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A ‘ . , l . .‘ , ‘ ‘ ‘ '\ MICHlGAN ST TE UNIVERSITY LIBRARIES Illlllllllll‘llll‘lMl! ll 3 1293 0091 831 This is to certify that the dissertation entitled MEASUREMENT OF ELECTROMAGNETIC PROPERTIES OF MATERIALS VIA COAXIAL PROBE SYSTEMS presented by Ching-Lieh Li has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engineering .r/C ’ fléflg Major professor Date 3/”7 Ms MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State A University A PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. I DATE DUE DATE DUE DATE DUE l WWI MSU I: An Affirmative ActiorVEquel Opportunity Institution cmmd MEASUREMENT OF ELECTROMAGNETIC PROPERTIES OF MATERIALS VIA COAXIAL PROBE SYSTEMS By Ching-Lieh Li A DISSERTATION Submitted to Michigan State University in panial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1 993 ABSTRACT MEASUREMENT OF ELECTROMAGNETIC PROPERTIES OF MATERIALS VIA COAXIAL PROBE SYSTEMS by Ching-Lieh Li This thesis presents two methods to determine the conductivity, permittivity and permeability of materials via coaxial probe systems. Using a frequency-domain network analyzer, the determination of the complex permittivity and complex permeability over a wide band of frequencies can be achieved simply through two input impedances measurements. In this thesis two coaxial TEM structures operating from 200 MHz to 2 GHz are analyzed. Two experimental setups of coaxial probe systems were designed to measure the electromagnetic (EM) properties of materials. One is the open-ended coaxial probe system, by which a nondestructive measurement of EM parameters can be easily achieved for various materials in practical applications. The other is called coaxial cavity system, which is suitable for liquid materials or lossy materials of thick dimensions. Experimental results of these two methods are satisfactory and they agree quite well. In the open-ended coaxial probe system, theoretical analysis predicts the excitation of surface waves, radiative waves and radial guided waves which can influence the quantification of EM parameters of materials. A careful study on these complex waves was conducted. On the other hand, a resonant phenomenon was observed theoretically and experimentally in the coaxial cavity structure. In order to investigate the effect on the characterization of EM properties near the resonant frequencies, the metallic wall loss of the cavity is calculated using Poynting theorem and perturbation method. Both methods were studied carefully by full wave analysis. Theoretical approach leads to an integral equation for the electric field at the aperture of the coaxial probe system. The method of moments is applied to transform the integral equation into a set of simultaneous algebraic equations so that the numerical solution for aperture electric field can be obtained. After the aperture B field is obtained, other quantities such as the input impedance and the EM fields inside the material can be calculated. The input impedance of the coaxial probe can then be used to determine the EM parameters of the material. A series of experiments was conducted to measure the input impedances of the coaxial probe systems, which are in contact with materials, using an HP network analyzer. An inverse algorithm is then used to determine the EM parameters of the materials from the measured input impedances. Various materials, which include low, medium and high permittivity materials, have been measured and the results were found to be satisfactory. ACKNOWLEDGEMENTS The author is sincerely grateful to his academic advisor Dr. Kun-Mu Chen who provided much of the guidance, counsel and encouragement throughout the course of this research. Special thanks are extended to the members of the guidance committee, Dr. E. J. Rothwell and Dr. C. Y. Lo for valuable suggestions, especially to Dr. D. P. Nyquist for his generous and unfailing support during the period of this study. Special thanks are also extended to Dr. J. Ross and Dr. J. Song for their kindly helps in the experimental and theoretical aspects, respectively. In addition, it is the author’s pleasure to thank his wife, Yuling, who offered thorough understanding, helping and encouragement. The research reported in this thesis was supported by Michigan Research Excellence Fund and in part by Boeing Airplane Company. TABLE OF CONTENTS LIST OF FIGURES .......................................... iii INTRODUCTION .......................................... 1 REVIEW OF WAVEGUIDE THEORY ........................... 5 2.1 Introduction of Hertzian potentials ........................... 5 2.2 Orthogonality properties of waveguide modes .................... 9 2.3 Reflection and transmission at a discontinuity interface for TM modes . . 10 OPEN-ENDED COAXIAL PROBE TO MEASURE THE PERMII'I'IVITY AND PERMEABILITY OF MATERIAL .......................... 21 3.1 Introduction ........................................... 21 3.2 Theoretical Study Using Full Wave Analysis .................... 24 3.2.1 Application of General Waveguide Theory to a Coaxial Structure . . 24 3.2.2 Integral Equation for the Aperture Electric Field .............. 26 3.2.3 Identification of Complex Waves ........................ 32 3.3 Analysis via Equivalent Source Treatment of Aperture Electric Field . . . 37 3.3.1 Wave Equations for EM Field Maintained by Magnetic Current Distribution ....................................... 37 3.3.2 Scattered Wave and Primary Wave Solutions of Wave Equation . . . 41 3.4 Numerical Simulation - Method of Moments .................... 57 POWER BALANCE FOR COMPLEX WAVE EXCITATION IN OPEN- ENDED COAXIAL PROBE STRUCTURE ........................ 65 4.1 Introduction ........................................... 65 4.2 Surface Wave Power for Open-Circuit Case ..................... 66 4.2.1 Derivation of Surface Wave Field Expressions ............... 68 4.2.2 Calculation of Surface Wave Power ...................... 72 4.3 Radiated Power for Open-Circuit Case ........................ 77 4.3.1 Power Relation in Spherical Coordinates ................... 77 4.3.2 Far Zone Field Derivation via Saddle Point Method ........... 81 4.4 Radial Guided Wave Power for Short-Circuit Case ................ 91 4.4.1 Radial Guided Waves inside Material Medium ............... 93 4.4.2 Calculation of Radial Guided Wave Power .................. 96 COAXIAL CAVITY SYSTEM TO MEASURE THE PERMITTIVITY AND PERMEABILITY OF MATERIAL ............................. 101 5.1 Introduction .......................................... 101 5.2 Theoretical Study Using Full Wave Analysis ................... 104 5.3 Equivalent Circuit Concept for Discontinuity ................... 109 5.4 Numerical Simulation - Method of Moments ................... 113 5.5 Computation of Metallic Wall Loss for Coaxial Cavity ............ 120 6. EXPERIMENTAL MEASUREMENT FOR VARIOUS MATERIALS ..... 130 6.1 Introduction ......................................... 130 6.2 Calibration Procedure for Experimental Setups .................. 134 6.3 Numerical Inverse Algorithm .............................. 142 6.4 Experimental Results via Open-Ended Coaxial Line Probe Method . . . 144 6.5 Experimental Results via Coaxial Cavity Method ................ 153 7. CONCLUSION .......................................... 163 APPENDIX: FORTRAN COMPUTER PROGRAM ................. 166 BIBLIOGRAPHY ......................................... 204 ii 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 LIST OF FIGURES A discontinuity interface between two different media. ............ 11 The reflection and transmission coefficient of a transmission line discontinuity, where Yu and Y,, are the characteristic admittance of ’ each line, respectively. .................................. 14 A discontinuity interface between two different media, where there are TM waves of amplitudes c,, b, coming from the medium of either side respectively. .................................. 16 A cascade connection of N+1 sections with TM wave incident. ...... 18 An open-ended coaxial probe terminated on a metallic flange placed against a material medium . A TM wave is excited inside the material ........................................... 20 Geometry of an open-ended coaxial line terminated on a large metallic flange placed against a stratified isotropic medium. ........ 22 Deformed contour integral in complex kc-plane for open—circuit case. ............................................... 35 Deformed contour integral in complex Ice-plane for short-circuit case. ............................................... 36 Planar waveguide excited by rotationally symmetric magnetic current distribution ...................................... 38 Reflection coefficient of an SR-7 type coaxial line placed against various homogeneous materials; frequency is 3 Ghz. ............. 64 Surface wave and radiative wave excitation in the open-circuit case of an Open-ended coaxial probe placed against a material layer. ...... 67 The power radiated into the material medium is evaluated over the surface of a hemisphere for an open-ended coaxial probe. .......... 79 Transformation to spherical coordinates, (a) in Ice-B plane (b) in p-z plane. .............................................. 82 Mapping of two Riemann sheets of the kc plane onto a stripe of the ul=o+jn plane. The shaded regions are the pr0per Riemann sheet. .............................................. 84 Deformation of contour C into steepest decent path SDP. .......... 89 The power balance verification for open-circuit case. The total power carried away from the aperture and the transmitted power of the incident wave, both normalized by the input power, agree quite well. ............................................... 92 The radial guided waves are excited in the short-circuit case of an open-ended coaxial probe placed against a layered material medium, where Pp, stands for the power flow in region 1. ......... 94 The power balance verification for short-circuit case of an open-ended coaxial probe. The radial guided wave power and the transmitted power of the incident wave, both normalized by the input power, agree quite well. ............................. 99 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Geometry of a coaxial line terminated on a coaxial cavity which accommodates an isotropic material layer medium ............... Two segments of a coaxial line with a short circuit termination and its equivalent circuit. ................................... Two different segments of coaxial lines with a short circuit termination and its equivalent circuit. ...................... Discontinuity capacitance per unit length, Cd", of a coaxial air line v.s. step ratio or. ..................................... Geometry of a coaxial cavity driven by a coaxial line. The complex Poynting theorem is applied to the volume V beyond the cross section z=-l,. ........................................ The power loss of non-perfectly conducting wall is normalized by the input power, where the filled material is water and acrylic, respectively .......................................... The power loss of non-perfectly conducting wall is normalized by the dielectric power loss, of which the filled material is (a) water, and (b) acrylic, respectively. ............................. The power loss P. of non-perfectly conducting wall when the cavity is filled with acetone. P, is normalized by (a) the input power, and (b) the dielectric power loss, respectively ............. Experimental setup of an open-ended coaxial probe system to measure the EM parameters of materials. .................... Experimental setup of a coaxial cavity system to measure the EM parameters of materials. ................................ Two input impedance measurements are made by (a) using open—circuit and short-circuit cases. (b) preparing two samples of different thicknesses, or (c) inserting some known material for the second one. ......................................... (a) The structure of the coaxial line probe setup. (b) Its equivalent ' two-port network. ..................................... (a) The structure for the calibration of the coaxial line probe using three shorting stubs. (b) Its equivalent circuit is described by the S parameters and one segment of perfect transmission line. ......... The structure of the coaxial line probe with a spacer of length 1... Its equivalent circuit is described by the 8 parameters and two segments of perfect transmission line. ....................... The permittivity e of acrylic via Open-ended coaxial probe (11:14,, assumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. ...................... The permittivity e and permeability u of acrylic via open-ended coaxial probe. (a) The relative permittivity. (b) The relative permeability. ........................................ The permittivity e of acrylic via Open-ended coaxial probe in the frequency range of 2~10 GHz ............................ iv 102 110 123 127 128 129 131 135 137 141 146 147 150 6.10 The permittivity e and permeability p of acetone via open-ended coaxial probe. (a) The relative permittivity. (b) The relative permeability. ........................................ 151 6.11 The permittivity e of Eccosorb-L522 via open-ended coaxial probe (p=u0 assumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. ...................... 152 6.12 The permittivity e and permeability u of distilled water via open-ended coaxial probe. (a) The relative permittivity. (b) The relative permeability. .................................. 154 6.13 The permittivity e of acrylic via coaxial cavity method (p.r=rro assumed). The effect of cavity wall loss is shown. (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. ......................................... 157 6.14 The permittivity e of Eccosorb—L322 via coaxial cavity method (11:11, assumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. ...................... 159 6.15 The permittivity e of distilled water via coaxial cavity method (p=uo assumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. ...................... 161 CHAPTER 1 INTRODUCTION ' The research described in this thesis deals with the measurement of electromagnetic (EM) properties of materials. Two methods are investigated in this thesis: The first method employs an open-ended coaxial line probe which consists of an open-ended coaxial line terminated on a metallic flange and placed against a material layer. The second method uses a coaxial cavity which is partially filled with material and excited by a coaxial line. The first method offers an easy, nondestructive measurement, and the second method is useful to measure thick lossy materials. The subject of EM property characterization of materials has attracted interests of many researchers in this field for a long time [1]. In particular, the nondestructive measurement [2]-[4] and simultaneous detennination of complex permittivity and permeability [5]-[7] have gained much attention during the last two decades. The free space methods to achieve nonintrusive measurement employ beams of EM radiation which are directed to pass through the object [8],[9]. These methods possess the advantage that the samples can be easily introduced into the measurement area, but the frequencies employed in these methods are typically in the minimeter wave or quasi- Optical range and relatively large material samples are usually necessary. For smaller material samples, the frequency domain transmission line techniques may be suitable, for which the measurement fixture consists of a waveguide or a coaxial transmission line which holds the sample. In particular, the coaxial waveguide structures have been widely used for EM parameter measurements [l],[10]. Usually the S-parameters of the transmission line are determined and then the EM parameters are calculated. Other structures including stripline [6], microstrip line [11] and closed cavity methods [12] have also been employed. One of the advantages for the setups is that the lower frequency measurement can be achieved with a TEM structure. The disadvantage is that the material samples need to be machined to fit in the fixtures. To achieve a nondestructive measurement in the frequency range from 0.2 GHz to 2 GHz, an open-ended coaxial probe terminated on a metallic flange is proposed. This geometry possesses the advantage of simplicity and a nondestructive measurement can be achieved easily by placing the probe against the materials. Attempts to measure the EM properties of materials via an open-ended coaxial probe are certainly not new [2]-[4],[13]- [16]. But to our best knowledge, no study has been conducted to characterize both the complex permittivity and permeability of material using an open-ended coaxial probe. And little work has been conducted to investigate the short-circuit case (material layer shorted by a metallic plate) of an open-ended coaxial probe [17]. Also the effect of complex wave excitation upon the characterization of EM parameters has not been addressed before. With the geometry of an open-ended coaxial probe, only a small sample area is illuminated by EM fields in the frequency band of interest; this method is not very sensitive nor suitable for lossy materials of large dimensions. In order to overcome the weakness of an open-ended coaxial probe scheme, a coaxial cavity method is developed to excite EM fields which penetrate into the material medium. In variation to the conventional cavity method, a single-frequency method to extract the EM parameters from the change in resonant frequency and Q-factor [18],[l9], the coaxial cavity is partially filled with material and the reflection coefficient of the incident wave to the cavity is measured over a wide frequency band. The coaxial cavity structure provides a well-defined reference plane for precision measurement and is suitable for low frequency range. Conceptually, the complex permittivity and permeability of the material can be determined by simply changing the length of the coaxial cavity. Chapter 2 provides theoretical background of the general waveguide theory. Hertzian potentials, employed to describe the EM fields of guided waves, are reviewed. The orthogonality properties of these guided waves are listed for the subsequent analysis of waveguide discontinuity problems. Also a concise derivation of wave matrices is given to analyze the reflection and transmission of TM waves incident upon a discontinuity interface. Chapter 3 is devoted to the theoretical and numerical analysis of the open-ended coaxial probe, which consists of an open-ended coaxial line terminated on a large metallic flange and placed against a layer of material. An integral equation for the aperture electric field is established by matching the tangential EM fields across the aperture. The method of moments is then applied to solve the electric field integral equation (EFIE) numerically. The theoretical analysis predicts the excitation of surface waves and radiative waves for the open-circuit case and radial guided waves for the short-circuit case in this structure. The concept of equivalent magnetic surface current was used to replace the aperture E field and an analysis on the radiation of this equivalent surface current in a layered medium is presented. This study offers a check to the study based on the EFI'E and also provides some physical pictures relevent to the problem. Chapter 4 studies the power balance for the open-ended coaxial probe by comparing the total power carried away from the aperture, in the form of surface waves, radiative waves and radial guided waves, to the transmitted power of the incident wave at the aperture. This provides a solid verification for theoretical and numerical results. The power associated with each complex wave is computed via Poynting vector, of which the EM field components of surface waves and radial guided waves are derived by Cauchy residue theorem, and the far zone field of radiative wave is evaluated by the saddle point method. The EM fields excited inside the material medium are found to be localized around the probe aperture for the frequency band of interest. This finding justifies the assumption of an infinite metallic flange in the analysis. Chapter 5 is devoted to the theoretical and numerical analysis of the coaxial cavity, which is partially filled with a material and driven by a coaxial line. The matching of the tangential electric and magnetic fields at the probe aperture results in an integral equation for the unknown aperture electric field. The method of moments is then applied to solve the EFIE numerically. It is observed that a resonant phenomenon occurs in this structure, and at the resonant frequencies the determination of the EM parameters of materials becomes ill-conditioned. The metallic wall loss effect of the coaxial cavity near the resonant frequencies is then investigated to help mitigate the ill-conditioned problem. Chapter 6 presents the experimental results on EM properties of various materials using both methods. The experimental setups and the calibration procedures for conducting accurate experiments are outlined. Two measurements on the probe input impedance are made for each frequency of interest using an HP network analyzer (HP87203). These measured input irnpedances are then used to determine the permittivity and permeability by some numerical inverse algorithm. The results on some known materials agree quite well with the published data. CHAPTER 2 ' REVIEW OF WAVEGUIDE THEORY The results of general guided wave theory are outlined in this chapter [21]. Definition of Hertzian potentials is first introduced, and the EM fields of guided waves along an uniform closed-piped system are provided in terms of suitable Hertzian potentials. Then the orthogonality properties for the guided waves confined by a perfectly conducting waveguide are listed, which are useful in the analysis of waveguide excitation and discontinuity problems. Specifically, the guided waves in coaxial line structures are analyzed in detail in subsequent chapters. The reflection and transmission of transverse magnetic(TM) waves incident upon a discontinuity interface from the medium of either side are also studied. The concept of wave matrix is then presented to handle the problem of EM wave propagation in a layered material medium systematically. 2.1 Introduction of Hertzian potentials Hertzian potentials are commonly used to describe electromagnetic (EM) fields in BM theory [22]. And there are in general two types of Hertzian potential, electric type Hertzian potential and magnetic type Hertzian potential. The electric Hertzian potential, fi‘, is defined such that the electric and magnetic fields in a homogeneous and isotropic medium are derived from it as follows: II=e-§-Vxfi a: ‘ azfi (2.1) E=V( vii.) - pe——‘ at’ with Vfi -pe 3211.23 (2.2) C 811 C where P is the polarization density vector associated with the sources and defined by =95, p=_v.p‘ (2.3) a: In the frequency domain and in a source free region, the relations become 5 = yfi.+v(v-fi.) = vaxfl, - (2.4) I? = jmerI-I‘ and Vzfi.+kzfi.=0 ’ (2.5) where k=m lie is the wavenumber in the medium. Similarly the magnetic Hertzian potential, II , is defined such that the electric and magnetic fields in a homogeneous and isotropic medium are derived from it, in frequency domain, as fellows: E = -jprxI_l,. (2.6) I? = Hfi,+V(v-fih)=vxvxfi,, and Vfipkzfifo (2-7) In studying guided waves along an uniform closed-pipe system, it is common to classify the waves into two basic types as follows [22]: 1. Transverse electric modes (TE modes) These modes contain magnetic field component but no electric field component in the direction of propagation. They are also referred as H modes and the field components may be derived from a magnetic type Hertzian potential having a single component along the axis of the guide. 2. Transverse magnetic modes (TM modes) These modes contain electric field component but no magnetic field component in the direction of propagation. They are also referred as E modes and the field components may be derived from an electric type Hertzian potential having a single component along the axis of the guide. Transverse electric modes ( E2 = 0, Hz 1% O ) The TE modes can be derived from a magnetic Hertzian potential by letting fic= 0 and fit =2 11‘. The electric field then has transverse components only and can be expressed as follows: E = -jm “VXflh (2.8) =jw pixvtII,‘ where . 6 Vt = v-ZE (2.9) Separating variables in II”, we write IIh=r|rhg(z), then the EM fields can be expressed in terms of the wave function wk as 2 s Hz=kcrvke 1“ Ht: xI‘V‘tyhc‘P‘ (2°10) E,= ¥Zh(ixfi,) with 2 2 V: ‘l'r+kc‘l’h=o (2.11) k:=I‘c+k2 where 2,51%; is the TE mode wave impedance. Transverse magnetic modes ( Hz a: 0, Hz = 0 ) The TM modes can be derived from an electric Hertzian potential by letting fih= 0 and 11:211.. The magnetic field then has transverse components only and can be expressed as follows: fi=jcoerfi. (2.12) = -jtoe£xV,II‘ where . a ‘ (7,4131. (2.13) Similarly separating variables in II‘, we write H‘=r|t.g(z), the EM fields can be expressed in terms of the wave function it, as lifting?”z E" = awry,“ (2'14) 11,: :Yfixfi) with Vfrp, + left, =0 (2.15) kz=F2+k1 where Y.=% is the TM mode wave admittance. 8 2.2 Orthogonality properties of waveguide modes Guided-wave modes inside perfectly conducting waveguides possess the following interesting and important mathematical orthogonality properties: 1. Longitudinal field components and/or wave functions of different modes are orthogonal. 2. Transverse field components of different modes are orthogonal. Mathematical statements of the orthogonality properties are listed here for later developments. Letting w", III” be the solutions for the n“ and In" E mode or H mode with n em, then the orthogonality properties are ffcs. “and! =0 (2.16) flaw. V.r.ds =0 or H H ds =0 f c, .. .. (2.17) [meeww and [Imam =11“, finds =o for TB modes or TM modes (2.18) [1:35-2:51de =ffcs.fitn.fitndg =0 ...foralln,m;mixedTEandTMmodes These orthogonality properties are exceedingly useful because with these a given arbitrary field can be expanded into a series of E and H modes in the analysis of waveguide excitation and discontinuity problems. 2.3 Reflection and transmission at a discontinuity interface for TM modes In this section the complex amplitudes of the reflected and transmitted waves are studied when a TM wave is incident upon a plane interface surface between two different media. Also the wave matrix is derived for waves incident upon an interface from the medium at either side to handle a general reflection and transmission problem. A discontinuity interface between two different media is considered first as shown in fig. 2.1. When a TM wave is incident on the interface, it is partially reflected and partially transmitted. The notations Erin, and Er stand for the electric fields of the incident wave, reflected wave and transmitted wave, respectively, as follows: 4‘1: E,= 13,011,112) +£e,,(u,.u2) 1e * '- 2.19 ER: [3301145) +i€zk(p,l,p2)] e I“ ( ) .. .. -r if [ermpw +ze,7(n1.u,)l e " where [1.1 and 112 are the transverse coordinates. And the associated magnetic fields are ~ -r _. -r 1?,= 11,011,112): “= 111,011.15” “ I'1‘ (2.20) " +1}: .. urinals,» .. -r .. .. -r fi1=hr(|11,|12)¢ 21:Y2'¢1(Fp”2)e It Yr°€x(1‘rr“2) c with Kn. are! A A A A A it: T (112111“11132)’Y.1(F2111'M1 '12) 1 (2.21) H. “’32 I2 yz= (azaj-fi1a1)2Y.2(|12111-111fi2) where '11, '12 are the unit vectors of the transverse coordinates. 10 51 ET —W——> _—4vvvv~——> W— ER 81’ “1 z = d 52' 112 medium 1 medium 2 Figure2.1 A discontinuity interface between two different media. 11 The boundary conditions for tangential E and H fields at the interface are £-E’(z=d-) =£-E(z=d:) i-fi(z=d-) Jae-41:) Application of the boundary conditions leads to .9 -PI 4 +" nu etc eke _ —o -rzd T. ‘Fd Pd -0 -I‘d Yl-(ZIe ‘-é’le‘)=Y2°ere ’ which can be simplified to -o 'r'd +" rye e, e eke _ -o -rzd YueIe —Y.1eke -YA2e,e (2.22) (2.23) (2.24) Since the boundary conditions on tangential E and H fields must be satisfied at every point of the interface, then clearly ER and 3,. must have the same transverse functional dependence as 5, such that we may define é'1 = Cl €o( "1’ 1‘2) in = b1 50(111, "2) 31' = ‘2 a“ “v “2) Substituting (2.25) into (2.24) gives Cr e434 + b1 er‘d = 62 e-l‘zd YAl cle'r‘d - YAlbler14= Yaczc’r’d The following operation, (2.26) *Ycz - (2.27), leads to (Ycz 'Ycr)cr‘-r14+(yc2 +Ycrwr‘r'd: 0 which is arranged as 12 (2.25) (2.26) (2.27) (228) 1: Yu‘Ya e—zrrdcl (2.29) Yul-”’32 Defining R to be the ratio of the reflected wave amplitude to the incident wave amplitude, we have bl = Ycl-Ycz £4134 (230) Similarly the following operation, (2.26)*YA1+(2.27), leads to 2 Yul cl e-rld = (Yet + ya) 62 e434 (2.31) or TE 3 = 2”» ere-rod (2.32) CI Yul +Y12 where T is the ratio of the transmitted wave amplitude to the incident wave amplitude. In case d=0, then R and T become R = M 2 R1 Yc1+Y02 (2.33) T= 2y“ 2 T Y31+Y¢2 12 such that the interface reflection coefficient R1 and transmission coefficient T12 have the same forms as those for a transmission line discontinuity as shown in fig. 2.2. The general reflection and transmission problem must include waves incident upon an interface from either side, which is studied next. Wave matrices The wave matrix method is usually employed to handle a general reflection and transmission problem in a systematic way. 13 Yel Ye2 at line 1 line2 # z = 0 R __ Yel-YeZ Ye] " r"??— , T = 2— el e2 Y¢1+Y¢2 Figure2.2 The reflection and transmission coefficient of a transmission line discontinuity, where Ye, and Ye; are the characteristic admittance of each line, respectively. 14 With reference to fig. 2.3, c,- denotes the amplitudes of the waves travelling to the right (i=l,2) and b,- denotes the amplitudes of the waves travelling to the left in either medium. When a TM wave of amplitude c, propagates from the left side of the interface, it is partly reflected and partly transmitted, which is also true for a TM wave of amplitude b2 propagating from the right side of the interface. Therefore in the left side, there is a total reflected wave propagating in negative 2 direction. And in the right side, there is a total transmitted wave propagating in the positive 2 direction. Letting b, and c2 be the amplitudes of the latter two waves, respectively, we can write r d .. -r as r d .. bre I ‘owv 1‘2) = R1 ”1‘ I ‘o(|"vl‘2)+ T21 b2e 2 9.211%”) (2 34) _ a - 4.. cze 134:0“qu = szzer’ 3,011. 112%: T12 etc I" €415,112) or bi = R1 {213461 + T21 A-(r,-r,)d b2 (235) c2 = R2 an"I b2 + T12 6:01-13) 4 cl A (2°36) where R1 = Ycl -Y32 _ R 2 Y¢1+Yc2 (2.37) _ 2r,1 = 2132 12 — r 21 —— Y11+Y02 Y02+Ycl Equation (2.36) can be arranged to be (:1 .-..- i e(I‘.-I‘,)d c2 - fl eu" +1.1)d b2 (2.38) T12 T12 Substituting (2.38) into (2.35) and after some manipulation, we have 15 -I‘,z . -r d; -r d; cle o cle ' e0 lecle ' ea w w w -F d, —r,d, > c e 2 e Rlcle e0 2 0 , Fir... . J ble e0 R b 1‘24A ‘ "I” blerrdéot 2 2e e0 \ w W I‘d. I'd, T21b2e 2 e0 bze ’ e0 81'1'11 £2,112 medium 1 z = d medium2 4‘22 3 C 22 Figure2.3 A discontinuity interface between two diflerent media, where there are TM waves of amplitudes c1 , b2 coming from the medium of either side, respectively. 16 b1: 51 e—(r‘| +1:sz + Trszr ’R1R2 {try-1'94 b (2.39) 2 2 T12 T12 Since using (2.37) it can be shown that leT21 -R1R2=1, then (2.39) becomes _1_ A-(r, 43).! b b. = fl e-fl‘: “‘2” C: + 2 (2.40) T12 12 Since Rl=-R2, (2.38) and (2.40) can be written in a matrix form as Cl ea‘r -l‘3)d R1603 +1.2” CZ A11 A12 CZ (2.41) l I b1 T12 R1 C --(I‘l 43):! e -(I‘, -I‘1)d b2 A21 A22 b2 where [A] is called a wave transmission chain matrix. With this wave matrix, we can relate the amplitudes of the forward- and backward- propagating waves in the output side to that in the input side such that for a cascade connection of N+1 sections as shown in fig. 2.4 we have ‘ . «Wan, -(I‘r1‘m>a ’ b1 ’ 1 720*” R1‘ e bN+l 421 A22 bmr Letting R” be the overall ratio of backward-propagating wave amplitude hi to the forward-propagating wave amplitude C}, then Ru can be derived as _ b1 _ A216N+1+A22bN+1 (2.43) bl ‘ - ' Cr AllcN+l+A12bN+1 There is an alternative expression for Ru that employs a recursive relationship of Ru as follows [23]: Since R1 = —R2, dividing (2.40) by (2.38) leads to 17 .2822: o>a3 SE. .23 2.388 :2 .3 530258 0388 < in 2:?"— i size Q m :2 ~ .— ‘It’lll e ~+Z NNTz I» O _+z 8:62: :2: 2+2“. e .s §.U men—1 _ 8:62: s. N . 1 a . w G N Q —.U m3)» 9 mien» _ 83cc:— —1 .—U 18 -b-1- = R1 e’(rl §r2)d 02 + e'(r‘ -I‘2)db2 cl en‘! -I‘3)d (Fl ‘43)de (2.44) = e-Zl‘ld R1 e bzlcz e + R1 e b2 / c2 c2+R1e 1‘24 134 +9 which can be written as _ e + e e la R,2 Therefore for a cascade connection of N+1 sections, we have the following recursive relationship (i=1,...,N): 'PM‘I 134‘: x», =6“?! R1‘ + ‘ KW“) . (2.46) -r, r e 1 ‘9+Rje ”WK, Utl) which offers another approach for computing Ru- Finally let’s verify (2.45) by an example to conclude this section. With reference to fig. 2.5, an open-ended coaxial probe is placed against a dielectric material layer with thickness d, which is investigated in chapter 3 in detail. For each TM mode inside the material and the air, we have V (2.47) and (2.45) reduces to -2111de = e-Zl‘ld Ya] - Yez (2.48) + YcZ which is exactly the same as (2.30). 19 medium 1 medium 2 coaxial line \ z=0‘ z=d Figure 2.5 An open-ended coaxial probe terminated on a metallic flange placed against a material medium . A TM wave is excited in- side the material. 20 CHAPTER 3 OPEN-ENDED COAXIAL PROBE TO MEASURE THE PERMITTIVIT Y AND PERMEABILITY OF MATERIAL 3.1 Introduction In this chapter the analysis of an open-ended coaxial probe system for the measurement of electromagnetic (EM) properties of materials is presented [10,23,24-26]. The geometry of this problem is an open-ended coaxial probe terminated on a large metallic flange placed against a stratified material medium as shown in fig. 3.1, by which a nondestructive measurement of EM parameters can be achieved. The goal is to relate the permittivity e and permeability p of the material to the input impedance of the coaxial probe, thus, by measuring the latter the former, e and p, can be inversely determined. Theoretically, this probe is analyzed as follows. An incident TEM mode to the probe is partially reflected by the discontinuity at the aperture and it also excites EM fields in the .stratified material medium. Additionally, higher order coaxial waveguide modes are excited near the probe aperture. The EM fields in the coaxial line part and in the material medium can be expressed in terms of modal functions, and the matching of the tangential electric and magnetic fields at the probe aperture will result in an integral equation for the, unknown aperture electric field. i 21 coaxial line d1 0 2: line terminated on a large metallic flange placed against a stratified isotropic Geometry of an open-ended coaxi medium. Figure 3.1 22 The electric field integral equation (EFIE) is investigated and it is found that certain singularities may occur in the integrand, which requires a careful analysis before a numerical scheme can be applied to solve this integral equation. A careful theoretical analysis for the structure of fig. 3.1 leads to the prediction of the excitation of surface waves and radiative waves for the open-circuit case, in which the material is backed by a dielectric (layer N is dielectric, usually air), and radial guided waves for the short-circuit case, in which the material is backed by a metallic plate (layer N is metal). The finding of these complex waves comes from the singularities and/or branch cuts of the integrand in the EFIE and can be identified via deformed contour integration in the complex plane. To determine the reflection coefficient of TEM mode at the probe aperture, the integral equation for the unknown aperture electric field is solved by the method of moments. After the aperture electric field is obtained, the reflection coefficient of the TEM mode or the input impedance of the probe can be determined in terms of e and p of the material medium. Conversely, if the input impedance of the probe is experimentally measured with the help of a network analyzer, e and p of the material medium can be inversely determined. Conceptually, for a simple medium two measurements are required to determine two unknowns e and p at each frequency of interest. Therefore by measuring the open-circuit input impedance and the short-circuit input impedance of the probe, the complex a and u of the medium can be uniquely determined. In section 3.2 the results of general guided wave theory are applied to solve the EM fields at both sides of the aperture, and an EFIE is derived by matching the boundary conditions across the aperture. Meanwhile the reflection coefficient of the incident TEM 23 wave is expressed in terms of the aperture electric field. The identification of complex waves in the structure is also discussed in this section via the Hankel transform representation of EM fields and a deformed complex contour integration. In section 3.3 the finding of complex waves excitation due to the singularities in integrand of the EFIE is verified using the concept of equivalent magnetic surface current for the aperture field. This offers an independent treatment other than the analysis of section 3.2. Not surprisingly, these results turn out to agree to each other in both sections. In section 3.4 the method of moments is applied to solve the EFIE. The TM eigenmodes of the coaxial line are chosen to be the basis functions for unknown aperture field and Galerkin’s technique is used to convert the integral equation into a set of simultaneous algebraic equations, from which the numerical solution of aperture electric field is obtained and the input admittance at the aperture is resolved. Finally some checkings for the correct numerical implementation are also given in this section. 3.2 Theoretical Study Using Full Wave Analysis 3.2.1 Application of General Waveguide Theory to a Coaxial Structure In this section the results of guided wave theory in chapter 2 are applied to a coaxial waveguide structure with a geometry as shown in fig. 3.1 [27]. Due to rotational symmetry, only TM modes are excited when a TEM mode is incident to the system. With the help of (2.15) for TM modes, the wave function \er satisfies a two-dimensional Helmholtz equation, Viv.+kfv.=0 ‘3'” subject to the boundary conditions of 24 rlv.(p =a)=0 (3.2) 11.0) =b)=0 where k: = k2+I‘2 and V3 is the transverse part of the V2 operator. In cylindrical coordinate V3 stands for l—a—(pi +1.1 such that (3.1) p 69 59 p2 at becomes 1 a _(p av.) Lazy. +kfv.=0 (3.3) p 69 69 p2 air: and by the separation of variables method the solution of 1|: is J .(k 0) =A J k —-—"——‘fl'l—Y k e’” (3.4) 11'... .1 .( cup p-) Y “a...“ a.) .( “PH where kc” satisfies JA(kMa)Yn(k”b)-JA(k"b)Yn(k”a)=0 (35) With the help of (2.14) the field components of TM modes can be expressed as E = 2 1'! er”: END. = *1‘3MV‘wCflfle‘ruz ' . k . l (3.6) ‘ k ,,.iJ.(k ,..p)- 7%,3—"21’. (k.,..p)1 = $12.24.; wan-z J (Irena) ‘r-‘L$AIJ,.1>)(kc,,..-Y—(A—7)Y,(k¢..p)l II,” = :YA(£ xi) r Jk , * “my ;.k< ...p>- §%2%Y.91"¢¢*r”'z —E%[Jn(kcnnp) 2611:3131 (ken-19 )] 25 These field components are used to represent the fields inside coaxial line in the next section. 3.2.2 Integral Equation for the Aperture Electric Field ' An integral equation for the aperture electric field is derived in this section by matching the tangential EM fields across the aperture. Because of the discontinuity at the aperture, higher order modes are excited near the aperture. The EM fields in both sides can then be expressed'as a sum and/or an integration of these higher order modes. For the case of TEM mode excitation, because the fields inside the coaxial line are q)- independent and the stratified medium is rotationally symmetric (which is a quite good approximation in practice), the EM fields scattered due to the discontinuity at z=0 also exhibit the rotational symmetry. Consequently, only TM modes can be excited, and TE modes can be shown to be zero. Fields inside the coaxial line Let’s first consider the EM fields in region 230 (inside the coaxial line), where there exists incident and reflected TEM mode and backward-propagating higher order TM modes. Letting R be the reflection coefficient of the TEM mode at z=0, the field components of TEM mode inside coaxial line can be expressed as follows [27]. if 5A0%(e”"" +Refl‘“) (3.8) if =$Ao—l—(e.jk‘z-Refl“) 1] t I where A0 is the incident TEM mode amplitude. 26 For higher order TM modes, only those modes with (tr-independent property (n=0) are excited as mentioned. By (3.4), (3.6) and (3.7) the TM mode field components are E;0n = €20.¢C0.e +r.-z A”. [10(kcoup) You“. ) Yo“: a 1119)] c1 e 13" ‘ I‘ 311...!” = «1- tour 0 For simplicity, omitting the index 0 in the above equations leads to E;' = :n‘p‘. eTr-z J (k a) +1: , =AA[J (kmp)-_°_fl'_.Y (kelp)]¢ " (3.10) 0 Y0(kcna) 0 5.1. = +1"..V.¢..e"" or more explicitly, _._ A . J (Irma) . , E... = +I‘..A..pk..uo(k..p>- ° mam. Y0(kcna) . (3.11) _._ , J (k a) I +r H=- ka -°"" Yk -‘ .. mad» “.1 o( ,..p) Yo(k,..a) o( ...p)]e Because the eigenvalue kc, satisfying (3.5), is discrete, the total transverse EM fields in region 250 are the sum of the TEM mode and all possible TM modes as follows: if MOM!” +R en") + p E A.81~(p)er" " "" (3.12) 1 - " I'M P7.=M,A—A(¢ lat-Rafi") "O 2 -P—‘Anst.(p)eP-‘ t ur-Om where 27 Yo(kma) ° "'9 (3.13) E can [11(kcup) Y0(kcara) —J0(k¢ll 0) Y1 (kc 111 P) 1 Thus the fields at the left hand side of the aperture are obtained by setting z=0' a..(p)=r,.k,.113(k,.p)- in (3.12) E.r(z=0') =fiAo-l—(l +R )+ t: 2311,31”) (3.14) P and ”APO? =$A.—1—(1 -R) -¢i> 515434;.) (3.15) mp 111-1 I". where 11,, e‘ are those EM parameters of the material filled in the coaxial line. Fields inside the material Next let’s consider the EM fields in region 220 (inside the material). As mentioned before the field components exhibit rotational symmetry and only TM modes can be excited inside the material, thus the Helmholtz equation for wave function ‘1’. becomes Vtz ‘1'. ”5311'. =0 (3.16) subject to 1|rA(p =0) being finite. The solution of it. can be easily shown to be 1|7A=DJn(kcp) with Yn(kcp) discarded (3.17) where kc =‘/I'° +ch2 is the continuous eigenvalue and kl is the wavenumber inside the j ”' layer of the material. Again only the rotationally symmetric term (n=0) gives the right solution, therefore with the help of (3.6), (3.7) and (3.17) the field components in the j“ layer of material are 28 E: =D,k:ro(kcp)e"1‘ if: = tI‘ijkcpJgkcpk‘r"! «.1: pa, 11(kcp)e ‘9‘ (3.18) 2 . 151‘ it, =¢YAIBlJl(kcp)¢ 1‘ m, . . where Yo): —IA—- rs the wave admrttance. I Since the eigenvalue kc is continuous in this case, the total fields should be expressed as an integration (instead of summation) of all the possible solutions. Thus the total transverse field inside the 1“ layer of the material (including if direction) is E, = a [0" are "ha” 31‘) 11(kcp)kcdkc (3.19) 1?, =6]; 13,11j (e "1‘41” 23‘) J1(kcp)kcdkc The quantity Rb! is the amplitude ratio of backward wave to forward wave of a TM mode at the interface 1 (between layer 1 and j+1) and it is shown in chapter 2 that there exists a recursive relationship (2.46) between Ru and R.“ such that Ru of layerj can be computed from that of neighboring layers. It can also be checked that the longitudinal component of electric field is a B - ‘ E. 31F] (e ”PR” e”")J.,(Ic,p)k§arrc (3.20) 0 1 The transverse field components at the right hand side of the aperture (inside layer 1) are obtained by setting z=0’ in (3.19): E,(z =0’) = a [0'31 (1 +11, l)Jl(kcp)k€dkc (3.21) 1?,(2 =0') = .5 [07.131 (1 -R,, )11 (kcp)kcdkc 29 where R“ of layer 1 can be computed from those of neighboring layers. Matching the tangential E fields and H fields at z=0, an integral equation can be derived for the unknown aperture E field. Equating (3.14) and (3.21) over the aperture of aspsb, we have 4%“ +R )+A':31A..8..(p) =i310 +R“)J,(kcp)kcdkc (3.22) and 40-53%(1 -R)-§lrm.1,sr,(p) in, 3,0 -R“)Jl(kcp)kcdkc (3.23) where Ym=£o—e-'- and Yul =j:el. n 1 Equations (3.22) and (3.23) can be cast into an integral equation for the aperture electric field via orthogonality properties. Let 8’(p) be the E field at z=0, i.e. Ep(z=0), then (3.22) becomes Ao'fii 1 +R ) + 21.4.84» = 8’(p) = [B1 (1 +Rbl)Jl(kcp)kcdkc (3.24) m- 0 Employing the orthogonality properties and an identity for the Bessel functions , (3.24) can be solved for the unknown amplitudes 240,24” and BI: Using . J]: k' d =ak-k' k { ,( ,9)“ cm 9 (a J/ c (3.25) b f8.(p)8.(p)pdp = O... L we have 30 A“ (1 +10 limb/(1)1119”p A... = [819)8..(p)pdp 8, (k.)- [81ml (k p)pdp (1%). Also from A, we have 1+R= b b from A.1n(-) «- a b 1 [81049 A0 ln('a’) a 1-R=2- (3.26) (3.27) Substituting these amplitude coefficients into (3.23), an integral equation for 81p) is derived as follows: [4,11 2- b from -2Ym R .(p)f8’(p)fl,. (9)de TI‘P Aoln(—)¢ 3'1 =IYR’ (“2’ [819)! (k 9)de th p)kdk cl (1+Rb:)¢ l c 1 c c c 01' A. ll =-l—l— bf8’(p)dp + .2 Y", 9! ,(p)f8’(p'.)8 (9)de 01119 “(pm(_ a)“ " (141,1) ” . . . + [Yfl— [3(p)J,(k.p)pdp Jl(kcp)k.dk. 0 (1+Rbl) a 31 (3.28) (3.29) Before solving (3.29) for the aperture electric field numerically, it is found that there exists singularities in the integrand of the integral w.r.t. kc. The understanding of the physical meaning for these singularities requires further study which is presented in subsequent sections. 3.2.3 Identification of Complex Waves The excitation of surface waves, radiative waves and radial guided waves in the open-ended probe structure is investigated in this section [28]. It is found that there exists singularities in the integrand with respect to k.- of the integral equation (3.29), which correspond to the excitation of surface waves for the open-circuit case and radial guided waves for the short-circuit case. The identification of these complex waves is made by examining the EM field expressions via a deformed complex contour integral and is verified in section 3.3 by using the equivalent source concept for the aperture electric field. 7 With the help of (3.19) and (3.26) the EM fields in layer 1 of the material are derived as .. l (e433 or“) b . . . . E9 =f< 1 R“ [81p )Ji(kcp )pdp Jl(kcp)kcdkc ° ) ( +12“) ° (3.30) - l (e'r‘z-R e132) 1: . ' ' ' H.=f* Y., ’1 [810)11099 )de J,(k.p)k.dk. O . (1+RM) a which can be viewed as the EM field representation via the use of inverse Hankel transform. 32 By recognizing that the expressions inside the braces of (3.30) are odd functions of Ice. an alternative field representation of EM fields in layer 1 via Hankel functionHlm is derived using the following formulas: J ( )_ Hi"(p> +11%) 1 " ' 2 (331) 11919) =Hf’t-p) and for an arbitrary odd function fat) we have ff(kc)Jl(kcp)kcdkc 0 = % fie.) [Hf’t- ,p>+Hf"(k.p)1k,dk. 0 = % ff(-u)Hl2’(up)(-e)(-du) . (332) 0 I .. +-2- {ftkglrf’rkrwe ] é- f(k.)H?’(k.p)k.dk. i Combining (3.31) and (3.32) gives the field representation in layer 1 as follows: Hi”(k.e>k.dk. -1-(e-P,1+Rmel‘,z) b A A A e, "2] (1+2!) [semen >de - -I‘,z_ 1‘1 ”6:1 Y1“ Rare t) 2__ ‘ (1+R“) (3.33) b fflp’flflgp' )p'dp' H?’(k.9)k.dk. With (3.33) it can be verified that the roots of 1+1!M =0 correspond to the surface wave pole singularities for the open-circuit case. This is demonstrated by converting the 33 real-axis integral to a deformed contour integral with a proper choice of branch cuts as shown in fig. 3.2 [22]. By Cauchy theorem the electric field can be deduced to -1 (“Walther“) . , . ) . 2ch (“R“) [81931th )pep H?(k.p)k.dk. (3.34) (e -I‘,z+Rbl em) 6 . . . ) 5 c. (1+R“) [ [ 819311059 )949 H? (k,9)k.dk. where C, stands for the contours around the surface wave poles and Cb for the contour along the branch cut. Also the roots of 1+RM =0 correspond to the radial guided wave pole singularities for the short-circuit case, which is depicted in fig. 3.3. For this case the electric field can be deduced to (e -l‘,z+R“e I‘ll) ':ZC (1+R“) H?’(k.p>k.dk. (335’ f 819311059 )9'49' where C, stand for the contours around the radial guided wave poles. Similar expressions can be derived for magnetic field also. In addition, via the formula of Hankel function Illa) the integral equation (3.29) can be rearranged as follows: A, ll =-l—l—f8’(9)d9 +31 Y“. 8 (9) f 8‘19”! (9)949 mp them! a)“ (3.36) (11 +_[Y(1cl—— Rb) H?)(kcp)kcdkc $.11) [3(9' )Jl(k.9' )9'49' The integral equations (3.29) and (3.36) are in the appropriate form for further development of the numerical solution of aperture electric field 81p), using the real axis integration for (3.29) and the integration along branch cuts for (3.36). 34 ci kc -plane fi fl» C? q t C CW ® ® 9 Cb C” Figure 3.2 Deformed contour integral in complex kc- plane for open-circuit case. 35 Figure 3.3 Deformed contour integral in complex Ice-plane for short- circuit case. 36 3.3 Analysis via Equivalent Source Treatment of Aperture Electric Field In this section the connection between the surface waves/radial guided waves and the singularities in the integrand of the electrical field integral equation (EFIE) for the open-circuit/short-circuit case is studied. The analysis in this section gives an alternative viewpoint and offers another approach to the problem other than the analysis of section 3.2, by which the physical phenomena are revealed clearly. For the excitation problem of TM modes along a planar dielectric slab waveguide by an annular magnetic current distribution as shown in fig. 3.4, it can be shown that the results do agree with that of the previous section if the aperture electric field is replaced by an equivalent magnetic surface current. For illustrative purpose an open-circuit case with a 2-layer material is considered in which the excitation of surface waves is well known [21,22]. The analysis of TM mode excitation by a two dimensional surface current source is presented. And by replacing the aperture electric field by an equivalent magnetic surface current, the excited EMfields and the integral equation for the aperture electric field are derived, which are exactly the same as that derived by using a full wave analysis in the previous section. Finally the equation leading to singular points of the integrand in EFIE (3.29) is converted into the exact eigenvalue equation of the surface wave modes. 3.3.1 Wave Equations for EM Field Maintained by Magnetic Current Distribution In fig. 3.4 a 3-D magnetic current distribution, .7”, situated over the plane interface between the material and the metallic ground plate is considered. in is assumed to be rotationally symmetric, and the notation if stands for the primary wave maintained by]; and if, E; for scattered waves due to contrast between different regions. Only TM 37 A 2 region (2) . £2 EV \QP :fip‘l‘fils jm=$Jm¢(P,Z) region (1), 81 F31;\ gVélp 2:0 -——>i) magnetic current distribu dddd modes with field components El, EA and H. are studied. The wave equations for EM field with magnetic source terms are formulated first in this section, then the solution of the wave equation for magnetic field is derived in next section and the electric field can be deduced from the magnetic field via Ampere’s law. . Firstly, let’s start with Maxwell’s equations for an electric source free region, where j=p=oA ' v-E =0 VxE = J. "jmpfi * vxii=jaer V. = £5 1 11 Taking curl operation on the equation of Ampere’s Law leads to VxNxI?) =jtoerE vw-fi) -v2ii =jdevxi V(%)-VZII=-jwe.7n+kzfi thus the wave equation for I? is V217 +13}? = V(&'-) +jmejm 11 Similarly taking curl operation on the equation of Faraday’s Law gives vx(vxE) = 47x], -jw .1va WW?) 4722‘ = -vxj,, +1121? thus the wave equation for E is vzi+kZE=VxIA or 39 (3.37) (3.38) (3.39) (3.40) (3.41) V21? +131? = WE!) flee]; P VZE+kZE = ijuI To solve the wave equation (3.39) for 11., let’s first recall the formula of Laplacian operator for a vector field in cylindrical coordinates: v2? = 9 (v27),+$(vzr),+2 = [17.0.4141 mm k 0 and the following identity for the Bessel function holds p'fJ,(Ap)I,(Ap'>MA=o(p—p') (3.51) 0 Substituting H;.(p,z)=ffi;,o,z).r,(tp)wt into (3.49) leads to 0 . a1 l a l -.t 32 2 ~: _ 3.52 “($4 36—9-7)Jl(lp)fl..(l.z)+(gz- +k.)H..(l.z)Jl(lp)}M1 -0 ( ) which can be simplified via the definition of 110.9) to £ {- mama Lona-$4911: .(uw (19)}141 =0 (353,) 01' . f {(_ +ki -Az)1?’,(t,z) }J (Apnea =0 (3’54) 0 Equation (3.54) gives (i -I‘i)1?:.=o (355) 322 where 4342—12. 42 The solutions of (3.55) are 330.2) = W:(A)e‘r" (3.56) With the help of (3.50) and (3.56) the scattered magnetic field in space domain is derived via the inverse Hankel u'ansform, : . $1}: H..(p.z) =fW:(A>e 1,0»di o and the electric field can be deduced via Ampere’s law as S 1 a l E =- —H “P j0€¢az ‘3. l jwe [Hagar-041mm; I 0 = efwfiotfl. e‘r-‘Jlo. (3)1611 0 where Z. =I‘¢l (jme‘) is the wave impedance. The scattered EM fields are then 15;p = if W:(A)Z. e‘r-‘Jlopndt 0 H;,= f nge‘r“ 1(1)»di 0 (3.57) (3.58) which are written out explicitly in different regions for further development as follows: Outgoing transmitted fields in region 2 are ‘ g; = +[W2‘(A)Zz (”11(2) ,3)di 0 11;, = fW;(A)e"'=‘J,(A (3)di 0 43 r ...... +2 direction (3.60) Outgoing and incoming scattered fields in region 1 are . l 5,“, = +fvv;(zt)zl (”13,01 mm 0 _ > ...... +2 direction 1H,}: fw;(t)e"1‘J,(Ap)MA ' ° ' 1 (3.61) 5;, = - f W,‘(A) z1 e*’*‘J,(Ap)AdA ° ) ...... -2 direction +I‘l H;.= [may alumna 0 Primary wave due to original magnetic current distribution The Green’s function technique is used to solve for the solution of the primary wave. The Green’s function for the wave equation due to a ring source is obtained first by the application of Hankel transform and Fourier transform [10,23]. Then the primary magnetic field is deduced from the Green’s function by the principle of superposition. Wave equation (3.48) is rewritten as follows, in region 1, 62 p 11 6 p 2 l p 62 p . _ __ __ _ = 3.62 aszu+p apH“+(kl p2)HM+&2HI¢ jute!” ( ) The Green’s function for an unbounded medium 1, due to a (b-independent ring source at p='p', z=z', is defined by 1.11 .kf-i.i (“Mg“): _ 50’ "P ”52-1) (3.63) 21tp The Hankel transform representation for Green’s function and the identity for the Bessel function are r G(p.p'.z) = [5(1.p'.z)11(lp)ldl ( _ 0 (3.64) p’ [1.0 pmopbm =a(p -p') L 0 Substituting (3.64) into (3.63), we have I {(62 _ +-'§" ’—)J (1000.9 .2) “ii- “‘kbalm'zfl 10' 13)}2'dA o 92 322 3.; a“ p (3.65) _ _ 1 ' . _ — Zt' {Jlapwlapndi 6(z 2’) which can be simplified via the definition of 11(Ap) to o 2 ~ ' 62 ~ ' f{'A 11(AP)G(Atp tz) +(—'2' +k11)G(}.,p :Z)J1(Ap) o a! (3.66) + 51:6(z -z’)Jl(}.p)Jl(l p') }Adt\ =0 01' f {(_ +1‘12 12)G(A.P .2) + —5(z- -z’)Jl(Ap) } 110. pud}. =0 (3.67) o This leads to a: - . __ l .. ($456014: ,z)- 271,09) 6(2 2’) (3.68) =-caa-Z) where -r§=kf-A2 and c=-21—J,(tp'). 1! Equation (3.68) for G(A,p',z) can be solved by exploiting the Fourier transform 45 Frtéi=[G'(A.p'.z)e"“‘dz "' (3.69) .. , =_1_' .. z G(A.p.z) ZnLFflGlehdn' and the Cauchy residue theorem of complex analysis. Skipping the details, the solution of (3.68) can be written as C '1‘! It ‘Z'l " 1, ' =_L__ (3.70) G( p.z) 21,1 Therefore the Green’s function maintained by a ring source is , .C 'rrll‘l'l G(p.p .z.z) =f-17P—J,(xp)m ° ' (3.71) . 1 {ml-z" JA'J}. MA 02“ 1 l((3)1(9) And the primary magnetic field maintained by J” is derived from the Green’s function by the principle of superposition as r111 1| 115.: f f Lids/(1196.) [1 E—J,(Ap')J,(rp)1dr (3.72) - "r111 '31 ffdpdz(-jmelJ")fe—2P——p1(Ap'1)J(Ap)MA Also the electric field is deduced via Ampere’s law as 1‘, lz-zl f I do "dz( my...) I (:21) —p'J up)! mom (”3) where :21 in 5‘}; stands for 46 +Zl for primary wave in +2 direction, z>z' (3.74) -Z1 for primary wave in —2 direction, zm P ' I'1 it"! I ' E... = I an (-jweK ,) f (+21) p'J,op)J,m a 2r! The summary of scattered fields and primary fields is given below for further development. Outgoing transmitted fields in region 2 are . l 5;, = + f W;(1)z2 (”"110 pm; 0 i ...... +2 direction (3°76) H5, = [W;(r)e"1‘1,(r p)).dl 0 Outgoing and incoming scattered fields in region 1 are ‘ Eip” +fM(A)Z.e""Jt(lp>m (3 77) o . i ...... +2 direction Hf, = fw;(r)e"".l,(rp)rdr 0 47 5;, = — f Wfo.) z1 e’r“Jl(Ap)AdA 0 ) ...... -2 direction (3°73) Hf, = [ii/{(1)5010 until. 0 1 Primary fields in region 1 are, letting z"-z', I‘llz-z' Hfa= -fdp( vine, K-¢)feT I191! (1931 (Mm-411 (3.79) I‘ll-2| E’.=fdp( -jwe K...) f (+21) Mamas)“; 21‘1 Application of Boundary Conditions to Determine Wz', W; and W1" The continuity of tangential EM field is applied to determine unknown amplitudes, Wz', W,‘ and W;, of the scattered fields due to the contrast from different regions. The total fields in regions 1 and 2 are ”26 = ”2‘9 529 =5th _ r + - Hit-”10+HIO+HIO (3.80) E19 =Efp +51", +Efp and the boundary conditions are B.C. at z=0 E19 =0 B.C. at z=d E29 =51, & Had!“ Since there are three unknowns and three boundary conditions, an unique solution is guaranteed. With the help of (3.76) to (3.79), the application of BC. at z=0 leads to 48 fdp'c 7'06, .,> f ( 49%— (>40le (mm +fwf(l)z,Jl(rp)ldr - fW,’(r)z,J,(Ap)MA (3'81) 0 o , " e-l‘1z' . = f - .>1f( due. K32—-..,) —p’J(Ap")dp- th- Wiml J=oplmo o This implies that the integrand of (3.81) is zero, or -W;+Wl‘ = -V1(0) (3.82) where (Pl: . V(0)= f: ( due. W) —p'1(lp)dp (3.83) With the help of (3.76) to (3.79), the application of B.C. at z=d for the electric field leads to e--I‘,(d 1') fw;z.e"""J (Ap>AdA= fdp'( 1'08, K...)f(2.)— p'J.(Ap')J.(Ap)AdA ., 1‘: (3.84) +fwf(r)zle""’J,(ap)rdr - fW;(l)Zler‘dJl(Ap)MA 0 0 or a b [W 2') (Z) (-jweK )— pJ (19m , l 1 I "'4’ (3.85) -W‘,z,e"1"+ w;(r)21e'r"’- W,‘(r)z,e*“" }Jl(}.p)M1 =0 and this gives 2 Wf— e-rld —W— 21 er‘d _WI -r,d_ -—1-Vl(d) (336) 22 1e22 49 where e-I‘lfl-z ) V,(d)=f:(-J'w€K ..l— —p'J.(Ap)dp’ I‘1 (3.87) With the help of (3.76) to (3.79), the application of B.C. at z=d for the magnetic field leads to . [PM '1) I W‘e 13"! (Aim-d1 [JIM-1'99...) f 9J.(lp')J.(Ap)I-dl Fl +fw;(l)c'P"Jl(l p)).d}. + [vi/{(1)3110 mm 0 0 01' e-I‘,(d -z) - b f f( 7'99. K...>— —p'J (Apldp 0 a "W'ze'r"+WI(l)e"‘"+ Wane” Mamet =0 and this leads to -W1’e -r‘d—W{er'd+W2'e "r”! = Vl(d) Equations (3.82), (3.86) and (3.90) can be used to solve for ”’1', W; . Adding (3.86) and (3.90) gives Z _ Z Z (-1+—‘>W;e “"4179872‘" 41-59mm 2 Z: -Z +8 _ Z2 -Ze - 22+er2+21¢ 01' -1212 we '"14- w; =R12e “v.60 50 (3.88) (3.89) (3.90) (3.91) (3.92) where Ru=(z,-z,)/(z,+zl) (3.93) Adding (3.82) and (3.92) results in -(1+R.2c'”‘l‘)W; = -V,(0) +Rne'r‘4Vl(d) (3-94) or V(0) «Hart‘s! (d) w; = 1 1 (3.95) 1 +Rue -zr,d From (3.86), we have .. R12 e “ZI‘IJV1(O) _R12 e ~I‘1‘yl(d) (396) l +R12e -2r,d W; = Wi'ylm) = Knowing WI, W1- , the EM field expressions in region 1 are now ready to be carried Ollt. Electromagnetic Fields in region I The EM field expressions in region 1 for a general 2-dimensional magnetic disk current 3' =$K“(p) are derived here. And the results are specialized to the case of an open-ended coaxial probe when the 2-dimensional magnetic current resides right on the metallic ground plane. The EM fields in region 1 are the sum of the primary wave and the scattered waves as follows. For the electric field, E19 =15?p +1.";" +15;" (3.97) where 51 'rrll‘il E".=fdp( ‘1031.¢) [(121)9J.(19')J,(19)MA 5:21] V 1(0)R :12“). eri‘J (A mat (3.98) 01+R:e'2r a; . _ f -Rue "’“thm Jae "“899 R1 _2r z1 cri‘Jlopndl 0 1+ 2e Substituting Vl(0) and Vl(d) in (3.98), the expressions for 5;; and EL, can be arranged in a form similar to El’ : (9.9+!) . Eip= -fd9( 1961 K'J‘Zl 1-2.1421 21.1 9J.(19)J,(19)Ad1 R" (3.99) are 'ere 1 e-I‘itz-z') . +fdpt-iwel K...)f RH... 21.1 91,091,1de and b '21‘14 erui‘) - . , e l , , . El, =fdp(-J°°31K~O)fl R12 12124134219 21‘ le(}.p)Jl(Ap)AdA ' (3.100) b el‘(z+z) . , 921‘ ‘ . +fdp(-jtoele¢)fl+ R12 “4:421: 21‘ le(}.p')Jl(Ap)l.dA Similarly the magnetic field in region 1 can be expressed as the sum of the primary wave and the scattered waves as follows H,,=Hf.+u;.+u;. (3.101) where 52 . e'rrll’zl ”f0: ~fdp( -j(n)el K'erzr 9J.(19')J.0-9)Ml =f V (0)-R ..e (”V (d) -r, o 1+Rue 2” - '-R '"I‘V 0 - IWV Hl.=f 12¢ 1() if: 1(d)e1pr'zJ(A )MA ‘1 (tp )tdl. (3.102) 1+R12e Substituting Vl(0) and Vl(d) in (3.102), the expressions for H1" and 111'. can be arranged in a form similar to Hf... + b ' o 1 e_r‘(z*z') ' Ht.=fdp(-ime.K..>f W 21. pJ.(Ap)J.(Ap>MA a 01+Ru¢ 1 (3.103) b ' -R 12‘ 21)" e-r,(t-z') ' +fdp(-1f ‘2 W 21. pJ.(Ap)J.(Ap)MA a o 1+Rue 1 (3.104) b o + fdp'(-ime.K..> f l anon/.0) own a 0 +R12¢-zrld 21"1 To utilize the above results for the EM fields of an open-ended coaxial probe i=0, z>0 is, assumed such that V.(0>= f’(-ioeK...;I.—.>19'J.(Ap)dp’ (3.105) 434 V(d)= f: Hagar—K...) —9'J(1-9)d9’ From (3.97) to (3.100), the electric fields can then be expressed as 53 b . "' c -l‘,z . Efp =fdp (-jweIKM)[Zia-PTpJIOp'fllQpMd}. a 0 + 9 ' '1_R12e'zrrd e-I‘Iz . 151,: fdp(-jne,lr”) f 1 z pJ,(tp)J,(Ap)td1 (3.106) a 0 + Rue-29.4 121‘l . b " 41'1" r - _ o . ”12‘ e l: l E", — [ dp (-Jwe1K") 10‘ 1+R12e.zr'4212rr pJIO. p')Jl(A p)AdA Letting R.M be Rue -2r,d’ the total transverse electric field then becomes - 9 + - E19 -E,P +5“, +15“, b - -p ' . 2 ’1 I a =fdp(-1tae,K”){ 1+szl‘zp‘ pJ,(tp)J,(rp)rdr a b o a 2R er‘z a +fdp(-jwe1K'.)£ 1 +13; z1 2r. pJ,(A p')J1().p)AdA (3.107) 0 b . -I‘z I‘z , jwe e ‘ +R ' =fdp(— P 121K...)f 1+Rble a 1 0 b1 97.09)].(19Md1 -I‘lz b 0 .e +Rb1erlz p l =fdp(-K...){ 1+3“ 9J1(19)I.(19)1d1 d Replacing the aperture electric field by an equivalent surface current, that is Elp(z=a’) = -K”, the electric field for an open-ended coaxial probe is derived as I‘ z I“: ‘ R e . “ pJ1(Ap')Jl(lp)AdA (3.108) 1+Ru . b _ _ 5,, =fdp'E,,(z=0*) f‘ a 0 I“: b [51,(z=0*)J,(19)9'd9' comm ‘ - ‘1‘]: _fe +Rwe 0 l +Rbl 54 Equation (3.108) is exactly the same as (3.30), which was derived by the previous approach, if the dummy variable A is changed to kc. Similarly from (3.101) to (3.104), letting 2' =0 the magnetic fields can be expressed as H".=fdp( 1’09. ..) f _’T"pyl(.p.,l(.p,.d. b I“ 213d e-I‘z 11.}: fem-jute.) f 41.14 ‘21, 9J.(19')J.(19)MA (3'10” 5 --_2R12¢ 2.9.4 er“ Hfs=fd9'(-wa.K...)f a“ 91.(19')J.(19)Kd1 .1+R.2e . 21‘ Since Ru is Rue 4134, the total transverse magnetic field becomes H1. =Hf. +11;0 +171". =fdpc—jwe. K.....)f1—+—2 R. ‘Z—F— p'JlQp'fllQpMdA I (3.110) [M -jme .K...)f 1 22—4 ;;—: p'J 41911.09)di bl - eg—I‘ ten: 1': =[dp'(-Y.. K...) [‘1 R’“ p‘J.()p)J.(Ap)m jun:1 , . where Y.1 =—I‘— IS the wave adrruttance. 1 Replacing the aperture electric field by an equivalent surface current, that is Elp(z=a*) = -K”, the magnetic field for an open-ended coaxial probe is derived as 55 tie-1‘12 P12 R 1‘ H10 =fdp YclElP(z=o‘)fel 1+RM 917.0. 9')J.(19)Ad1 (3.111) - -_RI‘,z l‘lz b e . R“ fE.(z=0*)J.(Ap)p'dp'401mm Equation (3.111) is exactly the same as (3.30), which was derived previously, if dummy variable A is changed to kc. It has been shown that the same EM fields are excited when the aperture electric field is replaced by an equivalent surface current. Consequently, the same integral equation for the aperture electric field can be derived. What is left to be verified is that the roots of 1+1!“ =0 correspond to the surface wave pole singularities in this TM open- circuit case with a 2-layer medium. With the help of (3.93) and the definition of RM, R“ is shown to be 62“" P2, I‘VE” 6‘ (3.112) 1‘2/ Pr+€2/ 8. where I“ =(Irf-kbll 2, i=l,2. The relation of 1 +11“ =0 can be rewritten as 1+RM=1+¢‘"I“ 1‘2/1‘1-e2/ 31 =0 (3.113) r./r.+e./ e. which can be arranged to be 2 d 1-e' 1" 93.5 (3.114) “(2134 I‘1 92 It can be easily check that the roots exist only in range of It: 3sz kr Thus letting ‘ (3.115) I‘.=JK.=i(kf-kf)3 56 then (3.114) becomes _:_=-.__ (3.116) l+e 2"“ 1K1 92 The left hand side is equivalent to jtan(K1d) so that 1‘ e tan(K,d) = fi (3.117) Equation (3.117) is the well known eigenvalue equation of TM surface wave modes for a dielectric slab waveguide. 3.4 Numerical Simulation - Method of Moments In this section the method of moments is employed to solve the EFIE (3.29) [29]. The TM eigenmodes of the coaxial line are chosen to be the basis functions for the unknown aperture field and Galerkin’s technique is used to convert the integral equation into a set of simultaneous algebraic equations, from which the numerical solution of aperture electric field is obtained and the input admittance at the aperture is resolved. Finally some verifications of the correct numerical implementation are also shown in this section. The unknown aperture electric field is expanded into a finite sum of TM eigenmodes of the coaxial probe as follows: 8 8(9) = 2 learn) (3413) no where , . Ci [11(kclp)Y0(kcia) -J0(kcla)yl(kclp) ] "’0 3.119 8349) =4 ( ) i=0 l . 9 57 and k“ satisfies the following eigenvalue equation [30] J 0(kfla) Yo(kflb) -Jo(kc‘b) Y 0(kda) = 0 (3-120) Substituting (3.118) in the integral equation (3.29) leads to (from now on lam”). for simplicity), as 2 1 l 1 l A —— = ——— 2 var a ' s . . . . (3.121) + 2 Yma.(p)f2 V. a.(p)a..(p)pdp n=1 41$ + fY..-(——— R“) f2V. a.(p)J.(kp"‘)pdp J.(k p)k.dk 0 (1 Ru). which is rearranged to be s 011., 9-1'0 '1. P 111(15): (3.122) + 2V. 2 Y... a ..(p)fa.(p)a.. (pm)... i=0 ‘m=l (1 R...) . . . +2V fY"(——l HR») [914p )Jl(kcp')pdp Jl(kcp)kctflcc I Galerkin’s method Galerkin’s method suggests using basis functions R’(p) as testing functions and does the following operation: Operating f (3.122) ca,(p)pdp .j =0. s 58 a system of linear equations is derived as follows: 2 b A..— f 819% =2V.——fa.(p)dp faipwp' i=0 1"a 0 ln(— (b!) (3.123) + '20 V’21Y... f 819))! (9)d9 fame... (p)p”dp +2Vdea—— R“) ..(1 T). f3,(9)-Mk.9)dp kcdkc [849'fl.(k.9')9'd9' with j = 1,...,S. Via orthogonality properties, (3.123) is simplified to be 2 b A—b 1n— °n. ,. (a) 1 =2V——— 6 In 1-0 n.1n(b)52’°'n( b)“ (2) (3.124) S + EViYmblR 6 (=0 and Y(——1 R“) b + ‘20 V f .. (1+.R .) f8fi9V.(k.9)d9 f aipv.(k.p)p'dp' kdk withj = 1,...,S. These simultaneous algebraic equations can be expressed as 4.11] = {[Y‘]+[Y"]+[Y‘]} [V] (3.125) with the matrix elements as 59 no (3.126) Y1: = ”fly!!! 61»: 51111 O (l—Rb ) b Yf= [Yd (1 T1). [819N.(k.9)d9 f 819')’.(k.9)9'd9' k.dk. 0 a Equation (3.125) can then be solved by a standard numerical subroutine for linear equations [31]. After solving 8’(p) from (3. 125), the normalized input admittance can be related to 8’(p) with the help of (3.27) [32,331, 2- .— l—R_ A .ln(lbl 4) £819)“ Y...= + (3.127) 1 R f: 3(9)d 9 A.ln(bla) Since 81p) is proportional to A0, A0 will be canceled in the expression of normalized input admittance. Setting A0 to 1 leads to b b KW) 9 2M;)-f3(9)d9 = ‘ (3.128) —— f; a.) p ‘ Aoln(1b/a) [8(9)d9 d -024 1nsz 0) yin Before presenting some results of the numerical simulation for (3. 125) some aspects about evaluation of the matrix elements are addressed next. 60 Evaluation of the matrix elements The computation for the elements of matrices Y ‘ and Y b is trivial. However the evaluation for the elements of matrix Y ‘, involving an infinite integration w.r.t. kc, need more care due to the slow convergence and oscillatory behavior of Bessel functions. Fast convergence of the integration w.r.t. k: can be achieved by the technique of adding and subtracting an asymptotic term to the integrand [34]. By observing I‘l-o kc & Ila-+0 when kc-ooo, we can rewrite 1’; as Y; =] $042“); 2" ”(k )4 la“ )J(k )‘d'kdk 1‘98; 0 P1(1+Ru) kc a ,(p 1 cp p a ,P 1 .99 P c c - b 1) (3-129) +f [aflpyl(kcp)dp [8199110593930 dkc O a a 5197300!" The integrand of Y; is now proportional to Ice-2 despite of the decaying factor of Bessel functions. Furthermore, 1’}:- involving a double integral can be reduced to a single integral by employing the following formula [20]. fZ.(a9)l,(99)949= p [132.(99)J.-.(B9)-¢Z.-.(a9)J.(139)1 (3-130) a2_pz with a #0 and Z, denoting JV and YR. The fast convergent integral ij- can now be evaluated numerically and the infinite integral range can be truncated at a reasonable value according to a preset accuracy. It may save many efforts if the asymptotic term m, a double integral with a slow convergence and oscillatory behavior, can be cast into some kind of closed form. Unfortunately asym need to be evaluated numerically. 61 Using the following formula the time consuming form for asym in (3.129) can be converted into a hypergeomeuic function with an integral range only from 0 to 1 instead of from 0 to infinity [48]: ." . -a'se/a 1:. ..el’ {11(k.P)J.(k.P)dk. ..2 I‘m/2) F12. 2.2.( p) l . 1 = 9 I‘(2) ./t' dt 2.2 I‘(3/2)I‘(1/2) . W 1.979).. EK(9'.9) with p’p}fil.(9)81.(9)999199 Using the following change of variables, x=(p+p')/ (5 y=(-9+9')l )/5 and with dxdy=dpdp', m becomes Try—n7 = fif’m” 2 [ff/2‘ K(9'.9)R,(9)8.(9)9 9'dy +12% K(p.p')a,(p)a.(p')p 621an + m -x+fib I . ’ («2+mtz [[6 ’“P +P)R.(9)R.(9)9 9d) + Effie K(p’p')ai(9)at(P')P 93.?de where p'=(x+y)/ (5 and p=(x-y)/ (5- 62 (3.131) (3.132) (3.133) (3.134) Equation (3.134) is now ready for numerical evaluation. Although asym involves a triple integral, the improper integral from O to infinity no longer exists. Furthermore the evaluation of the asymptotic term can be performed once for all cases because it only depends on the dimensions of the coaxial line. After all these considerations about the evaluation of the matrix elements, (3.125) can now be calculated numerically by any kind of computer code. Computer program and numerical results A fortran computer program was written for the numerical evaluation of (3.125) and (3.126) such that the aperture electric field and other quantities can be calculated when [1,6 of the material are given. The validity of the computer program is verified by computing the reflection coefficient of a SR-7 type coaxial line radiating into a homogeneous material medium of complex permittivity at 3 GHz, the geometry in the study by D. K. Misra [35] and J. R. Mosig [36]. The reflection coefficient as a function of z=d-j e", the material permittivity, is computed for various permittivities alongthe constant 8' and constant 8' lines. Our numerical results as shown in fig. 3.5 compare quite well with the existing ones. So far the "forward problem" has been solved correctly and the "inverse problem", the final goal, for determining p and e of the material via measurements of the input impedance of the coaxial probe will be addressed in chapter 6. 63 i—-l— Mosig etal. o o o Misra ct a1. + + + This theory a” constant line 1. 0.8 0 6.4 3.2 16] -1 > Rr 1 b O . 2 C 0 4 e’ constant 8 line H O , 16 ‘1 Figure 3.5 Reflection coefficient of an SR-7 type coaxial line placed against various homogeneous materials, frequency is 3 GHz. 64 CHAPTER 4 POWER BALANCE FOR COMPLEX WAVE EXCITATION IN OPEN-ENDED COAXIAL PROBE STRUCTURE 4.1 Introduction The power balance is verified in this chapter for an open-ended coaxial probe system by comparing the total power carried away from the aperture to the transmitted power of the incident wave at the aperture [38]. Also the EM fields excited inside the material medium are found to be localized around the probe aperture when the frequency is lower than 2 GHz. With the complex wave excitation for an open-ended coaxial probe identified in chapter 3, it is possible to identify various contributors to the total power carried away from the aperture. The power contributors are surface waves and radiative waves for the open-circuit case and radial guided waves for the short-circuit case. The power carried by each wave can be expressed in a Concise form using Cauchy residue theorem or saddle point method. so that it can be evaluated more efficiently [39]. In addition, the excitation of the complex wave is found to be strongly dependent on the frequency. In section 4.2 the surface wave power of the open-circuit case is evaluated. The EM field components of each surface wave mode are derived. The associated power carried in the radial direction can be calculated using the Poynting theorem. 65 In section 4.3 the radiative wave power of the open-circuit case is evaluated. Since the wave is directed into the half space, the radiative wave power can be computed over the surface of a hemisphere in the far zone. The far zone fields are derived and then simplified using the saddle point method so that the associated power can be computed more efficiently. In section 4.4 the radial guided wave power of the short-circuit case is evaluated. The EM field components of each radial guided wave are derived using a deformed contour integral. The radial guided wave poles are investigated and it is found that there exists propagating and evanescent radial guided waves. The power of the prOpagating wave is then computed using the Poynting theorem. 4.2 Surface Wave Power for Open-Circuit Case In this section the total power of the surface waves excited by an open-ended coaxial probe system in the material layer is studied while the power of radiative wave through the material layer is investigated in the next section. For illustrative purpose we consider fig. 4.1, an open-ended coaxial probe placed against a two-layer lossless medium. Since the surface wave modes are orthogonal [21], the total power of surface waves is the sum of the power associated with each surface wave mode. The surface wave is known to propagate laterally, therefore no real power is carried away in positive 2 direction, and the power is directed in radial direction in both regions 1 and 2. The EM field components of each surface wave mode inside the material layers are computed first using a deformed contour integral, and then the associated power is evaluated numerically. 66 region 2 .., . . » .-u -;o_'. -~ .,~ -- '.-..L .‘ ~.'.‘;'~ . .. _ _.~ -. ".. ‘_ _ '.--'- , _..,_._-&...,.._ , _ _ A, . - ~ . . c - Pr2 Pr2 Figure 4.1 Surface waves and radiative wave excitation in the open-circuit case of an open-ended coaxial line probe placed against a material layer. 67 4.2.1 Derivation of Surface Wave Field Expressions Surface Wave Fields in Region I With the help of (3.19), (3.20) and (3.26), the EM field components in region 1 of the material medium can be expressed as [f 8’(p wk p ”)pdp )(e ”..“+R,,e“‘>l(k p)k dk [3’02 )1 (k p >de )(e “512,136.! (k p)k dk (4.1) “1 “:91 o (b = .1. l ' ' ' ' 'Pt‘_ 1"1 2 E. {1‘1 (“R“) [819 )Jl(k.p)pdp](e Rue ‘)J,(k,p)k,dk. In order to identify the surface wave fields, the EM field components in (4.1) are rewritten, with the help of (3.30) and (3.31), as follows: . b l 1 0 I I a - 591:5]. 1 + [Ig(p )Jl (kcp )pdp )(8 I‘ll +Rbler!z)H?)(kc p)kcdkc ‘. d l . l (b ‘ r r (2) (42 - I I I I - It. 1 . H.1- ELY“ 1 +12“ \[815011059 )pdp} («2 Rue ‘)H, (kcp)k,.dk. ) u (D W 1 1 . . . . - 5,, =- f — 1 f 819 mm )9 dp (e “‘-R,.e“‘)Hé”(k.p)kde. 2 -. Pl 1+.ij [a ) It was proved in chapter 3 that the surface wave poles satisfy 4134 T2, Pr'ezl 6, =0 (4.3) 1+RM =l+e l‘le‘l+ezl e:1 With reference to fig. 3.2, it is shown that using the deformed contour integral with proper choice of branch cut, E”1 in (4.2) becomes 68 1 (e -l‘lz + R91 arts) 6 ' ' ' , (2) ":5 C (1+R. ) f$(p)J,(k,_p ”‘19 H, (kt-Plkcdkc ' I a (4.4) (e 'P“+ er“) b , ' , , J";- C. (132:) 1.319)“ch )Pdp 111(2)(kcp)kcdkc l a ' where C‘" stands for the contour around the surface wave poles and Cb for the contour along the branch cut. Similar expressions can be derived for Her and 1?21 also. It is found that the surface wave poles are simple poles. Thus by letting It” denote the m“ surface wave pole, the surface wave field component E, in region 1 can be derived by the residue evaluation of the first integrand in (4.4) as following 5 5,, = '“i [fflp'flfikmp'wflp'] (e '1‘“ +Ra19r‘5flfz’(kwp)km (4.5) . 1 11m k Jew | Similarly the surface wave field components for H. and Ez in region 1 are D H.1=-n1' Y.,[ I Slp'fl1(k,,.p')p'dp'] (e '1‘“ -R.,er“)H? ’(kmpkm. . l lrrn [kc—k kc‘kqa 1+Rbl W] (4.6) b 5,] = -1r1' -1-,1-[f819')l,(k,,.p')p'dp'] (e'r“-R.,er“)H32’(k,,.p)k§,. 1 a lim 1 kc-k' 1+1!” rk.-k,,..1 Introducing notations Q and L, (4.5) and (4.6) are written as following, 69 Epl = H“:- -njoLk.,..(e“‘“ +R,, er") Hf’rkwp) 1:11:10sz (e'r'z —R,, J") Hf’awp) (4.7) E =-nj— 0ka ..(e ”*‘-R.,,e"“)Hé”(k,..p) where Q! 819')! (k p')p'dp' i 1 "" (4.8) L lim k-k 3.4:... .( , q...) Surface Wave Fields in region 2 With the help of (3.19), (3.20) and letting Ru=0, the EM field components in region 2 are it: 5592;1fo -rzzJI(kcp)kc&c 0 where B2 is the unknown amplitude of each TM mode. With the help of (2.32), we have where .81 is expressed in (3.26) as ~ .. .. - _ t 4.9). H.=¢Ha=¢fY.sze Meme. ( o 512 = [1.32 e -P’zJo(kcp)kc2&c 0 P2 “B-2 : 21’“ coy-rod E T (4.10) Br Ya! + Yd B (k,)= J k d (4.11) 1 (1 RMISTP) ( 9» p 70 Equation (4.9) then leads to b [2(p')‘,l(kc P.) p'dp'] T8 -rzz‘,1(kc p)kcdkc (4°12) 592: [11+R“ H.2= r", ([8193ch pm)pdp]Te r"! 1,0: p)k die 0%! 1 + 1R“ (4.13) :1 1 . . . . .13. 2 a, { 1,2 1+ RH [[8112 )J.(k.p)pdp)re J.(k.p)k.dk. In order to identify the surface wave fields, the EM field components in (4.12) and (4.13) are rewritten, with the help of (3.30) and (3.31), as b [8195110, 9') p'dp'] T e'r"H1‘2’(k,p)k.aUc, u—t um N ll NIH H + b iROitfiig' :3 7... .N bl (b ) ~ fap')1,(k.p')p'dp' Te‘r=‘H,“’(k.p)k.dk. (“4) a: u um N u NIr-d Ml.— \ ~ [3(p')Jl(kcp') p'dp' Te-rzzllom(kcp)kc2dkc \0 l :1 | ... y—a + 2% The surface wave field components E92,}!2 and Ez2 can then be derived in a similar way as in region 1. By the residue evaluation of integrand in (4.14), we have b k -k =- ° ' ' ' ' ‘le (2) cm .nJIZKp)Jl(kmp)pdee H1(kmp)kmlim 1+sz 1: -km "LI-Ru b ”of-fi1Y.zf3’(p')J,(k,,.p')p'dp'Te’”=‘H,<2>(kpumum (415) d b l l o o I -rz kc- =- — J k d T 2Hm]: k2,. lirn nirz[3(p),(.,.p)p p e ..(,” p) m... 1 R: 71 01' EM=-1thLTlc e’rz‘Hfz’am p) 11.2%an QLTk Me'rz‘afz’a p) (4.16) E12: '7‘} F2- Q L Tkmz e 43; Hy)(kmp) where Q and L have been defined in (4.8). With the field components in both regions, as given in (4.7) and (4.16), for each individual surface wave mode, the associated power is calculated next. 4.2.2 Calculation of Surface Wave Power To calculate the power associated with each surface wave mode, the following relationship is used, p: f 112,“? x fi'1-ds’ (4.17) S 2 Since I? has only H. component while E has E, and Ez components, we have 1 . . . . _. ' P:E fstJzEpH. - 115,111,] ~ds (4.18) With the help of (4.7) and (4.16), it is found that EPH; is pure imaginary in both regions, therefore the real power flows only in the p direction, which is consistent with the characteristic of the surface wave. Equation (4.18) then becomes OZR P=-;-‘[IR[-E H;]pd¢dz ”limit: R-E[ 11‘] am +1.2} -E H‘ d d (“9) 200 up 12££J ,2¢2]P¢Z aPPI-l-sz where P” and Pp2 are the radial power flows in regions 1 and 2, respectively. 72 Let’s compute the radial power flow in region 2 first. Since the surface wave poles I: exist in the range of [k2 , kl], 1‘2 in (4.16) is real and the surface wave fields can m be expressed as H;,= n j 1;", Q‘ L‘ 1" kg" 6”“ Hf’"(kwp) . 1 - r21:2 = -1tj F, QL kap, e l‘1‘1=I,"’(k,,,,,.,p) where Ycz=jtoezlPT Therefore we have . “'ij -2r ' 121211.242 r 2’ lorrkwlzkW e 1‘ Hf’ampm?’ (kwp) 2 Using the following Wronskin, I ' 2 J,(z)Y, (z) - Y,(z) J, (z = — M the product of the two Hankel functions in (4.21) can be reduced to ml” (kg... p) 11911.” p) =J' [Jo(k,,... p) -i Yo(k.,.. 9)] [1104... p) +1Y1(k.,.. 9)] = [11%- P) Yoa‘a. P) 'Jo(k.,.. ”7'1“.” p)] +j[ ...... ] = 2 +. 00000 “WP J[ l where the imaginary part [...] doesn’t contribute to the real power. With (4.21) and (4.23), we have ‘21}: . we, 1245:2152] = -2n W IQLTkmF ‘ p 2 With (4.24) P»2 becomes 73 (4.20) (4.21) (4.22) (4.23) (4.24) n 032 . -2r sz=—f21t WIQLTkWFHe “612]“ 2 1:2 me:2 2 =— — QLTk e 2 IPZIZI ‘F‘l (4.25) Next, let’s compute the radial power flow of the surface wave in region 1. With I“ 21°61 in (4.7) the surface wave fields become 11;, = «116011;. (e ”"‘ -R.. e”“)'H.‘”"(k,,..p> 1 (4.26) E“ = - xi}? on; (e "’1‘ -R,, 3"")113’0, p) l where pl is real and R01 has been derived in (2.48) as 4134 Ycr'Ycz = {21914 I‘2, 51-182]!!! 3 {21’1“ Bfi (4.27) Rm: e . Ycl + Yd lepl+]€2/€l I) '1'ij where P2 is real, u=l‘2/Bl and Kf=ezlel. Using (4.27), we have e'Jarz _Rbl ejpnl ”411,: _e—2}B1d ”‘1'": (111,: . 4 28) v +JK3 ( ' 1K3 coswd-z) - u sinB,(d-z)] - d ‘19; =2j u+j , The substitution of (4.28) in (4.26) leads to Ele;l=4n2 7M1 |QLkw|2km 1 a? WK: [Kfoosmd—o-u sans,(d-z>]’Hf’(k,.p)11911:...» (4.29) 74 With the product of the two Hankel functions being simplified, as in (4.23), equation (4.29) leads to R¢[E21H¢1]='sn:—: III‘IQL enzl 112:]? (4 30) 21 [KfcoslillGI-z)-11sin;;1(d-z)]2 ; Therefore, P pl becomes PPl --f8%e—1IQLkPPPPI f [chosfll(d—z)vsin[31(d-z)]dzd¢ (4 31) 02 After some manipulation, the integral with respect to z in (4.31) can be expressed as d 1 [K3 ensue—ow sum-ore 0 . _ (4.32) =K:£ l-r-oosfildsmBl +uzd1 1-oosBl dsin_fl_ ——d]+ +fiudM ‘ 2 31d 2 B d 2B1d and) Equation (4.31) then becomes 81:29:41 IQLlc P1= 2 pl U+K r ml: f(d) ‘ (4.33) Note that with (4.25) and (4.33), it is clear that PP is independent of p. This is consistent with the fact that surface waves excited in this structure are cylindrical waves. Combining (4.25) and (4.33), the radial power flow associated with each surface wave mode is PP =PP P+P2 4.34 =8" ”6‘ Ion“ f( 4+) — -—:3IQer PPPeIZ (”‘2‘ ( ) B] D ZP'I'K‘ with 75 r2= 1.3.4; ‘31:ka -k:pu L=lim l kart“ l +Rbl (kP 4:4“) _ P2 (4.35) b o =f8(p')J,(k.,.p')p'dp' and f(d) is defined in (4.32). Finally it is noted that the numerical computation of Q requires the information of aperture electric field 81p). The numerical solution of 81p) can be obtained by eigenmode expansion as presented in section 3.4. From (3.118) and (3.119), we have s 3(9) = 2 V3.01) . (4-36) M where 8,(p) is defined in (3.119). Then after some manipulation, the quantity Q in (4.35) can be expressed as b o=f8(p')J,(k.,.p')p'dp' : b b 1;: VP [3,001, (km p')p'dp' We [11 (kg... p')dp' (4.37) r C k Jo(k,.,a) 03ml?) 1005.1?) - V ‘9’" £1 —JP(kPP_Pa)) M ‘ (k?! -k:") kc: + VP k—l— you” a) —JP (km 12)] 0PM 76 The total surface wave power is the sum of the power of individual surface wave mode, as given in (4.34). The other contributor to the total power carried away from the aperture is the radiative wave which is considered next. 4.3 Radiated Power for Open-Circuit Case In this section, the radiated power contribution of the open-circuit case for an open- ended coaxial probe is studied. With reference to fig. 4.1 in the previous section, the total power carried away from the aperture consists of the surface wave power and the radiative wave power. Because of the orthogonality relationship between the surface wave and radiative wave [21,37], the total power can be expressed as the sum of the surface wave power and the radiative wave power, the former has been presented in the previous section. To compute the radiative wave power, the radiative wave should be evaluated along the branch cut for the sake of consistency. But for simplicity and computational efficiency, the far zone fields is evaluated via the saddle point method instead, since it can be shown that the far zone fields can be approximated by the radiative wave fields [40]. 4.3.1 Power Relation in Spherical Coordinates As given by (4.14) in previous section, the EM fields in region 2 are - b . - l l I I I I -r z (2) EPZ‘ELTRn[£g(PV1(kCP)PdP T9 2 H1 (kcp)kcdkc . (4.38) _ 1 1 .2 - 5..'l.‘.Y‘21+Ravr b [81:25.11 (ka') p'dp') Te'r"H1‘” (kcp)k,dkc b zrp'w (k p')p‘dp' Te'r=‘H“’(k mm 1 c 0 c c c 1'1 1 E =_ _ ‘2 2£P21+Rb1 77 With reference to fig. 4.2, the total radiative wave power carried away from the aperture is computed as follows: P, = PP, +PP2 = [S éRJExI-i'] as =fs éRJr-(Exii‘flds (4.39) where S is the surface of the hemisphere with radius r and P" is the radiated power in region i, i=1 ,2. Because the material is assumed to be lossless, the total power is independent of the spherical radius r. In order to compute the power more efficiently, the radiative wave at great distance is considered, where r is sufficiently large. When r-wo, it is clear that P is zero. This leads to r1 P, =PP2=%fo%]:"RP[f - (szfi;)]r2sin6d¢d0 The electric field in region 2 has 2 and p components as 13", = a PP, +213}, Using the following relationship between different coordinates a = sine f + case 0 '2 = case r -sin0 0 the electric field in (4.41) becomes Ez=fEr2+éEoz where . EP2 =Estm0 +Ezzcosfl E02 =Echos0 -Ezzsin6 78 (4.40) (4.41) (4.42) (4.43) (4.44) Figure 4.2 The power radiated into the material medium is evaluated over the surface of a hemisphere for an open-ended coaxial probe. 79 With H2= -¢H ‘2, the substitution of (4.44) in (4.40) leads to l 3- 23 I . PP=- 2 R 5,211,, rzsmfldtbdfl 2]" I" '[ J (4.45) =1: fPfoI-rzsinzezau;,+rzsrno coseEqug,]de With p=rsin0, if r approaches infinity at far zone with 0 #0, then p approached infinity also. This leads to 3 ‘ -1(k.p-—x) H10)“, 9) - 2 e ‘ \ “ch (4.46) F— - -2 2 gimp 4) Hf’flcp) - \ 1:ka Substituting (4.46) in (4.38), the EM fields at the far zone can be expressed as 5P2= [111(k) (‘3‘ ("were .. (4.47) 5.41.4.) H.2=fh.p "dp The far zone fields in (4.47) can be evaluated more efficiently using the saddle pornt method (also known as the method of steep decent). It is observed that the far zone fields 80 in (4.47) take a form of a generic integral as I(k,)= fh,(k,)e“‘2‘e""c°dkc (4.49) where h,(kP), i=1 to 3, is defined in (4.48). 4.3.2 Far Zone Field Derivation via Saddle Point Method [21,39] To apply the saddle point method, the integral of (4.49) must be first placed in an appropriate form as follow: With I‘zejfl , we can transform the (9,2) and (15,9) planes to spherical coordinates. With reference to fig. 4.3, we have p=rsin6 (450) z= rcosO and . kfikzsm'l' (451) B =k2°oswlv with . ¢=0 +111 (4.52) dkc=kzooswdt|1 This leads to [3“ka = kzrcosflcosw +k2rsin68im|r =kzrcos(IIv-0) (4.53) Equation (4.51) represents a mapping of complex kc plane into a strip of the complex 1|: plane. The two Riemann sheets in Ice plane are mapped into a connected strip with 21: along the 0 axis as follows: 81 A e , - p (b) l3 A \II "2 kc (a) Figure 4.3 Transformation to spherical coordinates, (a) in kc -B plane (b) in p-z plane. 82 With (4.51), we have 1‘; =J°B =J'k2008(0 tin) (4.54) =lczsino sinhn +ij oosa coshn On the proper sheet of the Riemann surface, we have R.{I‘2}>O and hence sina sinhr. > o (455) which corresponds to the range of {oz[-1t,0], n<0 (4.56) o:[0,1t] , n>0 Similarly for the improper sheet of the Riemann surface, we have o:[-1t,0] , n>0 (4.57) a:[0,1t] , n<0 This mapping is illustrated in fig. 4.4 where the shaded regions are the proper Riemann sheet. Also with (4.51), we have kfikzsin‘l' =kzsino ooshn +jkzcoso sinhn (4.58) or R.{kc}=kzsino ooshn Inlkc1=kzooso sinhn (4.59) With (4.59), the four quadrants of the proper Riemann sheet on Ice plane are mapped into the regions designated by P,, i=1,...,4, as shown in fig. 4.4. Also the real axis of kc plane is mapped into the contour, denoted by C, passing from (-1t/2,-~) to (rt/2,00) in the 4: plane. 83 in w — plane I I P1 PP4 = —1t -—1t/2 " . 7 If 1t/2 C P2 P3 I I Figure 4.4 Mapping of two Riemann sheets of the kc plane onto a stripe of the w = o+jn plane. 84 With (4.52) and (4.53), the generic integral (4.49) can be rewritten in spherical coordinates as 1(6) =fh(1|1)e """°°"" ‘°’ k2 coswdq: (4.60) C Let’s consider the following general integral, [(11) = fg(w)e‘fl“’)dm = fg(to)e“(°’ e’”“"dw (4.61) C C in which flu) =u(m) +jv(c.)) is a complex function of the complex variable t.) anda is a large positive number. This integral can be approximated based on the properties of flu) at the stationary point (or called saddle point) too, implying f' ((1)0) =0. The concept is to deform the contour C to pass through “’0 and the orientation of the contour is directed along the steepest decent path (SDP), where v=v(mo) is constant and u(wo) is maximum. Then significant contribution to the integral comes from this steepest decent path near (.10, because a u(w) decrease rapidly away from “’0 along the SDP and destructive oscillation is avoided due to the constant phase v(mo). By Taylor’s series expansion about the saddle point, we have fun) -f(«»,) + éf'woxw - 6,)2 (4.62) =f( 6.) + imam +20) where f.(wo) =F9K (4.63) (m - (no) =Re" This leads to 85 14(6)) -u(mo) +%FR2cos(E +26) (4.64) v(w) -v(too) + %FR2sin(E +26) Along the path . {+26 =11: (4-65) we have = — .1— u(w) u( (no) 2 FR2 (4.66) V((1)) =V( m0) and m: (4.67) e aflw) =6 ‘fl°o) e- 2 It is clear that am“) in (4.67) has a constant phase, in addition, it has a maximum at (.10, (R=O) , and decreases exponentially away from (.10. Therefore (4.65) defines the steepest decent path about the saddle point 660. Hence we have =-_€,3_ ’ (4.68) With the change of variable in (4.63) and assuming 6 unchanged, we have do.) =e1‘dR and (4.61) becomes, [(11) . amok» [W 8(06 ,Reu) ‘7" am (4.69) where SDP is the steepest decent path. For a sufficient large a , the contribution to the integral in (4.69) mainly comes from in? the SDP near the saddle point “’0 since e 2 decreases rapidly away from (no. This argument prevails even on the limiting case, where a approaches infinity, such that «5173’ e 2 is extremely small for R490, then we have 86 . J .2 1(a)=g(mo)e"‘°°’e” fe 2’ dR (4.70) 21: afloo) ,5 = — e e (I «F 80%) The generic integral (4.60) in spherical coordinates can now be converted into a closed form using (4.70). Comparing (4.60) to (4.61), we have a =k2r g(¢)=kzh,(rlr)cosv (4.71) f(l|') = -1' 008W -9) = I401!) +J'V(¢) This leads to u(r|r) = -sin(o -0) sinhn (4.72) v(1|r) = -cos(o -0) ooshn and f'(¢)=jsin(w -6) (4,3) for) =jcos(\lr -6) The saddle point is obtained from f'(t|to) =0, then we have ‘1’, = 0 (4.74) This gives for.) =j F =1 (4.75) 5 = E b = --_ t __ 87 and f(‘|’,) = ‘j u (we) = 0 (4.76) V(rlv,,)=-l The steepest decent path is determined by equating v(r|r) to v(r|ro) in (4.72) and (4.76) as cos(o -0) coshn =1 (4-77) The sop is depicted in fig. 4.5 and it is clear that =-:PE should be chosen. Therefore, with (4.70) the generic integral in (4.60) becomes - , 11 I(r)= 3i h,(k,)k,eosoe "‘2 e 4 (4.78) kzr where h‘(kc), i=1 to 3, is given in (4.48) with kc=k2sin6 P2 =1' 5 =1": 0080 (4.79) r‘1 =‘/kf 4:} = (Mania -kf Thus the EM fields in (4.47) becomes , ,_ _ r :- Ep2=\ilrhl(kc)k1 eose e ”5 e 4 E = —2fl ”20¢ )kz cosO (”Veg (4°80) 12 \ k2? c 21: -) r I": H”: £711,091; oose e ‘1 e 4 with h,(kc), i=1 to 3, defined in (4.48). Hence, using the saddle point method, we are able to derive the far zone fields in (4.80) by deforming the contour C to SDP. However in deforming the contour, some of 88 w - plane _ surface wave pole 0' -n/2 n/2 e + M? leaky wave pole Figure 4.5 Defamation of contour C into steepest decent path SDP. 89 the poles in h‘(kc) may be encountered. When this happens, the contour should be warped as in fig. 4.5. Thus the contribution to the far zone fields from the integration around these poles need to be included There are tivo kinds of poles, the surface wave pOIes and the leaky wave poles, the former locates on the pr0per Riemann sheet and the latter on the improper Riemann sheet. From fig. 4.5, it is shown that these poles are encountered only for a range of 6 greater than some critical angle 0‘. Also even when these poles are encountered, it can be shown that the contribution from either surface waves or leaky waves decays exponentially as r- no. Therefore the far zone fields can be approximated by the radiative wave fields. Substituting (4.80) and (4.48) in (4.45), it can be verified that the radiative wave power is independent of r. Therefore by letting r-m, the evaluation of the far zone fields via saddle point method becomes exact, not just approximate, and the accuracy of the power evaluation can be preserved. Power Balance Verification The total power Pr carried away from the aperture is the sum of the surface wave power and the radiative wave power as mentioned before. Physically, P, should be equal to the transmitted power of the incident TEM wave, which is shown as follows: The TEM wave in its transverse dependence can be expressed as it" = 6 l P (4.81) [i=6 L M and the incident power of the TEM wave is obtained as 90 (4.82) Since R is the reflection coefficient of the TEM wave at the aperture, the transmitted power P, is P, = (l - |R|2)PP . (4.33) By solving the simultaneous algebraic equations (3.125), the numerical aperture electric field is determined and the reflection coefficient is obtained. Then the transmitted power P: can be computed using (4.83) and the total power Pr carried away from the aperture can be computed using (4.34) and (4.45). The quantity Pr is compared to P, as shown in fig. 4.6, where a=0.31020m, b=0.7l45cm, d=lcm, 81=380 and €¢=£o are assumed. They are almost equal as expected. Also it is found that the excitation of surface waves and radiative waves is strongly dependent on the frequency. Especially at the low frequency range, less than 2 GHz in this case, the transmitted power is so small that the EM fields excited inside the material media are localized around the probe aperture. This finding justifies the assumption of the infinite metallic plate in the full wave analysis, and provides a physical reason for the validity of the quasi-static analysis [14,35]. Similar phenomenon is observed for the short-circuit case in next section. 4.4 Radial Guided Wave Power for Short-Circuit Case In this section the total power contribution due to radial guided waves excited in the short-circuit case of an open-ended coaxial probe system is studied. 91 Normalize Power 03' +++ PT I Pi 0.8- ° ' ' Radiative wave power/Pi ' . °°°°° Surface wave power / Pi ' ° Frequency (GHz) Figure 4.6 ThePpower balance verification for open-circuit case. The total power earned away from the aperture and the transmitted power of the incident wave, both normalized by the input power, agree quite well. 92 For illustrative purpose, we consider an open—ended coaxial probe placed against a single layer lossless material medium as shown in fig. 4.7. Since the radial guided waves are orthogonal, the total power carried by radial guided waves can be computed by summing up the power associated with each individual mode. The radial guided wave poles are studied and it is found that there exists propagating and evanescent radial guided waves. It’s obvious that the power is carried away in radial direction. The EM fields for each radial guided wave are derived first using a deformed contour integral, after that the power of a propagating wave can be evaluated numerically. 4.4.1 Radial Guided Waves inside Material Medium The general field expression (4.2) in region 1 is valid for either open- or short- circuit case: b f8p‘dp'] Pp] —> z: Figure 4.7 The radial guided waves are excited in the short-circuit case of an open-ended coaxial probe placed against a layered material me- mmmm%flm®mmemwmmML 94 This leads to Rb) = _e'2rrd (4.86) It is found that the radial guided wave poles satisfy 1+RM=1-e'2r‘d=0 (4-87) Letting I‘l ejBl, we have 91:1(k3'k: >0. Equation (4.87) then becomes 1 _e'12514= e120”: _ e‘lzprd= O (4.88) 01' 4.89 [51:61»:met ( ) where m = 0, l, 2, 3, It is found that the propagating radial guided wave poles exist in the range of [0,k1] such that from (4.89) we have = 2_2 = 2_1132 (4.90) where It” is the In" radial guided wave pole and m goes from 0 up to M< kid/1r. , Also there are evanescent radial guided waves when km locates along the negative imaginary axis, which decay exponentially in p direction and carry no real power flow at all. With reference to fig. 3.3, it is shown that using a deformed contour integralEpl in (4.84) becomes ml; "““Rhe’o 2 c. (1+R,,) b [I 8195146969'49' Hfz’(k.p)k,dkP (4.91) where C, stands for the contour around the radial guided wave poles. 95 Similar expressions can be derived for Her and Ezl’ also. It can be shown that the propagating radial guided wave poles are simple poles. Therefore the radial guided wave field Ep1 can be derived by the residue evaluation of the integrand in (4.91) as b , Epl = -1tj(f3’(p')ll(kwp')p'dp'] (e 4"! +Rbrer‘z)Hra)(kcn-P)kw (4.92) . l ‘ tel-14:1: 1 +1?“ [kc-km'] Introducing notations Q and L,, (4.92) becomes EPl = -1tj 1‘,(21.,Icw(t:“"z +Rbler'z)H1m(kwp) (4°93) where Q has been defined in (4.8) and L, is Similarly the field components H.1 and Ezl of a radial guided wave are hr,l = -1tj(j(o el)QLPkm (Km 42,, er“)H1(2)(km) (4.95) an = -n 101.32,, (e "'1‘ -R,,1er")Hf)(kq_P) 4.4.2 Calculation of Radial Guided Wave Power The power associated with each radial guided wave is computed using (4.18). Since the power is directed in radial direction only, Ppl is calculated from (4.19) as p=p= , pt R.I-E..H;tlpd¢dz (4.96) NIH OS“. 0"? With l‘1=jfl1 and (4.95), the field components of the radial guided wave become 96 11;, = uj(-jt0 e,)Q‘L,‘k;,,(e”’!‘ —R“J"‘)’H,W(kw) (4.97) Ezl = _anL‘k;'(¢'lfltl_RPPJD11)H:2)(k¢m) After some manipulation, we have e-mz -R,,e”“ P can: "-211.41 ‘15.: = e-mtdz cos 5P (4.1) (4.98) Substitution of (4.98) in (4.97) leads to Eula, =41t2(-j 6 e1) IQLPlcm-Iz Itmeoo2 0101 —z)H((,2)(kq. p)H1m.(km p) (499) With the product of the two Hankel functions as simplified in (4.23), (4.99) gives R,[EP,H;,] = —8 1t mellQL’kwlz oos2 Bl(d-z)—l- (4.100) p Substituting (4.100) in (4.96) results in 21: 4 pp =% P! 81: t.se,|(21.,kwl2 { oos2 0,(d—z)dzd¢ (4.101) Note that Ppl in (4.101) is independent of p , which is consistent with the fact that radial guided wave is a cylindrical wave. After some manipulation, the integral with respect to z in (4.101) can be expressed as " 8(d)=fcoszfll(d-z)dz 0 4.102 d , if [3, =0 ( ) = d . —2- , 1f ”1‘0 Therefore (4.101) becomes PP = 8 1:20 eIIQLPkal’gM) (“03) with 97 BP=Vk12-k;. L = lim i l 5' 5“». I11 1+Rbl (kc'kw) (4.104) b Q=f8’(p')J,(k,,.p') 939' Also the quantity Q can be computed numerically using (4.37). With (4.103), the total power carried away by radial guided waves is the sum of the power of each individual propagating wave: U P," = 3.30 8 1:20 e1|QLPka|Zg(d) (4.105) Power Balance Verification The radial guided wave power Pro». is the total power carried away from the aperture, which should be equal to the transmitted power of the incident wave due to power conservation. The total radial guided wave power Pro. in (4.105) is compared to the transmitted power PP of (4.83) as shown in fig. 4.8, where a=0.3102cm, b=0.7145cm, d=1cm and 8=3€° are assumed. They are almost equal as expected. Also in fig. 4.8 the excitation of radial guided waves is shown to be strongly dependent on. the frequency. The kinks in the figure correspond to the occurance of the next high order radial guided wave mode. It is noted that in the low frequency range, less than 2 GHz in this case, the transmitted power is so small that the EM fields excited inside the material medium are localized around the probe aperture, as in the open-circuit C386. 98 Normalize Power 0.9 0.8 0.7 0.6 0.5 J. 2 4 6 8 10 12 14 16 18 20 Frequency (GHz) Figure 4.8 The power balance verification for short-circuit case of an open-ended coaxial probe. The radial guided wave power and the transmitted power of the incident wave, both nomialized by the input power, agree quite well. 99 Therefore when an open-ended coaxial probe is used to measure the EM properties of a thick material layer at'the low frequency range, the scheme of using open-circuit input impedance and short-circuit input impedance to quantify both complex permittivity e and permeability p fails , because these two irnpedances are almost identical. To overcome this difficulty we can either prepare two samples of the same material with different thicknesses, or use a coaxial cavity structure to contain the material, which is studied in chapter 5. With these schemes the complex permittivity 8 and permeability u can be uniquely determined as long as two different input irnpedances of the probe can be measured for the same material. 100 CHAPTER 5' COAXIAL CAVITY SYSTEM TO MEASURE THE PERMITTIVITY AND PERMEABILITY OF MATERIAL 5.1 Introduction In this chapter the analysis of a coaxial cavity system for the measurement of electromagnetic (EM) properties of materials is presented [21]. As mentioned in chapter 4, when an open-ended coaxial probe is used to measure the EM properties of a thick material layer, the EM fields excited inside the material layer are localized around the probe aperture. Thus, the scheme of using open-circuit input impedance and short-circuit input impedance to quantify both complex permittivity e and permeability u will fail, because these two impedances are almost identical. One method to overcome this difficulty is to let the center conductor of the coaxial probe extend into the material medium so that the EM fields are extended deep into the materials. To accommodate this arrangement, a coaxial line is terminated on a coaxial cavity with a movable backwall as shown in fig. 5.1. When the cavity is partially filled with a material, two input impedances of the coaxial line can be measured by setting the cavity backwall at two different locations. From these two input impedances the complex permittivity e and permeability p of the material can be determined. Theoretically this cavity system is analyzed as follows. An incident TEM mode to the probe is partially reflected by the discontinuity at the aperture and it also excites EM 101 coaxial line ’ 0 0 J; .. J TbIaT c adjustable <—— h —> \ shorted JP end plate Figure 5.1 Geometry of a coaxial line terminated on a coaxial cavity which accommodates an isotropic material layer medium. 102 fields in the material layer medium inside the cavity. Additionally, the higher order coaxial modes are excited near the probe aperture. The EM fields in the coaxial line part and in the material medium inside the cavity can be expressed in terms of modal functions. The matching of the tangential electric and magnetic fields at the probe aperture will lead to an integral equation for the unknown aperture electric field. To determine the reflection coefficient of TEM mode at the probe aperture, the integral equation for the unknown aperture electric field is solved by the method of moments. After the aperture electric field is obtained, the reflection coefficient of the TEM mode or the input impedance of the coaxial line can be determined in terms of e and u of the material medium. Conversely, if the input impedances of the coaxial line are experimentally measured with the help of a network analyzer, e and p of the material medium can be inversely determined. It can be observed that a resonant phenomenon occurs in this structure, and at these resonant frequencies the determination of p and e of materials becomes ill-conditioned. The metallic wall loss effect of the coaxial cavity near the resonance frequencies is then investigated to help mitigate the ill-conditioned problem. In section 5.2 the results of general guided wave theory are applied to solve the EM fields at both sides of the aperture, and an EFIE is derived by matching the boundary conditions across the aperture. Meanwhile the reflection coefficient of the incident TEM wave is expressed in terms of the aperture electric field. In section 5.3 an equivalent circuit concept for a coaxial line discontinuity is presented, by which the physical meaning of input impedance at the aperture of the probe is clearly revealed [41,42]. Also, the validity of the results based on full wave analysis is verified by comparison with that of equivalent circuit concept. 103 In section 5.4 the method of moments is applied to solve the EFIE. The TM eigenmodes of the coaxial line are chosen to be the basis functions for the unknown aperture field. Galerkin’s technique is used to convert the integral equation into a set of simultaneous algebraic equations. After the numeriCal solution of aperture electric field is obtained, the input admittance at the aperture is resolved. In section 5.5 the power loss due to a non-perfectly conducting cavity wall is studied and several numerical results are presented. The analysis is based on the energy conservation (complex Poynting theorem) and perturbation approach. In general, this analysis can be applied to a cavity of any Shape. 5.2 Theoretical Study Using Full Wave Analysis Integral Equation for Aperture Electric Field An integral equation for the aperture electric field is derived in this section by matching the tangential EM fields across the aperture. Because of the discontinuity at the aperture, higher-order modes are excited in the coaxial line near the aperture in addition to the incident TEM mode. In addition, the EM fields excited inside the cavity can be expressed as a sum of coaxial cavity modes. For the case of TEM mode excitation, because the fields inside the coaxial line are ¢-independent and the coaxial cavity with materials is rotationally symmetric the fields excited due to the discontinuity at 2 =0 also exhibit the rotational symmetry. Consequently, only TM modes are excited, and TE modes can be shown to be zero. With the total transverse EM fields inside the coaxial line in (3.12), and the fields at the left hand side of the aperture 2 =0’ already given in (3.14) and (3.15), only the EM fields inside the coaxial cavity need to be solved. 104 Fields inside coaxial cavity For the fields in region 220 (inside the cavity and material), the field components exhibit rotational symmetry as mentioned before and only TM modes can be excited in the cavity. For the a“ TM mode the field components in the j “ layer of material are derived from (3.6) and (3.7) in the form of (3.11). By letting tn —4 a, we have l J(k a) 0 P E": I‘ D “I: J k -Ly—y 1: cm: 5‘ ; I. "p CC[ 0( CUP) Y0(kcaa) 0( c'p)] “3 - " ' 10(kcaa) ' 61‘ 1 H“ = -Jmchj¢ ¢ kc‘ [J0(kc¢ p)-Y—(k—7)Yo(k“ p)]e I. 0 ca Combining the forward and backward waves in the z-direction leads to in = fi B]: (e ‘1}.¢+ij. ‘13.!) as (p) .jme fita=¢ P I ja %.{ (e'PI-‘—R,,.e“'-‘)a.(p)} The transverse field components at the right hand side of the aperture (inside layer 1) are obtained by setting z=0‘ in (5.6): Elam =6 [8,,i(1+x,,,)+ x B,.<1+R,,, )8.(p)r p “=1 (5.7) B - B £l(r-R,,,)+jtoe,s 1' 711 "1 Ia Ii,(z=0*)=| 'II: II '31 ->1“l<- Ill ->|°‘|<- Figure 5.2 Two segments of a coaxial line with a short circuit termination and its equivalent circuit. 110 _ z! -29 (5.19) R- 21-20 where z, = 21 j mplzl‘ (5.20) and z = fl ln(bla) z = h ln(b/a) (5.21) 1 e1 21: ' " ea 21: Next, a coaxial structure consisting of two segments of coaxial lines with different outside conductor diameters is studied. In fig. 5.3, Yd stands for the shunt admittance at the discontinuity, the physical reason for the shunt capacitance is due to the fringing fields (higher-order TM modes) at the discontinuity. Because of the discontinuity at the aperture, the longitudinal component E2 is needed to satisfy the boundary condition, thus higher order TM modes are excited. If these TM modes are not supported in both coaxial lines, and the shorted end is located sufficiently far away from the discontinuity, then the shorted end has no influence on the TM modes, of which the wave admittance is jmel jme1 y = = m. I‘ 2Jomcu (5.22) 1“ kcza -k: with c, =e,/ k3, -kf . Thus the wave admittance Ym of each TM mode corresponds to a Shunt capacitance. With Y: = -j Yleotflilll), the input admittance Y1 is computed by 111 K—O—H Zn 21 J’. _ b = Yd r Yi k- 11 ->| Figure 5.3 Two different segments of coaxial lines with a short circuit termination and its equivalent circuit. 112 Y,=Y,//Y; awed-momma (5°23) awed-nomad,» where CdaYdljw = 2 C“. C And the reflection coefficient is R: I"? (5.24) 1+y, with Viz; ”Cd" 1”““1’1’ . )5 (5.25) Y0 Y0 If the material is lossless and the coaxial cavity is made of perfect conductor, then y, is pure imaginary and lRl=l, because no power is carried away and no conducting loss due to the cavity wall exists. 5.4 Numerical Simulation - Method of Moments In this section the method of moments is applied to solve the EFIE (5.18) [29]. The TM eigenmodes of the coaxial line are chosen to be the basis functions for unknown aperture field and Galerkin’s technique is used to convert the integral equation into a set of Simultaneous algebraic equations, from which the numerical solution of aperture electric field is obtained and the input admittance at the aperture is resolved. Finally some verifications for the numerical simulation are also given in this section. 113 The unknown aperture electric field is expanded into a finite sum of the TM eigenmodes of the coaxial line as follows: 8 3(9) =50 K349) (536) where ' C,[Jl(kdp)Yo(kda) -Jo(kda)Y1(kdp) ] i=0 a‘(p) =) (5.27) i=0 _1_ , P and kc! satisfies the following eigenvalue equation [30] J.(k,.a)Y,(k,,b)-J,(kc,b)Yoatda) =0 (5.28) Substituting (5.26) in (5.18) leads to (from now on assume 11,4110 for simplicity) 2A ”S . . 01-11 1 [intent “a p ngpln(2)¢"o a + b 111- 1 5 . . —— Rm [Harman 1+R C 1-0 '11 P 1110 111(2) 4 . (5.29) .. " s + 2 rumour: n8,(p')a,,(p')p'dp' and 41-0 . b . 1081 1_Rbll s a I c 0 +2 81 2V9! at d .413, 1mm “(9);. . ,(p) .(p)p p which is rearranged as 114 24.1 s 11 1 1' . . °—=2V, ———f ,(p)dp 7|, p #0 flapln(_), a s ( .. " + 2 V! ii 1+2“ 16 [3105949, #0 \“I P In 1n(;) 4 (5.30) / . S b + 2 V, 2 Ym8.(p)f8.(p')8.(p')p'dp'] 1-0 :”.I 8 i=0 \3311‘1. 1+Rbll b (p) f 8,1p')st.(p')p'dp'J Galerkin’s method Galerkin’s method suggests using basis functions 81(p) as testing functions and does the following operation, b f (5.30) =8,(p)pdp .j=0---S C such that a system of linear equations is derived as follows: :o‘ni—fa (mapfaxp )dp I’Rbr l b l, 1 ' 5.31 +2V ° 8(1))«19 atom ( ) 1-o ‘ “—11+Rblo]n(£)[ j if ‘ a S a +2 V.[ 2 meaxpm, (p)dpf8(p',,)at (9")pdp] n-l PS 1931161. ”(.413 HR...” New(p)dpfa,.(p')a(p"')pdp] 115 Via the orthogonality properties (5.11) and (5.12), (5.31) is simplified to be _’ 1 b 4.414251) .-3K—1n(;)5).5 01], a 1-0 7], “‘th— 1 l-Rblolnw/a) )0 to i=0 ‘11,1+R.,, lute/a) (5.32) +2V BY“, 6 1.1.5 i=0 n=1 5 +EV Rmfa, (9)3 .(p)dpffl,(p'.)$ (9)9619 t-O 6'1 JhI‘ 1+1 +¢aRbl withj = 1,...,S. These simultaneous algebraic equations can be expressed as 4,111 = {1Y‘1+1Y‘1+1Y"1+1Y°1}1V1 (533) with the matrix elements as 2 b I) = —1n(;) 510 1 b 193’ = —1n(;) 6,, 6,0 ___l_1 -2Rbloln (bl 0) 11 1mm ln(c/a) ’° ‘° :2? (5.34) :5. =2Ym6 6 III In tun . . ' 1- 19‘:= zm’ PR“:fa,,(p)a (p)dpfa(p'.)a (pm)... Equation (5.33) can then be solved by a standard numerical subroutine for linear equations. After determining 8’(p) using (5.33), the normalized input admittance can be related 116 to Z’(p) with the help of (5.16), l b 2--—— 81 )d 1.1;: A0 ln(bl 0)]; p p (5.35) “R f: “(9) p yin: 4, 111(15/ a) Since 8’(p)'is proportional to A0, A0 will be canceled in the expression of the normalized input admittance. Setting A o to 1 leads to b b _ y)”: 1) MW of: “PM " [erode Before presenting the results of the numerical simulation for (5.33), some aspects about the evaluation of the matrix elements are addressed next. Evaluation of the matrix elements The computation for the elements of matrices Y J, Y‘ and Y b is trivial. But the computation for each element of matrix Y ‘ involves an infinite series which need to be truncated according to some preset accuracy. Also in (5.34) two integrals need to be computed for each term in the series of 19:, and it is quite time consuming. Fortunately, using the following formula each term in the series of Y]: can be converted into a closed form [20], fZ,(u p)Z,(B 9)de = “2‘: ,-,(B p)-aZ,_l(a p)Z,(B 9)] (5.37) where a tip and ZP denotes JP and Yv. To derive a closed form for each term in the series of Yfi, let’s consider the following integral, 117 b f8,(p)3.(p)dp b b =N. Y.(k,.a)fst,(p)1,(k,.p)dp —J,(k..a) f 8,0») Y,(k..p)dp where aflp) is, from (3.13), at, = Cj[Jl(kcjp)Yo(kcja) dodger/41:46)] (5.38) (5.39) With (5.37) and after some manipulation the first integral in (5.38) becomes, 5 f8,(p)J,(k,.p)pdp b = If [11(de)Yo(kcja)-Jo(kcja)Yl(kcjp)]Jl(k“ 9)de C k J k a = —’ “ 1 1.0:... b>——°( " )-J. (kg-kipkq '1 J,(k,,b) Similarly using (5.37) the second integral in (5.38) becomes b f$,(p)Y,(k,.p)pdp b = 111119608068-J.(k.,a)¥,(k,,p)1130,,»de C k“ 160‘.- a) __1_ 3 yam“ b) 1 (kg—1.3,», ,, 100‘ch -Yo(kc .0) Substituting (5.40) and (5.41) in (5.38) leads to b I a,(p)8.(p)pdp=N.C,F., a where 118 (5.40) (5.41) (5.42) F k“ [J ( b)Yo(k“a) -J,(k,,a)rr,(k,,b)] (5.43) “’ (It,2 ,f-kpkd :11: Similarly we have b . f 8MP) as “’)de =N¢ ClFai (5-44) where k 210kc( CC 'm?%1—7_C:Z)P”(k b)r,,,11t a) -,,J1k,,,a)¥1k,b)1 15.45) at With the help of (5.42) and (5.44), Y; in (5.34) becomes e l- Y;;= 2 ’w ’ 12"“ Nigcjrflrq (5.46) "1 P111+Rb1¢ The above formulation can be numerically evaluated efficiently. After all these considerations about the evaluation of the matrix elements, the simultaneous algebraic equation (5.33) can now be solved by any kind of computer code. Computer program and numerical results A fortran computer program was written for the numerical evaluation of (5.33) and (5.34) so that the aperture electric field and other quantities can be calculated when a and 11 of the material are given. The validity of computer program is first verified by computing the impedances of (5.20) for various materials. The second verification is made by comparing the shunt capacitance at the discontinuity with that of the equivalent circuit, which was Shown in the work of J. R. Whinnery [42]. The results are shown in fig. 5.4, in which the shunt capacitance per unit length 119 0.1 Discontinuity capacitance C d, (pf/cm) Figure 5.4 Discontinuity capacitance per unit length, C d’, of a coaxial air .9 O 00 0.06 ’ PCD O A 0.02 ' *** J. R. Whinnery This theory line v.s. step ratio a. 120 CP',(a) is introduced as I CA“) = (5.47 ) 2 1: ac” with a =fl as the step ratio. c-a The results of our calculation match well with that of [42] and it shows that the "forward problem" has been correctly solved while the "inverse problem", the final goal, for determining a and p of the material via the measurements of input impedance of the coaxial line will be addressed in chapter 6. 5.5 Computation of Metallic Wall Loss for Coaxial Cavity In this section, the metallic wall loss of the coaxial cavity is studied [27]. The analyses described in previous sections assume perfectly conducting cavity, that is not true in practice. The power loss due to the non-perfectly conducting wall may influence the input impedance and the reflection coefficient, which then, in turn, influences the characterization of the EM parameters of the material filling the cavity. The analysis for metallic wall loss is based on the conservation of energy (Poynting theorem) and perturbation approach. Let’s first recall the complex Poynting theorem for a closed volume V with surface P. =P4 +1'211)(UuI - U.) +PP (5-43) with p =flz-rdv 2 V a (5.49) P44743381) V 121 and (5.50) The notation P. stands for the power supplied by an active source in V, Pd for the dielectric power loss in V, and U. and U P for the stored magnetic and electric energy, respectively. Also P. stands for the power transfer over boundary surface S. For the structure in fig. 5.5, we have .750 beyond the cross section z=-l,, at which all higher order TM modes decay out and only the TEM mode exists. By letting V and I be the voltage and current associated with the TEM wave, the P: term can be rewritten as --l *1 .~ 1 r - 1 PP- PfCPa'i'xH ) zds+fM2E'( 8x1? )ds' = -.l. - l . 1 5.51 2V1 +£1.11sz K ds ( ) where w (5.52) R. = ‘lL’fp/Oc Hence (5.48) becomes 2,. (5 53) TIIF=P4 +1211) (Un- U.) +(l +})PPP, - In case of perfect conductor, PPP=0, and we have z,=2P4*’2‘°(U-‘"¢’ 15.54) I1 F 122 ~+ TEM Rm R0 V —> 11: bIaT Zin = -11 Z = 0 Figure 5.5 Geometry of a coaxial cavity driven by a coaxial line. The complex Poynting theorem is applied to the volume V beyond the cross section =-11. 123 The current I can be derived according to transmission line theory as 1= (1’ +1-)L,_,P _ 1 “1‘1 _ “13111 73V"? We ) (5.55) V; _ =_Z: (Jail! _ Rue 131,1) where R1. is the reflection coefficient of the voltage and Zc is the characteristic impedance of the coaxial line. Since the transverse dependance of electric field of the incident TEM mode is assumed to be 8’(p)=l, we have p VJ = ffflpflp =1n(bla) 55“) Therefore (5.54) leads to the normalized input impedance zh=%=zzc PdgflGUJm-U.) c 1112(2)“ _ Rh e-flp’l'r (5.57) In case of non-perfect conductor, PW 1130, a modified input impedance is derived via similar steps. The perturbation approach assumes that the aperture EM fields are not disturbed by PW so that the current I in (5.55) is unchanged. This leads to Pd +121.) (Um - U.) + (l +j)Pw b 4201 In2 — 1— e ‘ ‘ (a)| R, F =2,“ch b (“DPW 1201 2_ __ ' 11 1n(a)|l Rue f C (5.58) Also the modified reflection coefficient becomes 124 I - 1 R... = 1f- (5.59) ze+1 The influence of wall loss PW on the reflection coefficient can then be obtained by the following steps: 1. Transform Ra at the aperture, obtained numerically, to RPP-I at z=-ll by RM =R e'flprlr (5.60) and we have = “Rb- (5.61) 1’36. 71» 2. Include the influence of PW on the input impedance and the reflection coefficient by (5.58) and (5.59). 3. Transform Rh back to Ra by R; =12“ 6,251,] (5.62) In these Steps the metallic power loss P19 is calculated numerically using (5.52), in which the surface current I? is related to the unperturbed tangential magnetic fieldH. and the value of 1.57X107 S/m for the surface resistance R. is used for the brass cavity wall. That is 121?: fiXfilz = |H¢|2 (5.63) The component H. can be expressed in terms of the aperture B field by substituting (5.14) and (5.15) into (5.6). As soon as the aperture E field is solved, H. and PW can be computed numerically. 125 Skipping the computational details, the results of the metallic wall lossPw normalized by the input power and the dielectric power loss Pd are shown next. Figure 5.6 shows PW, normalized by the input power, for acrylic and water, where the cavity of length 1" is fully filled with the material and the publiShed value of e for each material is used for computation. The peaks shown in the figure occur at resonance frequencies. Also fig. 5.7 shows the same Pw as normalized by Pa for acrylic and water. Even though the metallic wall loss is small compared to the input power, it may be large when compared to the dielectric power loss, especially for low-loss material at low frequency range, thus influencing the dielectric constant characterization. Figure 5.8 shows PP, for acetone (e,=20, £50.05) normalized by the input power and dielectric power loss, respectively. It shows that the power loss of metallic wall is important in the EM characterization of the low-loss and high-permittivity materials. 126 g 0.02 - - PC C _e M T Y Normalized by input power 0.01 0.005 - "mhm o 1 1 J J 1 1 0 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 ’. 4’. . "mfimmw' Frequency (0112) Figure 5.6 The power loss of non-perfectly conducting wall is normalized by the input power, where the filled material is water and acrylic, respectively. 127 000 g v v y Y r ‘ r 008= . a 0.07 )- water . .2 {i 0.06 - . = .0 1,3 0.05 )- _ . .11 .2 Q 0.04 - . 3‘ E i 0.03 " “ E 2 0.02 e . 0.01 = . 0 1 1 A L A 1% 1 L 0 0.2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 Frequency (6112) (a) 0.55 . . . . . . . - . 0.5 _ 4 .. 0.45 h = E “‘1 1c .3 0'4 i" ‘1. m -1 i. o 0.35 L , . = (P E . E 0 3 r . """" ~ P i Q , ’ ' >~ ‘, P-’ \P g 0.25 ’— “ a" ‘ \.\ -‘ ‘3 0.2 )- ~. . O I .I z 0.15 P ‘ 'l «4 0.] )' 9‘ ..... i- "1P4 0.05 1 - i 1 i A i . A i. 0 0.2 0 4 0 6 0.8 1 1 2 1 4 1 o 1 8 2 Frequmcy (0H2) (b) Figure 5.7 The power loss of non-perfectly conducting wall is normalized by the dielectric power loss, of which the tilled material is (a) water, and (b)acrylic, respectively. 128 0.12 Y ' V 1' V V Y Y Y 0.1 ~ - acetone 9 8 I 1 Nonnelindbyhputpower .0 8 0.02 - j 0 l 1 A A 4 A A A A 0 0.2 0.4 0.6 0.8 l 1.2 l .4 1 .6 l .8 2 Frequency (GI-(z) (a) 0.8 . . . . . a r . f 0.7 .- » -4 a acetone § 0.6 r- 4 § 8. 05 ~ - S 0.4 r- _ .1 .B‘ a 0.3 - - 0.2 - ~ 2 0.1 - o ' . l . . J 4 . l 4 O 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 Frequency (GI-l2) (b) Figure 5.8 The power loss PW of non-perfectly conducting wall when the cav- ity is filled with acetone. Pw is normalized by (a) the input power, and (b) the dielectric power loss, respectively. 129 CHAPTER 6 EXPERIMENTAL MEASUREMENT FOR VARIOUS MATERIALS 6.1 Introduction In this chapter, the measurement procedures to obtain the complex permittivity e and permeability p of materials are discussed and the results for 8 and p of several materials are presented [44]. Using either the open-ended coaxial line probe or the coaxial cavity system that were analyzed in previous chapters, the complex a and p can be measured with a network analyzer connected to the measurement setups as shown in figs. 6.1 and 6.2. As mentioned before, two measurements of the probe inputiimpedance are needed to determine both 8 and 11. For the open-ended coaxial probe, as shown in fig 6.3, this can be achieved by using open-circuit and short-circuit cases, or preparing two samples of the same material with different thickness. Another alternative is to measure an unknown material by inserting some known material (usually air) between the aperture and the unknown material. For the coaxial cavity system, this can be achieved by setting the backwall of the cavity at two different locations. The results of these two measurements are used for solving the simultaneous equations for e and p by numerical inverse scheme. This procedure is then repeated for each frequency of interest. 130 .mEtBaE we E80882. SE 05 2388 9 E893 Boa 3530 895.5% 5 he 9:8 _ScoEtomxm ~62sz Bah—«SW “1950: 9 ‘ 880580 33 .md 3 Al 058 .<.z “98:80 on». .md 0:: be 3550 EE 3 Babe—a x836: owns: 58% coco“ _SoE 131 20:02:: go £20883 Em 05 050008 8 E82? .3300 3:88 a 00 9:8 _ScoEtoaxm ~605me Saba—a x850: 9 880580 093 dd 8 Al 038 .<.Z 580550 09: .md lv 0:: :0 3580 EE 3 Iv honing 0:950: .50an coca IV % 3:0qu 565?: 033-20 03:2: 132 JUL JUL metallic plate (a) Zin] (b) Zm ._ . known material I {1.3 I: Zinl (C) Figure 6.3 Two input impedance measurements are made by (a) using open-circuit and short-circuit cases, (b) preparing two samples of different thick- nesses, and (c) inserting some known material for the second one. 133 In section 6.2 the calibration procedure for the measurement setups is discussed [43,45]. The scattering parameters for the equivalent two-port network between the probe aperture and the measurement reference plane of a network analyzer can be determined via the calibration procedure. With these scattering parameters the measured input impedance at the reference plane can then be converted to the input impedance at the probe aperture. In section 6.3 the Newton’s iterative method is presented [46,47]. Newton’s method is used to solve the simultaneous nonlinear equations, which are resulted from the two measured input impedances. In sections 6.4 and 6.5 the measurement procedures and the measurement results of various materials via the open-ended coaxial probe and the coaxial cavity method, respectively, are presented. The limitations for the use of these probes are also addressed. 6.2] Calibration Procedure for Experimental Setups In this section, the calibration for the experimental setups is studied. General effects of the GR. type connector, the teflon bead and the phase shift of the transmission line, which are located between the probe aperture and the measurement reference plane, are described by the scattering parameters of an equivalent two-port network as shown in fig. 6.4. By determining the scattering parameters of the two-port network, the measured input impedance and reflection coefficient at the reference plane of a network analyzer can be converted to the input impedance and reflection coefficient at the probe aperture, or vice versa, as follows [43]: 134 ; GR. Type Bead Connector ‘ Pr be Refegence Plane Aperture Network Analyzer (21) R1 R2 0— »———0 Port 1 D [S] > Port 2 o— ——o g ! Reference Plane Probe of Aperture Network Analyzer (b) Figure 6.4 (a) The structure of the coaxial line probe setup. (b) Its equivalent two-port network. 135 R2: R1 -8“ (6.1) (R1 -Sll)SZZ+SIZS21 01' __ SIZSZIRZ +5 - 11 (6.2) l 'Szsz R, From (6.2) it is clear that only three combinations of scattering parameters $11,822 and 812521 need to be determined. Determination of the scattering parameters using 3 shorting stubs In order to determine the scattering parameters of the two-port network, three shorting stubs of different lengths 1,, 1,, and I, are fabricated. These shorting stubs are designed to completely fill the space between the inner and outer conductors as shown in fig. 6.5. The scattering parameters between the terminal plane 1 (the measurement reference plane) and the terminal plane 2 (the shorted end at z’=0) are determined, and the segment of the coaxial line between z’=O and the probe aperture is treated as a perfect transmission line. With the help of (6.2), the scattering parameters between the measurement reference plane and z'=;0 can be determined experimentally as follows: With reference to fig. 6.5, R1“ , R: and Rf denote the measured reflection coefficients at the terminal plane 1 with the three shorting stubs filled in the coaxial line. 12;, R; and R; denote the theoretical reflection coefficients at terminal plane 2. Then from transmission line theory, we have 136 Metallic Shortin g Stub GR. type Connector Measurement Reference Plane (a) Figure 6.5 (a) The structure for the calibration of the coaxial line probe using three shorting stubs. (b) Its equivalent circuit is described by the S parameters and one segment of perfect transmission line. 137 _la ' 3 > z' = 2 +10 Measurement 2' ___ 0 Reference Plane R, ; R2 R0 6____ a 1' Perfect TL. [8’] . , l .1 Terminal Terminal Probe Plane 1 . Plane 2 Aperture (1)) Figure 6.5 (continued) 138 x;=-1 R: = _e'flWa (6.3) R; = _e-JZN“ where fl=2n/10, (“flu-lb and (“flu-(c. Substitution of the measured Rf, R,” and Rf and (6.3) in (6.2) leads to s' ' " . R:=_12_‘S?LR1+511 (6.4) l"$22K: where 0t=a, b and c. If the lengths la, 1,, and [c are chosen such that these equations in (6.3) are independent, the scattering parameters can be solved to be S. = (Rf-Rbckf-Rb—(Rf-Rhtkf-efi 2’ (Rf-Rhmf-Rz‘mé-(Rf-RMRE-RDR; c b S's-RV"?1 1 s' " 1 s' ‘) ' (65) 12 21’ c b( " 22R2)( ' 22R2 ' ‘32 ' c SIZS'ZIR; 511=R1'—'—.—: 1'522R2 Because the amplitudes of R: , a=a,b and c, are one for all shorting stubs, the measured information is carried by the phases. To assure that the choice of la, 1,, and I, does lead to 3 different phases over a wide band of frequencies, it is necessary to make more than 3 measurements with these shorting stubs located at different positions randomly. In practical situation, up to 6 measurements were made to assure a good solution for the scattering parameters. 139 Since the scattering parameters for the two-port network between the measurement reference plane and z’=0 are determined, and the segment of coaxial line from z’=0 to the probe aperture can be treated as a perfect transmission line, the transformation of the reflection coefficient from the measurement reference plane of a network analyzer to the probe aperture can be made in 3 steps as follows: 1. Measure Rl using the network analyzer. 2. Transform Rl to R2 at z’=O plane by (6.1) = R! ”S“ (6.6) (R1 "510322 +5'12521 3. Transform R2 to Rd at the probe aperture as R. = Rzetzw. (6.7) Calibration for the Coaxial Line Probe with a Spacer A In case a liquid material is measured, it’s important to prevent the leakage of the liquid into the probe and assure a flat surface of the liquid at the aperture. To accomplish this, a spacer made of teflon is made to completely fill the space between the inner and outer conductors. It presents a flat surface at the probe aperture as shown in fig. 6.6. The length of the spacer should be large enough that all the higher order TM modes decay out before reaching the z=-ld interface. Otherwise the reflected higher order modes will complicate the theoretical analysis inside the coaxial line. It is still possible to do the calibration for this new structure using 3 shorting stubs in a similar manner as before. If the spacer is inserted into the coaxial probe after the reflection coefficient measurements of the 3 shorting stubs are made, then the scattering 140 OR. type Teflon Connector Bead 7! R,e i Measurement ‘ Spacer Reference Plane E 3 + i -10 -Id 0 Z 1 > 0 z' = 2 +10 R1 R2 R a S ; Perfect ‘ Perfect :. [ l 2 TL. , TL. .3..— 4 e : 20 l 2. Terminal Terminal Probe Plane 1 Plane 2 Aperture Figure 6.6 The structure of the coaxial line probe with a spacer of length 14. Its equivalent circuit is described by the S-parameters and two segments of perfect transmission line. 141 parameters of the two-port network between the measurement reference plane and z’=0 can be obtained, and the coaxial line between z’=O and the probe aperture can be treated as a perfect transmission line with two segments filled with different materials (air and teflon). Therefore the transformation of reflection c0efficient from the measurement reference to the probe aperture can be achieved. 6.3 Numerical Inverse Algorithm With the calibration procedure described in previous section, the measured input impedance at the measurement reference plane is transformed to the probe aperture plane. The input impedance at the probe aperture is an implicit function of e and u as well as the thickness of the material, etc. Conceptually, two equations are needed to solve for two unknowns e and it. Thus the complex 3 and u can be inversely determined by some numerical scheme if two input impedances are measured. In this section, Newton’s iterative method of finding the roots- of nonlinear equations is presented for the sake of completeness [46,47]. Let’s consider two general equations as f(x.y)=0 (6.8) 8(x.y) =0 At each iterative step of the algorithm, the objective is to find h, and hz so that (6.8) is satisfied approximately as follows: f(x +h1 :y +h2) .0 (6.9) 8(x+h1.y+hz)-0 142 By Taylor’s expansion and with the series truncated in 1st-order, (6.9) becomes .21.“ .3_f . f(x.y) axh‘ 6th 0 a a (6.10) _Qh _Q :0 g(x.y)+ax 1+0th . or l2: at 6x 32’ ”I = _ few) (6,11) Q 25 *2 any) 6:: By This leads to the solution for h, and h; as '5’! if 1'1 h1]=_ 6x 6y f(x.y) (6.12) h: is: g 800') _6x 3;», If (6.9) is satisfied according to some preset accuracy then the iterative process is stopped and the final values of x and y are the roots of the equations. Otherwise, the searching of new h, and h2 is repeated. Application of this algorithm to determination of e and u is as follows. With two measured input impedances, Z” and Zm, as shown in fig. 6.3, the values of e andu are searched so that the theoretical input impedances at the probe aperture, ZW and Z , match the measured ones. Therefore (6.8) becomes f(e.u)=Z,,,,(e.u) -Z.,,, =0 (6.13) 8(e.u)=lm(e.u)-Zm=0 With the above algorithm, we can determine the correct values of e and u that lead to the two measured input impedances. 143 6.4 Experimental Results via' Open-Ended Coaxial Line Probe Method In this section, we’ll present the results of complex permittivity e and permeability u of various materials measured by an open-ended coaxial probe. With reference to fig. 6.], a 14mm coaxial air line (a=0.3102cm, b=0.7l45cm) is connected to a 16"X16" metallic flange. The metallic flange is surrounded by an acrylic wall in order to accommodate liquid materials. The following procedures are the measurement steps for e and it using an HP8720B network analyzer [44]: 1. Set up the network analyzer. 2. Do the calibration for the network analyzer. 3. Do the calibration measurement for the fixture using the shorting stubs. 4. Calculate the 8 parameters for the equivalent two-port network of the fixture. 5. Put the material inside the fixture. 6. Measure the reflection coefficient. 7. Calculate the reflection coefficient at the probe aperture. , 8. Compute e and 1.1 using numerical inverse algorithm. 9. Plot 8 and it. Through these steps the EM parameters of four different materials were measured. These include a low permittivity material (acrylic), medium permittivity materials (acetone and Eccosorb-LsZZ) and a high permittivity material (water). The frequency range coversfrom 0.2 GHz to 2 GHz. It was found that for the open-circuit case the metallic flange is large enough to act as an infinite plate as the theory assumes. But for the short-circuit case, this assumption of infinite plate was valid for high loss material only. For low loss material, the radial guided waves are reflected from the edge of the flange, and a resonant phenomenon 144 occurs due to the high Q characteristic of the short-circuit structure. This suggests the use of open-circuit case only for low loss materials. Also it was found that the contact of the measured material with the open-ended coaxial probe is very important. An unintentional air gap may cause considerable inaccuracy in the final results. In addition, a good calibration for the fixture is essential for consecutive measurements. With three calibration stubs of different lengths (l,=5.84cm, 1,,=3.89cm and lc=l.88cm), a good calibration can be obtained for the frequency above some critical frequency fc, about 0.5 01-12 in our case. For the frequency below f,, a good calibration is not easy since the wavelength is so large that these shorting stubs do not provide well-separated phases. The critical frequency fc is then limited by the length of la. The difficulty in calibration at low frequency may be overcome if the coaxial feed line and shorting stubs are made longer. It is observed that there exists an ill-conditioned property in this method at the low frequency range in the determination of EM parameters of materials. That is, a small inaccuracy in the measurement data can lead to a large error in the final results of EM parameters of the materials. Figures. 6.7(a)—(b) show the measured permittivity e of acrylic when the permeability 11:11,, is assumed. Two measurements are shown using the setup of fig. 6.3(c): one with an air layer of 2.3mm between the probe and the material and the other without an air gap or a direct contact between the probe and the material. The results from these two measurements are very consistent (6,526, and 8,50), and the differences are within 4 percent for 6, over the most part of frequency. Using these two close measurements, the numerical inverse scheme generates simultaneously both 6 and p as shown in figs. 6.8(a)-(b). Figure 6.8(a) shows good results for 8 but the results for p are 145 3.2 r . ~ . . . . - . without air layer 2.8 l- ~.-.-.-.- with 2.30mm air layer - real part of relative permittivity 2.6 — 2.4 l‘ « 2.2 l- a 2 A A - A A A A A A 0 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 Frequency (GHz) (a) 0.4 V I V V V V Y 0.2 r- q 2.“ IE 5 0 - E 8. .‘z’ -0.2 L . E 35 without air layer E- 414 t -.-.-.-.- with 2.30mm air layer ‘ Z‘ a -E '0.6 "' -t 3 .E -0.8 ~ 4 -1 A A A A A A A A A 0 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 Frequency (0112) (b) Figure 6.7 The permittivity e of acrylic via open-ended coaxial probe (15110 assumed). (a) The real part of relative permittivity. (b) The imagi- nary part of relative permittivity. 146 3.5 I T Y T I r f I 1 3 ,- WM 2.5 *- 4 3‘ 2 " .1 IE _ real part .2 ,5 _ . . , E -.-.-.-. imaginary part 8. .2 l- . .5. 0 h 0.5 - «4 0 h- ................................ - ‘ _. .05 » - -1 1 J L i 0 0.2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2 frequency (GHz) (21) Figure 6.8 The permittivity e and permeability u of acrylic via open-ended coax- ial probe. (a) The relative permittivity. (b) The relative permeability. 147 b -5 g. g 40 > '3 .5! 2 45 -20 .‘ A A A i A J A L 0 02 DA 06 03 l 12 L4 L6 L8 2 .MmmquGHa (b) Figure 6.8(b) (continued) 148 unstable as shown in fig. 6.8(b). This is probably due to the ill-conditioned property of this method to simultaneously determine 8 and p for low permittivity materials for the low frequency range and difficulty in system calibration for the low frequency range. In addition, fig. 6.9 shows the results of the permittivity of acrylic for the frequencies up to 10 GHz. The values are within a reasonable range for all the frequencies, a slightly larger deviation found in the frequencies above 5 GHz may be due to the use of the GR type connector or the effect due to the excitation of complex waves. Figures 6.10(a)-(b) show the permittivity and permeability of acetone. The results were obtained from two measurements of two samples of different thicknesses, 1.27mm and 2.54mm, with the setup of fig 6.3(b). The results for complex permittivity are good (6,521, and 850), but the results for complex permeability again show the ill-conditioned property of this method at low frequency, as 11:11,, is expected. As mentioned earlier, difficulty in system calibration for frequencies below fc may also contribute to the poor results for complex permeability at low frequency range. ' Figures 6.1 l(a)-(b) show the measured permittivity e of an absorbing material (Eccosorb-L522) when the permeability 11:11,, is assumed. Since Eccosorb is a high loss material, the two measurements were made by using open-circuit and short-circuit setup as shown in fig. 6.3(a). For a sponge-like material as Eccosorb, it is hard to secure a very good contact with the coaxial probe. Especially it was found that the contact with the small center conductor of the probe is very important. Experimentally, a small spot of silver paint was painted on the surface of the Eccosorb to ensure a good contact with the probe. Two set of results generated from the open-circuit and the short-circuit cases are reasonably good (both 8, and e, vary with frequency), but they are not very consistent. During the experiment, it was found that the shape and the EM properties of the material 149 3.5 2.5 ’WVMMA/VM 1.5 - . I g relative permittivity \ I \ .v\ o ..... o‘-‘- o'- 4‘ "‘ -‘. ’ s 4‘ I O I \ I \ O ‘ ." ..... - " ‘~” ‘\ o, ‘ ' """" l ‘\’ """"" ’ ‘,’ \. I ~ 1. 10 to “i M 0.. q oo 0 frequency (0H2) Figure 6.9 The permittivity e acrylic via open-ended coaxial probe in the fre- quency range of 2~10 GHz. 150 20P\———J\\/\/\/\/\/—N 15" c4 10]- El ii 0 -, - . -5. . _, 0 1 2 3 4 5 6 7 8 9 10 frequaley(GI-Iz) (a) 4 relative permeability -1 - 1 _2 : A A A A A A 0 l 2 3 4 5 6 7 8 9 10 frequency (G112) (b) Figure 6.10 The permittivity e and permeability u of acetone via open-ended coaxial probe (a) The relative permittivity. (b) The relative per- meability. 151 real part of relative permittivity imaginary part of relative permittivity Figure 6.11 18 I’ 1 r v v 1 f T7 v lo» 3 . short-ck! open-ctr 10- ‘5 s '- ‘9 Q ‘t- . Q - ‘s - ...... Q.- -- --- -.- O .......... ..... ...... I») 0 0:2 0:4 0.6 0.8 1 1:2 1:4 l.6 1:8 freqmcy (0112) (b) The permittivity e of Eccosorb-Ls22 via open-ended coaxial probe (u.=|.toassumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. 152 change when it is pressed due to its softness. Because of this reason it is difficult to obtain simultaneously 8 and p of this kind of materials. Figures 6.12(a)—(b) show the permittivity and permeability of distilled water. The results were obtained from two measurements of two samples of different thicknesses, 1.1mm and 2.3m, with the setup of fig 6.3(b). The measured results are (e, 578, and i5-2~--8) and (11,51, and p, 50) which are considered to be quite satisfactory. It is noted that the ill-conditioned behavior of the method was not observed when the material is of high permittivity type. 6.5 Experimental Results via Coaxial Cavity Method In this section, the results of EM parameters of various materials measured by a coaxial cavity system are presented. With reference to fig. 6.2, a coaxial cavity (cavity radius 5") driven by a 14mm coaxial air line (a=0.3102cm, b=0.71450m) was fabricated. The center conductor of the coaxial cavity was made removable in order to facilitate the calibration of the 14mm coaxial line. The measurement steps in section 6.4 can be applied to the coaxial cavity method as well. Through these steps the EM parameters of three different materials were measured. These include a low permittivity material (acrylic), medium permittivity material (Eccosorb—Ls22) and high permittivity material (water). The freqUency range covers from 0.2 GHz to 2 GHz. It was found that the contact of the movable shorting backwall with the cylindrical wall is very important In the experiment, aluminum foils were placed between them to achieve a better contact. In addition, a good calibration for the 14mm coaxial line, as 153 80 I 60 ~ .1 2: __ real part E, -.-.-.-.- irmgary part .= 40 . . E 0 G- 20 i' '1 0 '- ., 4A 0 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 frequency (GI-Iz) (a) Figure 6.12 The permittivity e and permeability u of distilled water via open-ended coaxial probe. (a) The relative permittivity. (b) The relative permeabil- ity. 154 2.5 I 1 T ' ' 1 2 I _ real part . -.-.-.-. imagary pan permeability 0.5 . . 0 - ........................ . .................................................... -0.5 - - _l A" A A A A A A A A 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 frequency (6112) (b) Figure 6. 12(b) (continued). 155 shown in fig. 6.5, is essential for consecutive measurements. With three shorting stubs of different lengths (la=5.84cm, lb=3.89cm and lc=l.88cm), the calibration can be made by removing the center conductor of the coaxial cavity. It was found that the calibration was quite good for the frequency above some critical frequency f, about 0.5 GI-lz in our case. For the frequency below f,, a good calibration is not easy since the wavelength is so large that these shorting stubs do not provide well-separated phases. The critical frequency fc is then limited by the length of la. The difficulty in calibration at low frequency may be overcome if the coaxial feed line and shorting stubs are made longer. Figures 6.13(a)-(b) show the measured permittivity of acrylic (852.6, and 8,50) 1 when the permeability 11:11,, is assumed. The effect of metallic wall loss is shown. The effect of the metallic wall loss on the imaginary part of e was expected to be considerable for low frequency range, but it was not evident because the results at this frequency range are not stable due to the poor calibration for frequencies below fc. Figures 6.l4(a)-(b) show the measured permittivity of an absorbing material (Eccosorb-L522) when the permeability p=po is assumed. The results are compared to that obtained by the open-ended coaxial probe method. The results are reasonably good in the high frequency range, while some inconsistency in the low frequency range may be due to the poor calibration for frequencies below f, and the ill—conditioned property of this method. Figures 6.15(a)-(b) show the measured permittivity of distilled water when the permeability 11:11,, is assumed. The results are compared to that obtained by the open- ended coaxial probe method. It was found that the determination of EM parameters is ill-conditioned at the frequencies near cavity resonance even though the effect of metallic wall loss is considered. 156 2.9 - with wall loss ‘ ........ without wall loss 2.7 ~ 2.6 ~ ' " 2.5 - . 2.4 - - real pan of relative permittivity 2.2 I l 2.1 ~ . 0 0.5 1 1 .5 2 frequency (61-12) (a) Figure 6.13 The permittivity e of acrylic via coaxial cavity method (14:11.0 assumed). The effect of cavity wall loss is shown. (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. 157 0.05 0 - - 3‘ IS A g -0.05 - 2 .3 j with wall loss “5 -0.1 F ........ without wall loss - 5 G. N 2. -0.15 - r -0.2 ' ‘ ‘ ‘ 0 0.5 l 1.5 2 frequency (6112) (b) Figure 6. 13(b) (continued). 158 40 T T I I Y 1 35 " .4 via cavity method :3 30 P via open-end coaxial probe method ‘ -: 'E 25 L 4 8. o .> 3 20 L « 9. 3:; 15 - 4 53.3. 10 _ . 5 " v d 0 J 0 0.2 0.4 0.6 0.8 l 1.2 1.4 1.6 1.8 2 frequency (6112) (a) Figure 6.14 The permittivity e of Eccosorb-LSZZ via coaxial cavity method (p.=p.o assumed). (a) The real pan of relative permittivity. (b) The imaginary part of relative permittivity. 159 -5 1 via cavity method ---- via open-end coaxial probe method L -10 r- 2: -15 ~ 272' -§ '20 "' , 8. 2 45 - 3 -30 r- 9 .' 3 -35 ~ an .‘ -40 ~ : 45 - -50 1 L 0 0 2 0.4 Figure 6. l4(b) (continued). 0.6 0.8 l 1.2 l .4 1.6 l .8 frequency (6112) 0)) I60 95 . . T 90 - -.-.-.-. by coaxial cavity ---- by open-end coaxial probe >s , ’ 'E 85 - E 8- 80 - (___ 5"". 3' .-. .' o o . .-" '.. ~.' ‘J ....... . - > N ‘ . g . ‘ . - g .. ...... 2 ------------------------------------------- a ------------- "t -------------- o I .2: 75 - S '53 70 ~ 65 P 60 1 1 L l 0 0.5 l 1.5 2 frequency (0H2) (a) Figure 6. 15 The permittivity e of distilled water via coaxial cavity method 01:11.0 assumed). (a) The real part of relative permittivity. (b) The imaginary part of relative permittivity. 161 -2 imag part of relative permittivity -10 -12 -.-.-.-. by coaxial cavity . .. "‘t~ --—- by open-end coaxial probe \ S \‘ . I .,o -_. \\:‘ S .O ‘ 'i . \‘. I '0‘ o 3 "‘\ ' \A ‘ “ l o ‘ \~‘ - \ I . s“ 0' s“ . ~ ‘b‘ . I . ' ‘. ~ ~ 3 '9 i | ‘s‘.' . at 9" ' I ~ ~. I t l I I s‘ ‘ ' . . l ' t I . g ‘s‘ v . t I o . ‘s‘ . . I . a. 9 9 I ‘0. o 0 _ I ‘s‘ ‘ I |.‘ I ‘ . .-. _ \' a b l 1 L 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 frequency (GHz) 0)) Figure 6.15(b) (continued). 162 1.8 2 CHAPTER 7 CONCLUSION ‘ This thesis studied two schemes of measuring the conductivity, permittivity and permeability of various materials using two coaxial probe systems over a wide frequency band (0.2 to 10 Ghz). The works of this thesis have some contribution to the EM characterization of materials, especially for the theoretical development of the open-ended coaxial line probe technique. The study of the coaxial cavity system also provides a new scheme of EM characterization for materials. Using an open-ended coaxial probe system, a non-destructive measurement can be achieved easily by simply placing the probe against the material medium. To overcome some weaknesses of the open-ended coaxial probe scheme, a coaxial cavity system was developed to excite EM fields which penetrate into the material medium. This system is suitable for measuring EM properties of liquid materials, but solid material samples need to be machined to fit in the cavity. Theoretical full wave analyses of these systems lead to an integral equation for the electric field at the aperture of the driving coaxial line. The method of moments was applied to transform the integral equation into a set of simultaneous algebraic equations so that the numerical solution for aperture electric field can be obtained. After the aperture electric field is obtained, other quantities such as the input impedance and the EM fields inside material can be calculated. The input impedance of the coaxial probe 163 is then used to determine the EM parameters of the material inversely when the input impedance is measured experimentally. For the open-ended coaxial probe placed against a material layer, the physical pictures are revealed clearly through the analysis of complex waves. For the frequencies at which no power is radiated or carried away by the surface waves, the excited EM field is localized at the aperture, thus, the assumption of an infinite metallic flange employed in the theoretical analysis is justified. This also provides the validity for the quasi-static analysis which has been used by other workers. However, at the frequencies where the radiative power and the surface wave power are significant, the influence of these complex waves on the characterization of EM parameters is important and the effect of finite flange should be investigated in the future. In addition, an efficient way to compute the radiative wave power was derived, and the analysis of the power balance also gave confidence to the evaluation of the radiative power from the open-ended coaxial probe. In the coaxial cavity system, it is found that at the resonant frequencies, the determination of the EM parameters of materials becomes ill-conditioned. The inclusion of metallic wall loss gives little improvement on the ill-conditioned problem, but it is important on the determination of the EM parameters for low loss materials, especially at the low frequencies. A series of experiments was conducted to measure the input impedances of the coaxial probe systems for various materials, which included low, medium and high permittivity materials. A good agreement between the experimental results and published data was obtained. Further improvements may be possible with additional investigation, which includes a better calibration technique with longer shorting stubs, minimization of the edge effect due to a finite flange on the radial guided wave excited in the open-ended 164 coaxial system when the material layer is shorted by a metallic plate. Finally, these systems may be modified and extended to measure EM properties of anisotropic materials. 165 APPENDIX FORTRAN COMPUTER PROGRAM ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc Program: EMparameters Class : Main program Call to: calibration, findkcn, cnewton, cjacobian, cnewtonfx, admittance, solveqns, polen, newtr, Besst, Bessfl, Romerg, fxvalue, quard, crbnfx, swpower, radialpower. Purpose: To compute the permittivity and permeability of an unknown material, which is placed against an open-ended coaxial line probe with a metallic flange. Items of interests - input admittance, aperture fields, complex wave power. Method: Newton’s method is used to inversely determine the permittivity and permeability of materials so that the theoretical admittances of the probe fit the measured ones. MOM is used to solve the integral equation for the aperture electric field via coaxial eigenmode expansion and Galerkin’s technique. Cauchy residue theorem and saddle point method are employed to calculate the complex wave power. Specifics: All units are in M.K.S. system. Program hierarchy for main program: EMparameters calibration findkcn cnewton cjacobian cnewtonfx admittance solveqns Program hierarchy for admittance subroutine: admittance polen newtr bessf0 romerg fxvalue, fxvalue2, fxvalue3 quard2, crbnfx solveqns swpower radialpower ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 166 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C c parameters and options are chosen and the experiment c data are read. Newton’s method is called to solve c the EM parameters. cccccccccccccccccccccccccccccccccccccccccccccccccccc This is PROGRAM PARAMETER (dim1=201,dim50=50) COMPLEX*16 caseepv(2,3),casemuv(2,3) COMPLEX*16 epv(65),muv(65),admit0(20) COMPLEX*16 zeros(4,diml),epi,cx(dim50),roots(dim50) COMPLEX*16 admitmeasured(dim50),y(2,diml) the main program, in which the initial EMparameters COMPLEX*16 updateadmit(20),cj REAL*8 tolerance,epdelta,datain3,datain4 REAL*8 a,b,fre,conv,datain1,datain2,pi REAL*8 datain0,frequency(dim1),scalefactor REAL*8 mu0,ep0,z(65),kcn(100),casez(2,4) INTEGER maxn,openckt,M,layerno,caseopenckt(2) REAL*8 x(20),p(20),accuracyr(20),accuracyi(20) REAL*8 absf,slope,wei(50,50).Xpt(SO,50) REAL*8 guessptr(2),guesspti(2) INTEGER cntrst5,cntrend5,step,step5 INTEGER unknownno,unknownep(20),unknownmu(20) INTEGER functionalmethod,expf1ag INTEGER asymlversion,asymZversion,cntr INTEGER versionfokntoinfi,cntrst,cntrend INTEGER rx,outputfxno,contourint,i,j,n INTEGER flagbessf,reevaluate,data1,data2 INTEGER swcontri,idxf,caseno,kcnf1ag,kcnno CHARACTER*12 filel(2) COMMON /paracasez/casez /paracaseno/caseno COMMON /paracaseepv/caseepv,casemuv COMMON /paracaseopenckt/caseopenckt COMMON /paraswcontri/swcontri /paraflagbessf/f1agbessf COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON /parareevaluate/reevaluate /para/tolerance,intno /paraabM/a,b,epi,M /parafre/fre,conv /paramuv/muv /para7/1ayerno /para9/epv,z /paraopen/openckt /parakcn/kcn /paraadmitmeasured/admitmeasured /paraunknownno/unknownno /paraunknownep/unknownep,unknownmu /parap/p /paraepdelta/epdelta /varabsf/absf /parafunctionalmethod/functionalmethod /paraupdateadmitlupdateadmit /paraasym1version/asymlversion,asymaversion /paraaversionfx2kntoinfi/versionfx2kntoinfi /paraoutputfxno/outputfxno /paracntrst/cntrst,cntrend,step /paracontourint/contourint /paraslope/lepe /xandw/xpt,wei cccccccccccccccccccccccccccccccccccccccc input the parameters and options cccccccccccccccccccccccccccccccccccccccc C 167 10 20 3O cj=(0.d0,1.d0) pi=atan(1.d0)*4.d0 mu0=4.d-7*pi epO=1.d-9/pi/36.d0 WRITE(*,*)’input caseno:' READ(*,*)caseno WRITE(*,*)'caseno=',caseno WRITE(*,*)'input openckt(1/O)? (of casel case2)’ READ(*,*)caseopenckt(1),caseopenckt(2) openckt=caseopenckt(1) ‘ WRITE(*,*)’input a,b,M,epi,fre,conv:’ READ(*,*)a,b,M,datainl,datain2,fre,conv epi=datainl+cj*datain2 WRITE(*,*)'input layerno:' READ(*,*)1ayerno DO 10 i=layerno,l,-1 WRITE(*,*)’input epv(',i,’)',' ,muv(',i,')=' READ(*,*)datainl,datain2,datain3,datain4 ,datain5,datain6,datain7,datain8 caseepv(1,i)=(datain1+cj*datain2)*epO casemuv(1,i)=(datain3+cj*datain4)*muO epv(i)=caseepv(1,i) muv(i)=casemuv(l,i) caseepv(2,i)=(datain5+cj*datain6)*epO casemuv(2,i)=(datain7+cj*datain8)*muO CONTINUE casez(l,layerno+1)=0.d0 casez(2,layerno+1)=0.d0 DO 20 i=layerno,2,-1 WRITE(*, *) ' input thickness ( ’ , i, ’) (of casel case2) ’ READ(*,*)datainl,datain2 casez(1,i)=casez(1,i+1)+datain1 casez(2,i)=casez(2,i+1)+datain2 z(i)=casez(1,i) CONTINUE WRITE(*,*)'input tolerance,intno' READ(*,*)tolerance,intno WRITE(*,*)'input epdelta’ READ(*,*)epdelta unknownno=0 DO 30 i=1,layerno READ(*,*)data1,data2 unknownep(i)=data1 unknownmu(i)=data2 unknownno=unknownno+data1+data2 CONTINUE WRITE(*,*)'input flagbessf,reevaluate:’ READ(*,*)flagbessf,reevaluate WRITE(*,*)'expflag,contourint’ READ(*,*)expflag,contourint WRITE(*,*)’input slope:’ READ(*,*)slope WRITE(*,*)’input kcnflag, no of point’ 168 READ(*,*)kcnflag,kcnno OPEN(2,FILE='xp.dat') OPEN(3,FILE=’wp.dat') READ(2,*)rx READ(3,*)rx DO 40 j=2,rx DO 40 i=1,j READ(2,*)xpt(j,i) READ(3,*)wei(j,i) 40 CONTINUE CLOSE(2) CLOSE(3) WRITE(*,*)’input cntrend5,cntrst5, :’ READ(*,*)cntrend5,cntrst5,step5 IF (kcnflag.EQ.l) THEN CALL findkcn(a,b,kcnno,'kcn.dat') ENDIF CALL rvectorbin=datain2 60 CONTINUE WRITE(*,*)’input filename' READ(*,*)file1(l),file1(2) WRITE(*,*)’fi1e1(’,1,')=’,file1(1) WRITE(*,*)’filel(’,2,')=',file1(2) DO 80 i=1,unknownno OPEN(i,FILE=filel(i)) READ(i,*)sca1efactor DO 70 j=1,dim1 READ(i,*)datain0,datain1,datain2 frequency ctr3 total —> tota13 fmm ->fmm3 These two subtoutines are called only when a triple c integration is encountered. Cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A subroutine describing the integrand of a integral c which is done by cromerg subroutine. Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc SUBROUTINE fxvalue(cx,y) COMPLEX*16 y,kN,rbn,cx,tota12,ckc,ckcpass,yc COMPLEX*16 zarg,jl,y0,yl,h20,h21,w,epv(65),epidummy COMPLEX*16 chkciab,cj0ijab,CjOkca,chkcb COMPLEX*16 k(65),cn2(0:50),gama1,gama2,sol(50) REAL*8 kcn(100),kc,beta1,beta2,jOkcab(65) REAL*8 zdummy(65),kcpass,a,b,x1evel,ylevel,x,upper REAL*8 kci,kcj,kcjpass,kcipass INTEGER fxnoZpass,ctr2,fxno,fxnoZ,iprint,asmees,M COMMON /para2/tota12,ctr2 /varxylevel/xlevel,ylevel COMMON /para3/fxno /para31/fxn02pass COMMON /paraasmees/asym¥es /para8/kN COMMON /varkcij /kcjpass , kcipass /varckc/ckcpass , kcpass COMMON /varcj0kciab/cj0kciab,CjOkcjab COMMON /paraabM/a,b,epidummy,M /para11/k COMMON /parakcn/kcn /paracn2/cn2 /parasol/sol COMMON /para9/epv,zdummy pi=atan(1.0d0)*4.0d0 fxn02=fxn02pass CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Asymptotic term evaluation CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC IF (asmees.NE.O) THEN x=DBLE(CX) IF (asmees.EQ.1) THEN upper:x-sqrt(2.d0)*a ELSEIF (asmees.EQ.2) THEN upper=-x+sqrt(2.d0)*b ENDIF IF (upper.LT.1.d-14) THEN y=0.d0 ELSE xlevel:DBLE(cx) ctr2=0 tota12=0.d0 fxn02=3 CALL quard2(0.d0,upper,fxn02) y=tota12 191 & & ENDIF GOTO 100 ENDIF kC=DBLB(CX) ckc=cx kcpass=DBLE(cx) ckcpass=cx kcj=kcjpass kci=kcipass . CALL crbnfx(ckc,rbn) IF (abs(ckc).NE.O.d0) THEN y=(1.d0-rbn)/(l.d0+rbn)/sqrt(ckc*ckc-kN*kN)*ckc-1.d0 zarg=ckc*a CALL besst(zarg,chkca,j1,y0,y1,h20,h21,iprint) zarg=ckc*b CALL besst(zarg,cj0kcb,j1,y0,y1,h20,h21,iprint) ENDIF IF (abs(ckc).LE.1.d-10) THEN y=0.d0 GOTO 100 ELSEIF (fxno.EQ.0 .or. fxno.EQ.4) THEN GOTO 4 ELSEIF (fxno.EQ.1 .or. fxno.EQ.5) THEN GOTO 5 ELSEIF (fxno.EQ.2 .or. fxno.EQ.6) THEN GOTO 6 ELSEIF (fxno.EQ.3 .or. fxno.EQ.7) THEN GOTO 7 ELSEIF (fxno.EQ.10) THEN GOTO 10 ELSE , WRITE(*,*)’report fxno error;fxno=',fxno ENDIF yc=(CjOkca—CjOkcb)/Ckc Y=Y*YC**2 GOTO 100 yc=(chkca-chkcb)/ckc y=y*yc*2.d0/pi*ckc/(kci*(kci*kci-ckc*ckc)1 . *(cj0kcb*chkciab-cj0kca) GOTO 100 yc=(cj0kca-cj0kcb)/ckc y;y*yc*2.d0/pi*ckc/(kcj*(kcj*kcj-ckc*ckc)) *(chkcb*chkcjab-cj0kca) GOTO 100 yc=2.d0/pi*ckc/(kcj*(kcj*ij—ckc*ckc)) *(chkcb*CjOkcjab-Cj0kca) y=y*yc*2.d0/pi*ckc/(kci*(kci*kCi-ckc*ckc)) *(chkcb*chkciab-chkca) GOTO 100 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Radiation wave power evaluation CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 10 X=CX 192 DO 50 i=1,M zarg=kcn(i)*a CALL besst(zarg,cj0kca,jl,y0,y1,h20,h21,iprint) zarg=kcn(i)*b CALL besst(zarg,chkcb,j1,y0,y1,h20,h21,iprint) 30kcab(i)=CjOkCa/chkcb 50 CONTINUE kc=DBLE(k(1))*sin(x) Ckc=kc betal:DBLE(k(1))*cos(x) beta2=sqrt(abs(k(2))**2-kc**2) gama2=(0.d0,1.d0)*beta2 gama1=(0.d0,1.d0)*beta1 IF (abs(x-pi/2.d0).LE.1.d-6) THEN rbn=0.d0 ELSE CALL crbnfx(ckc,rbn) ENDIF zarg:kc*a CALL besst(zarg,cj0kca,jl,y0,y1,h20,h21,iprint) zarg:kc*b CALL besst(zarg,cj0kcb,j1,y0,y1,h20,h21,iprint) W=0.d0 DO 200 i=1,M W=W+cn2(i)*sol(1+1)*kc/(kcn(i)**2—kc**2)/kcn(i) & *(chkcb*jOkcab(i)-Cj0kca) 200 CONTINUE W=W*2./pi W=W+(Cj0kca-Cj0kcb)/kc*sol(1) cccCcccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c The following if is to take care of the end pt, .c x=pi/2.d0, for which theta=90 deg s.t. beta2->0.d0 and c cos(x) —> 0.d0 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc IF (abs((epv(2)-epv(l))/epv(l)).LT.1.d-3 .and. & abs(x-pi/Z).LT.l.d-6 ) THEN y=1.d0/abs(k(2))**2 y=y*(sin(x)**3*abs(k(1))+sin(x)*cos(x)*beta1) y:y*abs( W/(1.d0+rbn)/2.d0 )**2 ELSE y=cos(x)**2/beta2**2 y=y*(sin(x)**3*abs(k(1))+sin(x)*cos(x)*beta1) y=y*abs( W/(l.d0+rbn)/(l.d0+gama1*epv(2) & /(gama2*epv(l))) )**2 ENDIF 100 RETURN END Ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C A subroutine describing the integrand of a integral c which is done by quard2 subroutine. ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 193 113 123 SUBROUTINE fxvalue2(x,y) COMPLEX*16 y,ckc,total3,zarg,j0,jl,y0,y1,h20,h21 COMPLEX*16 ycl,yc2,yc3,yc4 COMPLEX*16 jlkcrp,ylkcrp,jOkca,y0kca,j1kcr,y1kcr REAL*8 yreal,x,r,rp,ratio,dumm REAL*8 kc,a,b,kci,kcj,xlevel,ylevel INTEGER fxn02,iprint,fxno,ptr,ctr3,dummy COMMON /varxylevel/xlevel,ylevel COMMON /paratota13/total3,ctr3 /para31/fxn02 COMMON /para3/fxno /varkcij/kcj,kci COMMON /varratio/ratio,dumm /pararadiusab/a,b ptr=fxno*10+fxn02 kc=kcpass Ckc=ckcpass ylevel=x rp=(xlevel+ylevel)/sqrt(2.d0) r=(xlevel-ylevel)/sqrt(2.d0) yreal=r/(rp*rp) ctr3=0 total3=0.d0 dummy=0 ratio=r/rp CALL romerg3(0.d0,1.d0,dummy) y=yreal*total3 IF (ptr.EQ.3) THEN y=y*2 . d0 GOTO 100 ELSEIF (ptr.EQ.l3) THEN GOTO 113 ELSEIF (ptr.EQ.23) THEN GOTO 123 ELSEIF (ptr.EQ.33) THEN GOTO 133 ELSE WRITE(*,*)'report fxno error fxvalue2 subroutine’ ENDIF zarg= kci*rp CALL besst(zarg,j0,jlkcrp,y0,y1kcrp,h20,h21,iprint) zarg= kci*a CALL besst(zarg,jOkca,j1,y0kca,y1,h20,h21,iprint) zarg= kci*r CALL besst(zarg,jO,jlkcr,y0,y1kcr,h20,h21,iprint) ycl=(jlkcrp*y0kca-j0kca*y1kcrp)*rp yc2=(jlkcr*y0kca-j0kca*y1kcr)*r Y=Y*(YC1+YC2) GOTO 100 zarg= kcj*rp CALL besst(zarg,jO,jlkcrp,y0,y1kcrp,h20,h21,iprint) zarg= kcj*a CALL besst(zarg,jOkca,j1,y0kca,y1,h20,h21,iprint) zarg= kcj*r CALL besst(zarg,jO,jlkcr,y0,y1kcr,h20,h21,iprint) yc1=(jlkcrp*y0kca—j0kca*y1kcrp)*rp 194 133 100 yc2=(jlkcr*y0kca-j0kca*y1kcr)*r y=y*(ycl+yc2) GOTO 100 zarg= kci*rp CALL besst(zarg,j0,jlkcrp,y0,y1kcrp,h20,h21,iprint) zarg= kci*a CALL besst(zarg,jOkca,jl,y0kca,y1,h20,h21,iprint) zarg= kci*r . CALL besst(zarg,jO,jlkcr,y0,y1kcr,h20,h21,iprint) ycl:(j1kcrp*y0kca-j0kca*y1kcrp) yC2=(j1kcr*y0kca-jOkca*y1kcr) zarg= kcj*rp CALL besst(zarg,jO,jlkcrp,y0,ylkcrp,h20,h21,iprint) zarg= kcj*a CALL besst(zarg,jOkca,j1,y0kca,y1,h20,h21,iprint) zarg= kcj*r CALL besst(zarg,j0,jlkcr,y0,y1kcr,h20,h21,iprint) yc3=(jlkcrp*y0kca-j0kca*y1kcrp) yc4=(jlkcr*y0kca—jOkca*y1kcr) y=y*(yc4*ycl+yc2*yc3)*r*rp RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A subroutine describing the integrand of a integral c which is done by cromergB subroutine. cccccccccccccccccccCCCCcccccccccccccccccccccccccccccccccccccc SUBROUTINE fxvalue3(cx,y) COMPLEX*16 cx,y REAL*8 u,u2,ratio,dummy COMMON /varratio/ratio,dummy u=cx u2=u*u y=(1.d0-u2)**2/sqrt(2.d0-u2) y=y/sqrt(l.dO-(ratio*(l.dO-u2))**2) RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A integration subroutine using quardrature scheme. CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE quard2(a,b,fxnumber) COMPLEX*16 save,fx,tota12,sum REAL*8 a,b,xx(50,50),ww(50,50) REAL*8 x(50,50),w(50,50),diffi,reltolerance,kc INTEGER j,intno,ctr2,maxidx INTEGER fxnumber,fxn02,idx,initialidx,idxmax COMMON /para/reltolerance,intno /xandw/x,w COMMON /para2/tota12,ctr2 /paraidxmax/idxmax COMMON /para31/fxn02 /varkC/kc fxn02=fxnumber 195 10 20 30 35 4O maxidx=11 initialidx=4 idx=initialidx DO 10 j=l,idx . xx(idx,j)=x(idx,j)*(b-a)/2.d0+(b+a)/2.d0 ww(idx,j)=w(idx,j)*(b-a)/2.d0 CONTINUE sum=0.d0 DO 20 j=l,idx CALL fxvalue2(xx(idx,j),fx) sum=sum+ww(idx,j)*fx CONTINUE save=sum idx=idx+2 DO 35 j=l,idx xx(idx,j)=x(idx,j)*(b-a)/2.d0+(b+a)/2.d0 ww(idx,j)=w(idx,j)*(b-a)/2.d0 CONTINUE sum=0.d0 DO 40 i=l,idx CALL fxvalue2(xx(idx,i),fx) sum=sum+ww(idx,i)*fx CONTINUE diffi=2.dO*abs(sum-save)/abs(sum+save) IF ((diff.GT.reltolerance).AND.(idx.LT.maxidx)) THEN GOTO 30 ENDIF tota12=sum RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A subtoutine for computing the coefficient of Rbn CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 10 SUBROUTINE crbnfx(ckc,rbn) COMPLEX*16 rbn,epv(65),ga(65),k(65),kratioZ,Rb2 COMPLEX*16 expterm, teml, tem2 , ij , ijml,muv(65) , Ckc, tem3 REAL*8 z(65),epvre,epvim INTEGER layerno,openckt,opentoair,i,j COMMON /paramuv/muv /para9/epv,z /paraopen/openckt COMMON /paraopentoair/opentoair /para7/layerno COMMON /para11/k DO 10 i=1,layerno epvre=DBLE(epv(i)) epvim=imag(epv(i)) ga(i)=cdsqrt(ckc**2-k(i)**2) CONTINUE Rb2=cdexp(-2.d0*ga(2)*z(2)) IF (openckt.EQ.1) THEN kratioZ=epv(l)/epv(2) Rb2=Rb2*(ga(l)-kratioZ*ga(2))/(ga(1)+kratiOZ*ga(2)) ELSE 196 20 Rb2=-Rb2 ENDIF ijml=Rb2 IF (layerno.GE.3) THEN DO 20 j=3,layerno kratioZ=epv(j-1)/epv(j) expterm=cdexp(-2.d0*ga(j-1)*z(j)) teml=ga(j-1)*(expterm+ijml) tem2=kratioz*(expterm-ijml)*ga(j) tem3=cdexp(-2.d0*ga(j)*z(j)) ij=tem3*(tem1-tem2)/(tem1+tem2) ijmlszj CONTINUE ENDIF rbn=ijm1 RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A subroutine for computing the surface wave power CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 100 SUBROUTINE swpower(sol,totalprho,toalplop2,wsave) COMPLEX*16 sol(*),epv(65),W,L,prh02,prhol,wsave COMPLEX*16f22,kr2,gama2,gar,k(65),cn2(0:50),epidummy COMPLEX*16 zarg,chkca,cj0kcb,jl,y0,y1,h20,h21 REAL*8 plop2(20),toalplop2,fre,dummyconv REAL*8 totalprho,tota1prhol,totalprh02 REAL*8 pi,z(65),omega,kcp(65).jOkcab(65),temp,a,b REAL*8 beta2,gama1,v,b22,kcn(100) ' INTEGER i,I1,kCOzerono,iprint COMMON /parafre/fre,dummyconv /para9/epv,z COMMON /paracn2/cn2 /parakc0array/kcp,chzerono COMMON /parakcn/kcn /paraabM/a,b,epidummy,Il COMMON /para11/k DO 100 i=1,ll zarg=kcn(i)*a CALL besst(zarg,cj0kca,j1,y0,yl,h20,h21,iprint) zarg=kcn(i)*b CALL besst(zarg,cj0kcb,jl,y0,y1,h20,h21,iprint) jOkcab(i)=cj0kca/Cj0kcb CONTINUE totalprho=0.d0 totalprhol=0.d0 totalprh02=0.d0 DO 300 m=1,chzerono pi=4.d0*atan(l.d0) omega=2.d0*pi*fre beta2=sqrt(DBLE(k(2))**2—kcp(m)**2) gamal=sqrt(-DBLE(k(1))**2+kcp(m)**2) gama2=(0.d0,1.d0)*beta2 gar=gama1/gama2 b22=beta2*z(2) 197 200 300 v=gama1/beta2 kr2=epv(1)/epv(2) temp=cos(b22)*sin(b22)/b22 IF (beta2.LT.1.d—15) THEN f22=kr2**2*z(2) ELSE f22=(kr2**2*(1.+temp)+v**2*(1.—temp))*z(2)/2.d0 f22=f22+kr2*v*(1.-Cos(2*b22))/(2.d0*beta2) ENDIF zarg= kcp(m)*a CALL bessf0(zarg, chkca, j1,y0,y1, h20, h21, iprint) zarg= kcp(m)*b CALL besst(zarg,cj0kcb,jl,y0,y1,h20,h21,iprint) W=0.d0 DO 200 i=1.Il W=W+cn2(i)*sol(i+l)*kcp(m)/(kcn(i)**2-kcp(m)**2) /kcn(i)*(chka*jOkcab(i) ~chkca) CONTINUE w=W*2./pi W=W+(chkca-chkcb)/kcp(m)*sol(1) L=exp(-2.*gama2*z(2))*( (gar-kr2)/(gar+kr2) *(-2.*z(2))*kcp(m)/gama2+ 2.*kcp(m)*kr2 *(1./gar-gar)/(gama1+kr2*gama2)**2 ) L=1.dO/L prh02=8.d0*pi*pi*omega*epv(2)*kcp(m)**2 *f22/(v**2+kr2**2)/(beta2**2)*(abs(W*L))**2 prhol=4.d0*pi*pi/gama1/beta2**2*omega*epv(2)**2 /epv(1)*kcp(m)**2*(abs(W*L))**2 /(1.d0+(gama1*epv(2)/epv(l)/beta2)**2) totalprho=totalprho+DBLE(prhol+prh02) totalprhol=totalprhol+DBLE(prhol) totalprhoZ=totalprhoZ+DBLE(prhoZ) plop2(m)=DBLE(prhol)/DBLE(prh02) CONTINUE IF (chzerono.NE.0) THEN toalplop2=totalprhol/totalprh02 ENDIF RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C A subroutine for computing the radial guided wave power CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE radialpw(sol,totalprho) COMPLEX*16 sol(*),epv(65),W,L,prh02,epidummy COMPLEX*16 cn2(0:50),gzZ,kr2,gama2,k(65) COMPLEX*16 zarg,CjOkca,Cj0kcb,jl,y0,yl,h20,h21 REAL*8 fre,dummyconv REAL*8 totalprho REAL*8 pi,omega,kcp(65),z(65) REAL*8 beta2,b22,kcn(100) REAL*8 jOkcab(65),a,b 198 INTEGER i,Il,chzerono,chzeronodummy,iprint COMMON /parafre/fre,dummyconv /para9/epv,z COMMON /paracn2/cn2 /parakc0array/kcp,chzeronodummy COMMON /parakcn/kcn /paraabM/a,b,epidummy,Il COMMON /para11/k pi=4.d0*atan(1.d0) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C 50 100 200 find the zeros CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC chzerono=int(abs(k(2)*z(2))/pi) kcp(l)=abs(k(2)) DO 50 i=1,kc02erono-1 kcp(i+1)=sqrt(abs(k(2))**2-((i)*pi/z(2))**2) CONTINUE kc02erono=kc02erono+1 DO 100 i=1,I1 zarg=kcn(i)*a CALL besst(zarg,chkca,jl,y0,y1,h20,h21,iprint) zarg=kcn(i)*b CALL besst(zarg,cj0kcb,j1,y0,y1,h20,h21,iprint) jOkcab(i)=Cj0kca/cj0kcb CONTINUE totalprho=0.d0 DO 300 m=1,kc02erono pi:4.d0*atan(1.d0) omega=2.d0*pi*fre beta2=sqrt