbRga2)k2 The total time response of voltages ¢1(t) and ¢2(t) across the capacitor Ct and the resistor Rb can be expressed as Re[¢1(t)] - k3exp(a1t) + khexp(a2t) + Re[¢i(t)] (A3.11) Re[¢2(t)] - klexp(alt) + kzexp(a2t) + Re[¢;(t)] where ¢i(t) and ¢:(t) are in Eqs. (A2.2). By using initial conditions ¢l(0) and ¢2(0) of Eqs. (A3.1) and Eqs. (A3 11), we have Re[¢l(0)] - Re[¢cm(ZCt/(2Cc + th))] - k3 + k4 + Re[¢1m] (A3.12a) Re[¢2(0)] - Re[¢bm(Rb/(zCb + Rb))] - k1 + k2 + Re[¢2m] (A3.12b) Since Rb (~ kn) << zCb (~ M0) as observed in Fig. 7.9 - Fig. 7.11, the following relation can be obtained ¢2(0) z 0 = k2 = -k1 - Re[¢2m] (A3.13) . rut—«r: 215 Now, using Eqs. (A3.10), (A3.11a) and (A3.12) we have k3 + k4 - (1 + Rg/Rb + CbRga1)kl - (l + Rg/Rb + CbRga2)(k1 + Rel¢2m1> - (CbRgal - CbRgaz)kl - (1 + Rg/Rb + chga2>Re[¢2m1 - Re[¢l(0)] - R8[¢1m] (A3.14) Hence k is found to be 1 k1 - (Re[¢l(0)] - Re[¢1m])/(CbR C R a gal - b g 2) + Re[¢2m](1 + Rg/Rb + CbRgaz)/(CbRga1 - C R a b g 2) (A3.15) The following equations are summary of the transient response of ¢:(t) and ¢;(t): ¢:(t) - k3exp(a1t) + kaexp(a2t) ¢§(t) - klexp(alt) + kzexp(azt) where kl - (Re[¢l(0)] - Re[¢1m])/(CbRgal - c R a + b g 2) Re[¢2m](1 + Rg/Rb + CbRga2)/(CbRga1 - CbRga2 216 k - -k - Re[¢ 2 1 2m] k3 - (1 + Rg/Rb + CbRga1)kl k4 - (1 + Rg/Rb + CbRga2)k2 and ¢1<0) - ¢lecC/1 ¢lm and ¢2m can be found in Eqs. (A2.5a) and (A2.5b). APPENDIX 4 QUASI-STATIC CURRENT FLOWING IN AN HETEROGENEOUS CONDUCTOR [35] Consider a conducting body immersed in a slowly time-varying electric field of jwt -+ -+ 4 E - E coswt - Re[E e o o o ] w : small (A4.l) There will be an induced current 3 inside the body, as evidenced by the movement of induced surface charges (Fig. A4.l). Thus, the conductivity (0) and the permittivity (e) of the body will affect the induced E field and the induced current 3 in the body. The current 3 can be calculated from 3 - (a + jwe)E - jw(e - ja/w)E - jweE (A4.2) where 9 - (e - ja/w) is the complex permittivity. The medium propagation constant 1 is defined as 7 - w[p?]l/2 - [wzpe - jwpa]l/2 (A4.3) where p is the permeability of the body. Let 2 be the physical dimension of the body, if lyil << 1, then the quasi-static approximation can be 217 218 ‘EQCOSUt Ecosut Fig. A4.l. A biological body is immersed ina slowly time-varying electric field. 219 applied to this problem as follows. Maxwell's equations for slowly time-varying fields can be written as v x E(¥,t) - -a§(?,c)/ac z 0 (A4.4) v x fi(¥,c) - aE(?,c) + e[aE(¥,c)/ac] If E(¥,t) - Re[E(¥)ejwt] and H(f,t) - Re[H(f)ert], then Eq. (A4.4) can be rewritten as v x E(?) - —jen(¥) z 0 (A4.5) v x fi(?) - (a + jwe)E(¥) - 3(?) Thus the following equations can be constructed v x E(¥) z 0 V ~ E(¥) - 0 if a and e are constant (A4.6) The equations for E field are the same as that in electrostatic case, therefore, the E field can be treated like an electrostatic field and solved using one of the electrostatic methods. Once E(r) is determined, the induced current 3 can be calculated from 220 3 - (a + jwe)E(?)ejwt (A4.7) It is noted that the phase of J(t) is different from that of E(t). From the quasi-static approximation of V x E- 0, (A48) a scalar potential V can be defined as E - -vv (A4.9) Since in Eq. (A4.6), V x E - 3 - (a + jwe)E, the following relation is achieved V - (V x H) - O - V - [(0 + jwe)E] - (a + jwe)V - E + V(a + jwe)'E (A4.10) If a and e are constants in the body, V(a + jwe) - 0 and hence v . E - 0 (A4.ll) Thus, Eqs. (A4.9) and (A4.11) lead to 221 -V - (VV) - O or V2V - 0 (A4.12) The Laplace equation is still valid in a source-free, homogeneous body for slowly time-varying fields. Now consider a concentric-sphere located in a homogeneous medium I?“ of complex conductivity 9 and immersed in a slowly time-varying electric field, as shown in Fig. A4.2. The inner core of the sphere has a radius a and a complex conductivity 9 while the outer shell of thickness (b - a) 1! has a complex conductivity 92. Under the quasi-static approximation, the primary potential Vp(r,0) associated with the primary incident field E0 is Vp(r,0) - -Eorcoso (A4.13) The secondary potentials due to the induced charges in and on the sphere are Vs(r,0) for b < r v:(r,0) for a < r < b (A4.14) v:(r,0) for 0 < r < a The total potentials are 222 Z 9 (r. 0. w) 31 9 Fig. A4.2. A conducting concentric-sphere located in a homogeneous conducting medium is immersed in a slowly time-varying electric field. 223 V(r,6) - Vp(r,6) + Vs(r,6) for b < r vl(r,o) - Vp(r,0) + v:(r,9) for a < r < b (A4.15) V2(r,0) - Vp(r,0) + v:(r,a) for 0 < r < a The possible solutions for V5, Vi and v: can be written as [35] s -(n+1) V (r,0) ngo Hnr Pn(cosfi) for b < r s 1 n -(n+1) V1(r,0) “:0 (Gnr + an: )Pn(eoso) for a < r < b s 2 n V2(r,6) - “:0 on: Pn(cosa) for 0 < r < a (A4.16) Thus, the total potential for these regions are V(r,0) - -E rcosfi + E H r-(n+l)P (c050) for b < r o n-O n n 1 n ~(n+1) Vl(r,0) Eorcoso +n§O (Gnr + Hnr )Pn(cosd) for a < r < b V (r 6) - -E rcoso + E GzrnP (c056) for 0 < r < a 2 ’ o n-O n n (A4.17) where P is the Legendre function and {H }, {G1}, {H1}, and {C2} are n n n n n constants to be determined by matching boundary conditions at the boundaries of r - a and r - b. 224 The boundary conditions at r - a and r - b are (l) V1(a,0) - V2(a,0) (2) Jr1(a,0) - Jr2(a,9) (3) Vl(b,0) - V(b,6) (4) Jr1(b,0) - Jr(b,0) where Jr1 and Jr2 are r-components of currents. Following these boundary conditions leads to (l) Gn - Gn + Hn A A A A l A A -3 1 (2) (or1 - 02)Eo - -(02 - 0‘1)G1 - (02 + 201)a Hl for n - 1 (92 — Ql)nan-IG: + [an + 91(n + 1)]a'(n+2)Hi - 0 for n - 2,3,.. (3) H _ Hl + G1b(2n+l) n n n A A A A 1 A A -3 1 (4) (01 - a)Eo - (a1 — a)Gl + (a - 201)2b H1 for n - 1 [91 + 9(n + 1)]bn'lG: + (9 - 91)(n + 1)b’(“+2)ni - 0 re: n - 2,3,.. By arranging the above relations, the following equations are obtained 225 - (G - 9)(2G + 9 ) (9 - 9 )(9 - 9)2a3/b3 G1 _ [ 1 l 2 l 2 l ]E 1 A A A A A A A A 3 3 0 (a1 + 20‘)(201 + 02) (a1 - 02)(al - a)2a /b (9 - 9)(29 + 9 ) - (G - 9 )(9 - 9)2a3/b3 + 39(9 - 9 ) G2 _ [ 1 1 2 l 2 l 2 1 ]E 1 (91 + 29)(2Gl + 92) - (91 - 92x91 - 9)2a3/b3 ° 39(9 - 9 )a3 H1 _ [ 2 l ]E 1 AAAA AAAA33o (01 + 2a)(201 + 02) - (01 — 02)(a1 - a)2a /b (91 - G)(2’a‘1 + 92) + (92 — 91x291 + 9)a3/b3 3 H1 - [ 3 3 ]b E A A A A A A A A o (01 + 20)(2o1 + 02) + (02 - al)(a1 - o)2a /b (A4.18) and G1 - H1 - 0 for n - 2, 3, ...... n n The total potentials can thus be expressed as (Pl(cosfl) - c056) V(r,9) - -Eorcosa + choso/r2 for b < r 1 l 2 . V1(r,0) - (-Eo + G1)rc050 + chosfl/r for a < r < b (A4.19) V2(r,0) - -Eorc050 + Gircosd for 0 < r < a 226 The induced electric fields inside the sphere can then be determined as E1(r,0) 52(r,9) A special case 9 - 9 1 2 and -VV1(r,0) - - 2(avl/ar) - 9(av1/ao)/r 1 1 1 1 3 . + 2Hl/r ) + 951n6(-Eo + C1 fcosHEo - G + Hi/r3) for a < r < b —VV2(r,9) - - 2(av2/ar) - 9(av2/ao)/r éeoso(so - 0%) + 931n9(-Eo + 0%) z(E - G o 1) for 0 < r < a (A4.20) of a concentric-sphere is a homogeneous sphere of which [(1 - 9/91)/(1 + 29/91)]b3eo [(1 - 9/91)/(1 + 29/91)]Eo (A4.21) [(1 - 9/91)/(1 + 29/91)]Eo 227 Therefore, the induced field inside the sphere is E1(r,0) - [3(9/91)/(1 + 29/91)](rcosfi - osin?)ao - 2[3(9/91)/(1 + 29/91)]Eo (A4.22) It is observed that the induced electric field inside a homogeneous sphere is uniform and in the z-direction which is the same as that in a dielectric sphere except the permittivity (e) of the latter is replacing the complex conductivity (9) of the former. 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