MSU LIBRARIES .—:—. RETURNING MATERIALS: PIace in book drop to remove this checkout from your record. FINES wiI] be charged if book is returned after the date stamped below. EM PROBING AND HEATING OF BIOLOGICAL BODIES WITH BARE AND INSULATED MICROPROBES By Abdolhamid Ghods A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1984 ABSTRACT EM PROBING AND HEATING OF BIOLOGICAL BODIES WITH BARE AND INSULATED MICROPROBES By Abdolhamid Ghods In the present research the schemes of using a microprobe for determining the electrical properties of biological bodies in vivo and for locally heating biological tissues are investigated, with the application to hyperthermia cancer therapy or other medical purposes. The relationship between the input impedance of the probe and the electrical parameters of the surrounding medium is used to determine the electrical properties of the medium, and the EM waves in the biological bodies maintained by the current on the probe are used to heat biological bodies locally. A detailed analysis of the bare microprobe in a conducting medium has been conducted and the electric field produced by the probe in the medium derived. Using the method of moments, Hallen's integral equation and electric field integral equation for the probe current are transformed into systems of simultaneous algebraic equa- tions which are then solved on a computer. A general theory for an insulated microprobe in a conducting medium based on lossy transmission line theory is presented. The current on the probe, the electric field in the medium maintained by the probe, and the heat pattern of the probe are found. Various equivalent terminal impedances for the insulated probe are intro- duced, and their effects on the current distribution and the heat pattern are investigated. Experiments have been conducted and the input impedances of bare and insulated microprobes in saline with various normalities are measured. The theoretical and experimental results are found to be in good agreement. ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to his major professor, Dr. K.M. Chen, for his guidance and encouragement throughout the course of this research. I also like to thank a member of my guidance committee, Dr. D.P. Nyquist, for his invaluable suggestions and guidance. I appreciate the help I received from the other members of my guidance committee, Dr. B. Drachman and Dr. D. Reinhard. Special thanks also to Mrs. Carol Cole for her skillful typing. ii TABLE OF CONTENTS LIST OF TABLES ....................... LIST OF FIGURES ...................... CHAPTER I. INTRODUCTION .................... II. THEORETICAL STUDY OF A BARE MICROPROBE IN A CONDUCTING MEDIUM ................ III. IV. 2.1. Hallen's Integral Equation for a Microprobe 2.1.1. Moment Method Solution 2.2. Electric Field Integral Equation for a Microprobe ............... 2.2.1. Magnetic Current Ring as the Driving Source ............ 2.3. Comparison of Different Methods ....... EXPERIMENTAL STUDY OF A BARE MICROPROBE IN A CONDUCTING MEDIUM ............... 3.1. Electrical Properties of Saline ....... 3.2. Experimental Setup .............. 3.3. Comparison of Theoretical and Experimental Results ............. APPLICATIONS OF THE BARE PROBE ........... 4.1. .Measurement of o and e ............ 4.2. EM Local Healing ............... iii 13 26 34 37 38 39 44 50' SO 56 iv CHAPTER V. INSULATED PROBES IN A CONDUCTING MEDIUM ..... 5.1. Theory of the Probe ............ 5.2.1. Parameters of the Transmission Line . . . . 5.2.2. Terminal Impedances of the Probe ..... 5.3.1. Input Impedance of the Probe ....... 5.3.2. Experimental Verification of the Theory . . 5.4.1. Current Distributions Along the Probe . . VI. APPLICATION OF INSULATED PROBE .......... VII. A USER'S GUIDE TO COMPUTER PROGRAMS USED TO CAL- CALCULATE THE CURRENT DISTRIBUTION AND ELECTRIC FIELD IN A BIOLOGICAL BODY INDUCED BY A BARE AND INSULATED MICROPROBE ............... 7.1. Programs for the Bare Probe ........ 7.2. Program HALIEQ ............... 7.2.1. Description of Input Variables and Input Data Files .......... 7.2.2. Example .............. 7.3. Program EFZI ................ 7.3.1. Description of Input Variables and Input Data Files .......... 7.3.2. Example .............. 7.4. Program Used for Insulated Probe ...... 7.4.1. Description of Input Variables and Input Data Files ........... 7.4.2. Example .............. VIII. SUMMARY ..................... BIBLIOGRAPHY ....................... 63 65 71 81 84 98 104 115 124 124 124 126 127 128 129 129 130 131 132 134 136 LIST OF TABLES Input admittance of an electrically short and thin bare probe in a conducting medium .......... Input admittance of the bare probe in the saline with normalities 0.1 N and 0.2 N ............. Electrical properties of Nacl solution ....... Parameters of lossy coaxial cable .......... Input admittance of electrically short and thin insulated probe in a conducting medium ....... Input impedance of insulated probe with various terminal i mpedances ................. The symbolic names of the input variables and the format specifications used in program HALIEQ The symbolic names of the input variables and the format specifications used in the first data file in program EFZI .................... The symbolic names of the input variables and the format specifications used in program INPIMP 25 36 4O 82 95 97 127 130 132 LIST OF FIGURES Evolution of geometry of a bare microprobe in a conducting medium .................. Geometry of the bare probe ............. Current distributions along bare probes in a con- ducting medium, obtained with Hallen's Integral Equation method ................... Current distributions along bare probes in a con- ducting medium, obtained with Electric Field Integral Equation method .............. Current distribution along a bare probe with various surface impedances when the probe is embedded in a conducting medium, obtained with Electric Field Integral Equation method .............. Input resistances of bare probes in the salines with different concentrations .............. Input reactances of bare probes in the salines with different concentrations .............. Equivalent magnetic ring model ........... Geometry of magnetic current ring when the field point is in the source region ............ Normalized electric field on the surface of the probe maintained by the magnetic current ring model . . . . Input impedance of the probe varies as the function of the length of the magnetic ring ......... (a) Experimental setup for the measurement of the input impedance of a probe with an E-H probe and a vector voltmeter (b) Setup for adjusting the phase of a vector voltmeter .................... Physical geometry of bare probe used in the experiment ..................... vi 19 20 21 22 23 27 3O 33 35 41 43 Figure 3.3 vii Comparison of theoretical and experimental results on the input resistance of a probe in various conducting media .................. Comparison of theoretical and experimental results on the input reactance of a probe in various conducting media ........................ Comparison of theoretical and experimental results on the input resistance of a probe in various conducting media ........................ Comparison of theoretical and experimental results on the input reactance of a probe in various conducting media ........................ A bare probe inserted in a tumor which is embedded in a biological body .................. Input conductance of a probe in various conducting media, obtained with Electric Field Integral Equation method and the driving point is modeled with a magnetic current ring .............. ,. . Input susceptance of a probe in various conducting media, obtained with Electric Field Integral Equation method and driving point is modeled with a magnetic current ring .................... Equi-power contours for a bare probe in the r-z plane. The power is normalized and the frequency of operation f = 600 MHz, 02 = 1.11 (s/m), €r2 = 76.7 Equi-power contours for a bare probe in the r-z plane. The power is normalized and the frequency of operation f = 600 MHz, 02 = 1.11 (s/m), er2 = 76.7 Evolution of geometry of an insulated probe in a conducting medium .................. Geometries for an insulated probe in a conducting medium ....................... a) Section (1) of the line b) Section (2) of the line (c) Common coordinate system for both sections 46 47 48 49 51 53 54 60 61 64 66 69 Figure 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 viii (a) Insulated probe in a conducting medium (b) Infinite cylinder ................ (a) Configuration of an insulated probe in a con- ducting medium (b) Equivalent circuit for terminal AA ....... Input impedance Zin = Rin + inn of symmetric insu- lated probes in a dissipative medium, y = 10.72 + j49.38 and ZC = 70.77 - j15.36 9. Both terminal impedances are assumed to be infinity ........ Input impedance Zin = Rin + inn of symmetric insu- lated probes in a dissipative medium, y = 10.72 + j49.38 and Z = 70.77 - j15.36 0. Both terminal impedances are assumed to be zero .......... Input impedance Zin = Rin + inn of symmetric insu- lated probes in a dissipative medium, y = 10.72 + J49.38 and 2C = 70.77 - j15.36 9. Terminal impedances Ze1 = 0 and Ze2 = a ........... Input impedance Zin = Rin + inn of.the insulated probe in a conducting medium versus permittivity of the insulator. Terminal impedances Z = Z = m . . e1 e2 Comparison of the input impedance Zin = Rin + inn of the bare probe with insulated probes with different terminal impedances ................. Configurations of insulated probes (a) Terminal impedances Zel and Ze2 very small (b) Ze1_very large and Ze2 very small ........ Comparison of theoretical and experimental results on the input resistance of an insulated probe with terminal impedances Zel = 0.0, Zez = 0.0 in various conducting media .................. Comparison of theoretical and experimental results on the input reactance of an insulated probe with terminal impedances Zel = 0.0, Ze2 = 0.0 in various conducting media .................. Comparison of theoretical and experimental results on the input resistance of an insulated probe with terminal impedances Ze1 = m, Ze2 = 0.0 in various conducting media .................. 72 83 87 89 91 94 96 99 .100 101 Figure 5.15 5.16 5.17 5.18 5.19 5.20 6.1 6.2 6.3 6.4 7.1 ix Comparison of theoretical and experimental results on the input reactance of an insulated probe with terminal impedances Z 1 = m, Ze2 = 0.0 in various conducting media .................. Current distributions along insulated probes with terminal impedances Zel = Ze2 = w, h = 15 mm and h2 is variable. The frequency 15 600 MHz ....... Current distributions along insulated probes with terminal impedances Z 1 — Ze2 = 0.0, h = 15 mm and h2 is variable. The Frequency is 600 Hz ...... Current distributions along insulated probes with terminal impedances Z 1 - m, Zez = 0.0, h1 = 15 mm, h is variable. The Frequency of operation is 6 0 MHz ....................... Current distributions along insulated probes with terminal impedances Z = 0.0 and leg = w. h = 15 mm and h is variable. Tie frequency of operation is 600 M 2 ....................... Current distributions along insulated probes with different terminal impedances ............ Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe Z = 64.5 - j13.4, 02 = 1.11 (s/m), erZ = 76.7, and tge frequency is 600 MHz .............. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe 2 = 64.5 - j13.4, oz = 1.11 (s/m), 5r2 = 76.7, and tge frequency is 600 MHz .............. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe Z = 64.5 - j13.4, 02 = 1.11 (s/m), ErZ = 76.7, and tfie frequency is 600 MHz .............. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe 2 = 64.5 - 313.4, 02 = 1.11 (s/m), erZ = 76.7, and tfie frequency is 600 MHz .............. Geometry of the bare probe used in HALIEQ ...... 103 107 109 111 113 114 119 120 121 125 CHAPTER I INTRODUCTION In recent years, electromagnetic radiation and propagation have played an important role in human life. Intercontinental satel- lite communication, radar detection, microwave technology, and the use of EM energy for medical purposes are only a few examples of the applications using electromagnetic energy these days. Many medical researchers have investigated the scheme of using EM energy to induce hyperthermia in biological bodies for the purpose of cancer therapy. It is known that when the temperature of a can- cerous tumor is raised a few degrees above that of the surrounding tissues, the adjoining chemo or radiotherapy becomes more effective intreating tumors [1]. Therefore, it is the objective of many researchers to find a noninvasive method by which to heat the tumor without overheating the surrounding tissues. Substantial progress was made in hyperthermia cancer therapy when Leveen et al. [2] used 13.56 MHz EM radiation to eradicate the tumors or to slow the progression of tumors in some cancer patients. Other researchers [3,4] have used EM waves at various frequencies to heat the tumors in animal bodies and reported sig- nificant tumor eradication. A successful analysis of the induced EM field inside an irra- diated body with an imbedded tumor, and design of an effective device for focusing EM energy in the tumor will depend on the knowledge of electrical properties of the tumor. Some researchers [5-8] have used an open-ended coaxial cable, or a very short monopole, or a symmetric probe [9] to study the electric field induced in the biological bodies and measure the electrical properties of various biological tissues. In the present research the techniques for determining the electrical properties of biological bodies in vivo and local heating of an imbedded tumor with an unbalanced monopole, which consists of a thin open-ended microcoaxial line with an extended center conduc- tor, are studied theoretically and experimentally. In Chapter II a theoretical analysis of a bare microprobe in a conducting medium is presented. The distribution of the current on the probe, and the input impedance of the probe are calculated numerically with different methods for various cases. The driving source is modeled as a delta gap or as a magnetic current ring. In Chapter III the descriptions of the experimental setup and the electrical properties of the saline are given. The input impedance of the probe was measured by the vector voltmeter. The experimental results are compared with the theoretical results. Chapter IV contains the applications of the bare microprobe. The heat pattern of the microprobe in the conducting medium and the methods for measuring the conductivity and permittivity of the conducting bodies are presented in this chapter. In Chapter V a theoretical analysis of an insulated microprobe in a conducting medium is given. The theory of lossy transmission line is used to investigate the current distribution on the probe, the input impedance of the probe, and the electric field produced by the probe in the medium. Various physical geometries of the insulated probes with different equivalent terminal impedances are introduced. The current distributions along and the imput impedances of the probes with various terminal impedances are compared in this chapter. A series of experiments were conducted to measure the input impedances of the insulated probes with various physical geometries. The theoretical results and the experimental results are compared. In Chapter VI the application of insulated probes for local heating is explained. The theory and the numerical results of the heat pattern produced by the insulated probes with various terminal impedances are given. A brief description of the computer programs used in this study to obtain numerical results is given in Chapter VII. CHAPTER II THEORETICAL STUDY OF A BARE MICROPROBE IN A CONDUCTING MEDIUM To measure the conductivity and the permittivity of a biolog- ical tissue in situe or for local heating of a biological body, a microprobe driven by an EM source can be inserted inside the tissue. The evolutional geometries for a bare coaxial microprobe in a con- ducting medium are shown in Figure 2.1. A coaxial microprobe con- sists of a microcoaxial line with an extended inner conductor. In a coaxial line the current is concentrated inside the line and the current on the inner conductor is equal and in opposite direction with the current on the inner surface of the outer conduc- tor. Assuming no accumulation of charges at the end of the outer conductor, the current at this end is continuous and the direction of the current on the outer surface of the outer conductor is as indicated in Figure 2.1(a). Inside the coaxial line the currents are equal and in opposite directions, therefore, their effects cancel each other. Only the current on the extended inner conductor and that on the outer surface of the outer conductor maintain an electric field in the medium. Hence the probe is equivalent to an asymmetric dipole as shown in Figure 2.1(b). Since the dipole is in a lossy ll (a) ’I\\ current amplitude ~’-‘ I ‘s ’ ~~‘,a—-‘ I\‘ I "~- \ I x I ‘\-r"‘\ ’/.- .— “"‘"\ A A [ :11 J (C) Figure 2.1. Evolution of geometry of a bare microprobe in a con- ducting medium. medium, the current decays so rapidly that the probe can be truncated as indicated in Figure 2.1(c) for the analysis. The probe in a lossy medium is studied first by the well known "Hallen's Integral Equation Method" and then followed by the "Elec- tric Field Integral Equation Method" (Pocklington method). Modeling the driving source in the probe is very important and it does depend on the physical geometry of the probe. In the electric field integral equation method the driving source is first modeled as a delta gap and later as a magnetic current ring. 2.1. Hallen's Integral Equation for a Microprobe The vector potential Az(z) on the surface of the probe main- tained by the current 12(2) on the probe satisfies the following where 21(2) is surface impedance of the probe and in general it is assumed to be non-zero. V0 is the applied voltage at the gap and 6(2) is a delta function. The solution to the above differential equation for Az(z) is given by A (z) = Ah(z) + A912) 2 z z where the homogeneous solution A2(z) is h_jk . Az(z) - - (0 [C1 cos_kz + C2 Sln kz] and the particular solution A:(z) is p five. 2.1.. . Az(z) = - to 1r-s1n klzl + 0 12(2 )2 (z )s1n k(z' - z)dz The general solution for Az(z) is . V Az(z) = - J—k-[Cl cos k2 + C2 sin kz +79 sin klzl +[ozlz(z')zi(z')sin k(z' - z)dz'] (2-1-2) 0n the other hand, the vector potential on the surface of the probe can be expressed as U hl I 9-.ij 1 Az(z) =‘4F -h2 Iz(z ) R dz (2.1.3) With the thin wire approximation, or with the assumption that the surface current on the probe can be approximated by a line current flowing along the axis of the probe, we can approximate R as R =‘Jiz - z')2 + a2 After combining equations (2.1.2) and (2.1.3) it gives h1 e-ij .k4 v0 -hZIz(z') T dz' = - Jw—u'l [C1 cos kz+C2 sin kz+-2— sin k|z| z . + f 12(2')Zi(z')sin k(z' - z)dz'] (2.1.4) 0 Equation (2.1.4) is called "Hallen's Integral Equation," which cannot be solved in closed form, but it can be done numerically. 2.1.1. Moment Method Solution In order to solve the integral equation (2.1.4), the moment method is used to convert the integral equation into a system of simultaneous linear algebraic equations with the probe current at different locations of the probe as unknowns [10]. The probe is partitioned into N segments, as shown in Figure 2.2. The unknown current on the probe I(z) can be expressed in terms of a sequence of pulse functions: N I(z) = Z InPn(z) n=1 where 1 Z s (AZ)n Pn(z) = 0 2 L (A2)n The boundary conditions (B.C.) state that 1(2) = 0 on the tips of the probe, hence I1 = IN = 0. After the substitution of I(z) in equation (2.1.4) it becomes [01.1 1*-1 - o o o \ N—I. ‘IL 0 (gonad) oT-I-H-I-l-I-l- zN z = --h2 ------- Figure 2.2. Geometry of the bare probe. 10 N-1 1 . 21 f W(z.z')dz' = - 11%?[C1 cos kz + C2 sin kz n n=2 . -h2 N-1 2 v . + 7‘1 sin klzl + Z : In 21(2')Sin k(z' - z)dZ'] (2-1-5) N=2 0 -ij where w(z,z') = 9-1—— Equation (2}1.5) is forced to be satisfied at N midpoints of N seg- ments. Hence there are a system of N simultaneous algebraic equa- tions and N unknowns (C1, C2, 12, I3, ... IN-l)’ Let us assume that the short part of the probe (hl) is made of a very good con- ducting material so that its surface impedance is approximately zero and the long part of the probe (hz) is covered with lossy material so that the surface impedance over this part of the probe is finite. We can express 21(2) as . O 0 5.2 5_h1 21(2) = The system of equations now becomes 11 N-l V um . ._ 0 . C1 cos k21+3E4TrZInA1n + C2 s1n kz1 -2— $111 klzl] n=2 N-l V C coskz + w" ZIA +C sinkz =- 0sinklzl 1 2 jEIn n 2n 2 2 7?. 2 n=2 N-1 mu c1 C°S kch+1 ” 3m: InANc+1 n " C2 5‘" kch+1 n=2 1 ch+1 . ' . vo + Z INc+1 0 s1n k(z - ch+1)dz = - 7r-sin kIch+1I N-l mu , . C1 cos kch+2 + W3 "2 InANc+2 "+02 Sll’l kch+2 n=2 1. j ch+1*“ 2 k( )d + Z sin 2' - z z' 0 Nc+2 i ch+2 V0 + Z sin k(z' -ch+z)dz' = - -2- sin klch+zi . zNC+1+A/2 (2.1.6) The surface impedance Zi of h2 in these equations is assumed to be constant. Also, the coefficient Amn is defined as r w(2.2')A m f n A (,')d'=i mn fumni’zz Z L 2Ln[.§5 +‘ ’1 + (245)2 :l-jka m = n 12 .52 m > No ' V _ 1 0 . - - Let us define Bm - - if'SI" klzml and Mmn - Amn + Dmn where Dmn - O for (m and n) 5.Nc and for other values of m and n . z 1 " . Z] 5111 k(z' - zm)dz' m = n zn-A/Z Zn+A/2 D = ( Zijf. sin k(z' - zm)dz' m f n z n-A/Z . 0 m - n < 0 With these definitions equation (2.1.6) can be expressed in the matrix form as - . . . l cos kz1 s1n kzli C1 1 31 cos kz2 sin kz2 12 82 Mmn = (2.1.7) IN-I Lcos kzN sin k2". - C2 .. LB". 13 where various matrix elements have been defined. The inversion of the equation (2.1.7) yields the unknowns: C1, 12, I3, ... IN-l’ C2. 2.2. Electric Field Integral Equation for a Microprobe Maxwell's equations in a lossy medium are 17-15(35) -- .m 17,631?) = mm Wm?) -- 33(7) + (Gambia?) + + v-B(r) = 0 Vector and scalar potentials in a conducting medium are ++ ' ++ 'ij A(r) =£%-d/pd(r') e R dv' (2.2.1) v + 1 + e"ij “Ms-my] p(r') R dv' (2.2.2) v where 6* = 8(1 - 12) (06 Electric field can be expressed in terms of scalar and vector poten- tials as Ed?) = - m?) - jam?) (2.2.3) Using equations (2.2.1) to (2.2.3) and the continuity equation v?3(F) + jmp(F) = 0, which gives the relationship between the 14 ++ current density J(r) and the charge density o(?), the electric field on the surface of the probe can be expressed as 52(2)“ fig-f [33r12(2') % 1p(z,z')+ k2I2(z')w(z,z') dz' " T (a.u The boundary condition on the surface of the probe states that the tangential components of the electric field at the inner surface and outer surface of the probe should be equal: 4.. a) (2.2.5) The electric field at the inner surface of the probe maintained by the source and the probe current is 2.ar=a=zhanu)-§u) (Lam where E:(z) is the driving electric field maintained by the source. The electric field at the outer surface of the probe maintained by the probe current and charge is given by (2.2.4). By equating (2.2.4) and (2.2.6), we have hl 3 , 3 . 2 1 u I 32712“ ) .33 w(z,z ) + k 12(2 )w(z.z ) dz '"2 2 =-1%§5— [21(2)Iz(z) - E:(z)] (2.2.7) Equation (2.2.7) is called the "Electric Field Integral Equation." In order to be able to solve the above equation, E:(z) should be 15 determined. The driving source should be modeled based on the physical geometry of the probe. First, the probe is studied with a delta gap model and then it is modeled as a magnetic current ring model. In the case of a delta gap '1/25 2 = o E:(z) = 4-V0f(z) and f(2) = 0 2 f 0 k where 26 is the length of the gap and V0 is the applied voltage across the gap. After substitution for E§(z) in equation (2.2.7), it is solved by the moment method. The probe is partitioned the same way as it was for solving Hallen's integral equation. The unknown current 1(2) is expressed as a sequence of pulse functions and after the substitution of this current in equation (2.2.7), it can be written as follows: N Z In [33—2 1p(z,z') 2226+] Z I nzkf 111(2, 2 ')dz' = z '=z+ (AZ)n n=1 n=1 2 33—3“— [z‘mxzm - Vof(z)] (2.2.8) where A z = z — a n .2- 2+ 3 2n +'% 16 and _ (z - z')(1 + ij) -§% w(2.2') = w'(2.2') = R2 w(Z.Z') . 0 Oizihl 21(2) = i [21 -h2_<_z_<_0 Equation (2.2.8) can be simplified as N I l 1 l I :2E:In{ 142m.z )dz +-;§-[w (zm.zfi)-v (zm.z:)] n=1 (Az)n = 347 [12(zmlzkzm) - vonzmn m = 1, 2, ... N 1 (2.2.9) This system of N simultaneous algebraic equations can be solved numerically. Equation (2.2.9) can be expressed in matrix form as follows: . ' 1 r l r I1 31) '12 B2 Amn = I B L L N 4 in N . Consequently 17 r r 1'1 ' 1 111 B1 = Amn I B i N d i. J . NJ where matrix [I] gives the current on the surface of the probe, and the matrix [B] is the electric field on the surface of the probe produced by the applied voltage. Matrix [A] is called the impedance matrix. Elements of matrices [A] and [B] are defined as . 0 m f Nc i- am A Amn =f w(zm.2')d2' + :1; [w'(zm.2_) - w'(zm.zfi)] (Az)n " for m 5_Nc, or m > Nc and m # n. Amn '4] w(zm.2')d2' + :1; [v'(zm.za) - ¢'(zm.Z:)J-%$1 Z1 (:32)" - for m > He and m = n. (2.2.10) a1 m 3DNc . a2 m > Nc 18 Some numerical results are shown in Figures 2.3 to 2.5. In Figure 2.3 the current distributions along bare probes of various lengths in a conducting medium, obtained with Hallen's Integral Equation method, are shown. In this case it is assumed that a1==a2 and h2 is variable. Since the probe is in a lossy medium, the cur- rent on h2 is decaying. In Figure 2.4 the current distributions along the same bare probes in the same medium, obtained with the Electric Field Integral Equation method, are given. Again, it is observed that the current on h2 is decaying. In both Figures 2.3 and 2.4 it was assumed that the surface impedance everywhere on the probe is zero. Figure 2.5 illustrates the current distributions along the bare probes with variable surface impedances on hz. It is seen that as the surface impedance increases, the current on h2 decays faster. Therefore, if we make 2‘ f 0, then the length of h2 has less effect on the input impedance and truncation of h2 creates less error. Figures 2.6 and 2.7 illustrate input resistances and input reactances of the bare probes of various lengths in the salines with normalities of 0.0, 0.1, 0.2, 0.3, and 0.4. In all cases h1==7.5 mm and a = 0.43 mm, but h2 is varied from 30 to 75 mm. In 0’ (0605'- input resistance changes moderately) when h2 is changed. However, the case of = 0.0, the input reactance changes rapidly (the for = 0.43 the changes on the input resistance and the input weoer reactance are much less than the case of = 0.0. For other 0 (0603'. > 0.43, the input impedance of the bare probe cases when weoer 153 saw: um=_ouno .e:_uos m:_uu=ucmu a cw manage .uozums :o_uo:aw Pocmouc~ m.=m__az .Acumcm—u>o3 cu uo~_—u5cocv u:_oa ac.>_cu seem mucoum_o mean mcopu meowu=n_cum_u acmcgau .m.~ mczm.u o.~- m..- o..- m.o- p ~.o n/oib\.f f i / o.m um; II:L/l..nluo .1111. II: x e m._uN:\‘/1.|11 /o/o/I// I]. r o_ o. u \ e o \ r ON ”.3, o._ u > \ N mpoe.co> u z . - _ s . s N o 1 z \. E: C.” n M \ e\m c._ u c .om .m n a. NI: m_m . c . o N o> h .1711 e N; .HW _; uh fit. . m 20 .uozuoe cowuoaam pucmou:_ u—upc u_cuum_m cu_; cos_ouno .e:_uoe acwuuaucou a c. monogq «can a=o_o m=o_u=apcum_u acaccau .v.~ «camel .Azumcm_o>o3 cu vm~ppa5cocv ucwoa a:_>wcn soc» oucaumpa m o.~s m.—1 o.—1 m.01 o «.0 ii/ 4 4 I/lol1c‘lOIIJVVIiIIU“ a o.~.~g ..z. - / I *l‘fi/I lllll‘ +l m..u~;.\‘ .144 “Po, o._ . o> Qpbmwbfl> M N; , ~.o u _; EEA%M u C —m a go ./ e\m 9.. u o < u_o> o._ u > NI: m_a u c on ) I ‘ ( 0 am . O> A * N; Mm” _; H_ is «five. ~+ OuN f 21. .852. =o_uu:am Focmwuc~ upm_m urcuuupu gun: uu=_uuno .53Fuoe m=.uu=u:ou a c_ uwuuunEo m. opoga wzu cog: moccanuae_ ouuucam mac—gu> ;u_: once: mean a agape =o_u=n_gum—u acacczo .m. m ~L=m_m u:_oa m=*>_cu soc» mucoum_a ”E: o:. 2- 8- 8- 8- 3- 9° 2 l 4 . iiilrl . 1.5.1.]: p p I, I I a ./ 2 . es m.o u a w E: o: u N; . 8.: 5.2:“... .8 .5 o; n o :2, o; n 0) ...? 88. u L 8:8 5.: a u c Ea 82 u .N m_na_cu>u N . n . . _ com Ea 37% Ar on (w‘lxl) 22 l h1 ! h2 2a E 111 = 7.5 m c h2 = variable 3? l] a = 0.43 m 2 (D 5 ° -o.9 (”finer 1x a I \ rUCOC 1.34 \ fl l' 20 <1 \_ a I'0.43 “Coer \ ° =1.33 mcocr 15 1) ='0.0 (0505': 100 Sun 3 . - T‘ "20"") 30 45 60 75 Figure 2.6. Input resistances of bare probes in the salines with dif- ferent concentrations. - 23 A 5 h1 = 7.5 mm 5 h2 = variable a = 0.43 mm -30.. -204 ° =o.43 NEOEr -10 4F ' O ___ 0 O /“’€0€r . \ fi‘— 0' _ -— - 0.9 Naoer 0 - 4/ “Fear-1°34 0“ / ° = 1.83 (0808'. A e t 4 30 45 60 75 h2 (mm) Figure 2.7. Input reactances of bare probes in the salines with dif- ferent concentrations. 24 becomes nearly independent of the length of hz. This phenomenon > 0.43, . . . . . G impl1es that when the probe 15 in a lossy medium, w1th weoer the characteristics of the probe mainly depend on the properties of the medium, not on hz. These characteristics of the bare probe can be used to measure the electrical properties of the lossy medium. For checking the consistency of the theory let us consider electrically short and thin bare probes. When a << h, ah << 1, sh << 1 the current distribution along the probe is approximately triangular. The input admittance Yin of a symmetric bare probe (h1 = h2 = h) in a lossy medium is known [11] to be .; o . ._ 'TI’h ._ Y1" ‘ G1" + JNC‘" ‘ Ln(h/3) - 1 (U + ch) (2.2.11) Table 2.1 shows the input admittance of the bare probe at different frequencies calculated from equation (2.2.11) and from theory developed in the previous sections, Hallen's Integral Equation method (HIEM) and Electric Field Integral Equation method (EFIE). The physical dimensions of the probe are h = 10 mm and a1 = a2 = 0.43 mm. The probe is in a lossy medium with o = 0.05 (s/m) and er = 7.4 50' The results shown in Table 2.1 confirm the consistencY of the theoretical methods developed in this chapter. 25 Table 2.1. Input admittance of an electrically short and thin bare probe in a conducting medium. Equation Frequency HIEM , EFIE (2.2.11) 10 MHZ 0.688 + j0.056 0.715 + 30.059 0.718 + 30.060 100 MHZ 0.689 + j0.565 0.717 + j0.588 0.718 + j0.601 600 MHz 0.759 + j3.53 0.789 + j3.67 0.718 + j3.61 26 2.2.1. Magnetic Current Ring as the Driving Source In the study of the probe it is very important to properly model the source region at the driving point. In the previous sec- tions the source region at the driving point was modeled as a delta gap and based on that model the current distribution along the probe was found. In this section, the source region is modeled by a mag- netic current ring. As shown in Figure 2.8, the gap is replaced by a circular magnetic current ring with a surface magnetic current, 2%, flowing around the probe. The relationship between the electric field at the gap produced by the applied voltage and the surface magnetic current is 'fié-fixE ' QJJE where h is the unit vector in the f direction. After substitution for h and E in the equation (2.2.12). we have %=_%$ _ mam The magnetic vector potential produced by the surface magnetic cur- rent is KMG) = 18;] 'IEMG'MGJF'MS' (2.2.14) SI where s' is the area of the source region. 27 ‘h—- h 4—- ._ ‘— <—- "V II I Nl< 00 N) 37‘) (AA —‘—J ____l. Ym=-/an‘ “_-/\ _y_/\ Km - r x ( 25 z) "‘_ VA Km--§-6-¢ ‘ Figure 2.8. Equivalent magnetic ring model. 1"?» "5) 28 Substituting the surface magnetic current from (2.2.13) in (2.2.14), the magnetic vector potential can be written as ennui?) Afih-%fif&'e.H as. any 5. R( ') Because of rotational symmetry the observation point is chosen at a = O. In a cylindrical coordinate system, R(F,F') can be expressed as follows: 2. R = [r2 + r - 2rr' cos 4' + (Z - Z')2] The unit vector 3' in terms of the unit vectors i and 9 in the car- tesian coordinate system is 8‘ = -i sin ¢' + 9 cos ¢' . (2.2.16) With (2.2.16), (2.2.15) becomes (+ )= 6 ['32]:0 e-ijCFfF') d 'd ' Am r va'4;[ sin a' R(F'F') a z AID] 211' e_ij(r Fl) ] + y cos ¢' +_+ d¢'dz' (2.2.17) _5' 0 RH. I") Since the integrand of the first term on the right hand side is an odd function of a', it integrates to zero. Also, at the plane of a a 0, 9 = I. The magnetic vector potential Ah(:) is 29 A 'ij(-Fs-Fl) Am“) = “i225"??? f cos 41'. e ds' (2.2.18) SI R(F,F') The electric field is related to the electric vector potential ++ ++ A(r) and the magnetic vector potential Am(r) by + ++__L.2++-l +- E(r) - jwe {vv + k ] A(r) e v x Am(r) (2.2.19) Since only magnetic current is present, 3(F) = 0. Therefore, + + _ ll 3 Ezir) ' "'E r ar (rAma) -'kR(F'F') v 1 a e 3 ’ =-§——-—-—— r cos a' ds' (2.2.20) 1T5 7‘ 8r [ Ll Rst-Vh) ] Let us define the integrand in (2.2.20) as F(r,z): e‘ij(-;s?d) F(r.2) = cos ¢' ds' SI R(F,F') The above integral cannot be carried out exactly, therefore, it will be evaluated numerically. When R + 0 the integrand is singular. This happens when the field point is in the source region. In order to evaluate the integral when the field point P(r, 0, t +-§) is in the source region (see Figure 2.9), the integral is separated into five integrals as stated below in (2.2.21). 30 AZ 0 ”fl -‘ ”' I Figure 2.9. Geometry of magnetic current ring when the field point is in the source region. 31 t 11' 6'3 kRVFs?’ ) F(r,z) = cos 4' +.+ d¢'dz' R(r,r') “8 2F e-ij(F,F') cos ¢ da'dz’ (2.2.21) R(F,F') The last integral in this equation which has a singular point is carried out analytically. The surface is approximated with the same area [12]. Therefore, 6 [“6 27 e-ij(+ “*') e-a‘kaFJ') f cos (1' +7" da'dz' =f cos¢' TF' ds' 1: _2€L RY‘.( 7 S' R(Y‘, ) ' r 32 The rest of the terms of F(r,z) can be found numerically and (2.2.22) should be added to it. The derivative of F(r,z) with respect to r, according to the finite difference, is aF(r,z) é F(r,z) - F(r - Ar,z) Bl" AY‘ After taking the derivative of F(r,z) numerically, its value can be substituted in (2.2.20) to determine the electric field 52(2) produced by the equivalent magnetic current ring at any point in the medium. The effect of the length of the magnetic current ring on the electric field produced by it is studied and it is shown in Fig- ure 2.10. In this figure the length of the current ring is assumed to be A9 = 1.0 or A9 = 2.0 mm, on a probe with a1 = a2 = 0.43 mm, in the medium with o = 1.1 (s/m) and er = 76.7 at 600 MHz frequency. The area under the curve is E-dl. In both cases the area under the curve is approximately equal to one, which is equal to the driv- ing voltage applied to the probe. The distribution of the electric field maintained by the magnetic current ring on the surface of the probe is close to a delta function. For this finding, it is reassumed that the delta gap model for the source region is an accurate model. The input impedance of the probe has been calculated with the magnetic current ring model for the driving source and using the electric field integral equation method. The effect of the 33 E2 (max) = 830.85 v/m a = a2 = 0.43 mm h = h2 = 15 mm 600 MHz 1.1 (s/m) e = 76.7 f Ez r Q '95” H II Ag = 2.0 mm _=i=dT—" Elk 3.0 220 1.0 o 1.0 2.0 3.0 2(mm) Figure 2.10. Normalized electric field on the surface of the probe maintained by the magnetic current ring model. 34 length of the magnetic current ring on the input impedance was studied, and the results are shown in Figure 2.11. In this figure the length of the ring is changed from 0.5 mm to 2.0 mm but no sig- nificant change in the input resistance or the input reactance is observed for this range of change. 2.3. Comparison of Different Methods In the previous sections different methods for solving the bare probe in a conducting medium were discussed. The input admit- tances of the probe with a1 = a2 = 0.43 mm, h1 = 7.5 mm, and h2 = 45 mm at 600 MHz frequency in saline with 0.1 and 0.2 normali- ties are shown in Table 2.2. The results in Table 2.2 indicate that the real part of the admittance (conductance) is in good agree- ment for different methods, but the imaginary part (susceptance) has some variations; this is caused by the size of partitions. 20 d Rin 10 ‘ / \ (L15 “s 35 ~~ ('— 5 -~~ ‘ " ‘ Figure 2.11. A f 0.5 1.0 1.5 2.0 Ag (mm) Input impedance of the probe varies as the function of the length of the magnetic ring. 36 Table 2.2. Input admittance of the bare probe in the saline with normalities 0.1 N and 0.2 N. Method 0.1 N Saline 0.2 N Saline EFIE with magnetic current ring 36.9 + j17.3 40.3 + j7.8 EFIE with delta gap model 35.1 + j21.2 40.4 + j11.8 HIEM with delta gap model 35.2 + j19.8 39.9 + j10.1 CHAPTER III EXPERIMENTAL STUDY OF A BARE MICROPROBE IN A CONDUCTING MEDIUM In order to verify the theory of a bare microprobe in a con- ducting medium, a series of experiments were conducted. The input impedance of the probe embedded in different lossy media was measured. One way to measure the impedance is to use a slotted line, VSWR meter, and Smith chart. This method has two limitations: first, at a low frequency a long slotted line is needed, and second, with a low resistance in the load the VSWR in the line is very large and a large VSWR can cause difficulties in using the Smith chart. Another way to measure the impedance is by a vector voltmeter and a V-1 (or E-H) probe, which is explained in more detail in Section 3.2. To simulate a lossy medium salt water was used, therefore, the electrical properties of saline, permittivity, and the conduc- tivity should be known. These quantities are strongly dependent on the frequency and the temperature. More about electrical proper- ties of saline is explained in references [13-15]. The experimental data in the above references are used in our study. 37 38 3.1. Electrical Parameters of Saline The electrical properties of pure water can be expressed as e = e' - J5", where c' = 802; and e" = 608;. The variations in real and imaginary parts of complex permittivity due to the change in frequency are indicated in equations (3.1.1) and (3.1.2). ES - 4.9 r + 4.9 (3.1.1) 1 + (wt) 8" = (85 - 4.9)wt (3.1.2) I 1 + (...)Z ('1 ll where w is the angular frequency, as and T are called the static dielectric constant and the relaxation time, respectively. In these equations a and T are dependent on the temperature. 5 After salt is added to pure water, the static dielectric con- stant (es) and the relaxation time (T) change with the salt concen- tration in the solution. 2' varies only due to the changes on T and e . e" varies not only due to the changes on T and 65, but 5 another term needs to be added in the following way: a ‘ 1 (3.1.3) = 1_-u_-_ e 8 32 J m where °i is called the inonic conductivity and it is due to Na+ and Cl' ions in the solution. The total conductivity is at = 01 + we" and 39 e = e' - g-(oi + we") (3.1.4) In Table 3.1 electrical properties of saline are shown. These results are obtained by interpolation for various normalities of saline. At 600 MHz frequency the dominant terms for conductivity and permittivity are the ionic conductivity and the static dielectric constant, respectively. With the increase in temperature, the con- ductivity of the saline increases while its permittivity decreases. 3.2. Experimental Setup The schematic diagram of the experimental setup is shown in Figure 3.1. The microprobe imbedded in a tank of saline is driven by a R.F. generator throughanE-H probe. The outputs of the E-H probe are connected to a vector voltmeter for the measurement of the input impedance. The E—H (V-I) probe consists of a section of transmission line with a short E probe and a small H probe. The E probe is a short monopole that induces a voltage proportional to the E field or voltage in the trnasmission line. The H probe is a small loop that induces a voltage proportional to the H field or the current in the transmission line [16]. The vector voltmeter has two channels, "A" and "B". If two signals are connected to channels "A" and "B", the amplitude of each signal and the phase difference between them can be measured by the vector voltmeter. By definition the imped- ance at any point in the circuit is the ratio of the voltage to the current at that point. Two signals from E and H probes can be Table 3.1. Electrical properties of NaCl solution. 40 20°C 30°C Saline _12 _12 Normal1ty 1(10 sec) Es oi(s/m) 1(10 sec) 85 01(s/m) O 10.1 80 0 7.5 77 0 0.1 9.92 78.2 0.889 7.44 75.2 1.044 0.2 9.74 76.4 1.778 7.38 73.4 2.089 0.3 9.56 74.6 2.667 7.32 71.6 3.113 0.4 9.38 72.8 3.556 7.26 69.8 4.178 0.5 9.2 71.0 4.44 7.2 68 5.22 41 Generator Vector Voltmeter A. 8 v y Figure 3.1. (a) (b) I V E-H Probe 50 9 SO 9 Probe b "A“ I080. Saline Generator (a) (b) Experimental setup for the measurement of the input impedance of a probe with an E-H probe and a vector voltmeter. Setup for adjusting the phase of a vector voltmeter. 42 measured by the vector voltmeter and their ratio is the impedance of the line at the V-I probe location. Physical geometry of the bare probe which was used in the experiment is shown in Figure 3.2. Before measurement several steps should be taken as follows: (1) The vector voltmeter should be calibrated. For this reason two equal signals are connected to channels "A" and "B" of the vector voltmeter and the phase indicator should be adjusted to show zero (see Figure 3.1(a)). The phase knobs should not be changed during the experiment. The vector voltmeter measures the phase difference between the signal connected to channel "8" with respect to the signal on channel "A". The amplitude of each signal could be measured separately. .(2) The E-H probe should be calibrated. A match load, for this case a 50 a load, is connected to the E-H probe and the voltage probe and the current loops on the V-I probe are connected to channels "A" and "B" of the vector voltmeter, respectively. We then let 50 = k._§_ A cw mnosa m we mocmpm_mmc page? asp co mapsmms .mueme_gmaxm ecu ~wuvumcomcu we com_gmaeou .m.m mczuwd Amen—mac suppeseez e.o m.o N.o ..o ..l .l I. o L. .. S .8 O C E 8 u N; l. 8. mg n r. atomzh as o._ n ma ..om as 88 u a .93. . . £2 08 u c (6)”‘21 + 47 .m_ums mcpauaueou mzopsm> cw once; a to mucmuomme page? on» no mapsmmg ”augmepgoaxo use —uowameoogu we :omwemasou .¢.m beamed Amcmpmmv xuwpmsgoz «.0 m.oo «.0 ~.o a d) .... ‘ as o.— u m< E8": 8.3"; xgoogpullll es m¢.o u a .axm . . N=2 com 1 e em mq 1:1 m... o—1 48 .mwume mcwuuzvcou mzopem> cw mnoca a $0 mucmpmwmmc bang? on» co mupzmme Faucme_gmaxm new Pmu_umsom;u mo compsmaeou .m.m meamwm Accepmmv xuwpwEgoz ¢.o m.o N.o P.o 1:11. 3 m 01 I 3 OP gr cm snooze N o as me u c . o . n _ axm . as m n 1 g I. on as o.~ n ma ea m¢.o u m NI: com u m .m0 nu ma hvl .... 0 49 .owume mcwuuaceou mzowgm> :F maoea a mo monogammc page? one go mu_=mme paucoepcmaxm vet pmu_umeomgu mo comwcanou .m.m mesmFd + O Am:_—mmv >u__wecoz e.o . 11ml . .5. Sum; 831:. .. » EEm¢.oum com .1111 :H EEO.Hu < .axu o o ~:z coon» mm a A 3... _ m... ml op- mp: (u)u1x CHAPTER IV APPLICATIONS OF THE BARE PROBE In the study of EM local heating of an imbedded tumor, it is very important to know the conductivity and permittivity of the tumor relative to that of the surrounding tissues. There are reasons to suspect that the electrical properties of the tumor may be dif- ferent from those of surrounding tissues because the blood flow to and tissue structure of the tumor are different from the normal tissues. A successful analysis of the induced EM field inside an irradiated body with an imbedded tumor and an effective device for focusing EM energy in the tumor will depend on the knowledge of electrical properties of the tumor. A bare probe can be used to measure the conductivity (0) and permittivity (e) of the biological bodies in vivo. A bare probe, when driven by an RF source, can also be used locally to heat the biological tissues. Figure 4.1 shows a bare probe inserted in a tumor. 4.1. MeaSurement of o and e In Chapter II it was shown that a coaxial probe in a lossy medium is equivalent to an asymmetrical dipole. The input admit- tance of a dipole is a function of the frequency of operation, physical dimensions of the dipole, and electrical properties of the 50 52 surrounding medium (i.e. permittivity, permeability, and conductivity of the medium). Permeability is defined as u = nour, which for a nonmagnetic medium gives ur = 1. Therefore, the input admittance of a fixed probe at a certain frequency in a nonmagnetic medium is only a function of o and e of the surrounding medium. The input admittance of a probe, Yin(o,e)==Gin(o,e)+-j8in(o,s), has a real part (conductance) Gin = fg(o,e) and an imaginary part (susceptance) Bin = fb(o,e). The input conductance and susceptance are functions of the conductivity and the permittivity of the medium. For finding a and c analytically from the conductance and the susceptance of the probe, the functions fg and fb should be known. These functions cannot be expressed in terms of simple func- tions, but the probe can be calibrated based on the theory developed in Chapter II. Input conductances and input susceptances of the probe are found for various values of o and c. There are two ways to use this information about the probe which lead to the determina- tion of the electrical properties of a biological tissue or a lossy medium. In the first method, two sets of curves are drawn. One set is the conductance and the other set is the susceptance of the probe versus the conductivity, where the permittivity is used as a param- eter. These sets of curves for a typical probe are shown in Fig- ures 4.2 and 4.3. The physical dimensions of the probe are given in these figures and the frequency of operation is 600 MHz. The 53 O O O to m Q' II II I L. S. $- 0) U (.1) 11.) :5 <5 C") N (gm) aaueionpuoo .mcwg “sneeze ewueemes e gum: empeeee mp uewee mcw>_ee ecu ecu eczema ee_pe:cm Fecmwuefi epewm eweueopm ;u_z eeeweuee .eweee mewpeeeeeu m:e_se> e_ mecca e we eeceueeeeee page“ .~.e weemwm As\mve o.m m.N o.~ m._ ow? m.o 0 on u 0 on u as me u N; as m.~ n F; L vow es o.F u me ow m 0 EE m¢.o n m Nxz cow 0 w :om em me at N 1:... ¢ .1111: .1 N 1. P 54 .mews beeceee emememee e new: empeeee m_ aspen mcv>wce ecu eegues :epueeeu peemeuem epewd eweuempm new: ememeuee .e_ees mewaeeecee m=e_se> :_ meece e we eeeeaememem “seem .m.e enamel re7 As\mve . . . . ‘1 D.Mp .' o N mpip 00—. mic Om H L AvOF 8 ... e 1 N as me 1 g .8 S u r. 2. SN 55 o._ u we om u L E 88 u e .e u e 1 .e u N=2 com 1 m on u 2 :0” mm me (s m) aouezdaosns 55 following example shows how electrical properties of a lossy medium can be measured. An experiment was conducted and the input admittance of a bare probe with the same dimensions as given in Figure 4.2 was measured. The conducting medium was 0.2 normal saline and the tem- perature was 25°C. The input admittance of the probe was measured to be Yin = 38.98 + j4.97 (mu). Draw two straight lines for Gin = 38.98 and Bin = 4.97 in Figures 4.2 and 4.3, respectively. From the possible values for er and 0 found in these curves inter- sected by the two straight lines, we can estimate the conductivity to be 2.4 (s/m) and the relative permittivity to be 65. The exist- ing values for 0.2 normal saline at 25°C is about a = 2.21 (s/m) and er = 74.8. Thus, the accuracy of this method is considered to be satisfactory. The second method for finding 0 and e is, after theoretically finding the conductance and susceptance of the probe for various possible values of o and e, the information is stored in the computer and then the input admittance of the probe imbedded in a medium with unknown 0 and e is measured. The measured values are fed to the computer and a computer program searches for the closest values for o and e of the unknown medium. Using this method the input admittance of the probe measured in the above example gives er==65.5 and o = 2.4 (s/m). The second method seems to give more accurate results. 56 4.2. Local Heating Another application of the bare probe is for local heating in cancer therapy by microwave hyperthermia. In this section the 'heat pattern of a bare probe in a conducting medium is being studied. In the previous chapter the current distribution along the probe has been determined. The electric field produced by the current on the probe at any point in the medium can then be found. The heat created by the electric field in the lossy medium is given by 131512. With the assumption that the current is only in the z direc- tion and with a rotational symmetry in our problem, the vector potential has only one component in the z direction and it is expressed as _ u h1 e-ij Az(r,z) =‘4; Iz(z ) R dz (4.2.1) '“2 and Ar(r,z) = A¢(r,z) = 0 . The relationship between the magnetic field and the vector potential is >+ + B = V x Therefore, Br(r,z) = Bz(r,z) = O 57 and aAz(r,z) B¢(r,Z) = ' 3r Maxwell's equation v x B(r,z) = u(0 + jms)E(r,z) implies that E¢(r,z) = O and 2 . a A (r,z) Er(r,z) = --i¥w——5%33———- (4.2.2) . 3A (r,z) 32A (r,z) Ez(r,z) = f; [% JF— + ——:—rf—] . (4.2.3) 2 Assuming that R = [r2 + (z - z')2] and the current on the probe as N 12(2') = IE: InPn(z') , n=1 the electric field in the r direction is expressed as jwu a2 e-ij Er(Y‘,Z) = - 7m h:ZIn P n(Z)e (12' or ' N h . 1 -JkR Er(r,z)= “ 41—72 In gaff Pn(z') :32-(9 R ) dz' 4nk _1 n-1'h2 Since jL-= - -§T-, then 32 82 58 I- 2 “2+ N . e-ij n 5,.(r.z) = - %21n[11k-§+—5)-—- "k n=1 z'=z_ n for -h2 5_z §_h1 (4.2.4) After substitution for the current and the vector potential in (4.2.3), we have 'hz "=1 32 [Ni ( ) - kR J + 1 P 2' dz' 3:2 _ n n R -h2 n-1 and then N e-ij e-ij - w 1.3.. - EZ(r’Z) 41rk Z In[ 7‘ 31‘] d2 +3-2- d2] n=1 (AZ) (Z (4.2.5) For large r or small r and z t (Az)n we can write -ij -ij e . _ e d], R dz - A R . (AZ)n (4.2.5) then becomes N . 2 2 . 2 2 -ij _ qu +2+k r 3 kr 3r e Ez(r,z)-- ZInFR—k T'J—T'j] R 411k ":1 R R R (4.2.6) 59 For small r and ze(Az)n, we have -ij / [ e__§_ dz' = 2Ln {-295 + [1 + (7;?)2] }- jkA (£12)In and -ij -1/2 % erz'=“A—2[1*(2AF)2] (AZ)n r 81‘ 2 -ij -3/2 a e . _ 2A 1 A. 2 A 2 7] R dz -;§[1+§(§F)]I1+(§F)] (A2) n Substitute the above approximate expressions in (4.2.5), and we have O '0111 N A A 2 '3/2 520.2) = 21:32 143111 + (72;) ] n=1 Numerically the r and 2 components of the electric field can be found in the medium. The heat produced by the field is given by 1311512 , where 1512 = 11:21? + |Er|2. Figures 4.4 and 4.5 show the equi-power contours for a bare probe in the r-z plane. Electrical properties of the medium and physical dimensions of the probe are also given in these figures. The driving voltages are V0 = 8.44 volts and V0==7.63 volts in Figures 4.4 and 4.5, respectively. The changes in the driving voltage resulted from keeping the input power equal in both cases (input power = 1 watt). The driving voltage is determined as follows: 60 9‘2 (mm) a1 = a2 = 0.43 mm 20 0 h]. - 7.5 m h2 = 30 mm Vo = 8.44 volts 10 A 0 J 4- r (m) 30 100 -10 -20 8.46 0.47 0.14 -30 Figure 4.4. Equi- power contours for a bare probe in the r-z plane. The power is normalized and the frequency of operation f= 600 MHz, 02 = 1.11 (s/m), er = 76.7. 61 .A 2 (mm) a1 = a2 = 0.43 mm h1 = 15 mm h2 = 30 mm VO = 7.63 volts ? s—JI- 30 r (M) Figure 4.5. Equi-power contours for a bare probe in the r-z plane. The power is normalized and the frequency of operation f = 600 MHZ, 02 = 1.11 (s/m), €r2 = 76.7. 62 . 1 * Power (input) 1 watt =-§ ReVOI 1 V12) = _ Re a o o w 2 (R1n + jX1n) and 2 2 1/2 _ Rin + Xin V0 ‘ (:2 ""7$fi?”' ) The power in these figures is normalized. The length of the probe h1 is 7.5 mm and 15 mm in Figures 4.4 and 4.5, respectively. In both cases most of the power is concentrated near the probe, along the z axis. As we see from these figures the heat pattern can vary with changing the length of the probe. CHAPTER V INSULATED PROBES IN A CONDUCTING MEDIUM In order to transmit energy into the conducting body and to avoid direct contact between the probe and the body tissue, the probe can be covered with insulated material. The insulated probe is used extensively for local heating in biological bodies. For heating applications, the insulated probe has two major advantages in comparison with the bare probe: (1) The input resistance of the bare probe is less than the input resistance of the insulated probe in a conducting medium, therefore, the insulated probe radiates more EM energy in the con- ducting medium. (2) The current on the bare probe decays rapidly along the probe, hence most of the EM energy radiated from the bare probe is concentrated near the driving point. However, in the insulated probe case the current does not decay rapidly along the probe, thus, the EM energy is distributed over a larger volume. Evolution of geometry of the insulated probe in a conducting medium is shown in Figure 5.1. Since the current flow is as shown in this figure and the surrounding medium is lossy, the probe is treated as a lossy transmission line. In this case the probe cannot 63 ////// /////////////A ///////////////J ‘ ///////////// #V W (C) \\ \\\‘ \\ .11 c—Jo CC C 1 I'D 01 ...a m < C —l 0 c .3 fi:\ 0.. 'U 1 O U. 0 do :3 \ 9’ n O 3 I 65 be truncated, therefore, equivalent terminal impedances Ze1 and Ze2 are 1ntroduced. 5.1. Theory of the Probe Since an insulated probe in a conducting medium can be con- sidered as a lossy transmission line, transmission line theory is used to analyze the characteristics of the probe [18]. The equiva- lent circuits of the insulated probe in a conducting medium in the form of lossy transmission line are shown in Figure 5.2. To start with, the equivalent transmission line of the probe is divided into two sections, where ZCl and Zc2 are the characteristic impedances of section (1) and section (2), respectively. The driving voltage V is divided into two voltages, V1 and V2. The input impedances of each section are (Zin)1 and (Zin)2, respectively. The input currents of these two sections are I1 and 12, but since the current at the junction is continuous, it requires that 1:12. This leads to the relation V1 = V2 (511) (Zin)1 (Zin)2 ’ ' ° The other relationship between V1 and V2 is V = V1 + V2 . (5.1.2) Using (5.1.1) and (5.1.2), V1 and V2 can be expressed as follows: 66 4. Zelé Zc1 V1 - section (1) section (2) (Zin)1 (Zin)2 Figure 5.2 Geometries for an insulated probe in a conducting medium. 67 r (Zin)1 V1 =V (mil + (Zin)2 ( (5.1.3) (Zin)2 (V2 =V(Zin)1+ (21102 The input impedance of a transmission line is ZL + 2C tanh [vh] 21” = Z1: 2C + 2L tanfi*[yh] (5°1°4) where y is the propagation constant of the line, Zc and ZL are the characteristic impedance and load impedance of the line, respec- tively, and h is the length of the line. Z Using the geometry of Figure 5.2 and defining tanh 01 = IQ Z el and tanh e = J2 , where Z and Z. are terminal impedances of 2 Ze2 e1 e2 two sections, the input impedance for each section of the line can be expressed as (Zin)1 = ZCl coth(y1h1 + 81) (5.1.5) (Zin)2 = ZCZ coth(y2h2 + 02) Substituting (5.1.5) into (5.1.3), V1 and V2 become IV = V ZCl c0th(ylh1 + 01) 1 2C1 COth(y1h1 + Bi] + ZCZ COth(Yzh2 + 82) 1 (5.1.6) 2 Zc1 cot‘hhlh1 + 01) + 2C2 cothhzh2 + 82) .V 68 The reflection coefficient of each section of the line is r _ Ze1 c1 _ Ze2 ' Zcz - Z -.____.._._,I' - 1 Ze1 + Zc1 2 Ze2 + Zcz After introducing the above notations, the voltage and current along each section of the probe will be found separately. A set of coordinate system is introduced for section (1) of the line as indicated in Figure 5.3(a). The voltage along the line can be written as 'Y S y S V1(s) = v+ (e 1 + r1e1> (5.1.7) At 5 = -h1 the voltage V1 = V1(-h1). Substituting for P1 in (5.1.7) we have V cosh e + sinh e v+=%- 1 1 (5L& cosh (ylh1 + 61) Using (5.1.7) and (5.1.8), the voltage along section (1) of the line becomes cosh (YIS - 61) V1(S) = V1 COSh (Ylhl + 61) (5.1.9) The current distribution along the line is + - S y S _ v Y1 1 11(5) - Tali" - I‘le ) (5.1.10) «‘— I1 2 i 'F e1jfi: Zc1 éy_v1 I -—a- 1 -h1 Si1 V (b) 1.. w 'hz w = 0 +— I1 <—-12 + .— Ze1 V1 V2 Ze2 - -F h1 z 44 -112 z = 0 (C) Figure 5.3. (a) Section (1) of the line. (b) Section (2) of the line. (c) Common coordinate system for both sections. 70 After substitution for V+ and II in (5.1.10) the current along section (1) of the line becomes v1 sinh (e1 - YIS) ZCl cosh (ylh1 + oi) 11(5) = (5.1.11) In Figure 5.3(b) the coordinate system for section (2) of the line is shown. The voltage and the current along this section of the line are cosh (wa - 92) v2(W) = V2 COSh (y2h2 + 82) (5.1.12) V sinh (6 - y w) 12(w) = 2 2 2 (5.1.13) 2C2 COSh (Yzhz + 927 So far there are two different coordinate systems for section (1) and section (2) of the probe. Choose one common coordinate system as indicated in Figure 5.3(c) and transform both coordinate systems to the new coordinate system as Z = s + h1 hi fl -W’hz After transformation the voltage and current along the probe are 71 cosh [11(z-h1) - 61] r V1(z) = V1 COSh [Ylhl + 01] (5.1.14) 1 for 0,: 2.5 h1 V1 sinh [81 - 11(Z-h1)] L I = . 1(2) ZCl cosh [ylh1 + 01] .(5.1.15) and 1 V (2) = V cosh [Y2(z+h2) + 92] (5 1 16) 2 2 cosh[y2h2 + 62]' ° ‘ l for 0.: 2.: -h2 ‘_ V2 sinh [82 + y2(z+h2)] ZC2 cosh[y2h2 + 82] (5.1.17) H N A N V I In these equations V1 and V2 are expressed in (5.1.6). 5.2.1. Parameters of the Transmission Line In Figure 5.4(a) an insulated probe in a conducting medium is shown. Region (1) in this figure is the outer conductor of the microprobe and it is usually made of a material with very high con- ductivity. Region (2) is the conducting medium with (02,52), and region (3) is the insulating material with (0d,ed). The insulated probe in a conducting medium is equivalent to a lossy coaxial cable with a complex propagation constant defined as y = /VZ = a + jB where Y is the admittance per unit length and Z is the total impedance per unit length of the line. For a coaxial cable Y is 72 II II A :: A f 11 II II (3) H I, 2a —-U [I ‘.—. a; ll 5 LI! {ill ll 11 1| 1 1 ... > 1 __V (2) conducting medium (1) (0 a) ,r ,\ 2’ 2 A 3°... 5.2.31 3 3 l (a) Lt 2a3 (b) / Figure 5.4. (a) (b) X )0 Insulated probe in a conducting medium. Infinite cylinder. 73 .< II 9 + 30C where Zno 206 = d and C = d g |n(a37a1) |n(a3/a1) and the total impedance per unit length is z=z"1+z;+ze where Z} and 2; are the surface impedances of the inner and outer conductors of the transmission line, respectively, and Z8 is called the external impedance of the line. For finding Z1 and 2; we assume the infinite cylinder geometry and that the current on the cylinder flows only in the z direction as indicated in Figure 5.4(b). Start with Maxwell's curl equations: +-+ v x E(?) -ij(r) Vxfifi) me+mnah. Since the cylinder is assumed to be infinitely long and there is rotational symmetry in the problem, we have +1+ +-+ 3E(r) = BHIF) = 0 32 32 +1+ +-+ Alglelgfilw 74 The electric field has only one component on the 2 direction and it is 2 aE(r) aE(r) ’- +12. new er a where k2 = m we - jwuo . The solution of the above differential equation for the inner con- ductor is E (r) = AJ k r) + BN 2 0(1 k" 0(1) where constants A and B can be determined by the boundary conditions. 0n the z axis (r = O) the electric field should be finite; this condition implies that B = 0, therefore, Ez(r) = AJO(k1r) On the surface of the cylinder at r = a1 we have Ez(a = AJ0(k a ) 1) 1 1 07‘ 75 E zl(a ) A = JoIk131) Substituting for A, the electric field inside the cylinder is J0(k1r) Ez(r)= E Z(a a1) i—BTEIEIY 0.: r.: 61 The current density on the cylinder becomes Jz(r) = 0 E (r) The total current in the cylinder is a1 2“ 21ra101 31(k1a1) = OIEZ(Y‘)Y‘dY‘d‘P = T Eaz( 1).] W 0 0 The definition of the surface impedance is the ratio of the electric field on the surface to the total current. For good conductors most of the current flows on the surface, therefore, 2. = Ez(a1) k1 00(k1a1) l .. 1 I2 211a101 J1(k1aI) The solution of the differential equation for Ez(r) in the outer conductor is Ez(r) = CHél)(k2r) + ougz)(k m, r) Boundary conditions for this region are: 76 (1) Electric field at infinity is Ez(r‘+ m) + 0; this implies that C = 0. Therefore 2 52(r) = 005 )(kzr) (2) At r = a3 we have E (a z 3) = DH02)(k2a3) 01" 0 = (:§(a3) H0 (kza3) The electric field in region (2) is Héz)(k2r) E (F) = E (33) 2 Z 2 H5 )(k2a3) The current in this region is 2w m (2) (2) H (k r) Zwa 0 H (k a ) 0 2 3 2 1 2 3 I = 0 E (a ) rdrd¢= - E (a ) 2 f0 fa 22 3 ”((32)(k233) E2 2 3 H(2) 3 The surface impedance for the outer conductor is then 2 _ ________ _ kg ”0 )(kzaa) 2 I ZFEFTF‘ (2) 3 2 H1 (k2 a3) ‘ For coaxial cables there is no radiation; therefore, r 77 In the transmission line theory the external impedance is due to the interaction between the currents on the conductors of the line. The external impedance Ze is defined as Ze = re + ije . e is zero and the external impedance becomes ze = jwle where e - 11. l - 2w ln(a3/a1) . It is assumed that the insulated material is nonmagnetic (u = 00). thus, we have u . e _ . 0 Z - Jw-§; ln(a3/a1) . The total impedance per unit length and the admittance per unit length of the probe have been defined for a general case. Now we will consider some special cases. Case (1). Iklall << 1, |k2a3l << 1 Small argument approximations for the first kind Bessel functions are . x2 J0IX) = 1 ' I?) 2 to £21134») 78 With these approximations the surface impedance of the inner con- ductor is _ 1 :gg - +380 Z. flanl 1 1 Small argument approximations for Hankle functions are Héz)(x) 5 1 + j $~ln-§% 11(2)”) £13} where y = 1.781. The surface impedance of the outer conductor in this case can be written as . 2 2 1 + j-— ln-————— 21 _ k2 n ykza3 2 - - 2nd 0 . 2 3 2 j-———-— uk2a3 For k2 =\/ -jwu02 , the surface impedance can be expressed as 79 Case (2 . lklall >> I, |k2a3| >> I The argument of Bessel functions are very large for this case and asymptotic approximations canibe used. In this case the first kind of Bessel functions are 2 n 11 Jn(x) s Ei-cos (x - 7;-+-Z) |x| >> 1 The surface impedance for the inner conductor then becomes . k Z1 = j 1 1 2"31°1 For k1 = ‘/ -j01110 , the surface impedance on the inner conductor is Zi = 1 mu 1 211a1 '23; (1 + j) Asymptotic approximations for Hankle functions are . .. 1x1 >>1 The surface impedance of the outer conductor for this case can be expressed as . k2 z' - j 2 2nd 0 80 Fora good conductor, k2= ‘/-jwuzoz , and the surface impedance of the outer conductor becomes i 1 “’“2 22‘21753‘ 25“”) The characteristic impedance of the line by definition is -2. 2 ZC - Y Y , where Z = r + jx, Y = g + ij . Substituting for Z and Y, the characteristic impedance becomes Z s r + jx C 94-ij The propagation constant of the line Y can be written as Y=a+JB=\f(Y‘+J'X)(9+J'wC) Squaring both sides of the above equation and equating the real and imaginary parts of both sides, we have 1 . 1/2 1/2 a = [.2 (rg - wa) +-%{ (rg - wa)2 + (gx + (urC)2 } I 1 1/2 1/2 a =l -%'(me - rg) +-%{ (rg - wXC)2 + (gx + er)2 } 81 The propagation constant of a lossy coaxial line is tabulated in Table 5.1. The inner conductor of the line is copper and the outer conductor is lossy medium with €r2 = 69 and -9¥-= 8.8. In this (DE: 2 table it is assumed that w = 7.16 x 108 sec'l, k2 = 41.5 (1 - j). Since the inner conductor is a very good conductor, the surface impedance of the inner conductor is very small. The surface impedance of the outer conductor and the external impedance of the “approx and Bapprox are calculated numerically. line for various values of a1 and a3 are shown. are taken from [19], but a and B exact exact 5.2.2. Terminal Impedances of the Probe, Zel and Ze2 In Figure 5.5(a) a configuration of an insulated probe in a conducting medium is shown. In order to investigate the character- istics of the probe it is necessary to know the terminal impedances of the probe at planeAA(Ze1) and at plane 88 (Zez). At plane 88 the equivalent terminal impedance is complicated. The current at this plane is not zero, therefore, this terminal impedance is not infinity. An equivalent circuit for the terminal AA is shown in Figure 5.5(b). In this figure when d = O the inner conductor is in direct contact with the outer conductor of the transmission line, or the conducting medium, therefore, Zel is very small. For d/(a2 - a1) << 1 and 2nd/A << 1 we have [20] a a a a T=_i_1_'|n £(%E]L+ Ill—23;) (5.2.1) 82 e.Nen + AH.me N.men + m.mm N.een + N.Nm m.HHHN + m.ee NMNfi + H.em WN a.emen Ne.mmmN m.eme ..emsh mm.emn 5N meeN.N HmNm.N eeeN.N enema.N acme.N edaxee emmN.e mNeH.e eeeHN.e memem.e mmHN.~ edaxea NHeN.N emm.N mmN.N mo.m meNN.HH xeeaaae eNN.e NNHH.e meN.e mmm.e mamm.m xeeeeea em.N Hm.m mea.~ NN.H memm.e Aeva. mme.e mNHm.e mNHm.e mNHN.e mNHm.e laden. .mpeee pewxeeu ammep we mgmueEegea .~.m mpeeh 83 I to generator . or . conducting medium 21 A ------ --- --- A (a) i?- f (b) Figure 5.5. (a) Configuration of an insulated probe in a conducting medium. (b) Equivalent circuit for terminal AA. 84 In the following example the terminal impedance (Z ) of an insulated e1 probe is found. Example: For an insulated probe with dimensions of a1==0.43nmu a2 = 0.96 mm, d = 0.1 mm, and the dielectric material with ad = 0, 8rd = 2.45, the terminal impedance (Zel) at 600 MHz frequency is B = 0.0065 YC and Z = JB = j0.0065 Y e1 C ' In our study the line is lossy, therefore,a1conductance should be parallel with the capacitor in the equivalent circuit. 5.3.1. Input Impedance of the Probe The input impedance of the insulated probe is a function of several parameters. It is a function of physical dimensions of the probe, the electrical properties of the conducting medium and the insulator, the terminal impedances, and the frequency of the operation. In Figure 5.3(c) the current at the origin is but from equation (5.1.15) the current at Z = 0 is given as 1(0) = Zc1 cosh [ylh1 + 61] (5‘3'1) 85 Substituting for V1 using (5.1.6) in (5.3.1), the current at the origin in 1(0) = sinh(y1h1+-61)sinh(y2h2+-ez) \I _ . . 'ZClcosh(y1h1+-91)Sinh(yzh2+62)+2C2 Slnh(Y1h1+-01)COSh(Y2h24'0§) (5.3.2) By definition the input impedance is Zin = T(%)" or Zin = Zc1 cosh (y1h1+61)sinh(y2h2+62)+ZC2 sinh(y1h1+-61)cosh(yzh2+-62) sinh(y1h1+-61)sinh(yzh2+-92) (5.3.3) In this study it is assumed that both sections of the probe have the same diameters for inner and outer conductors, and the same dielectric material, therefore, tflua characteristic impedances ZCl = ZC2 = ZC and the propagation constants Y1 = Y2 = y. With these assumptions the input impedance can be written as COSh(Yh1+ 01)Sinh(yh2+ 62) + Sinh(yh1+ 91)COSh(yh2+ 92) NF zC Sinh(yh1.+ el)sinb(vh2+ 62) (5.3.4) The input impedance of the probe for three different cases of terminal impedances are considered as follows. 86 Case (1). In this case it is assumed that both terminal impedances of the probe are infinity, Ze1 = Zéz = 00; this implies _ _ -1 the input impedance of the probe becomes 0 = 0. Substituting for 91 and 62 in (5.3.4), sinh [Y(h1 + h2)] 2‘" = Zc sinh [yhl] sinh [yhz] (5°3'5) For a symmetric probe, h1 = h2 = h, (5.3.5) becomes Zin = 22C coth [Yh] (5.3.6) or 2i” ‘ 22c 2222 (3:2) : g3:n(駣)) If ZC is pure resistive, for 8h = nW/Z, n = 1, 2, 3, ... the input impedance of the probe becomes pure resistive. In Figure 5.6 the input impedance of the insulated probe in a dissipative medium as a function of the probe length is shown. In this figure it is assumed that h1 = hZ’ a1 = a2 = 0.47 mm, a3 = 0.96 mm, and the frequency of operation is 915 MHz. The elec- trical properties of the insulator and the conducting medium are = 0.0 (s/m), e = 1.37 and 02 = 0.88 (s/m), e 2 = 42.5, respec- Od rd tively. For short insulated probes the input impedance is mostly I“ capacitive; the same phenomenon is observed in an electrically short bare probe. Xin (Q) 87 50 v 300 Rin (Q) ‘100 o -200 n J 1 cm = h -300 + Figure 5.6. Input impedance Zin = Rin + inn of symmetric insulated probes in a dissipative medium, y = 10.72 + j49.38 and Zc = 70.77 - j15.36 9. Both terminal impedances are assumed to be infinity.‘ 88 Case (2). In this case it is assumed that both terminal 1 m = jn/Z. impedances are zero, Ze1 = Ze2 = 0, 81 5'52 = tanh Substituting for 91 and 82 in (5.3.4), the input impedance of the probe becomes sinh [y(h1 + hz] 2‘" ‘ zc cosh [yhljcosh [yhz] (5.3.7) For a symmetric insulated probe h1 = h2 = h, the input impedance can be expressed as Zin 22C tanh [yh] (5.3.8) or sinh (2ah) + jsin (28h) C cosh (Zah) + cos (28h) ZZ Zin With the assumption that Zc is pure resistive, for ab = nn/Z, n = 1, 2, ... the input impedance of the insulated probe is pure resistive. Figure 5.7 shows the input impedance of the insulated probe in a lossy medium as a function of the probe length. The terminal impedances of the probe Ze1 and Ze2 are assumed to be zero. The physical dimensions of the probe and the electrical properties of the insulator and the lossy medium are the same as in case (1). In this case for an electrically short probe the input imped- ance is not mostly capacitive, and the input resistance is larger than the input resistance of case (1). 89 Xin (Q) 200 h 100,, -100r -2004 Figure 5.7. Input impedance Zin = Rin + inn of symmetric insulated probes in a dissipative medium, y = 10.72 + j49.38 and Z = 70.77 - j15.36 9. Both terminal impedances are assumed to be zero. 90 Case (3). In this case it is assumed that one of the terminal impedances is zero and the other infinity. (a). The terminal impedances Ze1 = 0 and Ze2 = m. Substi- tuting for 61 and 92 in (5.3.4), the input impedance of the probe becomes cosh [y(h1 + h2)] Zin ‘ Zc cosh [thj’sinhtyhzj (5°3°9) (b). The tenninal impedances Ze1 = m and Ze2 = 0. The input impedance of the probe is cosh [7(h1 + h2)] 2‘" ‘ Zc sinthhIJCOShfyhzj (5°3'10) Now consider a symmetric probe with h1 = h2 = h. The input imped- ance of the probe becomes 22 Zin C coth [ZYh] (5.3.11) OY‘ sinh [4ah] - jsin (43h) C cosh [4ah] - cos (43h) 22 Zin For 8h = nw/4, n = 1, 2, 3, ... the impedance of the probe is pure resistive (2C is a pure resistance). In Figure 5.8 the input impedance of the insulated probes with the terminal impedances of one zero and the other infinity are shown. In this figure the dimensions of the probe and the 91 Xin (Q) 50.. 300 * Rin (o) -100‘, 1' 1 cm -200., Figure 5.8. Input impedance Zin = Rin + inn of symmetric insulated probes in a dissipative medium, 7 = 10.72 + j49.38 and ZC = 70.77 - j15.36 a. Terminal impedances Ze1 = 0 and Z e2 =m. 92 electrical properties of the insulator and the conducting medium are kept the same as that in case (1). The input impedance of the insulated probe is a function of the electrical properties of the insulator. In Figure 5.9 the input impedance of the probe versus the dielectric constant of the insula- tor is shown. Both terminal impedances of the probe are assumed infinity in this figure.' Physical dimensions of the probe are a1 = a2 = 0.43 mm, h1 = 15 mm, h2 = 45 mm, and the frequency of operation is 600 MHz. The probe is immersed in a conducting medium with o = 1.86 (s/m), er = 76.3, and the insulator has °d = 0 and a variable permittitivity. Changes in the input reactance is more significant than the input resistance of the probe. For an electrically short and thin insulated probe in a dis- sipative medium we have ah << 1, 8h << 1, 33 << h and the input admittance of the probe has been found [11] to be 2 onh Y1” = G + M ‘ ln(h/a3) - 1 P2 + (1 + Y)2 + jw Tn(a3/a2) P2 + (1 + y) (5.3.12) 2 where P = weoer and Y = Er jn(a3/a2) 93 The input admittance of the insulated probe obtained from (5.3.12) and that from the theory developed in this study are compared in Table 5.2. The input susceptances are in good agreement, but there are some deviations in the input conductances. Mostly the devia- tions in the input conductances come from the fact that the theory developed in this study is not accurate for short insulated probes. In this table the probe is symmetric with h1 = h2 = 15 mm, a1 a2 = 0.43 mm. The electrical properties of the insulator are °d 0.0 (s/m), 8rd = 2.25, and the electrical properties of the conducting medium are variable. In Figure 5.10 the input impedance of the insulated probe with three different terminal impedances are compared with input impedance of a bare probe. The dimensions of the probes are; a1 = a2 = 0.43 mm, h1 = 15 mm, h2 = 45 mm, and the frequency of operation is 600 MHz. The electrical properties of the insulator are od = 0 (s/m), 8rd = 2.25, and the probes are immersed in the saline with various normalities. The input resistances of the bare probe and the insulated probe with terminal impedances Ze1==Ze2=<=° are very close, but their input reactances are different. So far it is assumed that both sections of the probe have the same radius (a1 = a2). Now we consider the case with a1 f a2, and find the input impedance of the insulated probe. In Table 5.3 the input impedances of the insulated probes with a1 = a2 and a1 # a2 are compared. The input resistances are very close in two different cases, but there is a constant shift in the input reactance. 94 Riin (o) .100 1. 50 Rin r 0.0 f = 600 MHz h1 = 15 mm h2 = 45 mm +r-100 al = a2 = 0.43 mm '4» -50 o = 1.86 (S/m) 76.3 ,_100 0 1.0 2.0 3.0 4.0 5.0 6.0 Erd Figure 5.9. Input impedance Zin = Rin + inn of the insulated probe in a conducting medium versus permittivity of the insu— lator. Terminal impedances Ze1 = Ze2 = m. 95 Table 5.2. Input admittance of the electrically short and thin insulated probe in a conducting medium. Yin (Theory) Yin (Formula) Frequency P =«fi% x 10'6 x 10'6 10 MHz 25.84 0.0005 + j73.3 0.18 + j73.3 10 MHz 52.94 0.0005 + j73.3 0.091 + j73.3 10 MHz 80.63 0.0005 + 373.3 0.06 + j73.3 10 MHz 109.73 0.0005 + j73.3 0.046 + j73.3 100 MHz 2.85 0.66 + j736.4 15.3 + j726.4 100 MHz 5.32 0.60 + j736.4 8.7 + 3731.4 100 MHz 8.06 0.57 + j736.4 6.0 + j732.3 100 MHz 10.97 0.56 + j735.8 4.55 4 j732.6 96 .(1) = Bare probe (2) = Insulated probe with Ze1 = Ze2 = 0.0 (3) = Insulated probe with Ze1 = m, Ze2 = 0 0 (4) = Insulated probe with Ze1 = Ze2 = w Xin (Q) Rin (Q) ...... Xin -————— Rin -250<» )250 -2004> #200 ~150‘- 0150 -100 .. (4) ———————————————— ..100 (2) _________ .N ‘50 " \‘\\\ i1 50 (4) (1)>'—‘%~_< (1) 0.0 ¢----M__~~ % 0.0 0 IN 0.2N 0 3N 0.4N Normality of Saline Figure 5.10. Comparison of the input impedance Zin = Rin + inn of the bare probe with insulated probes with different terminal impedances. 97 m.oo~ N.Nomn “.mfi ¢.©oflh “.0H u NmN .3 u HaN mo.m m.oo~ m.m~mn m.mmm c.2NHh m.mm~ " NmN .3 u HaN mo.m H.m_- «.mmn H.08N m.mh m.oo~ u NaN .o u fimN mo.m m.oo~ ¢.momw m.mH m.Nlo m.mH u NmN .8 u HaN Hm.H m.oom m.ommn m.m- “.mufiw m.m- u NmN .e u HmN Hm.H N.ofi- m.m~w N.~mm m.H¢n m.mm~ n NmN .o " HmN Hm.“ m.oo~ m.mmmn “.mm mm“ w.m~ u NmN .e u HmN m~.o N.~oN N.mmmn 0.253 mafia c.N~H u NmN .8 u HmN m~.o «.0H- m.~¢fi N.mmH N.mmh w.mmH u NmN .o " HaN mu.o Amocouumme cwv as me.o n mm as me.o ".mm u Hm mmucmcmasu mm." a mucmgmdmwo as H.o u Hm .mcwsgm» .mucmcmas_ uaacm .m.m mpnah 98 The shift in the reactance is due to the change in the admittance per unit length of the transmission line. The admittance per unit length is Y = g + ij, where g and C are inversely proportional to the ln(a3/a2). In this case 9 is zero, since od = 0, and only C is affected. 5.3.2. Experimental Verification of the Theory In order to test the theory developed in this study, a series of experiments were conducted to measure the input impedance of the insulated probes with various terminal impedances in different con- ducting media. To realize the terminal impedance of the probe Ze2 = m in practice is very difficult, but the terminal impedance Ze1 can be made very large. In Figure 5.11(a), since the probe is in direct contact with the conducting medium (the outer conductor) the terminal impedances of the probe, Ze1 and Z92, are very small. 0n the other hand, in Figure 5.11(b) the terminal impedance Ze2 is again small but the terminal impedance Ze1 is very large. The experiments were carried out only for these two cases. The measurement was conducted by the vector voltmeter, and the conducting medium was saline. The experimental procedures have been explained in more detail in Chapter,III. In Figures 5.12-5.15 the physical dimensions of the probe are a1 = a2 = 0.43 mm, a3 = 0.96 mm, h1 = 15 mm, and h2 = 45 mm. In Figures 5.12-5.15 the theoretical values and the experimental results of the input resistances and the input reactances of the insulated probes are compared. The terminal impedances in Figures 5.12 and 5.13 are to generator 2a2__,. _‘_ Ze2 Insulator Conducting medium H \ Z e1 (a) 14‘ to generator .Ei E? Ze2 /’ ; $4.... Insulator / S h2 ; / / / / :1 C: Conducting medium ‘ F / / zj / / h / 1 / E j 281 (b) Figure 5.11. Configurations of insulated probes. (a) Terminal impedances Zel and Ze2 very small. (b) Ze1 very large and Ze2 very small. 100 Rin (o) A .o Experiment --'Theory ' l. 300 f = 600 MHz 0 O 0200 O *r100 O f = 900 MHz 3 ‘ * 5 4fi 4.- 0.1 0.5 1'0 normality Figure 5.12. Comparison of theoretical and experimental results on the input resistance of an insulated probe with terminal impedances Ze1 = 0.0, Ze2 = 0.0 in various conducting media. 101 Xin (o) . - Experiment at 600 MHz +n+ Experiment at 900 MHz Theory 0+100 0 * =600 MHz 4+0 + 7=9oo MHz -100 i r 3 ‘ f 4— 0.1 0.5 1.0 normality Figure 5.13. Comparison of theoretical and experimental results on the input reactance of an insulated probe with terminal impedances Ze1 = 0.0, Ze2 = 0.0 in various conducting media. 102 Rin (a) o 0 Experiment _ Theory l»300 fi' f=600 MHz 4) 200 o 1* 100 ._Z 0 . __; f=900 MHz L ; : t i i 3 3 A f $- 0.1 0.5 1.0 normality Figure 5.14. Comparison of theoretical and experimental results on the input resistance of an insulated probe with ter- minal impedances Zel = w, Zez = 0.0 in various cond- ducting media. 103 Xin (9) f o 0 Experiment at 600 MHz +'+» Experiment at 900 MHz ___ Theory _200“ f = 600 MHz 0 O f = 900 MHz -100]? X ‘t \: A L A A _ L t f v 0.1 0.5 1.0 normality ’- Figure 5.15. Comparison of theoretical and experimental results on the input reactance of an insulated probe with terminal impedances Ze1 = w, Ze2 = 0.0 in various conducting media. 104 assumed to be Ze1 = 0 and Ze2 = 0, but in Figures 5.14 and 5.15 the terminal impedances are Ze1 w and Ze2 = 0. From these results it is observed that the agreement between theory and experiment is very good for the case of Ze1 = w and Ze2 = 0, and for the case of Ze1 = 0, Ze2 = 0 the agreement is still considered to be satisfac— tory. These results confirm the validity of the theory developed for the insulated probe in this chapter. 5.4.1. Current Distributions Along the Insulated Probe The current along each section of the probe can be found by substituting for V1 and V2 in (5.1.15) and (5.1.17) using (5.1.6). The current can be expressed as 11(2) = V sinh(y2h2+-ez)sinh[(y1h1+-el)-iy12] 0 < z < “1 (5.4.1) and I2(Z) = Sinh(y1h1+ 81)Slnh[(yzh2 + 92) + 722] V ZCl cosh(y1h1+-el)sinh(72h2+-ez)+-ZC2 sinh(y1h1+-eilcosh(y2h2+-ezl 412 _<_ z i o ' (5.4.2) 105 If we assume both sections of the probe to have the same radius (a1 = a2) and the same dielectric material, then ch = Zc2 = 2c and 71 = Y2 = y. Equations (5.4.1) and (5.4.2) can be simplified to V S‘inh(yh2 + 82) sinh [(7111 + 81) - yl] 11‘“ = T sinhTy(h + h )1 (“-3) c 1 2 0 §.Z §_h1 and sinh (yh + e ) sinh [(yh + e ) + yZ] I (Z) = l- 1 1 2 2 (5 4 4) 2 Z sinh [7(h + h )] ° ‘ c 1 2 -h 5,2 5,0 The current along the probe for different terminal impedances is found as follows. Case (1). In this case Ze1 = Ze2 = m, 81 = 82 = 0. Substi- tuting for 81 and 62 in (5.4.3) and (5.4.4), the current along each section of the probe becomes Slnh [yhz] Slnh [yhl - yZ] _ V 11(2) " z— sinh mh + h )1 (”-5) c 1 2 0 §_Z i h1 and I (Z) = . (5.4.6) 2 '7; Sinh [y(h1 + hi7] 106 For a symmetric insulated probe h1 = h2 = h, the current along the probe can be further simplified to v sinh [y(h - lZJl] _h < z < h (5.4.7) [(2) = 22c cosh (yh) -— -— In Figure 5.16, the current distributions along the insulated probe with the terminal impedances Ze1 = Ze2 = m are shown. In this figure it is assumed that a1 = a2 = 0.43 mm, a3 = 0.96 mm, h1 = 15 mm, and h2 is variable. The electrical properties of the insulator and the conducting medium are “d = 0, erd = 2.25, OZ: 1.1, €r2 = 76.7. The current at the tips of the probes are zero due to infinite terminal impedances. It is observed that the current does not decay along the probe, and for the short probe h1 = h2 = 15 mm, the cur- rent has a triangular distribution which is similar to the current distributions along a short bare probe. Case (2). Both terminal impedances are zero, Ze1 = Ze2 = O, implying 91 = 92 = %§-. Substituting for 91 and 92 in (5.4.3) and (5.4.4), the current distribution along each section of the probe is V cosh (yhz) cosh [y(h1 - 2)] 11(2) ‘ 72' sinh [v(h1 + h2)] (5'4°8) 0 :_Z §_h1 and 1137 8 .N22 com mp zucmzamcw mgh .mpampgm> mp N; new as mg" as .3 u N Nu #8 N mmucmumasp ch_agme gap: monoga umumpzmc_ mcopm mcopgzapcumpn acmggau .o~.m mczmwd ES N E- . 8- .3- cm. 2.. o 2 _l - w, i / £491 84.9.... 9491 23.:3 g 108 V cosh (vhl) cosh [y(h2 + 2)] 12(2) ‘ T sirm [yuil + (12)] (55-4-9) C For a symmetric probe, h1 = h2 = h, the current along the probe is simplified to V cosh [v(h - 121)] 1(2) ='2Z; sinh (Yh) -h2 5'2 j'h 1 (5.4.10) The current distributions along the probes with tenninal impedances of Ze1 = Ze2 = 0 are shown in Figure 5.17. The physical dimensions of the probes and the electrical properties of the insulator and the conducting medium are the same as that for Fig- ure 5.16. On the tips of the probe the current is very large because the terminal impedances are assumed to be zero. Case (3). In this case one of the terminal impedances is zero and the other is infinity. (a). The terminal impedances are Ze1 = w and Ze2 = 0 cor- responding to 81 = 0 and 82 = %§-. Substituting for 61 and 92 in (5.4.3) and (5.4.4), the current along each section of the probe is \/ cosh (yhz) sinh [y(h1 - 2)] 11(2) = 72' cosh [7(h1 + h2)] (5'4'11) 0:Z_<_h1 and m .sz coo mF Aucmacmgd ugh .anmFgm> m_ N; van as mFu F; .o.o .uN Nu E N 88335 F2253 5.; mono: 33:35 ago? 2033.2»va 29:3 .2. m 9595 109 ...N- 8- 8- on. 2- o 2 1‘ .l 0 o h . o 0 25 N . .m 4/0 cm 4 N; .. S N N l\ 2 .. 9.... 2 u s 3 u .....v .mF SN II; 2.35 110 V sinh (yhl) cosh [y(h2 + Z)] 12‘“ = ‘2‘; cosh fy(h1 + h2)] (“'12) “h2:Z<0 For a symmetric insulated probe, h1 = h2 = h, the'current in each section of the probe becomes _ v osh h) s' h (h- [2!) 11(2) — 7: C LY cosngg) J (5.4.13) 0 §_Z §.h and ._ V sinh ( h) cosh [ (h - JZ])] 12(2) - 7; J cosh (sz1 (5.4.14) -h2 5_Z §_0 In Figure 5.18, the current distributions along the insulated probes are shown. The current at the tip with infinite terminal impedance is zero and at the tip with zero terminal impedance the current is very large. (b). The terminal impedances are Ze1 = 0.0 and Ze2 = w cor- responding to 91 =-%g and 02 = 0.0. Substituting for 81 and 62 in (5.4.3) and (5.4.4), the current along each section of the probe is sinh (yhz) cosh [y(h1 - Z)] __1’_ 11(2) - Zc cosh [v(h1 + h2)] (5'4'15) 0 5_Z i h1 and 111 .sz ewo ...F 5.5223 .3 35:3: 2..— .0322: «F NF. :5. 3 u FF. .o.o u NmN ... u N 32.335 355.3 5:. 33.5 232:2. 93: 2033238 2523 .mF.m 9:6: GE N MN- 8- we- cm. 2.- o 2 m 2 u N; III' 8 u N; lo. 3 u N; It .2 _ fl 2 u N; I! 8 an n N; 11' NEE: LN : 112 V cosh (yhl) sinh [y(h2 + 2)] 12(2) =‘Z; cosh [mi1 + 52)] (5°4°15) -h2 §_Z 5_0 For a symmetric probe, h1 = h2 = h, the current on each section of the probe becomes ._ V sinh (yh) cosh [7(h - IZI)] 11(Z) -‘z; COSh (th) (5.4.17) 0 5_Z 5_h and _ V cosh ( h) sinh L (h - 12D] 12(2) - 22- Y cosh (ngl (5.4.18) -h §_Z §_0 In Figure 5.19, the current distributions along the insulated probes with Ze1 = 0.0 and Ze2 = m are shown. The dimensions of the probe and electrical properties of the insulator and the conducting medium are the same as that of case (1). The current distributions on the probes with various terminal impedances are shown in Figure 5.20. The current along the probe changes with the change on the terminal impedances. Therefore, the heat pattern produced by the EM energy delivered by the probe changes with the change on the terminal impedances of the probe. 113 NE: o3 PF .532on No 35:3: 2: 522.3) x N; 25 E: .2 u E .... u MEN 2; ad u N $2.335 25.53 5:: mono...“ .5qusz 95: 2.5532»an 29:3 .26. 8.53“. 2.5 N 2- om- 3- on- 3- o 2 11‘ I! d 3.219 8.9.1 3.9.1.. 8.9... a; “ F: m“ cm .3 .om 1m...” F 25:: 1.141 .mmucaumaeF Fachgou «cogoNNFu gan monocn umFoFamcF mcoFm mcoFuzaFcumFu acocgau .om.m ocamFu mwmv N mm- om- m5- . on- FF- o FF Fwy E .. ... Fey . oF .mF a . NmN .o.o u F5N AQV o.o . N...N .s « F0N Amy a . NaN .3 . FmN ANV .ON o.o . N..N .o.o . F5N AF. AFV A')(z-z')(3+3jk2 R-kng) araz K(z, Z )= R4 K(Zaz'). 117 a2 (z-z')2(3+3jk2R-k§R2)-R2-jkzn3 3:2- K(Z,Z ) = R4 K(Z:Z ) 9 The electric field components given by equations (6.1.3) and (6.1.4) become . 1 " (r- a3 cos ¢')(3+3jk2R- Kng) 12(2')(z-2') 4 41rk2 -h -n R 2 . d¢' . K(z,z ) 7?? dz } (6.1.5) and . h1 " (z-z')2(3+3jk2R- kgnz) E (n2) = - 3“?” I (2') z 2 z 4 4wk R 2 'hz "Tf R2 + jk2R3 - k§R4 d¢ - 4 K(z,z') — d‘z R TT (6.1.8) For a very thin probe we can assume that all the current on the . 1 probe is concentrated on the z axis, therefore, R = [r2+-(z-’z')2] , and the components of the electric field can be expressed as h . 2 2 1 r(z-z') (3+3Jk R-kR) Er(r‘.2)=- j-Luz’l‘ 12(2') 4 2 2 K(z,z')dz' 411k R 2 h2 (5.1.7) 118 and '11 “1 (z- z' )2(3+ 3jk2R- kng) E2032) = - 41—"? 12(2') 4 41rk R 2 -112 R2 + jk2R3 - k§R4 - 4 K(z,z')dz' (6.1.8) R The components of the electric field are found numerically and the heat produced by the EM energy in the medium is obtained from %02IEI2, where 11:12 = 15212 + IErlz- In Figures 6.1-6.4 the equi-power contours for insulated probes with various terminal impedances are given. The electrical properties of the insulated material are od = 0.0 (s/m), ard==2.25, and the physical dimensions of the probe are given in these figures. In Figure 6.1 the terminal impedances are Ze1 = Ze2 = m; in this case most of the heat is concentrated along the z axis and around the driving point. In Figures 6.2 and 6.3 it is assumed that one of the terminal impedances is zero and the other is infinity; in these cases the heat is concentrated mostly near the driving point of the probe and the tip with zero terminal impedance. In Figure 6.4 both terminal impedances are zero, and the heat is almost uniformly distributed around the probe. In general the heat produced by an insulated probe in a con- ducting medium is distributed in a larger volume in comparison with 119 2 (mm) Ze1 = Zez = m 20 a1 a2 = 0.43 mm a3 = 0.96 mm , h = 15 mm Zel 112 = 30 mm 0 :u- 30 r (mm) [' 30 L1 Figure 6.1. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe ZC = 64.5 - j13.4, 02 = 1.11 (s/m), 8r2 = 76.7, and the frequency is 600 MHz. 120 Ze1 = w, Ze2 = 0.0 a1 = a2 = 0.43 mm a3 = 0.96 mm 20 d- h1 = 15 mm h = 30 mm Ze1 2 g o .: 1 41 .. 10 20 30 r (mm) 100 ‘ 7.9 3.9 1.8 30 ‘ ’ Ze2 Figure 6.2. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe Z = 64.5 - j13.4, 02 = 1.11 (s/m), €r2 = 76.7, and the frequency is 600 MHz. 121 +‘Z (mm) ze1 - 0.0, ze2 m a1 - a2 - 0.43 mm 20‘? a3 = 0.96 mm 111 = 15 mm 100 h2 = 30 mm 36.3 15.3 8.04 \ 1 l /) lb 26 36 r (mm) e2 Figure 6.3. Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe Z = 64.5 - j13.4, oz = 1.11 (s/m), Erz = 76.7, and the frequency is 600 MHz. 122 42mm) Ze1=Ze2=00 a1 = a2 = 0.43 mm a3 = 0.96 mm 20 v h1 = 15 mm h2 = 30 mm Ze1 30 ‘ .Ze2 Figure 6.4. 100 37.3 16.2 8.8 Equi-power contours for an insulated probe in the r-z plane. Characteristic impedance of the probe Z = 64.5 - j13.4, 02 = 1.11 (s/m), Erz = 76.7, and the frequency is 600 MHz. 123 the heat produced by the similar bare probe, and for an insulated probe the heat pattern can be changed by the changing in the ter- minal impedances. CHAPTER VII A USER'S GUIDE TO COMPUTER PROGRAMS USED TO CALCULATE THE CURRENT DISTRIBUTION AND ELECTRIC FIELD IN A BIOLOGICAL BODY INDUCED BY A BARE AND INSULATED MICROPROBE This chapter is divided into two sections. The first section briefly explains the computer programs used to determine the current distribution and input impedance of the bare probe immersed in a conducting medium by solving Hallen's Integral Equation Method (HIEM) or Electric Field Integral Equation (EFIE). The second part gives explanation about the program used to find the parameters and the characteristic impedance of the insulated probe imbedded in the conducting medium. 7.1. Programs for the Bare Probe Moment method is used to solve integral equations for both cases HIEM and EFIE. The probe is divided into N segments, and the current is assumed to be constant along each segment. The geometry of the bare probe is given in Figure 7.1. 7.2. Program HALIEQ This program solves Hallen's integral equation for the bare probe. Given the necessary data the program solves equation (2.1.4) 124 125 II Section (I) 0 1 0‘1 0 .Section (2) Section (3) i Figure 7.1. Geometry of the bare probe used in HALIEQ. 126 for the current distribution, and the input impedance of both sym- metric and asymmetric bare probes. The main program contains a subroutine and a function as explained below. CMATPAC is a subroutine that solves the system of linear equations by Gauss's elimination process to determine the unknown currents on the probe. GRE is a function that computes Green's function. 7.2.1. Description of Input Variables and Input Data Files The data deck is composed of two files with each file having only one card. The names of the variables used in this program and the format specifications are given in Table 7.1. The variables in data files are defined below. First data file--accommodates the following variables: N is the total number of segments on the probe. M1 is an even integer; it is the number of segments on the sec- tion (1) and section (2) of the probe. M2 is the number of segments on the section (3) of the probe. Second data file--contains the following variables: FREQ is the frequency of operation in Hz. RRI specifies the radius of the section (1) of the probe. RR2 is the radius of the section (2) of the probe. 21 specifies the surface impedance of the probe in section (2) and section (3); it is assumed that the surface impedance of section (1) is zero. ZIG specifies the conductivity of the conducting medium. EPR is the relative permittivity of the conducting medium. 127 Table 7.1. The symbolic names of the input variables and the format specifications used in program HALIEQ. File Card . Variable Number Number Columns Name Format 1 1 1-3 N 13 4-6 M1 13 7-9 M2 13 2 1 1-12 FREQ F12.1 13-21 RRI F9.5 22-30 RR2 F9.5 31-39 ZI F9.1 40-45 ZIG F6.2 46-50 EPR F5.1 51-56 HH1 F6.1 57-62 HH2 F6.2 HH1 is the length of the section (1) of the probe. HH2 specifies the length of the section (2) plus section (3) of the probe. 7.2.2. Example Let us assume that the probe has the dimensions of h1==7.5 mm, h2 = 15 mm, and the diameters of the two sections are equal (a1 = a2 = 0.43 mm). The frequency is 600 MHz, and the electrical properties of the conducting medium are 0 = 1 11, er = 75.7. The surface impedance 21 = 100. Following is the list of numbers and corre- sponding variable names should provide in file no. 1 and file no. 2. 128 Numbers Variable Name Columns File No. 1 024 N 1-3 016 M1 4-6 008 M2 7-9 File No. 2 600,000,000.0 FREQ 2-12 0.00043 RRl 15-21 0.00043 RR2 24-30 100.0 ZI 35-39 1.11 ZIG 42-45 76.7 EPR 47-50 7.5 HH1 53-55 15.0 HH2 58-61 The numerical results are presented after the listing of the program HALIEQ. 7.3. Program EFZI This program solves EFIE for the bare probe given in equa- tion (2.2.7). Given the necessary data this program finds the current distribution and input impedance of both symmetric and asymmetric bare probes. The subroutines and functions used in the main program are as explained below. CAMTPAC is explained in HALIEQ. MAGRIN is a subroutine that finds the electric field on the sur- face of the probe, based on magnetic current ring as the driving source. SIMCON is a subroutine that calculates the first integral in a double integral. 129 FCT is a function that provides the integrand of SIMCON. SIMCOP is a subroutine that calculates second integral in a double integral. FCTP is a function that provides the integrand of SIMCOP. GRE is a function that computes the Green's function. 7.3.1. Description of Input Variables and Input Data Files The data deck is composed of two files, with each file having only one card. The names of the variables used in the first data file and the format specifications are given in Table 7.2. The second data file has the same variables and formats as the variables in the second data file in HALIEQ. The information about variables in the first data file is explained below. First data file-~accommodates the following variables: N is the total number of segments on the probe. M1 is the number of segments on section (1) of the probe. M2 is the number of segments on section (2) and section (3) of the probe. NG specifies the number of subdivisions the driving point segment will undergo in the partitioning process of the probe. ITEST is either one or two; when it is one the driving source is assumed to be a delta gap, and when it is two the driving source is modeled as a magnetic current ring. 7.3.2. Example Let us assume that all the conditions are the same as mentioned in example 7.2.2, but the surface impedance is zero everywhere in this case. The second data file is the same as explained in example 130 Table 7.2. The symbolic names of the input variables and the format specifications used in the first data file in program EFZI. File Card Variable Number Number Columns Name Format 1 1 1-3 N 13 4-6 M1 13 7-9 M2 13 10-12 NG I3 13-15 ITEST I3 7.2.2, except we have zero for surface impedance. The list of numbers and corresponding variable names in file no. 1 are given below. Numbers Variable Name ' Columns File No. 1 024 N 1-3 008 M1 4-6 016 M2 7-9 001 N0 10-12 002 ITEST 13-15 The numerical results are presented after the listing of the program EFZI. 7.4. Program Used for Insulated Probe This section briefly explains the computer program used to determine the parameters and the input impedance of the insulated probe, with various terminaI impedances. The probe is treated as a 131 lossy transmission line. The symbolic name of the program is INPIMP. The main program contains a subroutine COMBES, which finds the Bessel's functions of first and second kind for complex argument and complex order. The subroutine COMBES is explained in more detail in [16]. 7.4.1. Description of Input Variables and Input Data Files The data deck is composed of five files, with each file having only one card. The names of the variables used in this program and the format specifications are given in Table 7.3. The variables in data files are defined as below. First data file--accommodates the following variables: H1 is the length of the section (1) of the probe. H2 is the length of the section (2) of the probe. Second data file--contains the following variables: 2160 is conductivity of the insulating material. EPRD is relative permittivity of the insulating material. Third data file--contains the following variables: ZIGZ is conductivity of the conducting medium. EPR2 is relative permittivity of the conducting medium. Fourth data file--contains the following variables: A1 is the radius of the conductor in section (1). Aé is the radius of the conductor in section (2). A3 is the outer radius of the insulator. Fifth data file-~contains the following variable: FREQ is the frequency of operation. 132 Table 7.3. The symbolic names of the input variables and the format specifications used in program INPIMP. File Card Variable Number Number Columns Name Format 1 1 1-8 H1 F8.5 9-16 H2 F8.5 2 1 1-6 ZIGD F6.2 7-12 EPRD F6.2 3 1 1-8 2102 F8.4 9-13 EPR2 F5.2 4 1 1-8 A1 F8.5 9-16 A2 F8.5 17-24 A3 F8.5 5 1 1-12 ' FREQ F12.1 7.4.2. Example The input impedance of insulated probe with a1 = a2 = 0.43 mm, a3 = 0.96 mm, h1 = 15 mm, and h2 = 45 mm is found by INPIMP. The electrical properties of the insulator and the conducting medium are Gd = 0.0, e d = 2.25, 02 = 1.11, and-er2 = 76.7. Following is r the list of numbers and corresponding variable names 133 Numbers Variable Name Columns File No. 1 0.015 H1 2-6 0.045 H2 10-14 File No 2 0.0 ZIGD 3-5 2.25 EPRD 9-12 File No. 3 1.11 ZIGZ 3-6 76.7 EPR2 9-12 File No. 4 0.00043 A1 2-8 0.00043 A2 10-16 0.00096 A3 18-24 File No. 5 FREQ 2-12 600,000,000.0 The numerical results are presented after the listing of the program INPIMP. 134 c...’...-O0.0......‘0.0lifitittfifiifitfit‘ttitttOOOOOOOOOOOOOO......OOOCO'. C THIS PROGRAM DETERMINS THE CURRENT DISTRIBUTION ON THE PROBE AND C INPUT IMPEDANCE OF THE PROBE BY SDLVING HALLEN.S INTEGRAL EQUATION. Citifltltttttt.....OC‘OC...‘.....tt...“..“Ctttttitt‘ttt...t.“......i. PROGRAM HALIEQ (INPUT.0UTPUT) COMMON/HALGRE/JK DIMENSION G(1oo.100).Z(1oo).s(1oo).03(3) COMPLEX G.K.P.F1.02.03.82.34.OET.A.E.E1.E2.E3.E4.zIN.a1.OK.GRE REAL MU.IREA.IIMA.LA ' READ 100.N.M1.M2 READ 101.FREO.RR1,RR2.ZI.ZIG.EPR.HH1.HH2 FREO=FREO/1.0E+6 PRINT 102.FREO.RR1.RR2.ZI R1=RR1 s R2=RR2 PI=4.o-ATAN(1.O) s vo=1.o MU=4.0*PI-1.0E-7 0ME=2.0*PI*FREO*1.0E+6 M3=M1+1 s M4=M1/2 s M6=M4+1 H1=HH1/1000.0 S H2=HH2/1000.0 S H3=H1 DEL1=(H1+H3)/M1 OEL2=(H2—H3)/M2 DEL3=(DEL1+DEL2)/2.0 ct.......‘t...C......‘titfiifiitfifitfi.tiflt......0‘...tt......tttttttlit... C PARTITION THE PROBE IN N SEGMENTS. c....‘lfitfifittitfiittti......fi.......‘ttflttfiitttfifiiiItittlfittttttttttI... 11 14 12 13 10 OO 10 dd=1.N IF(uu-M3) 11.12.13 IF(dd.EO.1) GO To 14 OEL=OEL1 s uu1=uu-1 Z(dd)=2(ddi)-DEL 3 GO To 10 OEL=OEL1 Z(dd)=H1-O.5'DEL 3 GO TO 10 OEL=OEL3 uuz-uu-1 . Z(dd)=2(dd2)-DE 3 GO TO 10 OEL=OEL2 qua-uu—1 z(dd)=Z(Oua)-DEL CONTINUE EP-1.OE-9'EPR/(36.ORPI) PRINT 93.H1.H2.ZIG.EPR PRINT 39.N.M1.M2 KaOMEccsoRT(nu-EP-CMPLx(o.O.1.o)-MU:ZIG/OME) ALPHAt-AIMAG(K) s BETA=REAL(K) PRINT 81.ALPHA.BETA UKICMPLX(O.O.1.O)'K CD...“..U.C.O‘"...Ct.....t...‘..tOttttfl.tfififitttlttltitttttitltfititttt C FIND THE DIAGONAL TERMS OF THE IMPEDANSE MATRIX. C'...‘....I.iO....fl‘i......fifi.........‘CtttttttfittititOttttittttttittt. DO 2 II-i.3 IF(II-2) 3.4.5 DEL=DEL1 R=R1 3 GO TO 6 DEL-DELi R-R2 8 GO TO 6 DEL'DELZ R=R2 HDEL-DEL/2.0 D-HOEL/R 01-2.0-ALOG(D+SORT(1.0+O‘t2.0)) DZ-Di-dK—DEL 03(II)-02 CONTINUE 1J35 88(4.0‘PI)/(OME'MU) BI=CMPLX(ALPHA.BETA) 82=B-Bi s 83=8tZI BdaCMPLX(0.0.°B3) N=N+1 CO‘COtOO‘OOi...‘...'¢’.fifififififitfi‘.‘...Ottttfifi.*.00tttififi0'Otttttlttfittt' C THIS SECTION FINDS THE ELEMENTS OF THE IMPEDANCE MATRIX. C...*.$.fififitittfitittfittttttiifittfitttttittfitlfittt‘fittttfitttttttttttttttt 15 30 43 42 41 44 45 40 oo 15 L-1.N G(L.1)=B2PCCOS(K¢Z(L)) G(L.N)=B2*CSIN(KPZ(L)) G(L.M)=-B2*(V0/2.0)*CSIN(K*ABS(Z(L))) CONTINUE E1'(1-CCOS(KtDEL1/2.0))‘84 E2=(1~ccos(K~OEL2/2.o))*84 M5=N~M4 OO 30 d-1.M5 OEL=OEL2 IF(d.LE.M4) OEL=OEL1 L=U+M4 5(U)=z(L)-OEL/2.o CONTINUE N1sN-1 oo 40 I-1.N oo 40 d'2.N1 IF(I.EO.U) GO To 41 OELsOELz IF(U.LE.M1) OEL=OEL1 R=R2 IF(I.LE.M4) R=R1 R3=SORT((Z(I)-Z(d)) *‘2.0+R**2.0) IF(I.CT.N4.ANO.U.CT.M4) GO TO 42 G(I.d)=GRE(R3)'DEL S GO TO 40 IF(d.GT.I) GO TO 43 L’I-M4 S LL3d-M4 S LPSLL+1 F=84'(CCOS(K*(Z(I)-S(LP)))‘CCOS(K*(Z(I)-S(LL)))) G(I.U)=GRE(R3)—OEL+F GO TO 40 IF(I.GT.M1) GO To 44 IF(I.GT.M4) GO To 45 du-1 G(I.U)=03(UU) 3 GO TO 40 UU=3 G(I.U)=OO(UU)+E2 GO To 40 0082 C(I.U)=OO(UU)+E1 CONTINUE CALL CMATPAC(-1.G.N.1.0ET.1.0E-200) AVIN-CABS(G(M4.M)) PRINT 111.AVIN.C(N4.M)I ZINs1/(61M4.M)) AZIN-CABS(ZIN) PHI=ATAN(AIMAG(ZIN)/REAL(ZIN))*180.0/(2.OPPI) PRINT 107.AZIN.ZIN.PHI PRINT 103 00 95 I1-1.N IF(11.EO.1) GO TO 19 IF(11.EO.N) GO To 19 AMPI-CABS(G(11 .M)) GO TO 21 1436 19 G(I1.M)=0.0 AMPI=o.o 21 PRINT 25.11.2(11).AMPI.G(I1.M) 95 CONTINUE c...‘t......t..t.....‘fittfitiiti‘tfitfi...!‘iittit.tfitt‘tllttttt......ttt. C READ AND wRITE FORMATS. CO0.0tttfifititttiiitltfl.‘Otttttfitttt0ttt$ttttttltttltttitttttttttttto... 25 PORMAT(1HO.10x.13.4x.cz=-.F6.4.4x.~AMP(I)=:.F8.4.4x.-I=-. c2(Fe.4.2x)) 39 PORMAT(1Ho.2ox.2HN=.13,4x.3HM1-.13.4x.3HM2-.13./) a1 FORMAT(1HO.2OX.*ALPHA=*.F8.2.4X.'BETA=*.F8.2./) 93 PORMAT(1HO.10x.tH1=t.Ea.4.4x.«H2=:.Ea.4.4x.tSIC=t.Ee.4,4x. c-EPR=:.F8.4./) 1oo FORMAT(313) 101 FORMAT(F12.1.2(F9.5).F9.1.F6.2.F5.1.2(F6.1)) 102 FORMAT(1HO.10X.PFREO='.F7.2.vMHZ*.4x.*AII,F8.5.4X.'83-.F8.5.4x. t-ZI-'.F8.1./) 103 FORMAT(1HO.135('+‘)./) 105 FORMAT(2(F14.8)) 107 F0RMAT(1HO.2OX.PABS(ZIN)=¢.F8.2.4X.*ZINS‘.2(F8.2.2X).EPHI3'. CF8.2./) 111 PORMAT(1Ho.2ox.oABS(YIN)=—.Ea.4.4x.~YIN--.2(Pa.4.2x)./) END c#‘.“.....tfititttlitfi3....t...‘..‘..tlt..fi..tfi$......ttfititttttttttltt . COMPLEX FUNCTION GRE(R1) CttttttfittttttttntttttCOOOCttttttlttttttttttttttttltttt-ttttats...at... COMMON/HALCRE/UK COMPLEX UK CREaCExp(-UK~R1)/R1 RETURN END . Ctfitttllfititttttttt...Ittlttttiit...ttttttttttttitttt......tttitttit... c SUBROUTINE CMATPAC SOLVES THE SYSTEM OF EQUATIONS BY c GAUSS-ELIMINATION PROCESS. Ctitttttltttttttttttatltttttttttttttttttttttttt-ttttttttOtittttottttttt SUBROUTINE CMATPAC(IUOB.A.N.M.DET.EP) DIMENSION A(1oo.1oo) TYPE COMPLEX A.8.DET.CONST.S 30 FORMAT(1X.42HTHE DETERMINANT OF THE SYSTEM EOUALS ZERO./ 11x.36HTHE PROGRAM CANNOT HANDLE THIS CASE.//) DET-1. NP1=N+1 NPM=N+M NM1sN-1 IF(IUOB) 2.1.2 1 DO 3 I-1.N NPI=N+I A(I.NPI)=1. IP1-I+1 DO 3 u-IP1.N NPU=N+U A(I.NPU)=O. 3 A(U.NPI)=O. 2 DO 4 u-1.NM1 C=CABS(A(J.J)) UP1-U+1 DO 5 IaUP1.N D-CABS(A(I.U)) IF(C-D) 6.5.5 6 DET--DET DO 7 K-U,NPM CMAOOOO1 CMAOOOO2 CMA00004 CMA00005 CMAOOOOG CMAOOOO7 CMAOOOOB CMA00009 CMAOOOiO CMAOOO11 CMA00012 CMA00013 CMA00014 CMA00015 CMA00017 CMA00018 CMAOOO19 CMAOOO21 CMAOOO22 CMAOOO24 15 14 16 17 18 11 10 19 13 12 137 DETa-DET DD 7 K=d.NPM B=A(I.K) A(I,K)=A(U.K) A(d.K)=B c=D CONTINUE 1F(CABS(A(U.U))-EP) 14.15.15 DD 4 I=UP1.N CONST=A(I.d)/A(d.d) DO 4 K=dP1.NPM A(I.K)=A(I.K)-CONSTPA(J.K) IF(CABS(A(N.N)-EP))14.13.18 DETsD. IF(IUOB) 16.16.17 PRINT 30 RETURN DO 11 I=1.N DET=DET~A(I.I) IF(IUDB) 10.10.17 00 12 I-1.N KaN-I+1 KP1=K+1 DO 12 L=NP1.NPM s-o. IF(N-KP1) 12.19.19 00 13 U=KP1.N S=S+A(K.d)tA(d.L) A(K.L)=(A(K.L)-S)/A(K.K) RETURN END 24 16 8 GOOOOOOO0.0 0.00043 0.00043 100.0 1.11 76.7 7 .5 CMAOOO27 CMAOOOZB CMA00029 CMAOOOSO CMA00031 CMAOOO33 CMAOOO34 CMAOOO35 CMA00036 CMA00038 CMAOOOBQ CMAOOO4O CMAOOO41 CMAOOOA2 CMAOOO43 CMAOOO45 CMA00046 CMAOOO47 CMAOOO49 CMAOOOSO CMAOOOS1 CMA00052 CMAOOOSB CMAOOOS4 138 FREQ= 600.00MHZ A= .00043 B= .00043 ZI= 100.0 H1= .0075 H2= .0150 SIG= 1.1100 EPR= 76.7000 N= 24 Ml= 16 M2= 8 .' ALPHA+ 23.37 BETA+ 112.51 ABS(YIN)= .0505 YIN=I .0434 .0258 ABS(ZIN)= 19.80 ZIN= 17.03 -10.11 PHI= -15.34 ++++++++++++++w+_++++.++++.+.++++.++++_+.+++ 1 Z3 .0070 AMP(I)= 0.0000 I= 0.0000 0.0000 2 Z= .0061 AMP(I)= .0117 I= .0110 .0038 3 Z8 .0052 AMP(I)= .0181 I= .0170 .0062 4 Z8 .0042 AMP(I)=l .0244 I= .0227 .0088 5 2' .0033 AMP(I)- .0303 I= .0280 .0115 6 Z= .0023 AMP(I)= .0360 I= .0329 .0146 7 Z3 .0014 AMP(I)= .0415 I= .0374 .0179 8 Z= .0005 AMP(I)= .0505 I= .0434 .0258 9 Z=-.0005 AMP(I)= .0522 I= .0454 .0258 10 Z=-.0014 AMP(I)= 7.0469 I= .0434 00178 11 Z=-.0023 AMP(I)= .0453 I= .0429 .0145 12 Z=-.0033 AMP(I)= .0435 I= .0419 .0116 13 Z=-.0042 AMP(I)= .0417 I= .0406 .0092 14 23-.0052 AMP(I)= .0396 I= .0390 .0071 15 28-.0061 AMP(I)= .0373 I= .0369 .0053 16 Z=-.0070 AMP(I)= .0348 I= .0345 .0038 17 Z=-.0080 AMP(I)= .0319 I= .0318 .0025 18 Z=-.0089 AMP(I)= .0288 I= .0287 .0015 19 Z=-.0098 AMP(I)= .0253 I= .0253 .0008 20 28-.0108 AMP(I)' .0216 I= .0216 .0002 21 Z=-.0117 AMP(I)= .0176 I= .0176 -.0001 22 Z=-.0127 AMP(I)- .0131 I= .0131 -.0003 23 Z-.0136 AMPUC)‘I .0085 I8 .0085 -.0003 24 Z=-.0145 AMP(I)= 0.0000 I= 0.0000 0.0000 139 LOU-CCU.........0......-......ICUUCUCCUCUCOO......‘UOCCCUUUUI......‘CUU THIS PROGRAM SOLVES ELECTRIC FIELD INTEGRAL EQUATION TO FIND CURRENT C DISTRIBUTION AND INPUT IMPEDANCE OF THE PROBE. COO.......‘O‘IIOOOOUIOOCO......U.‘.....‘CCOCCCOIOCOOOO......00000...... C 13 11 12 10 PROGRAM EFZI(INPUT.OUTPUT.TAPE1) DIMENSION 06(4).B(60).Z(60).G(60.61).EZ(60) COMPLEX K.JK.D2.03.DG.DT.B.82.£3.64.E7.G.ZIN.A.EZ.GRE.8A.ALR -.UUK COMMON/EZIMAG/A1,A2.UK.DELG COMMON/EFCRE/UUK REAL LA.MU READ 100.N.M1.M2.NO.ITEST READ 103.FREO.RR1.RR2.ZI.ZIG.EPR.HH1.HH2 FREO=FREO/1.0E+6 H1=HH1/1000.0 H2=HH2/1000.0 PRINT 101.FREO.RR1.RR2.ZI.ZIC.EPR.H1.H2 PRINT 39.N.M1.M2.NG.ITEST PI=4.o-ATAN(1.0) s v-1.0 MU=4.0'PI'1.0E-7 OM£=2.0-PI-FREo-1.0E+6 M3=M1+1 s M4-M1+NC s Ms=M4+1 s NC=M1+1 s NP=N+NC A1=RR1 s A2=RR2 DEL1=H1/M1 DEL2=H2/M2 DO 56 INN-1.1 DELG-0.001 DEL3=(DEL1+DELG)/2.o DELa-(DEL2+DELG)/2.0 H1=HH1/1ooo.o+0.s-DELG H2=HH2/1000.0+O.5-DELC Do 10 d-i.NP IFIU.LE.M1) GO TO 11 IF(U.CT.M4) GO TO 13 DEL=DEL3 da-U-1 ZIU)=2(03)-DEL s 60 To 10 OEL=OEL2 IF(U.EO.MS) OEL=DEL4 02'0-1 2(d)'ZIUZ)-OEL S GO To 10 IFIU.E0.1) GO To 12 U1aU-1 2(d)'Z(dii-OEL1 3 GO TO 10 2(U)=H1-DEL1-o.5 CONTINUE EPsEPR-1.DE-9/(36.o-PI) K=OME-CSORT(MU‘EP-CMPLXI0.0.1.0)-MU'ZIG/OME) UKaK-CMPLXID.0.1.O) UUK-UK ALPHAa-AIMACIK) s BETAsREAL(K) PRINT 81.ALPHA.BETA LA'2.0'PI/BETA PRINT 57.EPR.ZIG PRINT 67.DELG COO...I......IO.....C..........CII......OICOOCOOCIt-CO......IOCOO...... FIND THE DIAGONAL TERMS OF THE IMPEDANCE MATRIX. C COCOIIOCOCO......‘C‘...’......‘OCCOCCC.........IIIOCCOUOUICOOOCOOIOOIOO DO 2 II'1.4 IFIII.GT.2) GO TO 4 A3-A1 S DEL'DEL1 IF(II.EO.2)OEL=DELG GO TO 7 47 45 44 48 140 A3=A2 DEL=DEL2 IFIII.EO.3) DEL=DELG HDEL=DEL/2.o O=HDEL/A3 D1-2.0vALOG(D+SoRT(1.0+Dtt2.0)) . D2=D1-UK-DEL R=SORT(HDEL*'2.0+A3'*2.0) RRE=1.0/R s RSQ=RRE**2.0 D3=-DEL'(RSO+UK1RRE)tGRE(R)/(K'*2) IF(II.LT.2) 07:0.0 . IF(II.GE.2) D7=CMPLX(0.0.~4.0*PI/(OME'MU))'ZI 06(II)=02+03+D7 CONTINUE 81=(4.0-PI)/(OME*MU) 82=CMPLX(0.0.-Bi) IF(ITEST-2) 44.47.47 CALL MAGRIN(NP.Z.EZ) DO 45 IR=1.NP 811R1=32tEZ(IR) GO TO 3 Do 43 NNa1.NP B(NN)=0.0 IF(NN.EQ.NC) B(NN)=BZ*(1.0/OELG) CONTINUE CItttfitiflfifitttttttltttltttttit‘fitltfittitttlttififi*tttflflfiiittittttttttl“ C THIS SECTION FINDS ELEMENTS OF THE IMPEDANCE MATRIX. C*.'.Itifiitttttttttittitfittitfitttlttfitt‘fltttiitttitttiit.‘fitfifltiiti‘.t‘ 3 55 DO 15 I=1.NP Do 15 UU=1.NP IF(I.ED.UU) GO TO 17 A3=A2 IF(I.LE.NC) A3=A1 R=SORT((Z(I)-Z(UU)) u2.0+A3nz.0) DEL=DEL1 IF(dd.GE.M3) DEL=DELG IFIUU.GT.M4) DEL=DEL2 HDEL=OEL/2.0 E1a2(I)-Z(UU)+HDEL s E2=Z(I)-Z(dd)-HDEL R1=soRT(E1::2.O+A3t-2.0) R2=SORT(E2‘*2.0+A3**2.0) F1=1.0/R1 s F2=F1**2.0 £3=-EiP(F2+deF1)-GRE(R1) P3-1.0/R2 s F4=F3t*2.0 E4=-E2*(F4+thF3)*GRE(R2) G(I.dd)'(E3-Ed)/(K“2)+DEL*GRE(R) GO TO 15 IF(I-NC) 31.32.33 081 S GO TO 34 d=2 S GO TO 34 d=4 G(I.Jd)=06(d) CONTINUE MP8NP+1 Do 55 L-1.NP C(L.MP)=B(L) CONTINUE CALL CMATPAC(-1.C.NP.1.DET.1.0E-200) AYIN=CABS(G(NC.MP)) PRINT 111.AYIN.G(NC.MP) ZIN-1.o/G(NC.MP) s AZIN=CABS(ZIN) 1J11 PHIaATAN1AIMA61ZINI/REAL(2IN))-180.O/(2.0'PII PRINT 107.AzIN.zIN.PHI PRINT 102 DO 95 I1=1.NP AMPI=CABS(G(I1.MP)) PRINT 23.I1.z(I1).AMPI.6(I1.MP) 95 CONTINUE 56 CONTINUE Ct...OIOCOCCtttt-ttttOCOOOCUOIDCOOOOOOOCOO6.6.6.0009...o-oovoo'oucoovoo C READ AND WRITE FORMATS. CO.........‘CIOOOCCOCOCO-......CO......COOOOOOOOCCIOOOUVOOOIt...'00.... 23 FORMAT(10X.13.4X.4(F8.6.4X)./) 39 FORMAT(1H0.20X.2HN=.13.4x.3HM1-.I3.4X.3HM2=.13.4x.3HN6=.13..2x. '°ITEST".I3./) 57 PORMAT(1H0.20X.4HEPR=.E3.3.4x.4HzICs.Fa.4./1 67 PORMAT(1H0.30X.5HDEL6s.Fa.6./) 81 PORMATI1H0.10X.6HALPHA=.E12.5.4X.5HBETA-.E12.5./) 100 FORMAT(5(I3)) 101 102 FORMAT(1HO.5X.SHFREO=.F6.2.4HMHZ .2X.2HA=.F7.5.2X.2HB=.F7.5.2X.4H ‘ZI'.F7.1.2X.4HZIG‘.F6.2.2X.4HEPR=.FS.1.2X.3HH1=.F7.4.2X.3HH2=.;7.4 ./) FORMAT(1HO.135(‘+‘)./) 103 FORMAT(F12.1.2(F9.5).F9.1.F6.2.F5.1.2(F6.1)) 107 111 FORMAT(1HO.2OX.9HABS(ZIN)'.E12.5.4X,4HZIN‘.2(E12.5.2X).4HPHI8. -E12.5./) FORMAT(1H0.20X.9HABS(YIN)=.E12.5.4X.4HYINs.2(612.5.2x)./) END C ....OOIIt...O......‘C......OCCI......‘IOCOOCCOCOUCOCCVCOOIOCOIOOOOO... SUBROUTINE MAGRIN(NP.Z.EZ) c ...-IOICCOOJOOCOCCCCC..C......‘O...O..........‘C'COOO0.100000......... 23 COMMON/MAGFCT/PI.ZZ.RO.ROP.EROP /MAGFCP/ZZI.RRO.RROP.EEdK COMMON/EZIMAG/Ai.A2.dK.OELG DIMENSION Z(60).F(4).EZ(60) COMPLEX K.UK.EEUK.SUM1.EZ.F.SUM4.SUM5.SUM6.PATCH REAL MU.LA PI=4.0‘ATAN(1.0) EEdK-UK DA-0.001-A1 XXsDELG/2.o PACT-1.0/(8.0-PI-Xx) XEND=XX ROP-A2 RROP-ROP x1--xx ERR-0.001 EPSxXX/1OO.0 EPORsEPS/ROP PATCHIPI~EPS*O.5-UK«PI10.25'1EP5-E2.O) DO 20 INDEx-1.NP 2282(INDEX) ZZI=ZZ DO 21 J-1.2 RO'O.9-A1-(2-d)'OA RRo-RO IF((RO.EO.ROP).AND.(INOEX.EO.NC)) GO TO 23 CALL SIMCDNIINDEX.X1.XEND.ERR.25.SUM1.NDI.R) F(JI-SUM1-ROPROP'2.O GO To 21 III1OO+INDEX CALL SIMCON(INDEX.EPS.XENO.ERR.25.SUM4.NOI.R) CALL SIMCON1II.0.0.EPS.ERR.25.SUM5.NOI.R) 41 21 22 20 59 142 FORMAT(1H0.50X.2(E12.5.4X.Ei2.5)./) F(U)=4.0«ROv(ROPtSUM4+RDP«SUM5+PATCH) CONTINUE EZ(INDEX)=FACT-(F(2)-F(1))/(ROoDA) PRINT 22.2(INDEX).RO.E2(INDEX) FORMAT(1H0.2OX.2HZ=.F9.6.4X.4HRAD=.F9.6.4X.3HEF=.F9.4.2X.F9.4./) CONTINUE CONTINUE END CutttttttfifitttttiItttttfitttttfittfifitfitlfifltfiiCtfitfittttt'ttittttCitO'Vtfiti SUBROUTINE SIMCON(INDEX.X1.XENO.TEST.LIM.AREA.NOI.R) Ctfiltltittiifiiiitfififittt-fititt‘t’fitfifitfifit“fififiifittttitttttilttttttfitiitt 31 50 COMPLEX AREA.OOD.EVEN.AREA1.ENDS.FCT NOI=0 00080.0 INTa1 v-1.0 EVEN=0.0 AREA1=0.O ENDS=FCT(INDEX.X1)+FCT(INOEX.XEND) H=(XEND-X1)/v DDD=EVEN+ODD X=X1+.5-H EVEN=0.0 DO 3 I=1.INT EVEN=EVEN+FCT(INDEX.X) X=X+H CONTINUE AREA=(ENDS+4.0*EVEN+2.0*000)*H/6.0 NOI=NDI+1 A3=CABS(AREA) IF((A3.LE.1.E-14).ANO.(NOI.LE.2)) GO TO 4 IF((A3.LE.1.E-14).AND.(NOI.GE.2)) GO TO 50 R=CABS((AREA1-AREA)/AREA) R=CA6S((AREA1-AREA)/AREA) IF(R-TEST) 32.32.4 RETURN AREA1=AREA INT=2*INT v=2.0-v GO TO 2 AREA=1.E-14 RETURN END cit.‘t...“...‘....$..CtttititttfiitiititttttCttfittittt.vttitttttttttt... COMPLEX FUNCTION FCT(INOEX.ZP) CttittltfittttilIitfittfittiltlthItittit......ttittttttttCttlttII-ibttfitt 41 COMMON/MAGFCT/PI.ZZ.RO.ROP.EPOR /FCTFCT/ZZP COMPLEX SUM2 ZZP=ZP IF(INDEX.GT.100) GO TO 41 CALL SIMCOP1INOEX.0.0.PI.0.001.25.SUM2.NOI.R) FCTSSUMZ RETURN CALL SIMCOP(INDEX.EPOR.PI.0.001.25.SUM2.NOI.R) FCT'SUM2 RETURN END Citattttltoat.totattttttut‘tttlttuc‘ttt-ntntattttttatcttttt.ottat...... SUBROUTINE SIMCOP(INDEX.X1.XENO.TEST.LIM.AREA.NOI.R) Ctitthiti‘.Otifitfiltfififitllt.‘l...‘..*..i..ttittO..I.¥tifi.tfifittlltttttttt 143 COMPLEX AREA.ODD.EVEN.AREA1.ENDS.FCTP NOI-O DDD=0.0 INTa1 v=1.0 EVEN=0.0 AREA1-0.0 ENDSaFCTPIINDEX.X1)+FCTP(INDEX.XEND) 2 H=(XEND-X1)/v DDD=EVEN+ODD X=X1+.5-H EVEN=0.0 DO 3 I=1.INT EVEN=EVEN+FCTP(INOEX.X) x=x+H 3 CONTINUE AREAs(ENDS+4.0-EVEN+2.0~ODD)-H/6.o NOI=NDI+1 A3=CABSIAREAI IF((A3.LE.1.E-14).ANO.(NOI.LE.2)) GO TO 4 IF((A3.LE.1.E-14).ANO.(NOI.GE.2)) GO TO 50 R=CA8$((AREA1-AREA1/AREA) IFINOI-LIM) 31.32.32 31 [F(R-TEST) 32.32.4 32 RETURN - 4 AREA1=AREA INT22'INT v=2.0-v GO To 2 50 AREA=1.E-14 RETURN END Ctlflfitt......‘OOOCCCOOC...-.........CI..UCCCI....-O....00.000.000.00... COMPLEX FUNCTION FCTP(INOEX.PHI) Cl...I...........IOIOIIOOCCCOOCVOtittttttltttttttfiC..-OOtttooctnnoo.ott COMPLEX GRE.EEUK COMMON/MACFCP/ZZI.RRD.RROP.EEUK /FCTFCT/ZZP COPH=COS(PHI) R1'SORT1(ZZI-ZZP)"2.0+RRO“2.0+RROP}*2.0'2.0'RRO:RROP°COPH) FCTP=GRE(R1)‘COPH RETURN END COOOCICDICCCCC......C...-.‘C..C.....UCOCIOOCOCCCOO'QOC'......CVIfififittt' COMPLEX FUNCTION GREiRR) C.........‘C.......C......I...'......‘C.........‘COOCCOOOUCICOOQVCCOCC' COMMON/EFGRE/UUK COMPLEX UUK 6RE=CEXP(-JUK-RR)/RR RETURN END 144 FREQ=600.MHZ A=.00043 B=.00043 ZI=).0 ZIG=1.11 EPR=76.7 H1=.0075 H2=.0150 N= 24 M1= 8 M2= 16 NG= 1 ITEST= 2 ALPHA= .23370E+02 BETA= .11251E+03 EPR= 76.700 ZIG= 1.1100 DELG= .001000 z= .007531 RAD= .000387 EF= .2553 -.0580 z= .006594 RAD= .000387 EF= .3715 -.0641 z= .005656 RAD= .000387 EF= .5723 -.O718 z= .004719 RAD= .000387 EF8 .9556 -.0817 z= .003781 RAD= .000387 EF= 1.7953 -.0957 z= .002844 RAD= .000387 EF= 4.0563 -.1174 z= .001906 RAD= .000387 EF- 12.6046 —.1570 z= .000969 RAD= .000387 EF= 72.4910 -.2482 z= -.000000 RAD= .000387 EF= 830.8496 -.3941 z= -.000969 RAD= .000387 8F: 72.4910 -.2482 2: -.001906 RAD= .000387 EF= 12.6046 -.1570 z= -.002844 RAD= .000387 EF= 4.0563 —.1174 z= -.003781 RAD= .000387 EF= 1.7953 -.0957 z- -.004719 RAD= .000387 EF= .9556 -.0817 z= -.005656 RAD- .000387 EF= .5723 -.0718 z= -.006594 RAD= .000387 EF= .3715 -.0641 z= -.007531 RAD= .000387 EF= .2553 -.0580 z= —.008469 RAD- .000387 EF= .1829 —.0528 z= -.009406 RAD= .000387 Era .1351 -.O484 z= -.010344 RAD= .000387 EF= .1020 -.0445 z= -.011281 RAD= .000387 EF= .0783 -.0410 Z: -.012219 RAD= .000387 EF3 .0607 -.0378 z= -.013156 RAD= .000387 EF3 .0474 -.0348 z= -.014094 RAD= .000387 EF= .0371 -.0321 z- -.015031 RAD- .000387 EF- .0290 -.0295 ABS(YIN)= .52670E—01. YIN= .48716E—01 .20023E—01 ABS(ZIN)= .18986E+02 ZIN= .17561E+02 -.72176E+01 PHI= -.11172E+02 +++++++++++++++++++++++++++++++++ .007531 .006594 .005656 .004719 .003781 .002844 .001906‘ .000969 -.000000 -.000969 -.001906 -.002844 -.003781 -.004719 -.005656 -.006594 -.007531 -.008469 -.009406 -.010344 -.011281 -.012219 -.013156 -.014094 -.015031 .009232 .015583 .021343 .026729 .031829 .036706 .041455 .046337 .052670 .049592 .048082 .046793 .045427 .043869 .042062 .039975 .037594 .034913 .031932 .028654 .025082 .021215 .017038 .012499 .007436 145 .009135 .015386 .021017 .026229 .031091 .035637 .039909 .044033 .048716 .047652 .047058 .046290 .045221 .043813 .042059 .039964 .037537 .034792 .031744 .028409 .024797 .020914 .016750 .012255 .007272 .001334 .002467 .003716 .005146 .006812 .008791 .011216 .014428 .020023 .013736 .009870 .006840 .004325 .002216 .000462 -.000965 -.002084 -.002910 -.003457 -.003738 -.003767 -.003557 -.003120 -.002459 -.001553 1x46 C...C...........O.It........CCOOCOIOOOOOOOOOI0.0...OIOOOOOOOOUOOOOOOUOC 6 THIS PROGRAM FINDS INPUT IMPEDANCE OF INSULATED PROBE. c0000....0....OIOCOOOOOOOOOOCOOOOOOOOOOOOOOOOOOQOOOOOOOOOOOOOOOOQOOOOOO PROGRAM INPIMP(INPUT.DUTPUT) DIMENSION 5(10).RR110).BURE(40).BUIM(40).YRE(12).YIM(12). SZINI4) ' COMPLEX 00(2).U1(2).GAMA(2).ZC(2) COMPLEX Yc.H02.H12.21I.z2I.ZEL.K1.K2.X.SHGH1.ER.ERR.D.KL.KD.ZI. ~GH1.GH2.GH3.GH4.2.ARG.CHGH1.CHGH2.SHGH2.TETA1.TETA2.ZIN REAL MU.L N'3 READ 71.H1.H2 PI=ATAN(1.)-4.0 2161-5.777E+7 EPR1-1.o AL-0.0 BE*0.0 READ 103.2IGD.EPRD PRINT 1O4.21GD.EPRD READ S2.ZIG2.EPR2 READ 100.A1.A2.A3 PRINT 1O1.A1.A2.A3 READ 106.FREO PRINT 106.FREO DME-2.0-PI-FREO EP1sEPR1-1.OE-9/(36.o-PI) EPDaEPRDo1.OE-9/(36.0-PI) MU=4.0-PI-1.oE-7 KT-CSORT(CMPLXIOM6-~2.O-MU~EP1.-DME-MUczIG1)) DO 7 u-1.1 IF(U.EO.1) A-A1 IF(U.E0.2) A-A2 x-K1-A UO(U)=CSORT(2.0/PIoX)*cc0S(X—PI/4.0) 7 di(d)=CSORT(2.0/PI'X)'CCOS(X-3.0'PI/4.0) EP2=EPR2-1.OE-9/136.0-PI) K2=CS0RT(CMPLX(DMEo-2.D-Mu-EP2.—OME-Mu-2162)) PRINT 33.K1.K2 DO 10 I-1.1 IF(I.EO.1) GO TO 1 AsA2 GO To 3 1 AsA1 3 6-2.O-PI-ZIGD/ALOG(A3/A) C32.O*PIPEPO/ALOG(A3/A) YCaCMPLx(G.OME-c) ARG=K2PA3 U=REAL(ARG) VaAIMAG(ARG) CALL COMBES(U.v.AL.6E.N.BURE.BUIM.YRE.YIM) HO2=CMPLXIBJREI1).BOIM(1))-CMPLX(0.0.1.0)'CMPLX(YRE(1).VIM(1)) H12=CMPLX(BJRE(2).BJIM(2))‘CMPLXI0.0.1.0)‘CMPLX(YRE(2).YIM12)) PRINT 31.UO(II.U1(I) PRINT30.H02.H12 ZiI=CMPLX(0.0.1.0)‘K1/(2.0‘PI-A'ZIGII 22I'(-K2‘HO2)/(2.0-PIPA3'CMPLX12162.0ME'EP2)'H12) ZE'(OME‘MU/(2.OOPI))~ALOG(A3/A) ZEL-CMPLX(0.D.ZE) ZI¢Z1I+Z2I RI-REAL(Z!) XI-AIMAG(ZI) PL-RI/(XI+zE) PsSORT(1.0+PL--2.0) 10 73 74 75 76 51 1147 FP=SORT(O.5*(P+1.0)) GPsSORT(O.5¢(P-1.0)) RD=CSORT(-Yc-2EL) . KL=KO*SORT(1.0+XI/ZE)*(FP-CMPLX(0.0.1.0)‘GP) ALPHAa-AIMAG(KL) BETA=REAL(KL) z=z1I+221+2EL R=REAL(2) L=AIMAG(Z)/OME HC=(OME*(R*C-L*G))/(OME**2.0-L¢C+R*G) FHC=SORT(O.5'(SORT(1.0+HC“2.0)+1.0)) PHC=SORT(O.5*(SORT(1.0+HC‘02.O)°1.O)) ZC(I)=SORT((OME*'2.O’L¢C+ROG)/((OME-C)¢*2.O+G*-2.O))‘(FHC- 'CMPLX(0.0.1.0)*PHC) PRINT 40.2C(I) PRINT 21.I.ALPHA.BETA GAMA(I)=CMPLx(ALPHA.BETA) CONTINUE IF(A1.NE.A2) Go To a GAMA(2)=GAMA(1) zc(2)-zc(1) DD 51 II-1.3 IF(II-2) 73.74.75 TETA1=CMPLX(0.0.PI/2.o) TETA2-TETA1 GO TO 76 TETA1-0.o TETA2=CMPLX(0.0.PI/2.0) GO TO 76 TETA2=0.0 TETA1=0.0 GH2=CEXP(GAMA(2)-H2+TETA2) GH1=CEXPIGAMA(1)~H1+TETA1) GH3=1.0/GH1 GH4-1.0/GH2 SHGH1=(GH1-GH3)/2.0 sHGH2=(GH2-GH4)/2.0 CHGH1=(GH1+GH3)/2.0 CHGH2=(GH2+GH4)/2.0 ZINIII)’(ZC(1)*CHGH1*SHGH2+ZC(2)'SHGH1*CHGH2)/(SHGHiPSHGH2) PRINT 25.TETA1.TETA2.ZIN(II) CONTINUE c...‘........ttttiitfi‘tfi....‘.0..t0.tttfittlitOttttttttifitt......ttifit‘. C C... 21 25 INPUT.OUTPUT FORMATS. tittttttitttttttittttttitttttttttittttttttttt'ttOttttttitttttitttfit FORMAT(1H0.2OX.2HI-.I2.4x.6HALPHA-.F12.5.4x.5H8ETA-.F12.5.//) FORMAT(1H0.2OX.¢TETA1-¢.2(F5.2.2X).2X.*TETA2=*.2(F5.2.2X).2X. t-ZIN-‘.2(F12.3.2X)./) FORMAT(1HO.2OX.4HH02-.2(E12.5.2x).4HH12=.2(E12.5.2X)./) FORMAT(1H0.20x.3HUO=.2(E12.5.2X).3HU1-.2(E12.5.2x)./) FDRMAT(1H0.20x.3HK1-.2(E12.5.2x).3HK2-.2(E12.5.2x)./) FORMAT(1H0.20X.25HCHARACTERISTIC IMPEDANCE-.2(F12.5.4x)./) FORMAT(F8.4.F5.2) FDRMAT(1H0.20X.5HZIG2s.F8.4.2X.5HEPR2-.F5.2./) FORMAT(2(F8.5)) FORMAT(5(F8.5)) FORMAT(1H0.2OX.3HA1-.F8.5.2X.3HA28.F8.S.2X.3HA3-.F8.5./) FORMAT(2(F6.2)) FDRMAT(1H0.20X.5HZIGD-.F6.2.2X.5HEPRD-.F6.2./) FDRMAT(F12.1) FORMAT(1HO.2OX.*FREOUENCYS‘.F12.1./) 0¢I~J 14 15 1O 13 11 12 1148 END SUBROUTINE COMBES(X.Y.ALPHA.8ETA.N.EURE.8UIM.VRE.YIM) DIMENSION BUREI40).BUIM(40).YRE(12).YIM(12) CALL BEGIN1X.Y.N.K.R) CALL dRECUR1X.Y.ALPHA.BETA.K.R.BJRE.BJIM) CALL USUM(ALPHA.8ETA.K.BURE.BUIM.SUMRA.SUMIA) CALL FACTOR1x.Y.ALPHA.BETA.O.R) CALL UNORM(K.O.R.SUMRA.SUMIA.BURE.BUIM) CALL YSUM (X.Y.ALPHA.8ETA.K.EURE.BUIM.ASUMR.ASUMI) CALL YGNU (x.v.ALPHA.8ETA.0.R.ASUMR.ASUMI.BURE.BUIM.YRE.YIM) CALL wRONSK (X.Y.BURE.BUIM.YRE.YIM) BUSO=BURE11)--2+BUIM(1)--2 IF(BUSO-.00000005) 14.14.15 CALL YSUMP(X.Y.ALPHA.BETA.K.8URE.BUIM.ASUMR.ASUMI) CALL YGNUPIX.Y.ALPHA.8ETA.0.R.ASUMR.ASUMI.BURE.BOIM.YRE.YIM) IF (N-1)1o.12.11 IF (N)13.12.12 CALL NEGN (X.Y.ALPHA.8ETA.N.BURE.BUIM.YRE.YIM) GO TO 12 CALL YRECUR(X.Y.N.BURE.3UIM.YRE.YIM) RETURN END CBES402 BEGIN SUBROUTINE PART 2 OF 16 SUBROUTINE BEGINIX.Y.N.K.R) SSOSXOOZ&Y002 KTEN=SORT(SSD)+20.0 NTEN-IABS(N)+10 M-MAXO(KTEN.NTEN) /2 K82EM+1 R - K + 1 RETURN' END CBES403 JRECUR SUBROUTINE PART 3 OF 16 SUBROUTINE URECUR1X.Y.ALPHA.BETA.K.R.BURE.BUIM) DIMENSION BURE(100).BUIM(100) RALPHA=R+ALPHA SSQsXOOZ+Y002 BURE(K+2)-O BUIM(K+2)-0 DURE(K+1)-1.OE—37 BJIM(K+1)'0.0 DOAIs1.K L1-K+1-I RALPHAaRALPHA-1.0 A-((2.0-x-RALPHA)+(2.o-BETAoY))/Sso B-((-2.0oY-RALPHA)+(2.0~8ETA-X))/sso BURE(L1)-(A-8URE(L1+1))-(8-BUIM(L1+1))-BURE(L1+2) BUIM(L1)-(aoBURE(L1+1)1*(A-BUIM1L1+1))-BUIM1L1+2) RETURN END CBES404 USUM SUBROUTINE 0 PART 4 OF 16 801 SUBROUTINE USUMIALPHA.8ETA.K.80RE.OUIM.SUMRA.SUMIA) DIMENSION BURE(100).BUIM(1OO) SUMRAa(BURE(3)'(ALPHA+2.O))-(BUIM(3)caETA) SUMIA-(BETAcBURE(3))+((ALPHA+2.0)-BUIM(3)) GRE-1.0 GIMnO 5-1.0 Door-5.x.2 s-s+1.o GREN'((GRE‘(ALPHA+S-1.O))-(BETAPGIM))/5 15 21 2O 1O 6 11 149 GIM=((GIM~(ALPHA+S-1.0))+(8ETA-GRE))/s GRE=GREN ALPTS=ALPHA+2.0'S GdR=GRE*BdRE(I) GdIIGIMthIM1I) GURI=GRE~BUIM(I) GJIR=GIMYBJRE(I) SUMRB=ALPTS*(GdR-Gdl)‘BETA‘1GJIR+GJRI)+SUMRA SUMIB=ALPTS'(GdIR+GdRI)-BETA‘(GJI-GUR)+SUMIA IF(SUMRA)15.21.15 IF(ABS((SUMRB/SUMRA)°1.0)-.OOOOOOOS)21.21.1O IF(SUMIA)20.11.20 IF(ABS((SUMIB/SUMIA)-1.0)-.00000005)11.11.10 SUMRA=SUMR6 SUMIA=SUMIB RETURN END CBES405 FACTOR SUBROUTINE PART 5 OF 16 SUBROUTINE FACTOR(X.Y.ALPHA.BETA.O.R) CALL LOGGAM(ALPHA+1.0.BETA.U.V) CALL COMLDG(X.Y.A1.81) A23ALPHAFA1-BETA*B1 BZ=BETAPA1+ALPHAEBi A28-A2 822-62 CALL COMEXP(A2.62.A3.63) A4-.6931471806«ALPHA 64s.6931471806«6ETA CALL COMEXP(A4.84.A5.85) A6-A3cA5-63-65 86=BSPAS+A3*BS CALL COMEXP(U.V.A7.87) O=A6tA7-86'87 R=BGFA7+A6987 RETURN END CBES406 COMLOG SUBROUTINE PART 6 OF 16 C COMPLEX LOGARITHM - BRANCH CUT ON NEGATIVE REAL AXIS SUBROUTINE COMLOG(x.Y.A.B) PI=3.141592654 A8.5‘ALOG(x-X+YPY) IF(X)5.1.4 B=.5cPI IF(Y)2.3.8 88-8 60 To a 8'0. GO TO 8 B=ATAN(Y/X) GO TO 8 B=ATAN(Y/X) IF(Y)6.7.7 B-B-PI GO TO a B=8+PI RETURN END CBES407 COMEXP SUBROUTINE PART 7 OF 16 SUBROUTINE COMEXP1X.Y.A.B) CsEXP(X) AaCcCOS(Y) 150 8-CvSIN1Y) RETURN END CBES408 UNORM SUBROUTINE PART 8 OF 16 SUBROUTINE UNORM(K.O.R.SUMRA.SUMIA.80RE.BUIM) DIMENSION OORE(1OO).BUIM(1OO) S'((SUMRA+BJRE(1))‘O)‘((SUMIA+BJIM(1))‘R) T'((SUMIA+BUIM(1))‘O)+((SUMRA+BURE(1))'R) IF(ABS(S)-A651T))100.101.101 101 TS-T/S Tsso=s~(1.o+(TS—-2)) 12 00131-1.K BOREN-(BURE(I)+BUIM(I)-TS)/Tsso BdIM(I)'(8dIM(I)-BdRE(I)-TS)/TSSO 13 BUREII)-BUREN GO TO 14 100 ST-S/T STSOsT'((ST"2)+1.0) 102 DO1031=1.K BUREN'(BJRE(I)‘ST+BJIM(I))/STSO BUIM(I)'(BJIM(I)‘ST-BdRE(I))/STSO 103 60RE(I)=BUREN 14 RETURN END CBES4O9 YSUM SUBROUTINE PART 9 OF 16 SUBROUTINE YSUM (X.Y.ALPHA.BETA.K.BdRE.BdIM.ASUMR.ASUMI) DIMENSION BURE1100).BUIM(100) A1-ALPHA-1.o A2-A1-1.o A3=A1+ALPHA AA-BETA-cz A5-2.O~A4 ABSO*(°A1)°'2+A4 GAMRE-((2.0+ALPHA)~(-A1)-A4)/A8so GAMIMa(6ETAo3.0)/Aaso ASUMR-GAMRE-BORE13)-GAMIM¢BJIM(3) ASUMI=GAMIM‘BURE13)+GAMRE‘BUIM(3) Ta1.o Do 500 I-5.K.2 T-T+1.0 81-2.O-T F1s81+ALPHA F2-A3+T F3-A1+T F5-T-ALPHA F6-A2+81 G1-F1-F2-As G2=1F2+2.0oF1)-BETA H1-G1-F3-62-8ETA H2=G29F3+GitBETA P1-F5cF6+A4 P2-(F5-F61-BETA P3-P1--2+P2--2 GRE-((H19P1+H2PP2)/P3)/T CIM‘11H2PP1-H10P2)/P3)/T TEMP--(CRE.GAMRE-CIM:GAMIM) GAMIM--(CIMcGAMRE+CRE~6AMIM) GAMRE-TEMP 65UMR=GAMRE~BURE(I)-GAMIM~BUIM(I)+ASUMR BSUMI-GAMIM-GUREII)+GAMR6-BUIM(I)+ASUMI IF(ABS((BSUMR/ASUMR)-1.0)'.OOOOOOOS)521.521.510 151 521 IF(ASUMI)520.511.52O .-. 520 IF(ABS((BSUMI/ASUMI)-1.0)-.OOOOOOOS)511.511.540 510 ASUMR=BSUMR ‘ 500 ASUMI=BSUMI 511 RETURN END CBES41O YGNU SUBROUTINE PART 10 OF 16 SUBROUTINE YGNUIX.Y.ALPHA.BETA.O.R.ASUMR.ASUMI.BdRE.BdIM.YRE.YIM) DIMENSION BURE(100).BUIM(100).YRE(SO).YIM(50) PI'3.141592654 TPI'2.0/PI QRE=TPI‘(O“2‘R“2) OIM=TPI‘2.0‘Q‘R DRE'ORE‘ASUMR'OIM‘ASUMI DIM=OIMPASUMR+ORE‘ASUMI IF(ALPHA)1.2.1 2 IF(BETA)1.3.1 3 CALL YzERO(X.Y.ALPRE.ALPIM) GO TO 720 1 PALPHA=PIoALPHA CDX=COS(PALPHA) SIX=SIN(PALPHA) EXY-EXP(PI'BETA) EXY1=1.0/EXY COSH-.5-(EXY+EXY1) SINH=.5‘(EXY-EXY1) DEN*(SIX‘COSH)*'2+(COX*SINH)*‘2 ERE=(SIXoCOX)/DEN EIM=(-COSH~SINH)/DEN A8503=2.0'(ALPHAot2+8ETAttZI ALPRE=ERE-((ORE'ALPHA+BETA-OIM)/ABSO3) ALPIM=EIM-((DIM-ALPHA-BETAOORE)/ABSO3) 720 YRE(1)=ALPRE'BURE(1)-ALPIM*BUIM(1)+ORE YIM11)=ALPIM*BORE(1)+ALPRE*BUIM(1)+DIM RETURN END CBES411 YzERo SUBROUTINE PART 11 OF 16 SUBROUTINE Y2ERO(X.Y.ALPRE.ALPIM) TPI-2.o/3.141592654 CALL COMLOG(X.Y.A.6) ALPRESTPI‘(-.1159315157+A) ALPIM-TPI-a RETURN END CBES412 wRONSK SUBROUTINE PART 12 OF 16 SUBROUTINE wRONSK(X.Y.BURE.BUIM.YRE.Y1M) DIMENSION BURE(100).BUIM(100).YRE(50).YIM(50) SSO'XP-2+Y#*2 TPI-2.0/3.141592654 AZRE'TPIPX/SSO AZIMI-TPI-V/SSO ZRE-BURE(2)-YRE(1)-BUIM(2)*YIM(1) ZIM=BJIM(2)'YRE(1)+BdRE(2)'YIM(1) azREsZRE-AZRE BZIM-ZIM-AZIM BdSOinRE(1)*‘2+BdIM(1)*02 C2RE=BURE(1)/Euso CZIM-(-BUIM(1))/BUSO YRE(2)-62REcC2RE-8z1M-CZIM YIM(2)-BZIMtczRE+BZREtCZIM RETURN 152 END CBES413 NEGN SUBROUTINE PART 13 OF 16 SUBROUTINE NEGN(X.Y.ALPHA.BETA.N.8dRE.8dIM.YRE.VIM) DIMENSION BURE(100).BUIM(100).YRE(50).YIM(50) L=IA85(N)+1 SSO-XP'2+Y"2 TX=2.0‘X TY=2.o-Y RALPHA-ALPHA A=(Tx-RALPHA+TYoBETA)lsso , a=(-TY-RALPHA+Tx-BETA)/Sso EURE(2)=A-BURE(11-8-BUIM(1)-60RE(2) BUIM(2)=8-BURE(1)+A-BUIM(1)-BUIM(2) YRE(2)=A-YRE(1)-8-YIM(1)-YRE(2) YIM(2)=B-YRE(1)+A-YIM(1)-YIM(2) IF(L-3)3.2.2 2 Do 1 I=3.L RALPHAaRALPHA-1.o A=(Tx-RALPHA+TYoBETA)/Sso B=(-TY-RALPHA+TXv8ETA)/sso GURE(I)-A-80RE(I-1)-8-BUIM(I-1)-EURE(I-2) BUIM(I)=8-BURE(I-1)+A-BUIM(I-1)-BUIM(I-2) YRE(I)=A'YRE(I-1)-B-YIM(I-1)-YRE(I-2) 1 YIM(I)'B'YRE(I-1)+A'YIM(I'1)-YIM(I'2) 3 CONTINUE RETURN END CBES414 YRECUR SUBROUTINE PART 14 OF 16 SUBROUTINE YRECUR1x.Y.N.BORE.6UIM.YRE.YIM) DIMENSION BURE(1001.BUIM(100).YRE(50).YIM(50) SSOax-~2+Y--2 TPI=2.0/3.141592654 AzRE=TPI~X/Sso AZIMa-TPI-Y/SSO L=N+1 ~ IFIL-3)3.2.2 2 DD 1 I-3.L ZREsBUREII)-YRE(I-1)-BUIM(I)-YIM(I-1) ZIM=BUIM(I)~YRE(I-1)+BURE(I)-YIM(I-1) 62RE=2RE-A2RE azIMszIM-AZIM BJSO=BJRE(I-1)"2+BJIM(I-1)"2 CZRE=BURE(I-1)/Buso CZIM=(-BUIM(I-1))/BUSO YRE(I)=BZRE-CZRE-BZIM-CZIM 1 YIM(I)aBZIMtCZRE+BZRE-CZIM 3 CONTINUE RETURN END CBES415 YGNUP SUBROUTINE PART 15 OF 16 SUBROUTINE YGNUP(X.Y.ALPHA.BETA.0.R.ASUMR.ASUMI.BURE.BUIM.YRE.YIM) DIMENSION BURE(100).BUIM(100).YRE(50).YIM(50) PI-3.141592654 TPIs2.0/PI ORE-TPIP(O"2-R'O2) OIM-TPI'2.0*O'R DREsORE-ASUMR-OIM-ASUMI DIMsOIMoASUMR+0RE~ASUMI IF(ALPHA)1.2.1 2 IF(BETA)1.3.1 3 CALL YZEROIX.Y.ALPRE.ALPIN) 1£53 GO TO 720 1 PALPHAaPI‘ALPHA CDX=COS(PALPHA) . SIX=SIN(PALPHA) ' EXYsEXP(PIcBETA) EXY1-1.0/EXY COSH'.5*(EXY+EXY1) SINH=.S*(EXV-EXY1) OEN=(SIXFCOSH)E*2+(COX*SINH)0'2 ERE=(SIXECOX)/DEN EIM=(-COSH‘SINH)/DEN ABSO3=2.O'(ALPHAP‘2+BETA*P2) ALPRE=ERE-((ORE0ALPHA+BETA*OIM)/ABSO3) ALPIM-EIM-((OIMoALPHA- ETAcORE)/A8S03) 720 TREaALPRE*BdRE(2)-ALPIM'BJIM(2)+DRE TIM=ALPIMPBURE(2)+ALPREEBJIM(2)+DIM ALPREI~(O‘X+R-Y)/(X-~2+Y¢*2) ALPIM--(X*R-OPY)/(X"2+Y0*2) YRE(2)=ALPRE*BJRE(1)-ALPIM*BJIM(1)+TRE VIM(2)=ALPIM—BJRE(1)+ALPRE*BJIM(1)+TIM RETURN END CBES416 YSUMP SUBROUTINE PART 16 OF 16 SUBROUTINE YSUMP(X.Y.ALPHA.8ETA.K.BURE.BUIM.ASUMR.ASUMI) DIMENSION BURE(100).BUIM(100) A1=ALPHA-1.0 A2-A1-1.0 A3=A1+ALPHA A4=BETAPP2 A5-2.0vA4 ABSO'I-A1)*‘2+A4 ROLDRE'((2.0+ALPHA)*(-A1)-A4)/ABSO ROLDIM'(BETA93.O)/ABSO RES1a-ROLDRE/2.0 VMs1--RDLDIM/2.o STOREIS.*(ALPHA-X+BETAPY)/(X**2+Y*t2) STOIM-3.‘(XPBETA-ALPHAPY)/(X"2+Y‘-2) REszs(RoLDRE-STORE~ROLDIMtSTOIM) VM52-1ROLDRE~5TOIM+RDLDIMoSTORE) ASUMR-RESithRE(2)-VM51*BJIM(2) ASUMR-ASUMR+RESZ‘BJRE(3)-VMS2*8UIM(3) ASUMIsVMS1-BURE(2)+RES1-BUIM(2) ASUMI=ASUMI+VMS2-BURE(3)+RES2*BJIM(3) T-1.0 DD 500 I-3.K.2 T-T+1.o 61-2.0-T F1=61+ALPHA F2=A3+T F3-A1+T F5-T-ALPHA F6=A2+61 Gi-F10F2-A5 62'(F2+2.0‘F1)‘BETA H1-G1-F3-G2-BETA H2-62tF3+G1:6ETA PIIFSPFB+A4 92-(F5-F6)*BETA P3-P1t‘2+P2¢‘2 CRE-((M1:P1+H2¢P2)/P3)/T CIM'((H2*P1-H19P2)/P3)/T 1541 TEMP:-(CREcROLDRE-CIMvROLDIM) RNEwIM--(CIMoROLDRE+CRe-ROLDIM) RNEwRE-TEMP REs1=(ROLDRE-RNEwRE)/2.o VMS1=(ROLDIM-RNEwIM)/2.D RES2=1RNEWRE'STORE-RNEWIM'STOIM) VM52=(RNewRE-STOIM+RNEwIM-STDRE) 6SUMR=RES1~60RE(I+1)-VMs1oBUIM(I+1)+ASUMR BSUMIIVMS1~BUREII+1)tRESIPBdIM(I+1)fASUMI 8SUMR=RES2-BURE(I+2)-VMs2-BUIM(I+2)+BSUMR BSUMI=VMS2PBJRE(I+2)+RESZ-BJIM(I+2)*BSUMI IF(ABS((BSUMR/ASUMR)-1.0)--00000005)521.521.510 521 IF(ASUMI)520.511.520 520 IF(ABS((BSUMI/ASUMI)o1.0)-.000000051511.511.510 510 ASUMR=BSUMR ASUMI=BSUMI ROLDIMsRNEwIM 500 ROLDRE=RNEVRE 511 RETURN END SUBROUTINE LDGGAM(X.Y.U.v) CLDGGAM LOG OF THE GAMMA FUNCTION OF COMPLEX ARGUMENTS FORTRAN II c THIS SUBROUTINE COMPUTES THE NATURAL LOG OF THE GAMMA FUNCTION FOR C COMPLEX ARGUMENTS. THE ROUTINE IS ENTERED av THE STATEMENT 6 CALL LDGGAMIX.Y.U.V) c wHERE X IS THE REAL PART OF THE ARGUMENT Y IS THE IMAGINARY PART OF THE ARGUMENT U 15 THE REAL PART OF THE RESULT v 15 THE IMAGINARY PART_OF THE RESULT DIMENSION H17) H(1)-2.269488974 H(2)-1.517473649 H(3)-1.o11523068 H(4)=5.2560646906-1 H(5)-2.523809524E—1 H16)-3.3333333336-2 H17)=8.3333333336-2 62-1.57O79632679 E8-3.14159265359 81=0.0 62:0.0 u-2 X2-X 4 IFIX)1.2.3 3 66=ATAN(Y/X) Tax--2 s B7-YPO2+T c REAL PART OF LOG T1-.5-ALOG(B7) IF(X-2.0)7.7.6 7 a1=a1+86 62-62+T1 x-x+1.o 0-1 60 To 4 6 T3o-Yo66+(T1-(X-.5)-X+9.1393853326-1) T2-86-(X-.5)+V0T1-Y T4-x T5--Y T1-B7 DO 3 I-1.7 000 155 T=H(I)/T1 T4=T~T4+X T5=-(T-T5+Y) 8 TI'T4“2+T5“2 T3=T4-X+T3 T22-T5-Y+T2 Go To (9.10).d 9 T3=T3-62 T2=T2-81 10 IF(X2)11.12.12 12 U=T3 v=T2 X=x2 RETURN 11 U=T3-E4 VaTZ-ES x=X2 RETURN C x 15 ZERO 2 T80.0 IF(Y)13.14.15 13 Ber-E2 GO TO 5 15 66-62 GO TO 5 C X Is NEGATIVE 1 E4-0.o 6580.0 IEGIO 16 E4=E4+.5*(ALOG(X*¢2+Y¢*2)) Es-ES+ATAN(Y/X) IE6=166+1 X=x+1.o IF(X)16.17.17 17 IF(MDD(IE6.2))16.4.16 18 ES=E5+E8 GO TO 4 14 PRINT 19.x2.v . 19 FORMAT(29H ATTEMPTED TO TAKE LDGGAM OF 2HXsF6.o.1X2HY8F6.O) CALL EXIT END .015 .045 0.0 2.25 1.11 76.7 .00043 .00043 .00096 600000000.0 156 ZIGD= 0.00 EPRD= 2.25 A18 .00043 A2= .00043 A3= .00096 FREQUENCY= 600000000.0 K1= .36992E+06 -.36992E+06 K2= .11251E+03 -.23370E+02 J0= .49863E+70 .52113E+70 J1= 52113E+70 -.49863E+70 H02= .86464E+00 .14723E+01 H12= —.11067E+01 .57375E+01 CHARACTERISTIC IMPEDANCE= 64.54555 -l3.05870 I= 1 ALPHA+ 7.66206 BETA: 37.87144 TETA1= 0.00 1.57 TETA2= 0.00 1.57 ZIN= 176.549 -56.234 TETAIB 0.00 0.00 TETA2= 0.00 1.57 ZIN= 163.186 -196.332 TETAI- 0.00 0.00 TETA2= 0.00 0.00 ZIN= 28.471 -98.405 CHAPTER VIII SUMMARY In this thesis we present some theoretical and experimental results on the study of a bare microprobe and an insulated micro- probe in a conducting (biological) medium. The dimensions of the probe are set to be very small so that it can be imbedded easily in a biological body. Since the current on the bare probe in a conducting medium is decaying rapidly, the probe can be truncated and treated as an asymmetric dipole in the analysis. For increasing the decay of the probe current, the probe can be coated with material with higher surface impedance. The current distribution along the probe, and the effect of the surface impedance on the current and the input impedance of the bare probe are found based on Hallen's integral equation and electric field integral equation. These equations are solved numerically by moment method. The driving source is modeled as a magnetic ring or as a delta gap generator. A series of experiments were conducted, and the input impedances of bare probes in various conducting media were measured with vector voltmeter and E-H probe. The agreement between the theory and the experiment was found to be satisfactory. 157 158 An investigation on the applicaton of a bare probe for mea- suring electrical properties of a conducting medium, and for heating a tumor imbedded in a biological body for the purpose of hyperthermia cancer therapy is conducted. The theory of lossy transmission line is used to solve the problem of an insulated microprobe in a conducting medium. The current on the insulated probe does not decay rapidly. therefore, equivalent terminal impedances are introduced. The current distri- butions along the probe for various terminal impedances are given. The input impedance of the probe is discussed and the input impedances of symmetric insulated probes with various terminal impedances are presented graphically. The heat patterns of insulated probes with various terminal impedances are shown, and the effect of the terminal impedance on the heat pattern is discussed. It is concluded that the heat pattern can be altered by changing the tenninal impedance of the probe. Finally, the computer programs used for finding the current distribution and the input impedance of bare and insulated probes with exampIes are given. In conclusion, the input impedance of the bare probe is more sensitive to the surrounding medium. therefore, it is suitable for measuring the electrical properties of biological bodies. 0n the other hand. the insulated probe can transfer more power in a biolog- ical body in comparison with the bare probe so that an insulated probe is a better device for local heating. BIBLIOGRAPHY BIBLIOGRAPHY . J.E. Robinson, M.J. Nizenberg, and M.A. MCGready, "Radiation and hyperthermal response of normal tissue in situ," Radiology, 113, 1, 195—198. 1974. H.H. Leveen, S. Napnick, V. Piccone, G. Falk, and N. Ahmed, "Tumor eradication by radiofrequency therapy," JAMA, 230, 29, 2198-2200, May 17, 1976. C.C. Johnson, H. Plenk, and C.H. Durney, "A 915 MHz exposure system for hyperthermia treatment of cancer," Presented at 1977 International Symposium on Biological Effects of Electromagnetic Waves, Airlie. Virginia. October 30-N0vember 4, 1977. A. Cheung. D. McCulloch, J. Robinson, and G. Samaras, "Bolusing techniques for batch microwave irradiation of tumors in far field." Presented at 1977 International Microwave Symposium, San Diego, California, June 21-23. 1977. E.C. Burdette, F.L. Cain, and J. Seals. "In vivo probe measure- ment technique for determining dielectric properties at VHF through microwave frequencies." IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-28, N0. 4, pp. 414-427, April 1980. T.N. Athey, M.A. Stuchly, and 5.5. Stuchly, "Measurement of radio frequency permittivity of biological tissues with an open- ended coaxial line: part I," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-30, No. 1, pp. 82-86, January 1982. M.A. Stuchly, T.w. Athey, G.M. Samaras, and G.E. Taylor, "Mea- surement of radio frequency permittivity of biological tissues with an open-ended coaxial line: part II," IEEE Trans. on Micro- wave Theory and Techniques, Vol. MTT-30, No. 1, pp. 87-90, Janu- ary 1982. G.B. Gajda, and 5.5. Stuchly, "Numerical analysis of open-ended coaxial lines." IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-31. N0. 5. PP. 380-384, May 1983. R.N.P. King, B.S. Trembly, and J.N. Strohbehn, "The electro- magnetic field of an insulated antenna in a conducting or dielectric medium." IEEE Trans. on Microwave Theory and Tech- niques. Vol. MTT-31, No. 7, pp. 574-583, July 1983. 1593 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. .160 R.F. Harrington, "Field Computation by Moment Methods," Macmillan, New York, 1968. 6.3. Smith. "A comparison of electrically short bare and insulated probes for measuring the local radio frequency electric field in biological systems," IEEE Trans. on Biomedical Engineering, Vol. BME—ZZ, No. 6, pp. 477-483, November 1975. M.K. Hessary. "Local heating of biological bodies with HF electric and magnetic fields," Ph.D. Dissertation, Michigan State University, East Lansing, MI 1982. J.A. Saxton and J.A. Lane. "Electrical properties of sea water," Wireless Engineering, October 1952. L.K. Lepley and M.A. Adams, "Electromagnetic dispersion curves for natural waters," Water Resources Research, Vol. 7, ND. 6, pp. 1538-1547, December 1971. K. Karimullah, "Theoretical and experimental study of the prox- imity effects of thin wire antenna in presence of biological bodies," Ph.D. Dissertation, Michigan State University, East Lansing, MI 1979. S.H. Mousavinezhad, "Implantable electromagnetic field probes in finite biological bodies," Ph.D. Dissertation, Michigan State University, East Lansing, MI 1977. R.N.P. King. "Transmission Line Theory," Dover Publication. Inc., New York, 1965. R.N.P. King, C.H. Harrison, "Antennas and Raves: A Modern Approach," The MIT Press, 1969. R.N.P. King, "Theory of the terminated insulated antenna in a conducting medium." IEEE Trans. on Antenna and Propagation, Vol. AP-12, No. 3. pp. 305-318, May 1964. N. Marcuvitz. "Naveguide Handbook,“ MIT Radiation Laboratories Series, No. 10, September 1950. “11111111111111 11111111111111111111s 3 1293 030613073