IMPLANTABLE ELECTROMAGNETIC FIELD PROBES IN FINITE BIOLOGICAL BODIES Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY SEYED HOSSEIN MOUSAVINEZHAD 1977 5‘“. L11.) 1( . . ,‘{ Y Michigan Statc WWWWWH 31239 niversity This is to certify that the thesis entitled IMPLANTABLE ELECTROMAGNETIC FIELD PROBES IN FINITE BIOLOGICAL BODIES presented by Seyed Hossein Mousavinezhad has been accepted towards fulfillment of the requirements for Ph.D. Electrical Engineering degree in and Systems Science //..//\ (1L Major professor Dam March 9, 1977 0-7639 ABSTRACT IMPLANTABLE ELECTROMAGNETIC FIELD PROBES IN FINITE BIOLOGICAL BODIES BY Seyed Hossein Mousavinezhad This thesis presents some theoretical and experi- mental results on the study of a dielectrically coated, small spherical probe used to measure the induced EM fields in conducting (biological) bodies of finite extent. The receiving and radiating characteristics of the insulated probe are determined as functions of the electrical parameters and geometry of a spherical con- ducting body. First, a general theory for a wire probe in a volume conductor is presented and the relation between the output of the probe and the induced electric field in the body is derived. The receiving properties of an in- sulated spherical probe immersed in a uniform electric field inside a conducting body are then discussed. An expression for the effective diameter of the probe is also derived. The expression for the input impedance of a dielectrically coated spherical antenna imbedded in a Seyed Hossein Mousavinezhad finite conducting body is formulated based both on the matrix equation method and transmission line theory. Finally, experimental results on the input impedance of insulated spherical probes and the measure- ments of the induced electric field inside conducting bodies are presented. The convergence problem of the theoretical input admittance and the computation of Hankel functions are also included in two Appendices. IMPLANTABLE ELECTROMAGNETIC FIELD PROBES IN FINITE BIOLOGICAL BODIES BY Seyed Hossein Mousavinezhad 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 1977 This thesis is dedicated to my wife, Susan, and our children Jamshid, Mehri,...? ii ACKNOWLEDGEMENTS The author wishes to offer his sincere gratitude to his major professor, Dr. Kun Mu Chen, for his assistance and encouragement throughout the course of this work and also for his editorial assistance in the preparation of the manuscript. He also would like to thank Dr. D.P. Nyquist, a membercflfthe guidance committee, for his many helpful suggestions and comments. I also want to thank Dr. Bong Ho and Dr. C.Y. Lo, other members of the committee for their time and efforts. Special thanks also to Mrs. Noralee Burkhardt for her skillful typing of the manuscript. This research was supported by 0.5. Army Research Office under Grant DAAG 29-76-G-0201. I would like to express my appreciation for this support. Finally, I would like to thank my family for their patience and c00peration. iii Chapter I II III IV TABLE OF CONTENTS INTRODUCTION GENERAL THEORY FOR AN IMPLANTABLE ELECTROMAGNETIC FIELD PROBE IMMERSED IN A FINITE VOLUME CONDUCTOR 2.1 A Wire Probe in a Finite Conducting Body 2.2 Spherical Antenna as a Probe AN INSULATED SPHERICAL PROBE IN A CONDUCTING BODY 3.1 Statement of the Problem and the Superposition Principle 3.2 Scattering from a Dielectrically Coated Sphere 3.3 Equivalent Circuit of an Insulated Spherical Probe in a Conducting Body 3.4 Some Theoretical Results on the Normalized Effective Diameter of the Spherical Probe DIELECTRICALLY COATED SPHERICAL ANTENNA IN A FINITE CONDUCTING BODY 4 1 Geometry of the Problem 4.2 Electromagnetic Field Solutions 4 3 Applications of Boundary Conditions 4 4 Matrix Equation Formulation of the Input Impedance Expression .5 Transmission Line Approach 6 Some Theoretical Results of the Input Impedance Computations 4 7 Radial Transmission Lines 4.8 Apparent Antenna Impedance and Capacitive End Correction iv Page 11 11 14 21 24 31 32 34 4O 46 55 64 81 88 Chapter VI SOME EXPERIMENTAL RESULTS 5.1 V-I (or E-H) Probe Impedance Measuring Techniques 5.2 Experimental Setup for the Measurement of the Input Impedance of a Dielectrically Coated Spherical Antenna in a Finite Conducting Body Comparison of Theory and Experiment Field Measurements using Insulated Spherical Probes mm #00 SUMMARY AND CONCLUSIONS APPENDICES A. Higher Order Modes and the Convergence of the Input Admittance Expression B. Computation of Hankel Functions REFERENCES Page 94 95 114 118 129 144 146 158 166 Table LIST OF TABLES Page Complex Normalized Effective Diameter of Small Spherical Antenna in Free Space (f = 600 MHz; 10 = 50 cm.) 26 . (2) Values of Hankel Functions Hn+l/2(kia) when a = 1 cm, f = 600 MHz (a/A0 = 0.02; 1 8i = 2.1 so; 50 18 free space permittivity 52 0 is free space wavelength) and Values of Hankel Function with Complex (2) _ Argument, Hn+l/2(kb)' when b — 1.5 cm, f 600 MHz (a/A0 = 0.03), k = w/uo ; _ _ ‘ 9.. = = -- E — e j w' e 70 to, o l m 53 8 Theoretical and Experimental Values of the Input Impedance of a Coated Hemi- spherical Antenna in a Finite External Medium. (a = 1.1 cm., b = 1.21 cm., c = 5.5 cm., f = 600 MHz, 6. é 3.0) (T = 20°c) 1r 122 Theoretical and Experimental Values of the Input Impedance of a Coated Hemi- spherical Antenna in a Finite External Medium. (a = 1.1 cm., b = 1.5 cm., c = 5.5 cm., f = 600 MHz, 5. é 3.0) (T = 20°c) 1r 123 Theoretical and Experimental Values of the Input Impedance of a Dielectrically Coated Hemispherical Antenna in a Finite External Medium. (a = 1.1 cm., b = 3.1 cm., c = 5.5 cm., f = 600 MHz, 5. é 3.0) (T = 20°c) 1‘ 125 vi Table 5.4 Page Experimental Input Impedance of a Dielectrically Coated HemiSpherical Antenna at Different Locations in a Finite Conducting Body (a = 1.1 cm, b = 1.2 cm., c = 5.5 cm., air = 3.0, Er = 77.898, 0 = 0.925 mho/m, f = 600 MHz; d is the Distance from the Center of the Antenna to the Center of the Conducting Body) 127 Experimental Input Impedance of a Dielectrically Coated Hemispherical Antenna at Different Locations in a Finite Conducting Body (a = 1.1 cm., b = 1.6 cm., c = 5.5 cm., Eir é 3.0, er = 77.898,<3= 0.925 mho/m, f = 600 MHz; d is the Distance from the Center of the Antenna to the Center of the Conducting Body) 128 Experimental Input Impedance of a Dielectrically Coated Hemispherical Antenna at Different Locations in a Finite Conducting Body (a = 1.1 cm., b = 2.2 cm., c = 5.5 cm., Sir é 3.0, er = 77.898,o = 0.925 U/m, f = 600 MHz; d is the Distance from the Center of the Antenna to the Center of the Conducting Body) 130 Illustrating the Convergence of the Input Admittance Series. (a = 1.0 cm., b = 1.5 cm., c = 5.5 cm., f = 600 MHz; a/lo = 0.02, Sir = 2.1, er = 77.9, o = 0.925 U/m, 260 is the total angular width, in degrees, of the gap) 156 Spherical Bessel Functions of Order 0, l and 2. These Results are Computed by using the Routine "COMBES". 164 vii Figure 2.1 LIST OF FIGURES Configuration of Probe in a Finite Heterogeneous Volume Conductor Equivalent Circuit for Probe in a Finite Heterogeneous Volume Conductor Insulated Spherical Probe in a Con- ducting Body Irradiated by an Incident EM Wave Dielectrically Coated Small Spherical Probe in a Uniform Incident Electric Field Inside a Biological Body Illustrating the Superposition Principle The Scattering from a Dielectrically Coated Sphere when Irradiated by a Uniform Incident Electric Field in the z Direction, Inside a Conducting Body Equivalent Circuit of the Dielectrically Coated Spherical Probe in a Homogeneous Biological Body Magnitude of the Normalized Effective Diameter of a Dielectrically Coated Spherical Probe Imbedded in Biological Body with Relative Dielectric Constant e r Normalized Effective Diameter of a Dielectrically Coated Spherical Probe as a Function of the Relative Dielectric Constant of the Conducting Body Dielectrically Coated Small Spherical Antenna in a Finite Biological Body. A Generator Maintains a Voltage V Across a Narrow Equatorial Gap of the Conducting Sphere. Spherical Coordinate Ssystem is also Shown viii Page 10 12 12 15 23 28 30 33 Figure 4.2 Page Enlarged View of the Gap Region of the Spherical Antenna. A Coaxial Line Connected to a R.F. Source, Maintains a Voltage V Across the Narrow Gap. Input Current I is Shown at the Edge where 6 = n/Z - 00 (280 is the Angular Width of the Gap) 41 Input Conductance (or Radiation Con- ductance) of a Spherical Antenna in Free Space as a Function of koa, where a is the Radius of Sphere and k0 = w/uoeo is the Free Space Wavenumber 66 Input Reactance of a Small Spherical Probe as a Function of the Radius of the Conducting Sphere 67 Theoretical and Experimental Values of the Dissipated Power due to EX, o/ZIEXIZ, as a Function of Y along x = 0.5 cm. Freq. = 2.45 GH, 0 = 5.934 U/m, e = 68.487 to. Salt Concentration = 0.5 normal 70 Theoretical and Experimental Values of the Dissipated Power due to Ex, o/ZIEXIZ, as a Function of Y along X 0.5. Freq. = 2.45 GHz, 0 = 5.934 U/m, a 68.487 5 . Salt concentration = 0.5 normal 0 71 Input Reactance of a Small Spherical Probe as a Function of the Permittivity of the Conducting Sphere 73 Input Reactance of a Small Spherical Probe as a Function of the Permittivity of the Conducting Body; b/a = l.l,...,3.0 75 Input Impedance of a Dielectrically Coated Spherical Antenna as a Function of the Conductivity of the Conducting Body 76 ix Figure 4.9a 5.1 5.3 5.5 Input Reactance of a Small Spherical Probe as a Function of the Relative Dielectric Constant of the Conducting Body Input Reactance of a Small Spherical Probe as a Function of the Relative Dielectric Constant of the Conducting Body. The Parameters are Relative Dielectric Constants of the Insulating Dielectric Shell The Geometry of Radial Transmission Line. ZL is the "edge" Impedance of the Spherical Antenna. The Gap Corresponds to the Medium Between the Conducting Plates (a) is the Junction Equivalent Circuit of the Spherical Antenna with an Edge Input Admittance of YIN° (b) is the Configuration of the Circular Parallel Plates of Area A which Approximates the Gap of the Antenna Cross-sectional (a) and Longitudinal (b) Views of the Cylindrical Coaxial Transmission Line. A Generator Main- tains a Voltage (or Potential Dif- ference) V Between the Inner and Outer Conductors Block Diagram for E-H (or V-I) Probe Impedance Measuring Device. All the Lines Shown are Standard GR 50 0 Coaxial Transmission Lines Test Line Together with E-H Block which Supports the Voltage and Current Probes Cross-sectional View of the V-I Probe Assembly, Electric Probe (b), and Magnetic Probe (c) The Calibration of the Phase Meter Used in the Vector Voltmeter Measurements. For the Configuration Shown the Phase Meter Should Read "Zero". X Page 78 80 82 89 96 102 103 105 107 Figure 5.6 5.7 Page Experimental Setup for Measuring the Input Impedance of a Dielectrically Coated Hemisphere in a Finite Conducting Body 116 Theoretical and Experimental Input Re- actance of a Dielectrically Coated Hemisphere in Free Space as a Function of Frequency 120 Configuration of the Dielectrically Coated Spherical Probe Loaded with a Microwave Detector. (Free Space Incident Plane Wave is also Shown.) 131 Dielectrically Coated Spherical Probe in the Upper Half Side of a TEM Trans- mission Cell. (Note: not drawn to scale.) Calibration Curve of the Probe is also Shown 133 The Intensity of the Square of Electric Field Inside the TEM Cell. The Spherical Probe was Located 6 cm Above the Center Conductor of the Cell. f = 320 MHz 135 The Output of the Dielectrically Coated Spherical Probe Inside a Finite Body Containing the 0.0 N Solution. The Frequency is f = 320 MHz 136 (a): Three Dimensional View of Tapered Anechoic Chamber. (b): Conducting Body Illuminated by Incident EM Wave 138 The Distribution of the Square of Electric Field Induced in a 16 cm x 12 cm x 4 cm Distilled Water Body as a Function of 2 Along Different Depths Indicated by the Parameter "2" 140 Distribution of the Square of the In- duced Electric Field 0.5 cm Above the Center of a Finite Conducting Body Con- taining 0.1 N Saline Solution (f = 500 MHz; 5 = 77.92, 0 = .897 mho/m) r 143 xi CHAPTER I INTRODUCTION In recent years, many researchers have inves- tigated the problem of Electromagnetic Radiation effects on biological systems and related potential health hazards. In order to understand the nature of the problem and to determine whether the radiation-induced effect is thermal or non-thermal, one needs to know the actual intensities of the induced electromagnetic fields inside the irradiated biological bodies. Experimentally, implantable EM field probes can be inserted into these bodies to measure the field intensities. In order not to perturb the actual field distribution in the body and to have a good resalution of the measurement, the probes are required to be electrically small. In this thesis, we present a study on an implant- able EM field probe which can be used to measure the induced EM field in a finite conducting (biological) body. Recent studies on the characteristics of some conventional probes used in conducting bodies have been reported [1 l. E 2]. [ 3]. In almost all these works, an infinite con- ducting body was assumed which neglected the effect of the boundary of the medium on the characteristics of the probe. In the present study, we consider a coated spherical probe in a finite conducting body, taking into account the boundary effects. In Chapter 2, we discuss some general properties of the probe in a volume conductor. We derive the re- lationship between the output of the probe and the in- duced electric field intensity in the irradiated body. The receiving characteristics of an insulated spherical probe is investigated in Chapter 3. A relation is derived for the effective diameter of the probe immersed in a uniform incident electric field inside a conducting body. In Chapter 4 we formulate the expression for the input impedance (acting as a radiating element) of a dielectrically coated spherical probe located at the center of a spherical conducting body. Two different approaches are discussed to obtain the series expression for the input admittance and some numerical results are presented. The end effects of the probe are also dis- cussed. Finally, in Chapter 5, we present some experi- mental results obtained in measuring the input impedance of the spherical probe in which a relatively new method of impedance measurement is introduced. A few examples of the measurement of the induced electric field in finite conducting bodies containing saline solution is also shown and compared to the theory. Two appendices, at the end of the thesis, discuss the convergence prob- lem of the series expression obtained in Chapter 4 and the numerical computation of Hankel functions used in the computer program. CHAPTER II GENERAL THEORY FOR AN IMPLANTABLE ELECTROMAGNETIC FIELD PROBE IMMERSED IN A FINITE VOLUME CONDUCTOR In order to measure electromagnetic field in- tensities induced in finite conducting bodies, appropriate field probes may be inserted in these bodies. In this chapter, we derive the relationship between the output of a wire probe and the intensity of the electric field at the location of the probe inside a volume conductor which is irradiated by an incident electromagnetic wave. After this, a simple spherical probe is proposed for further study because, an exact analytical solution exists for such a probe. 2.1. A Wire Probe in a Finite Conducting Body Consider a conducting body of volume V with electrical parameters 5(f), 0(f) and irradiated by “0 a non-uniform EM wave with an electric field intensity Ei(§), as shown in Figure 2.1. The induced electric field E(§) inside the body, in the absence of the probe, can be theoretically obtained based on the Tensor Integral Equation method developed originally by Livesay and Chen [4]- ZL = load impedance El(f) IO = terminal (load) current VO = ZLI0 = terminal voltage I(s) = I0f(s) = induced probe curren [f(0) = 1] EH?) Voltage Measure- ment F + + Device Probe 5(r), 0(r),u0 Figure 2.1. Configuration of Probe in a Finite, Heterogeneous Volume Conductor. in Figure 2.2. Equivalent Circuit for Probe in a Finite Heterogeneous Volume Conductor. When a probe is introduced into the body, E(f) induces a current I(s) on the probe. (5 measures the distance along the contour F of the thin probe as shown in Figure 2.1.) This current maintains its own secondary field Ep(¥) at any point in the body. Assuming the linearity, the total electric field at any point can be expressed as Et(r) = E(r) + Ep(r). (2.1) We aim to find the relation between output voltage of the probe vo(¥) and the induced field E(E) at the probe location. Using the boundary condition that the tangential electric field vanishes at any point on the surface of the perfectly conducting probe, one can write, S-Et(s) = V0(s)6(s) = ZLI06(s) (2.2) where ZL is the load impedance and S is a unit vector along the contour P. Assuming that I(s) = Iof(s) as the induced current on the surface of the probe, we can multiply both sides of Eq. (2.2) by f(s) and then integrate along P to get frf(s)s-E(s)ds + frf(s)s-Ep(s)ds = ZLI0 (2.3) (note that f(0) = l). The second integral on the left hand side is proportional to input current I0 and we can define the internal impedance of the probe as _-_1_ “3* Zin — Io frf(s)s Ep(s)ds . (2.4) This is equal to the input impedance to the probe when it is used as a radiating element. Substituting (2.4) in (2.3), we have Z A v06?) = f; f f(s)s-E(s)ds (2.5) Z. (r) + Z P in L where the relation V0 = ZLI0 has been used. Equation (2.5) is the general relation between the output voltage of the probe V0(?) and the electric field at the probe location in the conducting body. If we define —> A-> Veq(r) — frf(s)s-E(s)ds (2.6) as an equivalent voltage source for the probe, and noting that §—-= Io, Eq. (2.5) can be rewritten as + —> Iozin(r) + IOZL = Veq(r) . (2.7) Equation (2.7) suggests an equivalent circuit for the probe in the conducting body as shown in Figure 2.2. In practice, we are interested in electrically small probes such that internal electric field at probe location is uniform. Thus Eq. (2.5) can be written as Z A vo(‘r’) é[ f fFf(s)sds]°E('f) . (2.8) Z. (r) + Z in L This is the desired relation which shows that the output of the probe is proportional to the electric field at the probe's location in the absence of the probe. It also shows that the proportionality factor is a strong function of the location of the probe in the body (i.e. f) and also of electrical parameters of the medium (i.e. E(f), 0(3) and no) at the probe location. We note that the current distribution function f(s) is, in general, a function of the parameters of the medium. At the same time, input impedance of the probe when used as a radiating element zin(;) is a function of not only location ; but of electrical parameters 2, o and “0' It is obvious that in order to measure the field intensity inside, say, a biological body, we need to have a loca- tion - and local parameter - independent probe. In other words, the equivalent circuit shown in Figure 2.2 differs from the conventional circuits for a receiving antenna, in which Zin is a strong function of location and local parameters. Also as hidden in Veq(f), the current distribution function f(s) is not constant as one moves the probe around in a heterogeneous body. The solution of the input impedance of a dipole or a loop type probe is not easily obtainable in a finite conducting body. Therefore, a simple spherical probe will be treated rigorously throughout this study because, an exact analytical solution is possible for this model. 2.2. Spherical Antenna as a Probe The problem which will be examined in the next few chapters deals with an insulated spherical antenna as an implantable probe in a finite conducting body. The problem is schematically shown in Figure 2.3. It will be shown that when the coated spherical probe is located in the center of a spherical homogeneous conducting body, a closed form analytical solution can be obtained. After this, we will study the receiving and radiating characteristics of a dielectrically coated spherical antenna imbedded in a finite biological body. 10 To voltage measuring device +i+ E (r) Figure 2.3. Insulated Spherical Probe in a Conducting Body Irradiated by an Incident EM Wave. CHAPTER III AN INSULATED SPHERICAL PROBE IN A CONDUCTING BODY As was mentioned in the previous chapter, a dielectrically coated spherical antenna may be used as a probe in a conducting body. In this chapter, we consider the receiving characteristics of an insulated spherical probe when illuminated by a uniform electric field inside a biological body. An expression will be derived for the effective diameter of the probe and some theoretical re- sults will be presented for the normalized effective dia- meter as a function of relative dielectric constant and conductivity of the conducting body. It should be noted, however, that the results of this chapter are partly based on the results of the input impedance of a coated spherical probe when used as a radiating element in a finite body. The.Latterresults are developed thoroughly in the next chapter. 3.1. Statement of the Problem and the Superposition Principle As shown in Figure 3.1, an electrically small sphere of diameter 2a, coated by a dielectric shell of 11 12 Biological Body E, C, 110 X Figure 3.1. Dielectrically Coated Small Spherical Probe in a Uniform Incident Electric Field Inside a Biological Body Figure 3.2. Illustrating the Superposition Principle i ‘1) p u N- .a. E SI . :C u” To. ‘1 r .,.‘ 13 radius b is imbedded in a biological body. Assume that an electric field E0 exists at the probe location and this E0 is uniform over the probe when the probe is small. There is an impedance ZL across a narrow gap of the probe and we aim to derive a relation between the induced voltage across ZL and the impressed electric field E0. The dielectric coating has a permittivity 6i and perme— ability “0' The electrical parameters of the body are e, o and ”0' The spherical probe is located such that the z axis of the rectangular coordinate system is per- pendicular to the plane of the narrow gap and the impressed electric field is in the z direction. Since the biological body is assumed to be linear, we can apply the superposition principle. This principle states that the total electromagnetic field present at any point outside the spherical probe is the sum of the scattered fields from the shorted probe (a coated solid sphere) illuminated by the impressed electric field, plus the field radiated by the coated spherical antenna driven by a voltage which is equal to the voltage drop across the load impedance at the narrow gap. This is illustrated in Figure 3.2 where arrows on the spheres show the directions of currents. The radiating antenna will be analyzed in the next chapter and its input impedance will be formulated. In this chapter, we will solve the scattering problem. 14 3.2. Scattering from a Dielectrically Coated Sphere An insulated solid sphere together with the Spherical coordinate system is shown in Figure 3.3. The incident electric field at the sphere is expressed as E = E z = E0 Cos e r - Eo Sin e 6 (3.1) based on the approximation of the field being uniform over a small Sphere. Time harmonic dependence of the form exp(jwt) is implied but not shown in the analysis. From the incident field of equation (3.1), we can see that there exists only the r— and 6- components of the electric field in dielectric region and the con— ducting body. Furthermore, all fields are independent of azimuthal angle T due to the rotational symmetry. The magnetic field associated with this uniform electric field can be shown, via Maxwell's curl equation, to be iden- tically zero. This implies that the effect of the magnetic field will be neglected at this stage. The following relations are true under the stated approxima- tions. = O, E = 0 and Hr = H6 = 0 . (3.2) Of course, there is a scattered magnetic field maintained by the current induced on the sphere by the uniform incident electric field. 15 Z (r,6.¢) I I I \ I I o r'.' ‘E I \ O 8 : b a . I . I - I ’y \ I . \ ' ' s ‘P \ | .’ ' .\\ | 2 ' I E:il, IJ'O ' \\ 6! OI U0 conducting body Figure 3.3. The Scattering from a Dielectrically Coated Sphere when Irradiated by a Uniform Incident Electric Field in the z Direction, Inside a Conducting Body. 16 There are two regions where we have to find ex- pressions for the total EM field components. To do this, we can start from the Maxwell's equationsenuiderive the Helmholtz Wave equation. The solution to this equation will be considered in more detail in the next chapter. Here, we just write down the tangential components of E and H fields in the two regions: For r 3 b, the scattered fields by the coated sphere are s _ 1 5+ H¢n(r,6) — Pn(Cos 8)H¢n(r) (3.3) s _ 1 + s+ Een(r,8) — Pn(Cos 8)an(r)H¢n(r) (3.4) where (2) A H (kr) Hs+(r) E n n+7 , k2 = w u 5 an r];- 0 u <£=€-j%,n=zfl (3.5) (2) 2+ (r) = '0 H n-%(kr) - 2- sn 3 H72)(kr) kr K n+8 H(1) and H(2) are Hankel functions of the first and second kind, respectively. E is the complex permittivity of the conducting body and z:n(r) is the TM mode wave impedance in this region. An, for integer n, is an un- known coefficient to be determined later from the boundary conditions. In Equations (3.3) and (3.4), only the 17 out-going waves are considered. This approximation neglects the reflection of the waves at the outer surface of the conducting body. This may be valid because of the losses in the medium. For a i r i b, the total fields in this region can be expressed as _ 1 - + cpn(r,e) — Pn(Cos 8)[Hwn(r) + H¢n(rfl (3.6) E n(r,e) = pi (Cos e)[z; (r)H; n(r) - z; (r)H; nm] (3. 7) where H(l) (2) (H (r) s Bn “n+5(k r) , H+ (r) E Can+g(kir) mn Vkir In Vkir LI 2 _ 2. _ _9 k1 — w 14061' n1 - Ei < (3.8) + Hr(12) (k r) _ “ Hn::fi(k r) 1r _ Hélifki r) n Zn“) = ‘3'” Hm ‘ 12‘? - K M115(k r) 1 Note that since this is a finite region, there exist both . . + . . . out-901ng spherical wave, H¢n(r), and incoming spherical wave, H;n(r). Bn and Cn are two other unknowns to be determined later. 18 Since there is a solution for each n, the actual E and fi fields are infinite sums of fields given by equations (3.3), (3.4) and (3.6), (3.7), i.e.; awn, e) = z°° n=l Hwn(r'e) (3.9) where H¢n(r,6) is as given by (3.6). Similar expres- sions can be written for other field components. Up to this point, we have introduced three un- known coefficients An’ Bn and Cn‘ To solve for these unknowns, we use the Boundary Conditions. The first boundary condition is that the tangential electric field vanishes at any point on the perfect con- ducting metallic sphere, i.e. _ l + + _ - - _ Een(a,6) — Pn(Cos 9)[Zn(a)qu(a) Zn(a)H(Pn(a)] — o (3.10) which is valid for all n and 6. The second boundary condition states that the tangential E and fi field components are continuous at r = b, or - + _ s+ qu(b) + Hcpn(b) — Hcpn(b) (3.11) + + - - _ + s+ _ Zn(b)an(b) - Zn(b)H¢n(b) — an(b)H¢n(b) E051n (3.12) Note that in writing the continuity of g field, the uniform incident electric field E0 in the conducting body is included in the right hand side of equation (3.12). 19 The notation associated with E0, i.e. l n = 1 61m = (3.13) 0 n # l is the Kronecker delta. The three unknown coefficients are the solutions of the following system: YnXn = Fn (3.14) where An E06m x = B , F = 0 (3.15) n n n C 0 n (2) (l) (2) 1 n+l/2(kb) + Hn+]/2(kib) - Hn+1/2(kib) + sn(b) Zn(b) Zn(b) .423 ib i (2) (1) H(2) Y = Hun/2m") ’Hn+1/2‘kib)Hn+1/2(kib) “ (ES A??? W. H(1) - (2) L- 0 Hn+15 (kia)Zn(a) Hn +15(kia) J The matrix equation (3.14) gives us non-zero solutions for An' Bn and Cn only when n = 1. That means there is only one term in the infinite series of equation (3.9). This simply is due to the fact that the incident electric field was assumed to be uniform. If, for example, the incident field was assumed to be a plane wave, the 20 solution would be quite involved and there would be infinite terms in the series solution of Equation (3.9). We can now define a = —- , bl = —— and cl = —— (3.16) such that the expression for the tangential a field on the surface of Sphere, as given in equation (3.6), can be expressed as (1) (2) H (k.a) H (k.a) H (a,e) = Sin e[ 3” 1 b + 3/2 1 c E (3.17) (P VE.a l VE.a l 0 l 1 Note that P%(Cos 9) = Sin 6. The unknowns b1 and C1 are solved from equations (3.16) and (3.15) as f Jkib b = l + - (l) + + (2) [231 (b)+z1 (13)] H3/2 (kib)+[zsl (b) —z1 (b)] a(a)H3/2 (kib) {c1 = 0t(a)b1 (3.18) - (1) - Zl(a)H3/2(kia) (1(a) : + (2) o \ Zl(a)H3/2(kia) Up to now, the magnetic field on the sphere is completely known. We are interested in the current on the sphere. The surface current on the surface of the sphere is given by 21 A K(e) = a x i = r x H¢(a,6)$ = -Hw(a,9)9 (3.19) where H¢(a,6) is as given by equation (3.17). This current is proportional to E0 and can be written as Ke(9) = -Hw(a,6) = Y(6)E0 (3.20) where (1) (2) Ii (k.a) Ii (k.a) Y(6) = _[ 3/2 1 bl + 3/2 1 c¥]sin e (3.21) Vkia Vkia with b1 and c1 as given by equation (3.18). Note that Y(9) has the dimensions of an admittance. 3.3. Equivalent Circuit of an Insulated Spherical Probe in a Conducting Body Referring back to superposition principle as de- picted in Figure 3.2, the total surface current on the insulated Sphere of Figure 3.1 is given by Kte(m = Ke(m +-Kg (m (3.22) where Ke(6) is found in the previous section and is given by equation (3.20), Ké(6) is the surface current on the spherical probe when it is driven by a voltage generator. The radiating problem will be solved in the next chapter. At this point, we write Ké at the probe gap or 9 = 90° simply as 22 . _ o _ V Ke(e — 90 ) — Zna Z. (3.23) in where V is the induced voltage (or the voltage drop) across the load impedance of the spherical probe, and Zin is the input impedance of the coated sphere when used as a radiating antenna in the same conducting body. The general expression for this input impedance Zin will be derived in the next chapter. The induced voltage across the load impedance is given by V = -ZLI = ~2na Kte(8 = 90 )ZL (3.24) Note the polarity of this voltage drop as shown in Figure 3.1. Substituting equations (3.23) and (3.24) into the equation (3.22), one gets; V V - —————— = ——————— + Y(e = 90°)E . Zfla ZL 2na Zin 0 After rearranging, it becomes V(Zin + ZL)/ZL = -2 anZinY(6 = 90 )E0 (3.25) where Y(e) is given in equation (3.21). Equation (3.25) suggests an equivalent circuit for the insulated spherical probe in a conducting body as shown in Figure 3.4. The equivalent driving voltage for the probe in Figure 3.4 is, Veq‘= -2na ZinY(6 : 90 )E0 . (3.26) 23 V eq Figure 3.4. Equivalent Circuit of the Dielectrically Coated Spherical Probe in a Homogeneous Biological Body. 24 Note that Figure 3.4 is similar to Figure 2.2 of Chapter 2 which is the equivalent circuit of a wire probe. In analogy with a wire probe, we can define an "effective" diameter of the probe as Deff = Veq/E0 = -2na ZinY(9 = 90 ) (3.27) Furthermore, this can be normalized to the physical diameter of the sphere, 2a, to give dimensionless normalized effective diameter as deff = Deff/Za = -fl ZinY(6 = 90 ) (3.28) Finally, before ending this section, we note that Veq is the voltage developed across the load impedance when ZL + w. Therefore, useful information can be obtained from the effective diameter of the probe. Some theoretical results of this parameter are shown in the next section. 3.4. Some Theoretical Results on the Normalized Effective Diameter of the Spherical Probe The expression for the normalized effective dia- meter of the spherical probe was derived and expressed in equation (3.28) of the last section. The results of this section are also based on the results of the input impedance of the spherical probe acting as a radiating element Zin’ which will be analyzed in the next chapter. 25 First, in order to compare the receiving charac- teristics of the spherical antenna to the other conven- tional probes (such as dipoles and loops), the effective diameter of a small sphere in free space is calculated. In Table 3.1, the real and imaginary parts (or magnitude and phase) of the normalized effective diameter for a small spherical probe in the free-space are shown. The frequency is assumed to be 600 megahertz which corresponds to the free space wavelength of 50 cm. The spherical probes considered are all electrically small. As can be seen from Table 3.1, for small spherical receiving antennas, Deff/Za = l/2 (3.29) is a good approximation. This means that Deff = a (3.30) which is the physical radius of the sphere. This is similar to small dipole type probes, where the effective length is one half of the total physical length. The theory for the dipole antenna as a receiving element, can be found in King's book [5]. Several examples were worked out for the insulated spherical probe of Figure 3.1. It was found that the effective diameter, in general, is a strong function of the relative dielectric constant of the conducting body. 26 Table 3.1. Complex Normalized Effective Diameter of Small Spherical Antenna in Free Space (f = 600 MHz; A = 50 cm.) 0 (2m) Deff/Za MAGNITUDE PHASE (DEGREES) 0.5 0.44738 -0.00478 1.0 0.44805 —0.03796 1.5 0.45078 -0.12655 2.0 0.45448 -0.29493 2.5 0.45902 -0.56393 3.0 0.46428 —0.95012 3.5 0.47010 -l.46559 4.0 0.47630 -2.11801 4.5 0.48270 -2.91084 5.0 0.48913 —3.84390 5.5 0.49541 -4.91383 6.0 0.50137 -6.1l478 6.5 0.50685 -7.43891 7.0 0.51170 -8.87698 7.5 0.51579 —10.41877 8.0 0.51902 —12.05351 8.5 0.52127 -13.77013 9.0 0.52247 -15.55751 9.5 0.52254 -17.40459 10.0 0.52144 —19.30041 27 Of course, it is also a function of the thickness of the coating (or b). The example shown in Figure 3.5 illus- trates the magnitude of normalized effective diameter for a small, coated spherical probe. The frequency is f = 2.45 GHz (corresponding to the free space wavelength of A0 = 12.24 cm) and the radius of the Sphere is a = 1.0 mm. The sphere is coated with a dielectric of dielectric constant e. = 2.1 (Teflon). The conduc- 1r 1 mmho/m. The curves are tivity of the body is o plotted for different thicknesses of the coating as a parameter. The independent variable is the relative dielectric constant of the conducting body. It is to be noted that although the effective diameter is a complex quantity, the imaginary part is usually small compared with the real part. AS can be seen from Figure 3.5, for smaller values of b/a (i.e. for thin coatings), there is a considerable variation in the effective diameter as the relative dielectric constant is changed. But for higher values of b/a (i.e. for thicker coatings), there is almost no variation in the effective diametercflfthe probe as the Sr is varied. This is not true, however, for lower values of gr, say, er < 10. Fortunately, for most cases of interest in biological bodies, er is greater than 10. The results obtained in this chapter are based on the geometry of a conducting body of infinite extent. wk? .nh '5N ‘\\ .~;\ 28 xaamofluuomamflo m mo umumEMHQ m>Huommwm wwNHHmEMoz map mo wpsuflcmmz .uo ucmumcou afiuuomamfio m>HumHmm cuflz wpom HEOHmOHOHm SH pooprEH mnoum HMOfluwzmm powwow .H w om on om om 0v om ON OH 0 AAA.) _ _ q . . 2 _ Exogee o.H u 6 .HH. H.m u .0 Nmo mv.m u w EE o.H n m 1 Hoo.H u mxn . Ho.H .hw H.H )4) (I) /llLV: 1 m H l!!!!!!) I!!! o N nmquMAo m>fluomwwm UmNHHmEuoz mo mpsuacmmz ‘ .m.m musmflh om.o om.o ov.o om.o om.o .yr I v-v ...( ash-a n. '1 I; 4 ll’ lJ‘ 29 However, as we will see later, the input impedance of the probe is quite independent of the electrical parameters of the conducting body and the probe location. When the dielectric coating is sufficiently thick, the effective diameter expression derived in this chapter may also be valid for an insulated spherical probe immersed in a finite body. Finally, in Figure 3.6, the normalized effective diameter of the probe is shown as a function of the dielectric constant of the conducting body for different values of the relative dielectric constant of the coating, 5. . It is seen in this figure that for lower values of 1r eir' the effective diameter remains almost constant for a wide range of relative dielectric constants of the con- ducting medium. Therefore, as far as the receiving characteristics of the insulated spherical probe are concerned, in order to have the output of the probe to be independent of the electrical parameters of the conducting body, a thick coating with low dielectric constant materials is appropriate. In the next chapter, the radiating charac- teristics of the coated Spherical probe will be studied as functions of the electrical parameters of the medium and the probe location in a conducting body. 30 .mvom mcfiuospcou on» no ucmumcou oauuomamfio m>flumamm on» no cofluocsm m mm maoum HMOfiumnmm pmumoo waamowuuowawwa m mo HmumEMAa m>fiuommmm pmNHHmEuoz .m.m musmflm u m cm on ow om ow om om OH o A!- u q T W . q 1 cm. E\0:EE H u o . om. Nam mv.m u m SE Ho.a n Q SE o.H u m 1 cc. \\\ . om. o.m o.N(l) )II (AIII\\ HA ) o.H u .u MN\mme 2... CHAPTER IV DIELECTRICALLY COATED SPHERICAL ANTENNA IN A FINITE CONDUCTING BODY AS was seen in Chapter 2, the output of an EM field probe immersed in a finite conducting body is, in general, a strong function of its Input Impedance when used as a radiating element. Furthermore in Chapter 3, the expression for the effective diameter of a dielectrically coated Spherical probe was found to be dependent on the input impedance of the radiating spherical antenna. It is evident that to understand the performance of an EM field probe in a biological body, the input impedance of the probe acting as a radiating element must be determined. In this chapter the expression for the input impedance of the probe iS determined as a function of the parameters of the body and the relative probe location inside the body. The theoretical results for the input impedance of a coated spherical probe in a finite conducting body were computed numerically with a digital computer and are Shown in figures. In the next chapter, the accuracy of these results is verified by experiments. 31 32 It is noted that the major difference between the problem studied here and the ones considered by other workers and available in the literature is the fact that the conducting body in the present study is finite. The selection of a finite conducting body increases the degree of difficulty in the theoretical analysis. How- ever, with the geometries of a Spherical conducting body and a Spherical probe, an exact solution is obtainable by the method of boundary value problem. In the course of solving the problem, the matrix inversion method was first applied without success. Later, a transmission line approach was employed to find the solution success- fully. 4.1. Geometry of the Problem Figure 4.1 Shows the geometry of the problem to be considered in this chapter. An electrically small Sphere of radius a is driven by a voltage generator which maintains a potential difference V across a narrow equatorial gap. The spherical antenna is coated by a dielectric Shell of outer radius b and dielectric constant 6i. This dielectric coating region is assumed to be almost lossless, i.e. 5i is a real quantity. The coated antenna is then imbedded in the center of a conducting body of radius c. The electrical parameters of the conducting body are e (permittivity) and o Figure 4.1. 33 éih>N Eolpo (free Space) II III biological body Dielectrically Coated Small Spherical Antenna in a Finite Biological Body. A Generator Maintains a Voltage V Across a Narrow Equatorial Gap of the Conducting Sphere. Spherical Coordinate System is Also Shown. 34 (conductivity). It is noted that for an exact solution to exist, the insulated Sphere should be located at the center of the conducting body. We aim to find an expression for the input impedance of a radiating, insulated Spherical antenna, which is electrically small and can be used as a probe, imbedded in the center of a Spherical biological body. This input impedance will be shown to be a function of all parameters involved, namely, a, b, C, 6i, 8, 0 and the frequency of the oscillating source, f. 4.2. Electromagnetic Field Solutions Due to the geometry of the problem, the usual spherical coordinate system (r,6,¢) is used as shown in Figure 4.1. There are three regions in which electric and magnetic fields are to be determined from the Maxwell's Equations. The two curl equations are: _) v x E = -jwu0H (4.1) v x E = ju)_E (4.2) in which E is, in general, a complex permittivity. The time dependence of exp(jwt) is understood. Due to the rotational symmetry, all field quantities are independent Of (pf i.e. -—- ( ) E 0 (4.3) 35 Also, due to the uniform excitation of the antenna around the gap, it is true that E = 0 (4.4) It implies that there is no (9 component of the electric field in any region. With conditions (4.3) and (4.4), it can be shown that the magnetic field has only the w com- ponent. From Equation (4.1), 8E +—1§_ _l.__£.A—..' + V X E ‘ E 3r (r38) E 80 ]¢ ‘ J“’“0H (4'5) Equation (4.5) Shows that E = H.$, i.e. there is only n w component of E field at any point outside the Spherical antenna. Taking curl of Equation (4.2) one gets V x V x E = ng V x E (4.6) in which the complex permittivity g is assumed to be independent of the location. Using Equation (4.1) in (4.6), one has V x V x E = w noeH (4.7) as 32 1 3 l 3 2 .....__. + __ —_ . _ ' + . = . arz (ram) r2 88[381n 8 3 (rBD51n 8)] w nucrfia 0 36 This partial differential equation is a key relation for the derivation of all EM field components in the different regions. It is written in a form to facilitate the solu- tion by the usual "separation of variables" technique. Let us now consider the 3 different regions as shown in Figure 4.1: Region I, a i r i b or the dielectric coating region. In this region the insulating layer has a real dielectric constant ti and Equation (4.8) can be written as 2 3 (rH __ 3_ 3r2 LP 1 l 8 . 2 _ ) + 7 38[§i—I-1—8- 56- (rH](051n 9)] + kirHLp - 0 r _ (4.9) where ki = w “Oei and k1 is the real wave number in this region. Solution to the above equation can be written as rHLp(r,0) = R(r)®(0) . (4.10) and Equation (4.9) is rewritten; 2 2 r d R l d l d . 2 2 _ Egr*666[sinea-61}(4.21) _ (2) <1) _ ern_l/2(kr)] + Dn[an+l/2(kr) er Note that E2r is omitted since it is not used in the determination of the unknown coefficients A B C n' n’ n and D . n Region III, r 3 c or the free space region. In this region, the wave number is real and given by k - 2 (4 22) ‘ w “050 ' and only outward traveling waves represented by Hé2)(kor) are present. Therefore, the tangential field components are given as 4O - L_ m 1 (2) Hlp(r,6) - /E anl EnPn(Cos 6)Hn+l/2(k0r) (4.23) and E3e(r,e) = - ——i——— z”_l EnP1(COS 6)[nH(ii/2(kor) wEOr/r n— n n (2) - Hn_1/2(k0r)] (4.24) This completes the derivation of EM field components in the three regions shown in Figure 4.1. There are five B C unknown coefficients A Dn and En for each D! n! nl integer n. To find these unknowns and obtain the ex- pression for the magnetic field on the metallic sphere (and thus the current), we use the boundary conditions on the tangential field components as outlined in the next section. 4.3. Applications of Boundary Conditions The boundary conditions state that the tangential components of electric and magnetic fields are continuous at r = b and r = c (see Figure 4.1). Moreover, on the surface of the metallic sphere (assumed to be perfectly conducting), E16(a,6) vanishes at any point except at the narrow equatorial gap. An enlarged view of the gap' region and a possible feeding system is shown in Figure 4.2. The angular width of the gap is 260 and is assumed to be very small (e.g. 260 is of order of 10 degrees or smaller). Mathematically, we write 41 Perfect conducting sphere ‘P -F 4— +- +--F 4' 4- +--F 4- 4— +--t 4- ’ I ’ ’ ’ + dielectric gap ’ , "" region *F \ § 26 0 v coaxial Transmission Line To R.F. Generator Figure 4.2. Spherical Antenna. Across the Narrow Gap. Shown at the Edge where 6 = n/Z - 6 is the Angular Width of the Gap). (9 Enlarged View of the Gap Region of the A Coaxial Line Connected to a R.F. Source, Maintains a Voltage Input Current = n/Z - 6 ) V I is (260 42 _ X _ E16(a,0) — a 6(6 n/Z) (4.25) On the other hand, from Equation (4.16) in the last section, we have 1 (2) E (a,e) = --—4L——-z” _ P (Cos e){A [nH (ki a) 6 weia/a n—l n n n+1/2 _ (2) (1) _ (1) (4.26) Multiply both sides of Equation (4.26) by “P;(Cos 6)sin e" and integrate from 0 to n on 6 to get o~sa Ele(a,6)P;(Cos 6)sin ede = -——l——— X: N{An[ nH‘ii/2(k. a) we. la/a _ H(2) (l) k. 1aH 1/2(kia)] + BnIan+l/2(kia) k. iaH(11/2(ki a)]} f Pm (Cos 6)P: (Cos 6)sin ede (4.27) where we interchanged the summation and integration opeations. We now use the following orthogonality rela- tions of the Associated Legendre functions: F f P1(Cos 6)P1(Cos 6)sin 6d6 = 0 ; m # n (4.28) 2n(n+l) 1 2 . _ . _ [Pn(COS 6)] Sln ede - W , m - n 0%: Equation (4.27) then becomes, 43 l . _ _ j .2m(m+l). Ele(a,6)Pm(Cos 6)81n Gde - weia/g 2m+l 0‘sfi (2) '{Am[mHm+1/2(kia) _ (2) kiaHm-l/2(kia)] (1) m+l/2 (l) + Bm[mH (kia) — kiaHm_l/2(kia)]} (4.29) Using Equation (4.25) for the tangential B field on the sphere, the left hand side of Equation (4.29) is evaluated as follows: 1 . _ y Ele(a,6)Pm(Cos 6)51n ede - a 5(9 - n/2)P;(COS 6)sin ede O‘wd o~sd W|< 1 13mm). (4.30) Therefore, Equation (4.29) is finally written as (after some rearrangements and replacing m by n) y1nAn + y2an = gnV (4°31) where ’ _ (2) _ (2) yln - an+l/2(kia) kiaHn_l/2(kia)