., “45.“. . . . War... lilUMm-l .. 2.8.. 1mm...» may. dun 2...; .. ; it I.“ :, $.31... v. .. hfiflw It... . .1... . Ava, tunic?! at s , . ‘1‘}. A. P 3 .0.- u1 4.4. £ ‘ 2..“ ’1‘? ‘ I‘ 'l $55.1» mm 31.1,! ‘1...“ In“; V 51. . ., fiznnfiflliu : .35.; i . .I1.3.u :1? 2. «2...... alibi...» um“: .52, . . ‘3 infifii‘ 1...) squnhs flu} .Jtflz {pig 2.3 In . I... II 43.x. 1.3 $3 @wfifi #3,... .5 ...~n...fi¢.w.7 ‘10. 1‘ s. 5.... 5' LIBRARY Michigan State University This is to certify that the dissertation entitled ELECTROMAGNETIC MATERIAL CHARACTERIZATION OF A PEC BACKED LOSSY SIMPLE MEDIA USING A RECTANGULAR WAVEGUIDE RESONANT SLOT TECHNIQUE presented by ANDREW ERIC BOGLE has been accepted towards fulfillment of the requirements for the Doctoral degree in Electrical Engineering flea/(2% Major Professor’s Signature /6flv;y 260 7‘ Date MS U is an affirmative-action, equal—opportunity employer PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/07 p:/CIRC/DateDue.indd-p.1 ELECTROMAGNETIC MATERIAL CHARACTERIZATION OF A PEC BACKED LOSSY SIMPLE MEDIA USING A RECTANGULAR WAVEGUIDE RESONANT SLOT TECHNIQUE By Andrew Eric Bogle A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Electrical and Computer Engineering 2007 ABSTRACT ELECTROMAGNETIC MATERIAL CHARACTERIZATION OF A PEC BACKED LOSSY SIMPLE MEDIA USING A RECTANGULAR WAVEGUIDE RESONANT SLOT TECHNIQUE By Andrew Eric Bogle Electromagnetic material cllaracterizz-ttion is the process of determining the com- plex constitutive parameters of a certain media. The motivation here is the devel- Opment of a non-(lestructive material characterization technique for a lossy, simple- media (linear. homogeneous and isotropic) backed by a perfect electric conductor (PEC). In order to extract both the permittivity and permeability of the media, two independent experimental interrogations of the material are required. For a rectan- gular waveguide the experimental interrogations take the form of either reflection or transmission coefficients. The experimental data is then compared iteratively to the formulation, resulting in the extraction of the complex crmstitutive [‘)arameters for the inertia. This (,lissertation will demonstrate, through the use of a transverse slot cut in the bottom wall of a rectangular waveguide, how waveguide scattering 1')arameters are used to effectively extract both the permittivity and permeability of a lossy simple- rnedia backed by a PEC. A two-dimensional Newton’s complex root search algorithm is used to iteratively solve for the complex constitutive parameters. Results for a magnetic radar absorbing material are compared to measuren‘ients using a partially- filled rectangular waveguide technique to verify the formulation. For Erica and Derby iii ACKNOWLEDGMENTS I would first like to thank my wife Erica. Her support, understanding and tolerance of me during this important period of our lives has been nothing short of amazing. I love her with everything I am and look forward to the rest of our lives together. Thank you to Leo for believing in and supporting me during my entire graduate career. His patience and guidance have been invaluable in achieving this goal. Thank you to Mike for all his time and effort. Especially on round table sessions and last second measurements, without which I would not be graduating. Thank you to Dennis, Ed, and Shanker for all their time, teachings and guidance. They made my time with the EM group a wonderful experience. Thank you to all the fellow graduate students over the years for helping me through the good, bad and extremely difficult times. Thank you to my parents! There unquestionable love and support over the years has been more than I could ever dreamed of. Finally, Thank you to all my family members for all their support! ll would also like to thank the AFOSR (through Dr. Charles Lee and the grants F-496—2002—10196 and FA9550-06-1-0023) for sponsoring this research, and the NSF CAREER program (ECS 0134236) for partial sponsorship of this research. iv TABLE OF CONTENTS LIST OF FIGURES ................................ ix KEY TO SYMBOLS AND ABBREVIATIONS ................. xii CHAPTER 1 ‘ Introduction and Background ........................... 1 CHAPTER 2 Rectangular Waveguide Radiating through a Transverse Slot into a Half-Space 6 2.1 Introduction ................................ 6 2.2 Geometrical Configuration ........................ 6 2.3 Field Representations ........................... 7 2.3.1 Equivalent Currents ....................... 8 2.3.2 General Field Relations ..................... 9 2.3.2.1 Total Fields in each Region .............. 10 2.3.2.2 MT Hertzian Potentials in terms of Equivalent Currents 11 2.4 Development of the Coupled MFIEs ................... 13 2.4.1 Development of the MFIDEs ................... 13 2.4.1.1 BS1 ........................... 14 2.4.1.2 BS2 ........................... 14 2.4.2 Solving the Second-Order Partial-Differential Equation . . . . 15 2.4.2.1 MFIE at BSI ...................... 16 2.4.2.2 MFIE at. BS2 ...................... 17 2.5 MOM Technique .............................. 18 2.5.1 Expansion of Fields In Terms of unknown Slot Voltages . . . . 18 2.5.1.1 Expanded MFIE at BSI ................ 19 2.5.1.2 Expanded MFIE at BS2 ................ 20 2.5.2 Application of the MOM Technique ............... 20 2.5.3 MOM Solution in Matrix Form .................. 25 2.5.3.1 Matrix Element Definitions .............. 27 2.6 Rectangular \Naveguide Scattering Parameters ............. 29 2.6.1 Reflection and Transmission Coefficients ............ 30 2.6.2 Analysis of Slot Dimensions ................... 32 2.6.2.1 Thickness of Slot ..................... 32 2.6.2.2 Length of Slot ..................... 33 2.6.2.3 Width of Slot ....................... 33 CHAPTER 3 Green’s Function for EM Field Within a PEC Parallel-Plate Envirornnent 3.1 Introduction ................................ 3.2 Geometrical Configuration ........................ 3.3 EM Fields and Helmholtz Equation for MT Hertzian Potential . . . . 3.4 Spectral Representation of Principal and Scattered Waves ...... 3.4.1 Principal W'ave Representation ................. 3.4.2 Scattered Wave Representation ................. 3.4.3 Total Wave Representation .................... 3.5 Computation of Spectral Coefficients .................. 3.5.1 Tangential Components ((1 = 17,31) ................ 3.5.2 Normal Component ((1 = 2) ................... 3.6 Dyadic Green’s Function ......................... 3.7 Physical Observations .......................... CHAPTER 4 Rectangular Waveguide Radiating through a Trai'isverse Slot into Parallel-Plate Waveguide Filled with a Simple Media ...................... 4.1 4.2 4.3 4.4 4.5 4.6 Introduction ................................ Geometrical Configuration ........................ Field Representations ........................... 4.3.1 Equivalent Currents ....................... 4.3.2 Total Fields in each Region ................... 4.3.3 MT Hertzian Potentials in terms of Equivalent Currents Development of the Coupled MFIEs ................... 4.4.1 Development of the MFIDEs ................... 4.4.1.1 BS1 ........................... 4.4.1.2 BS2 ........................... 4.4.2 Solving the Second-Order Partial-Differential Equation . . . 4.4.2.1 MFIE at BS1 ...................... 4.4.2.2 MFIE at BS2 ...................... MOM Technique .............................. 4.5.1 Expansion of Unknown Slot Voltages .............. 4.5.1.1 Expanded MFIE at BSl ................ 4.5.1.2 Expanded MFIE at BS2 ................ 4.5.2 Application of the MOM Technique ............... 4.5.3 MoM Solution in Matrix Form .................. 4.5.3.1 Matrix Element Definitions .............. Rectangular W'aveguide Scattering Parameters ............. vi 38 38 38 38 40 42 43 43 44 45 47 48 49 51 51 52 52 53 54 56 56 56 57 58 58 59 61 61 61 62 62 66 67 71 4.6.1 Reflection and Transmission Coefficients ............ 71 4.6.2 Analysis of Material Properities ................. 72 4.6.2.1 Relative Permittivity and Permeability ........ 72 4.6.2.2 Material Thickness ................... 73 4.6.3 Analysis of Slot Dimensions ................... 74 4.6.3.1 Thickness of Slot ..................... 74 4.6.3.2 Length of Slot ..................... 75 4.6.3.3 \Nidth of Slot ...................... 75 CHAPTER 5 Results ....................................... 87 5.1 Experimental Setup ............................ 87 5.1.1 Validation ............................. 88 5.1.1.1 Radiation into a Half—Space .............. 88 5.1.1.2 Signal Attenuation ................... 89 5.2 Complex Constitutive Parameter Extraction .............. 90 5.2.1 Extraction Validation ....................... 90 5.2.2 Tested Samples .......................... 91 5.2.2.1 MagRAM ........................ 92 CHAPTER 6 Conclusions ..................................... 105 6.1 Suggestirms for Future Work ....................... 105 APPENDIX A Maxwell’s Equations and Hertzian Potentials .................. 108 A1 Introduction ................................ 108 A2 Maxwell’s Equations and the Wave Equation for E and H ...... 108 A3 MT Hertzian Potential .......................... 109 APPENDIX B Properties of Rectangular W’aveguides and Cavities ............... 111 B.1 Introduction ................................ 111 B2 Waveguide Modes ............................. 111 B.2.1 Geometrical Configuration .................... 112 B22 Modal Analysis (TEI Modes) .................. 112 B.2.2.1 y-axial direction .................... 113 B.2.2.2 z-axial direction .................... 115 B3 Green’s Functions ............................. 117 33.1 Rectangular Waveguide ...................... 117 B.3.1.1 Slot Excited Modal Fields ............... 118 8.3.1.2 Determination of Modal Expansion Coefficients . . . 118 vii B.3.1.3 Green’s Function due to a Transverse Slot ...... 122 13.3.2 Rectangular Cavity ........................ 122 APPENDIX C Solutions to Admittance Matrix Elements .................... 126 C.1 Waveguide Matrix Elements ....................... 126 C2 Cavity Matrix Elements ......................... 128 C21 881 - Self Terms ......................... 129 C22 BS1 - Coupled Terms ....................... 132 C23 BS2 - Self Terms ......................... 134 C24 BS2 - Coupled Terms ....................... 135 C3 Halfspace Matrix Elements ........................ 136 C31 772,2 74 722 .............................. 137 C32 7712 = 72.2 .............................. 138 C4 Parallel-plate Matrix Elements ...................... 141 C41 Spectral Integral Analysis .................... 142 C.4.l.l 1} Analysis ........................ 143 C412 5 Analysis ........................ 151 BIBLIOGRAPHY ................................. 161 viii Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 3.1 Figure 3.2 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 LIST OF FIGURES Geometrical Configuration: Rectangular waveguide radiating through a transverse slot into a half space ............. Geometrical Configuration: Top view of the transverse slot cut in broad wall of rectangular waveguide ................. Equivalent currents for slot electric-fields at BS1 and BS2 Pulse function expansion for the MOM technique Effect of the slot wall thickness on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHZ). Effect of the slot length on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). Effect of the slot width (using slot length to width ratios) on the magnitude (dB) Of the waveguide scattering parameters vs. fre- quency (GHZ). 0000000000000000000000000000 Geometrical Configuration: Parallel-plate waveguide filled with a simple media Physical Observations of the waves within a parall('—*.l—1.)1ate waveguide Geometrical Configuration: Rectangular waveguide radiating through a transverse slot. into a parallel—plate waveguide filled with a simple media ............................ Effect of the real component of the relative permittivity on the magnitude (dB) of the waveguide scattering parameters vs. fre- quency (GHZ). Effect of the imaginary component of the relative permittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). ........................... Effect of the real component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. fre- quency (GHZ). ............................ Effect of the imaginary component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHZ). ........................... Effect Of the material thickness of the real component, of the relative permittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). Effect of the material thickness of the imaginary component of the relative permittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHZ). ix 22 35 36 76 77 78 79 80 81 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 5.1 Figure 5.2 CI! 00 Figure gjl 4.. Figure . Figure 5..) V“ 03 Figure . . Figure 5.7 CI! 00 Figure Effect of the material thickness of the real component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). .................. Effect of the material thickness of the imaginary component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). ............ Effect of the slot wall thickness, for a parallel-plate region loaded with FGMI25, on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). .................. Effect of the slot length, for a parallel-plate region loaded with FGMI25, on the magnitude (dB) of the waveguide scattering pa- rameters vs. frequency (GHZ) ..................... Effect Of the slot width (using slot length to width ratios), for a pai'allel—17)1ate region loaded with FGMI'25, on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). . . . Experimental Setup: Rectangular waveguide radiating through a transverse slot into a half space .................... Experimental Setup: Rectangular waveguide radiating through a transverse slot into a finite parallel—plate region loaded with Ma- gRAM. ................................ Comparison of the formulation results and experimental data, us- ing the magnitude (dB) of the waveguide scattering parameters, for the radiation of the signal into a half-space vs. frequency (GHz). Con‘iparison of the formulation results and experimental data, us- ing the phase (degrees) of the waveguide scattering parameters, for the radiation of the signal into a half-s11)ace vs. frequency (GHz). The attenuation (dB) of a wave traveling 6 cm in a parallel-plate waveguide, for various values of the real component Of relative per- mittivity, at 8.2 GHZ vs. the imaginary component Of the relative pern'iittivity ............................... The attenuation (dB) Of a wave traveling 6 cm in a parallel-plate waveguide, for various values of the real component of relative per- mittivity, at 12.4 GHZ vs. the imaginary component of the relative permittivity ............................... The attenuation (dB) of a wave traveling in a pai'z-1.lle.l—plat.e waveg- uide, at 8.2 and 12.4 GHZ, loaded with FGMI25 vs. distance (cm). Relative permittivity and permeability extracted using generated acrylic data that has a 15 percent wavelength difference between the two material thicknesses ...................... 82 83 84 86 94 96 97 98 99 100 101 Figure 5.9 Figure 5.10 Figure 5.11 Figure B.1 Figure B.2 Figure B.3 Figure C.1 Figure C.2 Figure C.3 Figure C4 Relative permittivity and permeability extracted using generated acrylic data that has a 20 percent wavelength difference between the two material thicknesses ...................... Comparison of the relative permeal‘nlity, for the hj'lagR,.Al\'l FGMl25, using a partially-filled rectangular waveguide method and the resonant antenna technique vs. frequency (GHZ) ...... Comparison Of the relative permittivity, for the MagRAM FGMl25, using a partially-filled rectangular waveguide method and the resonant antenna technique vs. frequency (GHz) ...... Geometrical Configuration: Rectangular waveguide (y-axially di- rection) ................................ Geometrical Configuration: Rectangular waveguide (rs-axially di- rection) ................................ Rectangular waveguide with slot discontinuity ........... Evaluation contour for UHP closure in the complex 7)-plane Evaluation contour for LHP closure in the complex n-plane . . . . Evaluation contour for UHP closure in the complex g—plane . . . Evaluation COIIIOUI' for LHP closure in the complex E-plane . . . . xi 102 103 104 117 KEY TO SYMBOLS AND ABBREVIATIONS BSl: Boundary S1 B32: Boundary S2 CV: Cavity dB: Decibels ET: Electric Type EM: Electromagnetic HS: Half-Space IE: Integral Equation IDE: Integro—Differential Equation LHP: Lower Half Plane MFIE: IVIagnetic Field Integral Equation MFIDE: Magnetic Field Integro—Differential Equation MagRAM: IVIagnetic Radar Absorbing Material MT: Magnetic Type lVIoM: Method of Moments NRW: Nicolson-Ross-W’eir xii NDE: Non-Destructive Evaluation PP: Parallel-Plate PEC: Perfect Electric Conductor UHP: Upper Half Plane VNA: Vector Network Analyzer WG: Waveguide xiii CHAPTER 1 INTRODUCTION AND BACKGROUND Electromagnetic (EM) material characterization is the process of determining the constitutive parameters of a specific medium. The parameters typically sought are the bulk permittivity and permeability, which are specific values that describe an EM materials susceptibility to becoming polarized or magnetized when exposed to either an impressed electric or magnetic field, respectively. For time-harmonic representa- tions. both of these constitutive paran’ieters can be complex values. where the real parts are related to the energy storage of the material, and the imaginary parts are related to the loss mechanisms that convert incident electromagnetic radiation into heat. As the number of applications for EM materials continues to grow, so does the need to accurately characterize them. A number of disciplines rely extensively on the characterization of EM materi- als, including stealth, integrated circuits and agricultural technologies. For example, stealth uses the permittivity and permeability to describe how effective a material is at absorbing an incoming radar signal [1H3]. Whereas agricultural technologies uses the conmlex constitutive I.)arameters to aid farmers in deciding when to harvest their crops [4]. A (:tonn'non thread to all of the ap1,)lications is the increasing demand for better accuracy to achieve the desired results. This has lead to new challenges and increased levels Of complexity for the material characterization field as a whole. NO matter what the complexity level though, the desired result is still the consti— tutive parameters. Two distinct evaluation methods, destructive and non-destructive, exist for extracting the permittivity and permeability of a specific medium. The ma- jor difference between the evaluation methods is how the sample is handled. For a destructive evaluation the sample may be altered to fit the desired testing device, whereas for the non—destructive evaluation (NDE) the testing device must be built around the existing operational environment Of the desired sample. Destructive eval- uations are typically measured in a small scale laboratory setting, since this generally keeps the cost low and the environmental control high, both Of which are desirable traits when attempting to characterize any material. NDEs are generally not afforded either of these luxuries, and must be done in the existing operational environment at significant cost. Still there is a need to perform NDES Of EM materials. The same basic steps in comparing the measured and theoretical data are usually used to find the permittiv- ity and permeability in either the laboratory or oywratirmal setting. First. a material sample or testing device is built in the machine shop to facilitate the desired evalu- ation. Then the testing device is connected to a vector network analyzer (V NA) in order to Obtain experimentally measured data. Finally, numerical algorithms are used to extract the complex constitutive parameters of the sample from the experimental data. One of the most widely used algorithms is the Nict.)lson-Ross-W’cir (N RW) tech- nique, because it has the advantageous ability to directly solve for the material param- eters in closed form [5l-l6]. However, for the NRVV technique to be properly applied, the test samples must be linear, homogeneous and isotropic with coplanar front and back surfaces [7]. Essentially this means the testing device must not excite higher order modes, which unfortunately is not always possible. Several NDE techniques have been developed to allow full-characterization Of comiluctor-backed materials [8]-[11]. Three techniques, free-space, coaxial probe, and rectangular waveguide probe, are the most prevalent characterization methods. The free-space techniques have some unique advantages when compared to the other meth- ods. For instance, the characterization may be done having no physical contact. with the nutterial sample. They may also be characterized as a function of incident angle or vertical and / or horizontal polarization [12l-[15]. However, they are not very effective outside the laboratory setting since significant errors are introduced when the nu- merous assumptions associated with the technique cannot be met in the operational environment. Probe methods tend to have less restrictions associated with them, thus making them better fits for the Operational environment characterization methods. To date, the majority Of work investigated involves just. single probe methods. These maybe grouped based on how they radiate electromagnetic fields: antenna probes; open- ended coaxial line prol;)es; and open-ended waveguide probes. Antenna probes are essentially a combination of free-space and waveguide techniques, thus they still have a number Of the drawbacks due to free—space component of the measurement [16]—[17]. Coaxial line probe methods make good NDE techniques since they have a wider measurement bandwidth and a higher accuracy than antenna probes [18l-[22]. The major drawback associated with these methods is the lack of ruggedness the coaxial lines have when used in an Operational environment. Also, it is very difficult to ensure that no air-gaps are present between the center conductor of the open-ended portion Of the coaxial line and the material sample. W’aveguide probe methods ultin‘iately offer physical and EM advantages over coax- ial probes. Physically, waveguides are more rugged than coaxial lines, making them smart choices for tests performed in operational environments. Since the accuracy between waveguide and coaxial probes is similar, the EM advantages of waveguide probes are in characteristics such as; wave impedance, radiation and linear polar- ization. Allowing for more accurate characterization Of low permittivity materials, deeper penetration in lossy materials, and measurement of anisotropy, respectively [23l-[40]. The potential drawbacks of waveguide probes include their limited bandwidth and bulky size at lower frequencies. The concern of bulk sized waveguides is negated by using higher frequency waveguides. For instance, the VVR90 X-band waveguide has dimensions 0.9” by 0.4”, which make it very practical for use as a hand-held device. The concern of limited bandwidth has been investigated and solved by dielectrically loading the waveguide, enabling Operation Of the given waveguide at lower frequency ranges [10]. Recently, dual probe (collimrar aperture) waveguide methods have been investi- gated [41]. These techniques are able to obtain multiple independent interrogations of the medium, without altering the experimental setup, which leads to the ability to simultaneously extract both Of the complex constitutive parameters. This ability is especially significant in NDEs where alternate experimental setups are extremely difficult to develop. The dual probe techniques also help save time, reduce cost and remove the error associated with multiple measurements. all by simplifying the overall procedure necessary in the operational environment. Slot apertures are another 1')ossibility for a collinear aperture waveguide method [42l-[45]. Extensive investigation of antennas involving slot apertures in waveguide walls has been done [Mil—[56]. H(_)wever, no investigation has yet been done to extend these antennas to a. material characterization technique. This leads to the motivation for this dissertation, develop a NDE technique to characterize an EM material per- manently affixed to a perfectly electric conducting (PEC) s1.1rface. More specifically, full characterization Of the complex constitutive parameters, permittivity and perme- ability, for a lossy simple-medium (linear, homogeneous and isotropic) backed by a PEC surface using a slot aperture cut in the bottm'n wall Of a rectangular waveguide. Clumter 2 presents the integral equation (IE) technique that finds the theoretical scattering parameters for a rectamgular waveguide radiating through a traI'isverse slot, with finite thickness, into a half-space (HS). This is a. well known antenna problem that will provide confidence in the waveguide (WC) and cavity (CV) portions of the formulation presented in Chapter 4, that will ultimately be used for extracting the complex constitutive parameters of the medium. Chapter 3 develops the Green’s function, which will be used in Chapter 4, for an EM field within a parallel-plate (PP) environment. Chapter 4 presents the IE technique that finds the theoretical scattering parameters for a rectangular waveguide radiating through a transverse slot, with finite thickness, into a lossy, homogeneous and isotropic medium backed by a PEC surface. These scattering parameters are then compared to experimental results in order to extract the desired results, permittivity and permeability. Chapter 5 shows results of the complex constitutive parameter extraction for a magnetic radar absorbing material (MagRAh-l). The extraction is done using a com- plex two—dimensional Newtons root-searching algorithm, since the measurement tech- nique developed violates the assumptions of the N RW technique. Chapter 6 presents conclusions and recommendations for future work. CHAPTER 2 RECTANGULAR WAVEGUIDE RADIATING THROUGH A TRANSVERSE SLOT INTO A HALF-SPACE 2.1 Introduction The formulation of a finite-thickness transverse slot centered in the broad wall Of a rectangular waveguide, radiating into half-space, is considered in this chapter. Specif- ically, the reflection and transmission coefficients for the rectangular waveguide are sought. This is because they are ultimately used to perform the complex constitu- tive 1_)arameter extraction in Chapter 5, where the half-space is replaced by the PEG parallel~plate structure that. is analyzed in Chapter 3. lUmlt—irstanrling this well—struliecl problem provides an in'iportant form of crmfidence since the problem has been pre- viously studied, well known results are available for referencing with the formulation presented here. The first step in the formulation is to develop representations of the fields in the three (.liffercnt regions (VVG, CV, and HS). Next, a set of coupled IEs are developed by satisfying the continuity Of the tangential fields at the two interfaces (boundary SI (BSI) and b(_)undary S2 (BS2)) between the three regions. The final step is to solve the coupled IEs using a method of moments (Mol\‘l) technique for the unknmvn slot voltages. which in turn identify the scattered fields in the rectangular waveguide and ultimately the reflection and transmission coefficients desired. 2.2 Geometrical Configuration Consider a rectangular waveguide, backed by an infinite ground plane, with a trans- verse slot cut in the l(')wer broad—wall as shown in Figure 2.1. The waveguide has fixed dimensions for width —a/2 < .1: < 0/2, height t < z < ()1 and length y] < y < yT, where ()1 — t = b and the width (a) is in general greater than twice the height (b). The slot, shown in Figure 2.2, has fixed dimensions for width —L < .1: < L, height 0 < z < t and length —l‘l"' < y < ll", where the height (t) corresponds to the wall thickness of the waveguide and infinite ground plane. BS1 is located at 7’1 : (.171, yl, t) and is the interface between regions I and II. Similarly, B82 is located at 7‘72 2 (.172, yg, 0) and is the interface between regions II and III. AZ 2 = b] If A 1 Region I: Waveguide PEC y=y1 y=-W y=W y=yT Z = t 1 . " g 4—- Region II: CaV1ty<> y z = 0 3 > y = 0 Region III: Halfspace Figure 2.1. Geometrical Configuration: Rectangular waveguide radiating through a transverse slot into a half space 2.3 Field Representations In order to develop the desired IEs, a representation of the fields in the three regions is needed. However, to generate the appropriate set of field equations a knowledge of the sources is necessary, thus the first section solves for the equivalent currents that. maintain the fields in the three regions. 2.3.1 Equivalent Currents To develop the equivalent currents for this formulation, a. discussion of the excitation of the geometrical configuration in Figure 2.1 is helpful. The system is excited by the TE 1’0 mode, which propagates down the guided-wave structure until it encounters the transverse slot in the broad wall of the waveguide, disrupting the current along the waveguide wall. An impressed field is then maintained in the slot aperture, inducing a voltage across the slot. that subsequently maintains scattered waves in the rectangular waveguide. Thus, it is the impressed slot-fields that lead to the equivalent currents used to represent the fields in the three regions of this formulation. y /\ 1=l,2 y=W1 y=O — >~ Ey, Hr, X ————> y='W1 I x=-L1 x=0 x=L1 Figure 2.2. Geometrical Configuration: Top View of the transverse slot cut in broad wall of rectangular waveguide Noting that the dimensions of the slot are long and narrow (i.e. ratio of slot length to slot width equals 1/ 20), then the electric-field in the slot is assumed to be strictly longitudinal (y—directed), which in turn assumes that the magnetic current in the slot is strictly transverse (x-directed). These assumptions mainly serve to simplify the overall complexity of the formulation and have also been proven very accurate [47]. Under these assumptions, the use of Love’s equivalence principle [57] leads to the following four equivalent magnetic currents as seen in Figure 2.3. K;,2 (72' j 352 Figure 2.3. Equivalent currents for slot electric-fields at B81 and BS2 Kg, (71’) = —a x 131+ (11’) = — (2) x 19E; (fi’) = w; (71’) (2.1) Kg, (71’) = —a x E,— (fi’) = — (—2) x 9E1— (fi’) = —i-Ey_1 (fi’) (2.2) K722 (772’) = —a x E; (73’) = — (2) x 19532 (73’) = mg, (a’) (2.3) Kg2 (73’) = —a x E; (172’) = - (—2) x 3913ng (13’) = ~5cEy‘2 (13’) (2.4) Where (2.1) corresponds to the waveguide equivalent magnetic current, (2.2) and (2.3) to the cavity equivalent magnetic currents at B81 and BS2 respectively, and (2.4) is the half-space equivalent magnetic current. Note that all the equivalent currents have only a :r-directed component. 2.3.2 General Field Relations Since all the sources are equivalent magnetic currents, the general field relations are those based on Maxwell’s equations due a magnetic source. These general field relations in terms of magnetic-type (MT) Hertzian potentials (details in Appendix A) are written as follows flfl=fiWVXflm we Hm=VWfiMW+Wflm we where the MT Hertzian wave-equation is _K (F) M1 (2.7) (V2 +k2)7? hm— — and the solution to this wave—equation in terms of the MT Hertzian potential may be written as 7?. (F)= (5,? (NF) - —’J——K"('l18’ (28) JW.“ H I I 9 I where G 1s the dyadic Green 8 function. 2.3.2.1 Total Fields in each Region Using the knowledge that there are only sic-directed components of the sources, and the principle of superposition [58], the magnetic field in equation (2.6) is now written in terms of only the $-component as follows Hz.(F)= (~82— +A )m... (F) (2.9) where 7r,” is the .lr-component of the MT Hertzian potential given as K7711?" (T ) (13’ (2.10) M ”hr (7:) Z ./S 0:12;: (Flf’) a with GE as the :r-component of the dyadic Green’s function and K m1. as the equivalent magnetic current for each specific region. This allows the total magnetic field in region 1 to be written as 3;:le )= H“ (F) + HXVGS (F) (2.11) where Hg” is the incident wave in the rectangular waveguide, and H305 is the waves scattered by the transverse slot in the rectangular waveguide. In a similar 10 manner, the total magnetic field in region 2 is H3§FJFFFF2(F 1:) HCV1(r 7)+ HCV2( F.) ~~for 1:1,2 (2.12) where HEVI is the waves scattered by the transverse slot at z = t in the cavity region, and HCV2 is the waves scattered by the transverse slot at z— — 0. Finally, the total magnetic field in region 3 is H§FFFFF3 21’ cos (k. 2)) cos ]k~r(sl— '— c)] for :1 < 251’ and the well known half-space Green’s function is G F -, e—jkOR 2 20 HS (7‘2IF2) — 27m ( - ) where the distance R is defined as R— . ,2? 1‘2 ~ 42 221 — (.12 — .12) + ((12 — 112) + (Q — s2) ( . ) 12 The wavenumbers and the Neumann’s number [56] for both the rectangular waveguide and cavity are given as Ot( )71' 1.1%,) = WOW...) 2 1,2,3, em.) = H.301) = 0, 1, 2, kghwl“) : k8 _ kghf) : kghf) + kghf) (2'22) and w) = 1 for (1......)= 0 (2.23) 2 for (3(u2g,cv) 71$ 0 respectively. Details of equations (2.18) and (2.19) are found in Appendix B. These field representations for the three regions are now used to develop IEs in the next section. 2.4 Development of the Coupled MFIES In this section, a pair of coupled 1133 are developed (for unknown slot—voltages (V1, W) at 881 and BS2 respectively) by invoking boundary conditions at the two interfaces. The continuity of both the electric and magnetic fields ensures a unique solution for the desired unknowns. The fact that. magnetic fields are used leads to the formulation being called a MFIE. 2.4.1 Development of the MFIDES Since the MT Hertzian potentials involve and integral over the slot apertures, the first. integral equations developed are actually magnetic field integro-differential equations (MFIDE). Standard techniques are then used to solve the resulting second-order partial-differential equations, ultimately resulting in the desired MFIEs. 13 2.4.1.1 BSl The MFIDE at BS1 is developed by invoking the continuity of tangential magnetic fields at the interface (2 = t) giving nggionl (7,1) : H;egi.1n2 (71) (2.24) then, substituting equations (2.11) and (2.12) into equation (2.24) leads to HFFF (F1) + H?“ (F1) = H? W (F) + H?” (F) (2.25) finally, substituting (2.10) into (2.9) and then (2.14-2.16) into the respective magnetic field representation, leads to the first MFIDE result (6.2 + kg) {/3 y ( )GWC('J‘1|F'1')dSi .. 1 WM) E. F’ , ‘f ———"’-2 (72 [Gov (7116’) dSé] = —H.F."F (F1) (226) 52 JWHU where GWC (T'iIF'i') = GWG (Filfi') + GCV (Filfi') (227) Note that the continuity of the electric fields across the two interfaces is enforced _ . _ .1. _ by maklng E51 = Ey1 and Egg 2 Ey2. 2.4.1.2 B82 Similarly, the MFIDE at BS2 is developed by invoking the continuity of tangential magnetic fields at the interface (2 = 0) giving nggionZ (r72) : Hiegion3 (73) (2.28) 14 where substitution of equations (2.12) and (2.13) into equation (2.28) gives H9102) + H9” (7‘72) = H515 (F2) (229) and finally, substituting (2.10) into (2.9) and then (2.15-2.17) into the respective magnetic field representation, leads to the second MFIDE result (92 2 E11(T‘_i’) A, y Y ,;-+ ,.—v/ I (0.132 + 0) {/51 ijO GCVWIH ”81 E ‘ .7! — / Man) (I) (155] =0 (2.30) 32 JF‘J/IU where GCH ('J‘2I'F2') = GCV ('7‘2l7'2') + GHS ('IEIF‘E’) (2-31) 2.4.2 Solving the Second-Order Partial-Differential Equation By inspection, it. is seen that the MFIDEs developed in the last section are integro— differential ermations. These integro-differential equations are converted into purely integral equations by superposing the homogeneous and inhomogeneous solutions to the second-order partial—differential equations [56]. Thus, further simplifying the complexity of the overall pr(;)l,)lem. The solution of the homogeneous equation is solved using the method of unde- termined coefficients [59], where sine and cosine functions are chosen as the comple- mentary solutions. To solve the inhomogeneous equation, the forced response to the inhomogeneous one-dimensional Helmholtz equation I \II (.1) = — /s (.r’) sin [1; (:1: — .r’)]d;r’ (2.32) (l is used as the particular solution [60]. The superposition of the complementary and particular solutions leads to the strictly integral equations desired. These MFIEs are complements of Hallen’s integral equation for a dipole antenna [56], given the fact that the slots are assumed to lie in a PEC plane. 2.4.2.1 MFIE at BS1 The integro—differential equation (2.26) is written as a? 2 (5:2— +ke) F101) = A101) where _, E1 1“], _, 2 F101) 2/ “FL—lam (7~1[-r1’)d5[ 51 quo Ey2 (T2,) _. _./ / — —,——G 7' 7‘ (15 /.S2 JWHO CV ( 1| 2 ) 2 is in the form of the MT Hertzian potential and A1 (7:1) = —H§.flc (Ti) = —A10 [(TF/a.)2 — A78] cos(7r:1'1/a)e_jky10'yl is the forcing term based on the .r-component of the incident. 111agnetic field. The solution to equation (233) is written as a C _. p 4 F191) 2 F1 (7‘1)+ F1 (7‘1) where the complementary solution is given as F? (F1) = 01 (a) (H111) + 01 (1)1) sin (Ho-.221) (2.33) (2.34) (2.35) (2.36) (2.37) and the particular solution, in terms of the forced response of equation (2.32) and the 16 x-component of the incident magnetic field from equation (2.35), is A10 [(7r/a)2 — 12(2)] e‘jklef/l $1 Ff) (17]) = — k0 sin [kg (:131 —— 571)] cos(7r:‘1‘1/a)dfl (2.38) 0 Solving the forced response integral, and performing some algebraic manipulation, gives the. following form of the particular solution F1P('r_i) = —A106_jk3’10y1 [cos (1:021) — cos (71.1'1/a)] (2.39) Finally, substituting equations (2.34), (2.38), and (2.39) into equation (2.36) and combining complementary solutions, leads to the MFIE at 881 Bill (73’) a -.I / / Ey‘l (7:3,) .. -+/ I —*,—-——G F , '7‘ I' (IS — ——,——-—G '1' 7‘ (IS [51 J'WHO ”Cl 1| 1) l 32 quo CM ll 2) 2 = C1 (yl) cos (1.10.171) + D1 (111) sin (160.131) + Aloe—fllyIOy1 cos (7121/11.) (2.40) 2.4.2.2 MFIE at BS2 Similarly for BS2, the integro-differential equation (2.30) is written in the form a? - - (a5 + 19(2)) F2 (72) = A2 (7‘2) (2-41) where a E- 1 Ti’ 3 3 F2 (7‘2) =/ —'l-/,—(—-)-GCV (T2l7'1’) (15] 51 leuo E. F’ ‘/ Alf—Flam (72172) (18!. (242) . 5.2 M110 and A2 (r5) = 0 (2.43) 17 The forcing term A2 is equal to zero since there is no incident field at BS2, thus the solution to equation (2.41) has only a complementary portion given as ,C _. - F2 (7‘2) = C2 (.112) COS (190172) + D2 (1(2) 8111 “7012) (2-44) Substituting equations (2.42) and (2.44) into equation (2.41), gives the MFIE at BS2 12.1 (Fi’) - -, , / E12 (72’) 1 , —‘—,———G , .r 711‘ (15 - -—'.——G . '7. 7 15 /SI jw/lo Cl ( )I 1) 1 S2 JWHO CH ('2l72)( 2 = C2 (.112) COS (A7012) + D2 (.92) Bin (19012) (‘2-45) 2.5 MOM Technique 111 this section, a MOM tecl'mique is applied to solve the MFIEs developed in the last section. The steps taken to apply the MOM technique, along with a summary of the solution, is dis(‘russed here in. 2.5.1 Expansion of Fields In Terms of unknown Slot Voltages In order to solve equations (2.40) and (2.45) using a MOM technique the slot electric— fields are expanded in terms of the unknown slot voltages. Based on the discussion in section 2.3.1, the slot electric—field is assumed to have only a longitudinal component. Using this assumption, it is appropriate to separately expand the slot electric-field into longitudinal and transverse components as follows Eyl (F1) = V1(4171)f1 (.111) f0" 1 = 1:2 (2-46) where the voltage rise across the slot width is interpreted as 41') w] v.(.~.)=_ / Emma): / E.,1(r1))dyz (2.47) W1 —W1 18 The distribution fl is chosen to ensure the integral over the y—dependence is equal to one. This is so the only contribution to the slot field, due to the expansion, is from the unknown slot voltage. The following expansion of the slot electric field is obtained W1 Eli/10:1) = V1061) = V1081) / dyzf1(y1) (2-48) . —Wl where a constant distribution is chosen leading to the function 1 f1 (311) = 2,, [,1 (2.49) 2.5.1.1 Expanded MFIE at BSl Applying the slot. electric-field expansion in equation (2.48) to equation (2.40) leads to the expanded version of the MFIE at BS1 L1 L2 /d-'FiV1 (33/1)KWC(5171»3/1l$’1)— [Fidel/2 (FE/2lKCV2(F’1="JIlFFI2) = Cl (it/1) COS ((170171) + D1(y1)sin(k.():1:1)+ AiOC—jkyloyl COS (Ml/O) (2-50) where the kernels at BSl are defined as . fl (Ll/,1) _. _./ K .“T. u», = / d;’—,-——-—'——G,r, I' 1‘ 2.51 uc(li.y1l11) 111 JWHO 11C( 1| 1) ( ) —-W1 W2 1 ( ') 312 .. ../ K :13, a." = / d-’—2_-—G, 7‘ 1'2 2.52 CV2(1y1| 2) W .112 3W0 cv(1| ) ( ) _ ’2 19 2.5.1.2 Expanded MFIE at BS2 Similarly, substituting the same slot electric field expansion from equation (2.48) into equation (2.45) leads to the expanded version of the MFIE at BS2 L1 I L2 / (mil/1(Fl’ilfi'CV1($2.y2l-1’i)— / d$2V2 (17,2) KCH ($2vaIF’I2) _L1 —L2 = C2 (LI/2) COS ((60172) + D2 (y2) sin ((60:62) (253) where the kernels at BS2 are defined as W1 . f1(y') . _. ACV1 (1‘2s1/2I1‘l) = / dy’ jwul) GCV (7‘2l7‘1’) (2.54) —I'l'1 K .1? ,' 1', = f d, 2_ yg G 7"” *1” 2.55 CH(2y2| 2) W 1123mm) CH(2|2) ( ) - 2 2.5.2 Application of the MoM Technique The coupled MFIEs in equations (2.50) and (2.53) are classified as inhomogeneous Fredholm IEs of the first kind, and are solved using a MOM technique. There are two steps associtated with the Mob/I technique. They are the expansion of the unknowns and the application of a testing operator, using an appropriate set of expansion and testing functions, respectively. The choice of these functions is usually strongly influenced by the pl'iysical and mathematical characteristics of the IEs. From the physical point of view, the expan- sion functions should closely model the behavior of the unknowns, so that a minimum number of expansions is necessary to obtain accurate results. In addition, a prudent choice of the'expansion and testing functions could allow the various integrals to be computed in closed form. It is not necessary to obtain closed—form integrals; how- 20 ever, they do significantly reduce the computational efforts. However, experience with the moment method has indicated that simple basis functions, which expedite the computation of the matrix elements, are usually a suitable choice [60]. The slot under analysis in this fornmlation was chosen to be resonant at the mid- frequency of the rectangular waveguide. Using the knowledge that the slot is the complementary problem of the strip dipole, a reasonable approximation for the be— havior of the slot voltage is a sinusoidal distribution. Thus, it would seem a sinusoidal expansion function is the best fit from a physical standpoint. However, based on the discussion at the end of the last. paragraph a pulse function expansion is likely a more convenient choice, and therefore is chosen in this formulation. Also for convenience point matching at the center of each cell is used for the testing operator. Applying the following point-matching testing operator 1r, 1.) / / dIldyld (3‘) — 'T‘lml) 6 (yl) for ml 2 1,2, ..., N) (2.56) —u)—Ll (l = 1,2 depending on which interface the testing is taking place at) to equations (2.49) and (2.52) res1.)ectively, leads to L1 L2 I I . I I I 2 , ,j / (infill/1 (.171) RWC ($11111 , 0].]?1) - / (LIIQVZ (1'2) ACV2 (I17,,1 , (H.112) FL1 FL2 : C1 (0) cos (1.0.1‘1ml) + D1 (0)sil1 (k().1'1ml) + A“) cos (7111",1 /a) for ~1111=1,2,...,N1 (2.57) 21 for B81, and L1 L2 ~,,I / , I .. ,_I I I , , ,I / (1.11)} (~51)ACV1 (lganOIJq) — / d232V2 (372)1(CH (132",2,0]12) _L1 . —L2 = Cg (0) cos (komgm2) + D2 (0) sin (koxgmg) for m2 =1,2,...,N2 (2.58) for BS2. These equations are then expanded in terms of the unknown slot-voltages, using the pulse-function expansion shown in Figure 2.4 and given as 9 (N ""2 1112 P. PN y=W IIIIIIIIIIIeA x y=-W __ L =L x— L Ax”?! x", x M Art XII/"T xn, 2 Figure 2.4. Pulse function expansion for the MOM technique 151(1)) = Z V1..,P1..,(IE1) (2.59) 22 where the individual segment length and location respectively are A1?) = QLZ/Nl ; 2:171) = —L1 '1' (711 — 1/2)A1’l (2.60) and the pulse function is defined as A1", A41?) 1 (LT-[“1 -- Tl) < 171 < ($1711 + T) plnl ('rl) : 0 othencise (2.61) Substituting the expression for the pulse—function expansion into equations (2.57) and (2.58) respectively, gives '1 I , I Z V1,,1 / drlkwc (1:1,m‘1,0|;171) nlzl . Ilnlfiégl 2 - . ,r ‘l r f _/ — Z I’QNQ / d'l‘ZBCl/Q (.I'IIHI,0[.I,2) 112:1 AI“) 1'2”? "T. ~ 2 )1 (O) COS (kol‘lml) ‘l” DI (0) SH] (ICU-T177”) + A10 COS (71171.,n1/a) for 7111 =1,2,...,N1 (2.62) 23 for B81, and AJ‘ 11‘ + —2—1 [\1 1711 Z )1”, / 111:1 £12.]. _,I , 1 (L1 ll‘CVl (.1 21722 . 0|.11) AI N2 I2112+ +72 _ 2 V2", / dlerCH ($27712 ’ Olly?) A1? 1211.2 — ‘2—2 : C2 (0) COS (kOIQHIQ) ‘l‘ D2 (0) Sin (ICOFTQInQ) for 711.2 2 1, 2, ..., N2 (2.63) for BS2. These equations represent the MOM solution, for the unknown slot voltages (V1,VQ). to the MFIEs of section 2.4. However. four constants (C1, 01, C2, Dg) still remain unsolved. These constants get evaluated by invoking the brmudary conditions VlLleo for 1:12 (2.64) which leads to the final expressions for the MOM solution AI N1—1I17’1+—2_l _ _l r H ,l :2 V1711 / (ll'll‘ll/C ('111111’0lJ1) "1:2AJ‘ [171.1 -71 + ..[P 3 N2—1 ‘1'?"2 , ,,I , , ”I _ Z L2“? / d--’J2I\CV2 (1'1”,1 , 0[.1.2) 712:2 AT 12112 _‘2—2 = C1 (0) cos (19051717721) + Dl (0) sin (Aromlml) + A10 cos (715131",1/(1) for 7711=1,2,...,N1 (2.65) 24 at B81 and lAI] A71_1 $1721+ 1’ ’ , i, Z V1111 / d4L1ACV1 (1227712,0l.l.1) 711:2 AI 1112.1_ 2 A12 [VG—1 IQI'IQ-i— 1.. r I . / — 2: lg”? / (1&2th (£2,772,0IJ‘2) Ar I2n2_——T?2 = C2 (0) cos (A:0.I:2m2) + D2 (0) sin (190372",2) for mg =1,2,...,N2 (2.66) at BS‘Z. 2.5.3 MOM Solution in Matrix Form Equations (2.65) and (2.66) are suinn’iarized in I'natrix form as EV Z a,,,.,.C,,,. = b,” for m = 1, 2, 3, N (2.67) 7221 where N 2: N1 + N2 and Alocos 7r.r1 . (L for m=1,2,...,N1 b7” : ( 771/ ) (268) 0 for m=N1+1,N1+2,...,N 25 is the forcing vector due to the incident :1:-directed component of the magnetic field. The unknown slot voltages and constants are defined as b Cm = i f 07' f.» for for for for m=1 m=N1+1 m=N1+2,N1+3,...,N-1 m = N 26 (2.69) and finally, a summary of the moment method admittance matrix elements r m =1,2,...,N1 — cos (1602:1771) for n =1 . m = 1,2,...,N1 — s1n(k0.r1m) for n = Nl m = Nl+1,N1+ 2, N — cos (k():1‘2m_1\. ) f or i l 72. 2 1V1 +1 . . m = r\’1+1,N1 + 2, ..., N — sin (A'().I'2m_\, ) for i l n 2 N m =1,2,...,N1 O for n = N1 + l, N (VI-772,72 : i m =N1+1,Nl +2,...,N () for n =1,N1 . m =1 2 N1 LT/C 3 i 7 l'rllJl. for 'n 22.3,...,1Nr1—1 m = 1 2. ....N1 ('1 ,. ’ ‘ " _’m.n—N1 f0, , n :i’V’1+2,N1+3,...,N—1 m = N1+1.N1+ 2 ...,N lm—N 1.71 for n =2,3,...,N1—1 _[CH for m =N1+1,N1+2,...,N "m — N1 ,n- N1 n 2N1+2,N1+3,...,N—1 \ (2.70) 2.5.3.1 Matrix Element Definitions The solutions for the various admittance matrix elements (details in Appendix C) are given here. For the first quat’lrant of matrix elements. the. combination of the 27 waveguide and cavity self terms at B81 is [H'C = [Vll’G + [CV11 (2.71) "7111.771 'nllJll "11.721 where WC _ 43' ‘7 .- ,. .- ,. 1w. - W Z r:— (A... (Jim. - a/2)l l’w (m. - 0/2)] .2 _.-. ; a =1,2,3,... ~sin(k.r7AI1/2) [8 “will — 1] for (2.72) ,6 = 0, 1,2, and ("“11 = ————.j . ———‘1" {Air 2:; ‘2) "11:"1 woLWZ A: kmk-yFkZP ‘1“\ ~-) 1/ -sin [161T (“ml — L)] sin [lg—BI. (“"1 — L)] .6652 (1.3,,1. W) sin (kyl. W) cot (art) (2.73) For the second quadrant of the matrix elements, the coupled cavity terms at 881 are . j 61* 1g) :10 12 Z — 2 1.712 ”11.122 W, , ,r ' VOLH ”2 I‘ kl“; 16W. A131. sin (kzrt) ~sin (NJ-AIAJ'Q/Q) sin [1.er (Ilml — 1.)] sin [Al-"I” (1'2”2 — LN .cos2 (kl/1‘ W) sin (Ad-UP 14/2) (2.74) 28 For the third quadrant of the matrix elements, the coupled cavity terms at BS2 are [C2 21C)” 21 _ [”2 "1 m) "1 ’w’lloLW 2 2k yrkT F0631“ rt) -sin (IchAml/2) sin [kl’f‘($27n2 — L)] Sin M[::P ($1711 — L)] .6052 (163,1. W) sin (km. W) (275) Finally, for the fourth quadrant of matrix elements, the combination of the cavity and halfspace self terms at BS2 is given as CH CV 22 HS [7712 712 _—l7n2.J712+17712 n2 (2'76) where “W 22 _ j 2: er cos (kth) 7772 772— . wuoLW W2 F kl.7 kyl. kzl. sm (kzrt) -sin (knew/2) sin [ls-$1. (62m2 — L)] sin (er ($2,112 — L)] .6052 (km. W) sin (163,1. W2) (2.77) and —'A7 ;I‘ —J“ W —jA.r2€ J 0‘ 2772.2 2722 1 ctr/[(2)277 [HS ‘12,",2 _T2n2 ”12,772: _ . AI . _1 2W, ( —"w 2 (72.1.1. (2%) . Am +W smh_1( , )] or m =77. 2 mg f 2 2 f 07" 771.2 yé 712 (2.78) 2.6 Rectangular Waveguide Scattering Parameters The objective of this section is to obtain expressions for the rectangular waveguide scattering parameters Sill”! and 53112.1). This is accomplished by comparing the ratio of 29 scattered to incident electric field intensities at specific interfaces in the rectangular wax-reguide ['58]. The effect of various slot dimensions on the scattering parameters is also discussed. 2.6.1 Reflection and Transmission Coefficients A general formulation for computing the reflection and transmission coefficients in a rectangular waveguide is as follows. First, the reflection and transmission coefficients are defined as r— 5”” ; T: 5”?” (2.79) where the ratio of scattered field intensity to incident field intensity for the reflection coefficient is given as [‘= E?“ T’ 41—3) (2.80) 7 E2710 ( ‘ y=y1 and the ratio incident and scattered field intensities for the transmission coefficient is given as [El-”(7(7) )+ Es+ (7’)“ y=yT T = .- _, (2.81) Eénc( 7‘ )ly=y1 where the incident and scattered fields are added together because both fields are Iiieasured at the transmissicm plane. The incident field intensity is then given by the T5160 mode of the electric-field. Based on the modal fields found in section B.2.2.1, this is defined as inc —,> . . ~ )—j}r y E; (I ): 431063.10 (1. ~) 1. 3110 (2.82) The scattered field intensities, based on the mode expansion fields of section B.3.1.1 are E: (7’) = — z B7.Ag.,e;7 (.132) ejkf/Vy y < —l’Vl (2.83) 7 30 E (Vin , where B7 and C7. are the mode expansion coefficients. Since the incident electric field is a known TE10 mode, along with the properties of mode orthogonality and band-limited guided wave structures, the only mode of interest in the scattered electric fields is also the T E10 mode. Thus, equations (2.83) and (2.84) are written in terms of the TE10 mode as EL: (7) = -310A:10€;10(1,2)ejky10y y < —W1 (2.85) E§(7)=C'10A;mezm (2,2)e‘jk3110y y>W'1 (2.86) where solving for the mode expansion coefficients B10 and C10 gives the desired scattered field intensities. Substituting (2.82), (2.85), and (2.86) into (2.80) and (2.81) respectively, leads to r = —Bwej"’3/102yl (2.87) and T = e‘ikyloly'T”3/Il [1 + C10] (2.88) the reflection and transmission coefficients in terms of mode expansion coefficients B 10 and C10 respectively. Now the mode expansion coefficients, given in erplations (B64) and (B55), are solved by substituting the slot voltages (detern‘iined by the solution to the MOM technique) into these equations and rearranging the solutions so that Alf A11 sin (Icy W1) 1 B10 2 C10 — If)? Z V1 (21,, )cos(k3710:1?1n ) (2.89) l/l/labujfloliylo ”1:1 1 1 where N1 is the number of slot partitions at B81. 31 Substitution of (2.89) into (2.87) and (2.88) gives respectively, the reflection coef- ficient r (3;, y) = 99102911 (:17) (2.90) where N Arlsin (kg. 1171) i 1 F'- =— ‘19 V(.?_ )(k .') 2.91 (I) l/V1rrbw110k.510 7212::1 1 “”1 COS 11011,,1 ( ) and the transmission coefficient T (1:, y) = e“1ky1(>(yT’yI) [1 — r (2)] (2.92) Note that the reflection coefficient in 2.90 is directly related to the slot voltage in 2.91, and the transmission coefficient in 2.92 is related by one minus the same slot voltage. This has physical intuition, in that if the slot voltage goes to zero (ie. the slot is filled with PEC), then the waveguide is restored and complete transmission of "the signal is obtained. 2.6.2 Analysis of Slot Dimensions The purpose of this section is to understand the effect of various slot dimensions (length, width and height) on the rectangular waveguide scattering parameters. 2.6.2.1 Thickness of Slot The thickness of a standard waveguide wall is approximately x\/ 15. Generally in electromagneties distances of this magnitude have negligible effects on the system. However, the slot thickness due to the waveguide wall thickness, even when the size is small compared to the wavelength, exerts a noticeable effect on the slot. admittance. Since the admittance of the slot is significant at non—resonant frequencies, the effect of the wall thickness is very noticeable on the scattering parameters over a given waveg— uide band. Figure 2.5 shows the effect of various slot thicknesses on the rectangular waveguide scattering parameters versus a frequency range of 8.2 - 12.4 GHz. Notice 32 the upwards shift in resonance frequency and the decrease in bandwidth (increased Q of slot (tax-ity) for the scattering parameters as the. slot thickness increases. These results mirror those found by Oliner [47]. 2.6.2.2 Length of Slot The length of a strip dipole is associated with its resonant frequency [55]-[56]. Since the rectangular slot is the complementary problem of a strip dipole, the slot length is therefore also associated with its resonant frequency. Figure 2.6 shows the effect of various slot lengths on the rectangular waveguide scattering parameters versus a f‘re-‘(plency range of 8.2 - 12.4 GHZ. A constant slot length to width ratio of 1/20 and a slot thickness of 3.25 mm were. used to ensure only the effects of the slot length are seen. Also. only 21 frequency points are used in both figures to save on computation time, a little accuracy is lost, but the overall concept is shown very well. The four slot lengths, 1.67 cm, 1.50 cm, 1.36 cm, and 1.25 cm, where chosen since they are resonant at frequencies of 9,10,11, and 12 GHZ respectively. The results show the resonances to be just slightly less than the desired value as expected [55], as well as a slightly higher power loss as frequency increases. 2.6.2.3 Width of Slot The dimensions of the slot are assumed to be long and narrow, with a ratio of one- twentieth chosen as a safe value for a.}')1')li(‘°ation purposes. Here the effect of various slot widths is studied by keeping the length of the slot fixed at. 1.50 cm. Figure 2.7 shows the effect of various slot length to width ratios on rectangular waveguide scattering parameters versus a frequency range of 8.2 - 12.4 GHZ. The four slot length to width ratios (1 /25, 1/20, 1 / 15, 1 / 10) are chosen, and it is seen that as the slot width grows, so does the coupling through the slot. However, it. is also seen that if the width becomes too great, the assumption of a strictly transverse directed slot electric-field begins to break down. Hence, there is a trade-off between coupling and 33 complexity at around a length to width ratio of 1 / 15. 34 Magnitude of $11 81 521 for Various Slot Wall Thicknesses vs. Frequency —-—sll -0mm g —-—521 - 0 mm v ----511 - 1 mm g ----521 - 1 mm :2 ------511-2mm 1% J-----521-2mm : -----511-3mm -30 . - _.- -- ----J521-3mm -35 .1.. . - a 7 . _ . . ,- .,, a, n '40 "i”_" TT—_ " l "a'___" T 'T 7 ‘ V T T j ‘ " '7‘" ‘Y' '1‘ '_‘ '_ Y _ 'Y _ I” " l 8.20 8.62 9.04 9.46 9.88 10.30 10.70 11.10 11.60 12.00 12.40 Frequency (GHz) Figure 2.5. Effect of the slot wall thickness on the magnitude (dB) of the waveguide scattering paranmters vs. frequency (GHz). Magnitude of $11 a $21 for Various Slot Lengths vs. Frequency —-—511 - 1.67 cm +521 - 1.67 cm ----511 - 1.50 cm -*--521 - 1.50 cm ----- $11 - 1.36 cm --°-- $21 - 1.36 cm -----511 - 1.25 cm ----J521 - 1.25 cm Magnitude (dB) '40 i t T T"”"T‘ 1 v v “7' _ -- 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 'I'_'T_ " T Y i T Y 7 Y ’I l Frequency(GI-lz) Figure. 2.6. Effect of the slot length on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 36 Magnitude of 311 8: 321 for Various Slot Length to Width Ratios vs. Frequency —-—s11-1/25 +321-1/25 --‘-°s11-1l20 --'-'821-1/20 ..._,S11_1/15 ...... $21-1/15 ----'s11-1/10 -"-"821-1/10 Magnitude (dB) '40 i r r r V r 1 r 7 8.20 8.62 9.04 9.46 9.88 f fl 1 T——" fl fir 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GHz) Figure 2.7. Effect of the slot width (using slot length to width ratios) on the magni- tude (dB) of the waveguide scattering parameters vs. frequency (GHz). 37 CHAPTER 3 GREEN’S FUNCTION FOR EM FIELD WITHIN A PEC PARALLEL—PLATE ENVIRONMENT 3. 1 Introduction In this chapter, the MT Hertzian potential dyadic Green’s function [61] is derived, for a general 3D current. source in‘u'nersed within a PEC parallel-plate environment. This analysis is utilized in Chapter 4, ultimately leading to the extraction of the constitutive parameters for a lossy PEC backed homogeneous, isotropic media in Chapter 5. Simple relationships are given to extend the MT Hertzian potential dyadic Green’s function to electric and magnetic field forms. 3.2 Geometrical Configuration Consider the PEC parallel-plate waveguide filled with a homogeneous, isotropic me— dia, and excited by an impressed magnetic source as shown in Figure 3.1. The waveg- uide has fixed dimensions for height —d < 2 < 0 and is infinite in extent in the .r, y-plane. The origin is placed at the center of the z = O PEC plate. 3.3 EM Fields and Helmholtz Equation for MT Hertzian Potential Various methods are used to identify the electric and magnetic field dyadic Green’s functions Ce (7:177) and Ch (f‘lf’) [62]-[63]. One method involves identifying Ce H —0 —9 and G h by directly solving the wave equations for E and H. This yields v21? + 128' = v x f... (3.1) . s _. _. 1 _. VZH + [12H :- jLUCJn), “i— ———V (V ' Jr”) (3.2) jet/'11, 38 N) Va, PE C Z=-dt V r Figure 3.1. Geometrical Configuration: Parallel-plate waveguide filled with a simple media. which have relatively complicated relationships between E, E and j. A second method involves using the MT Hertzian potential 7?}, as an intermediate step (see Appendix A for further details), which is advantageous since it produces a simpler relationship between 7?}, and f This relationship is known as the MT Hertzian potential wave equation and is given as (3.3) (r—) (——) —o where G e and C h are then identified by solving (3.3) for 77'), and then computing E and If using E = —jw,u.V x 7?), (3.4) Another motivation for using the MT Hertzian potential is that 7?}, is less singular than E or H. Thus, the MT Hertzian potential Green’s function is derived, and is extended to E and If forms with equations (3.4) and (3.5) if desired. 39 3.4 Spectral Representation of Principal and Scattered Waves In order to identify the MT Hertzian potential Green’s function, a solution to the differential equation (3.3) is needed. This is done using the method of superposition where the solution is the addition of a complementary solution and a particular so- lution. The particular solution represents the principal wave 7?}? emanating from the magnetic source in in unbounded space, and the complementary solution represents the waves 77,“? scattered from boundaries at z = 0, —d in the absence of the source. The total solution for 7?}, is therefore 71,, = If + 75,? (8.6) where 77)]: and 7?}? satisfy the respective MT Hertzian potential wave equations f V277}? + k27ffpl_ — —j(,:':1 (3.7) V27172 + A2712”: 0 (3.8) and k2 = 0.12116. Equations (3.7) and (3.8) are then decomposed into three separate scalar equations each as follows .1 Jr V2711... (C) +1527?) P “(7) — ———"l” (4) (3.9) 1%” V773. (17+k'2703. (0 =0 (3.10) where a = :r,y, z. The solutions of these equations are found using the Fourier transform domain method. It is apparent that the structure is invariant along the :1: and y directions, thus prompting transformation of those variables using the 2D 40 Fourier transform pair be) “(PA (3.11) 00 OO 1 / /< 2702 --00 —OO 00 00 = / / \P(f)e_j’\'77dzrdy (3.12) -oo—oc where X = if + (2)2) (=> A2 = X . X = 52 +712). 7": .23.): + gy + :32 and d2/\ = dgdn. No transformation with respect to :5 is applied so boundary conditions may be enforced later at :5 = 0. ——(1. Applying the Fourier transform differentiation theorem, equations (3.9) and (3.10) become 82 ~ _. ~ _, jllla (X: Z) $71111; (A, Z) — pQWfa (A, Z) = —_]T(,d—#—_ (3.13) (92 s -' 2~S * $732.2 (A12) _ p "ha (M) Z 0 (3'14) 8 00 71,]; (X2) : / /7r}Pa (F)e jX Fdntdy (3.15) ‘—0C_ QC 7r}?w (F) e_jX'Fd;rdy (3.16) (I :12 3:12 A “>4 N \__/ || Jma (F) e_j)‘"dedy (3.17) .1510 (X z) = The general form of the solution to equations (3.13) and (3.14) is investigated in é\8 3\8 | 8l\8 8\8 sections 3.4.1 and 3.4.2. 41 3.4.1 Principal Wave Representation The spectral-domain representation of the principal wave is obtained by solving equa- tion (3.13), as dictated by the method of superposition. Since there are no boundaries present, the solution to this second-order partial differential equation is valid for all unbounded space. The solution to equation (3.13) is well-known and the details may be found in [7']. A quick (werview of the steps involved is presented here along with the solution. The first step involves the transformation of the last spatial variable 2: into the spectral domain. Then, along with the aid of the remaining differentiation theorem, solving for 77F}; gives ( 72' = 3.18 2.. m (4? +222) ( ) -P (- J22... (A, c) ~ A, C) Now, the inverse transform is taken to return to the complex A—plane, ensuring that the z-variable is present. for implementing the boundary conditions. After solving the complex C-plane analysis using Cauchy’s Integral Theorem [64], the desired result for the representation for the principa1 wave in the complex A-plane is 71)]; (X25) 2 GP (Kg: 3’) jma.(;\.,z,)dz' (3.19) In this, the spectral-domain principal wave MT Hertzian potential Green’s function is given by GP (Sink!) = E——2p—— (3.20) ~_7 v 11 with :5 as the field point, 2’ as the source point and p is the spectral-domain propa- gation factor. 42 3.4.2 Scattered Wave Representation The spectral-domain representation for the scattered waves is obtained from equation (3.14). The well—known solution of this second-order partial differential equation is a): (X, ,2) = n": (X) 8‘!” +11,- (X) (W (3.21) where Wat, are the complex amplitude coefficients of the up and down traveling re- flected waves. 3.4.3 Total Wave Representation The spectral-domain representation of the total wave is obtained by superposing the results from the previous two sections. This leads to the following result I —p 2—2 ~ .~ __ ~P ~S _ 8 Jma J 3+ —pz , ,r— )1): r ., I "v where functional dependence in the above equation has been dropped for notational convenience. Noting that the distance in the exponential for z-dependance integral has a sign change based on whether the observer location is greater or less than the source location, given as I z — 25’ . . . z > 3’ .3 — 2 = (3.23) 7’ — v 7 < 25’ allows equation (3.22) be written resrwrrtively as 22),“, = Lye—P: + 121/32?!” + It}: 221’: ...z > .~.’ (3.24) 22,“, = Va—epz + Wye—P3 + Ware!” ...2 < z’ (3.25) 43 where V; 2 V3: (X) = / eipzl Jma (5:13,) (13’ (3.26) are associated with the up and down traveling waves launched by the source. This also helps simplify the notation when implementing the boundary conditions. To obtain a unique solution for the spectral-domain MT Hertzian potential, six boundary conditions are enforced to solve the the six unknown spectral-domain coefficients. These spectral coefficients are determined in the next section. 3.5 Computation of Spectral Coeflicients To determine the spectral coefficients W3, boundary conditions are enforced on the total wave representation of the spectral—domain MT Hertzian potential. Since the boundaries of the parallel-plate structure are PECs, the well—known relationship that the tangential electric field equals zero on the surface of the PEC [58] is used at the boundaries 2: = 0, —d. Leading to the expansion of equation (3.4) in terms of the tangentiaf components E .1: and By as (97th~ 871% fan}, (97th “I“ — -— ' E :: —" l ‘1" ay 32 ’ y 1““ L 82 a2: Now, using the educated conjecture that any Jma, for a = :13, y, 2, leads only to the same mm, and enforcing the previously stated boundary conditions at a PEC inter— face, the following set of boundary conditions on the spectral-domain MT Hertzian potential are found 02 A: ”h; = 0 (3.29) 44 where equation (3.28) is the tangential boundary conditions, and equation (3.29) is the normal boundary condition. These boundary cmiditions are implemented in sections 3.5.1 and 3.5.2. 3.5.1 Tangential Components (02 = ;r,y) Enforcing the first tangential boundary condition (at the z = 0 interface) by substi- tuting (3.24) into (3.28) leads to the following expression 1)(—VO+ —W'(;L+IVC:) =0 ---a=x,y (3-30) where solving (3.30) for If}: gives 11;,- = v;,+ +117; ---(1 = .2. ,2) (331) Then, implementing the second tangential boundary condition (at the z = —d inter- face) by substituting (3.25) into (3.28) results in p (Va—e—Pd — l’lx’jepd + age-Pd) = 0 ---a = .2. y (332) where solving (3.32) for NHL gives (1 u": 2 24M (V; +115) ~22 = .2. y (333) Now, the substitution of (3.31) into (3.33), combined with solving for If": , leads to e—Pd V.’ + vt W21 = 2222(1 32—de ) “e = I y (334) 45 followed by substituting (3.34) into (3.31), and solving for W; , giving r— — d 7+ ,')(1 Va. 6 p + Va (1 VII/a = epd __ e—pd ~02 = 13y (3.35) where the tangential spectral-domain coefficients W; are only in terms of known coefficients Vai. All that remains to find the tangential spectral-domain MT Hertzian potential is to substitute (3.34) and (3.35) into (3.22) and manipulate the expression into its final form, given as 0 ~ —0 ~ _. I~t _, I Jma (A22) , 7Tb“ (X2) = fdz G (Apalz) W212 (3.36) -—d where G" is the tangential spectral-domain Green‘s function Q. A 94 L: N‘ V II Q2 22 A >4 ;z|z’) + Gst (X;z|z’) (3.37) (3P (X; zlz’) = ’—— (3.33) I I I I — z—z +a’) — )(:+: +21) — )(—2+: +d) — ' (—:—~z —(1) c p( + e I + e I + e p 2]) ((,.pd _ (0—H!) ll C38? (X; :I:') (3.39) This form of the Green’s function is very convenient for obtaining physical insight into its behavior (discussed in section 3.7), however, another form is presented in section 3.6 that is more practical for use in Chapter 4. 46 3.5.2 Normal Component (0 = 2:) Enforcing the first normal boundary condition (at the z = 0 interface) by substituting (3.24) into (3.29) and solving for It"; leads to W; = —VZ+ — W; (3.40) Then. implementing the second normal boundary condition (at the z = —d interface) by substituting (3.25) into (3.2.9) and solving for If": this time gives W: = —e“2Pd (v; + W;) (3.41) Now, in the same manner as the tangential components, substitution of (3.40) into (3.41), combined with solving for ”if, leads to 2:211 (-21; + 2:) .’+ _ ' H 3 —— epd _ e—pd (3'42) followed by substitution of (3.42) into (3.40), and solving for V1707, giving Vie—pd — V..+ cm W"; = “ “ (3.43) epd _ e—pd where the normal spectral—domain coefficients WZi are only in terms of known coef- ficients V}. Again, similarly to the tangential components, all that remains to find the normal spectral-domain MT Hertzian potential is to substitute (3.42) and (3.43) into (3.22) and manipulate the expression into its finally form, given as M 0 j (X ) .. - _. ., "lg 2 22),, (X2) = /d.~.’G” (X215) ————-—2-12’ (3.4.4) 47 where G” is the normal spectral-domain Green’s function G" (X; zlz’) 2 GP (X;z|z’) + GS” (X: zlz’) (3.45) which is split into principal GP and scattered C's" portions as follows GP (X; zlz’) 2: e____ (3.46) ~S _. , €—p(:—z’+d) _ €—p(z+zl+d) + e—p(—z+z’+d) _ €—p(——z—z’—d) G ”(A;2|:)= , ,, ’ 2]) (epd _ e—pd) (3.47) 3.6 Dyadic Green’s Function The purpose of this section is to present. a more compact and practical version of the MT Hertzian potential dyadic Green’s function for a PEC parallel—plate region. The lack of pl’lysieal insight upon first looking at this form is the only consequence of expressing it in this manner. Since, in practical use the MT Hertzian potential is used in the space-domain, the dyadic Greens function presented is also in the space-domain. The space-domain MT Hertzian potential is represented as 22,, (2*): /V dV’Z‘" (21,2); (3.48) (——) _ where G is the space-domain dyadic Green's functlon [63], and IS represented as follows E? (FIF) 2 7G0 (FIF) = .ifG” (FIF) .i' + 32G” (71F) ,1] + 500 (11F)? (3.49) (22V 1 I 00 3C ._‘ d 4 Ga (Fifi) = 1 / f C?“ (X; :I:') (221304 >(12A (3.50) 00—00 48 and G“ is the spectral-domain Green’s function, given as 45—4,», A, 4.; ) :1: coshp (11+ 2: + 2’) 2p sinh (pd) coshp d — ’) = ( (3.51) where the :1: refers to the tangential (+) and normal (-) cases respectively. It is noted that the dyad has only diagonal entries since no coupling between sources occurs. This is because the boundary condition at a PEC requires only the same potential component as the source component to satisfy the boundary condition. 3.7 Physical Observations In this section, equations (3.37) and (3.45) are analyzed to gain physical insight into the behavior of the waves in the PEC parallel-plate region. It is noted that the exponentials in both (3.38),(3.39) and (3.46),(3.47) respectively, are the same, thus only the independent terms are analyzed. Referencing the geometry in Figure 3.2, consider first the wave that travels directly from the source point :5’ to the field point 2 while traversing a distance of z — 2’. Since this wave does not reflect off either of the PEC walls it is associated with the principal wave in equations (3.38) and (3.46). Next, the second wave reflects off both of the PEC walls while traveling a distance of z — z' + 2d, thus, it. is associated with the first term of equations (3.39) and (3.47). The third wave is associated with the second term of the same equations since it only experiences one reflection and travels a distance of z + :5, + 2d. Similar discussions show that the fourth and fifth waves are associated with the third and fourth terms the same equations as waves 2 and 3. Note that since waves 2—5 scatter off the top (3 = O) and / or bottom (2 = —d) plates they are associated with the scattered waves. 49 Z """""""" AL 1 2 3 Z, "d It. .422. a: f... ..., is“..- _ -42: 3| Figure 3.2. Physical. observations of the waves within a parallel—plate waveguide 50 CHAPTER 4 RECTANGULAR WAVEGUIDE RADIATING THROUGH A TRANSVERSE SLOT INTO PARALLEL-PLATE WAVEGUIDE FILLED WITH A SIMPLE MEDIA 4.1 Introduction The formulation for a finite—thickness transverse slot centered in the broad wall of a rectangular waveguide, radiating into a parallel-plate waveguide filled with a homo- geneous, isotropic media, is considered in this chapter. Specfically, the reflection and transmission coefficients for the rectangular waveguide are sought. This is because they are ultimately used in Chapter 5 to perform the complex constitutive parameter extraction on the media in the parallel-plate waveguide. A majority of the formula— tion provided next mirrors the formulation provided in Chapter 2, thus not all details are repeated herein. The first step in the formulation is to develop representations of the fields in the three different regions (W'G, CV, and PP). Next, IEs are developed by satisfying the continutity of the tangential fields at the two interfaces (B81 and BS2) between the three regions. The final step is to solve the coupled IEs using a MOM technique for the unknown slot voltages, which in turn identify the scattered fields in the rectangular waveguide and ultimately the desired reflection and transmission coefficients. 4.2 Geometrical Configuration Consider a rectangular waveguide, on top of an infinite PEC parallel-plate waveguide filled with a homogeneous, isotropic media, with a transverse slot cut in the bottom wall as shown in Figure 4.1. 51 AZ 2 = bl I L 3 Region I: Waveguide PEC y=y1 y=-W y=W y=yr z =t 4 , A . (T «Region H: Cavity? y z = O ‘ > (#25 ) y = 0 Region III: Parallel- late Z = 'd ) v - . > Figure 4.1. Geometrical Configuration: Rectangular waveguide radiating through a transverse slot into a parallel-plate waveguide filled with a simple media 4.3 Field Representations In order to develop the desired IEs a representation of the fields in the three regions is needed. To generate the appropriate set of field equations a knowledge of the sources is necessary. The equivalent currents that maintain the fields in the three regions are therefore needed. 4.3.1 Equivalent Currents The equivalent currents necessary to represent the fields in the three regions for this formulation, are exactly the same as those developed in Section 2.3.1 for the formulation in Chapter 2, and are given as K;IUY)=1E;(fi) (4n K231 (III) —jEy_1 (III) (41-?) I232 (72’) = 22;, (12’) <4 3) . ____ 2,7,, (232’) = -IE;2 (22’) (4.4) where (4.1) corresponds to the waveguide equivalent magnetic current, (4.2) and (4.3) to the cavity equivalent magnetic currents at B81 and BS2 respectively, and (4.4) to the parallel—plate equivalent magnetic current. Note that all the equivalent currents have only an .r-directed component. 4.3.2 Total Fields in each Region Using the general field relations from Chapter 2 (reference section 2.3.2), the principle of superposition [58], and the knowledge that the equivalent currents have only an r—directed component, the magnetic field is written in terms of only the :r-component as follows H.(2)= (70: +1. 2)2,,, (2) (4.5) where 27),, is the I—component of the MT Hertzian potential given as Km. (,2) juJ/l (15" (L 222.12) = /g 02.01%) (4.6) with Gm as the .r-component of the dyadic Green’s function and Km}; as the equivalent magnetic current for each specific region. The total magnetic field in region 1 is then written as follows H""”’“I< )= HI""(2 )+ HIIGS<2 ’i) <4 7) £17 . where H5?“ is the incident. wave in the rectangular waveguide, and HB'C'S is the waves scattered by the transverse slot in the rectangular waveguide. In a similar manner, the total magnetic field in region 2 is H5‘9‘0’°’I<2=) HEI1<22)+ HSVI<22) for 1:1.2 (4.8) 53 where Hg V1 is the waves scattered by the tranverse slot at z = t in the cavity region, and Hg]? is the waves scattered by the transverse slot at z = 0. Finally, the total magnetic field in region 3 is H.11’9I"”'II(«2~2) = H5P1222) (4.9) HPP where 1s the waves scattered by the transverse slot 111 the parallel plate region 4.3.3 MT Hertzian Potentials in terms of Equivalent Currents All that remains to fully represent the fields in each region, is to define the MT Hertzian potentials associated with the total fields in equations (4.7-4.9). This is accomplished by substituting equations (4.1-4.4) into equation (4.6) and defining the Green’s functions for each region, leading to EJr (11’) 1105 !_/__l . I G [S 4.10 77hr 712/51 llG( ’71,) —_(_jW/1() l ( ) E— (17],) CV1 ~ 42 -‘/1__ 1’ 7r = —G I '1‘ 1'1 , db (4.11) 15+ (13') CV2 2 ~ ..2 ”U? ' 7r 1" = G V 1‘ 1‘2 , (15' (4.12) 12, ( 1) 52 C (II ) WHO 2 2— (2') PP —. ~ ..2 y? 2 ,, ..., Z -2; 222-2 —-——_——ds’ (4.13) where (4.11—4.12) is observed at either BSI or BS2. The Green’s function for the rectangular waveguide due to a transverse slot is given as , _, _. 6"; . . GH’G (1‘1]1‘1’) = 2115 Z, W sm [1112», (T1 —— 0/2)] sin [1411.7 (..1’1 — 0/2)] - 1 ,e_Jky’-\f 54 ”IT/1] ~12 2 1 - I2 2’ t 414 C05 37(231— ) COS 23A, (~1_ ) ( . ) The Green’s function for the rectangular cavity is given by ‘r kgr 5111(k3F(T(-) -sin [1211.14 (.111 — a,C-/2)] sin [3711“ (517] - tic/2)] .eos [391“ (111 — bc/2)] cos [kl/1‘ (y; — (Jo/2)] (4.15) cos [kZF (z) -— 09)] cos (kzrzf) for z) > 2] cos (13:er) cos [11731. (2] — 0(2)] for :1 < 2] 022121127) = 7:2 and the Green’s function for the parallel-plate waveguide from Chapter 3 is 3c. 00 (p ,0 0222 (23122) =/ f 212 (2)““1‘0’92 (4.16) x 002]) sinh (pd) and the spectral wavenumber is p2 = A2 — k2 = £2 + 172 —— k2 (4.17) The Neumann’s number for the waveguide and cavity regions respectively is 1 fO’I‘ (301,901,) 2 0 _ 1,7,.) 2 (4.13) 2 for 1(9) 22 0 and the wavemunbers for both the waveguide and cavity regions respectively are (r . . 7r ‘ (It'gxm) k’ = .2 , , , : 1’ 2, 3g 0.. 312.1“) 2112222 2..) “(11922) , 3 I (1119, cv)7r k7 . — O, 1, 2, '0- Hh I): bfwg 2(1)) fi( wg,CU) (”(1.1") 0 ”(121‘) “”(ml‘) ”(“1213 ( ) C51 C31 Details of equations (4.14) and (4.15) are found in Appendix B. These field represen- tations for the three regions are now used to develop IEs in the next section. 4.4 Development of the Coupled MFIEs The MFIES at B81 and BS2 are developed in a. similar manner as those developed in Chapter 2 (reference section 2.4). 4.4.1 Development of the MFIDEs MFIDEs are developed first. followed by solving the second-order partial-differenital equations, to arrive at the desired MFIEs. 4.4.1.1 BSl The MFIDE at 1381 is developed by invoking the continutity of tangential magnetic fields at the interface (3 = 1) giving nggionl (73) : H;egi0112 (r1) (4.20) then, substituting equations (4.7) and (4.8) into equation (4.20) leads to Hi7)d,,,—*(( 1)+HWGS(7‘ 1:) HCV1(T1>+HCV2(T-i) (4.21) finally, substituting (4.6) into (4.5) and then (4.10—4.12) into the respective magnetic field representation. leads to the first MFIDE result 292+ (( (El/1] , — ———QG dS (0131—21178) (51 lelo ” C (21!, 1 ) 1 E. . _ / ——’{? (T2 )GCV (21W) 2135} = —H;’2"C(r‘i) (4222) 52 Mile where GWC (Til‘r‘i’) = GWG (Till’i') + GOV (Til‘l‘i') (423) 56 Note that the continuity of the electric fields across the two interfaces is enforced by making E+ yl =Ey1 and Ey+2— — y2. 4.4.1.2 BSZ Similarly, the MFIDE at BS2 is developed by invoking the continutity of the tangential magnetic field at the interface (3 = ) giving Hwy-170122 (7—2) : H7:pgi0113 (7‘3) (424) .T I where substitution of equations (4.8) and (4.9) into equation (4.24) gives HCV1(T2)+ HCV7‘72( 2:) HPP(,,r 2) (4,25) Substituting (4.6) into (4.5) and then (4.11-4.13) into the respective magnetic field representation, leads to 692 -53—i (fl/) .2 a (07 + kg) [S ‘WGCV (‘7‘2l’rlil d5; “ 1 a? EMT?) s .. + (j + 7‘3)/S '—'U-,———GCV (T'2l'r'2') 075% ' 2 ()1 MILO 09' , ( E:2(’2 ) 2 (ii-2 + kz)/ —-‘-’.—-2-—GPP (T2172 ) (152 (426) .L 32 J71)” By adding and subtracting kg to equation (4.26) and doing some simple mathematical manipulations, the MFIDE is written as 82 Ell ( ) I +k —"/——1--G d3 (0132+ 8){/51 MM) CV (film) 1 E w E ,_. _ f8 y2j(72 )GCP (7‘2l7‘2 ) (15%} :- kQL MGPP (rah-é!) (13% (4.27) 2 . w :2 Jam 57 where k2 = k2 — kg (4.28) and _. _. ~G.’ 7777', G Ff" GCP("‘2I7‘2’) = m 5702' 2) + PP (”2| 2) (4.29) This form is useful when solving the partial-differential equations in the next section. 4.4.2 Solving the Second-Order Partial-Differential Equation The technique used to solve the second-order partial—differential equations in this section is also used in Chapter 2 (reference Section 2.4.2). 4.4.2.1 MFIE at BSl The integro-differential equation (4.26) is written in the form a? s s (575 + ‘3) F101) = A101) (430) where 4 E1 7'“, a a F1.(7‘1)=/S MGM/F (7‘1IT‘1’) (155 1 ijO E 7"" — f ——y?( 2 7cm (fil'ré’) dsa (431) 52 Jud/1.0 is in the form of the MT Hertzian potential and A1 (Ti) = —H§i¥"'(" (7‘3) = —A10 [Ur/(1)2 - ‘12)] C05 (775151/(L)€_jky10y1 (4-32) is the forcing term based on the i1‘1cident :r-directd magnetic field. Solving this partial differential equation in terms of a complementary and 1')articular solution (details in 58 section 2.4.2.1) and then superposing the two solutions leads to the MF IE at BSl E111 (743,) _. I ’ / E3]? (73,) I I ————‘. G r r” (13 — —. G "”7 15 [91 Jw/I'O HC( 1| 1) 1 32 quo CV (71l7‘2)( 2 = C1 ("g/1) cos (1.0111) + D1 (yl) sin (A5011) + 241()e_j"'y1()y1 cos (nail/a) (4.33) 4.4.2.2 MFIE at BSZ Similarly for BS2, the integro-differential equation (4.27) is written in the form a? .. .. (513 +163) F2 (7‘2) 2 A2 (7‘2) (4.34) where E I” F2 (722) = f _______.,'11 i 1 )ch (F217?) dsi 51 JWHO E 2 7‘2’ _. _. —/ -—-————y ( )GCp (’rgl’rgl) (185 (4.35) 52 M is in the form of the MT Hertzian potential and Ey? (7:2,) Age-5) =1}:2 / ——G p raw-3’ (18’ (4.36) 32 J01,” P ( | l 2 is the forcing term that was generated by added and subtracted kg to equation (4.26) The solution to equation (4.34) is written as F. (a) = F? (a) + Ff (a) (4.37) where the complementary solution is given as F20 (7‘2) = C2 (3120) COS (1.10:1:2) + D2 (3120) sin (760172) (4-38) 59 and the particular solution, in terms of the forced response in (2.32) and the forcing term in (4.36) is .._‘ m x __ F.2P(I‘72)=E/2 d5 EJJ 72)] / (1A6 )5. (€J"(yJ'/JJCO*‘1 Solving the forced response integral, and performing some algebraic manipulation, gives the following form of the particular solution _ 00 oo - ,I ' _ 7 p g k2 .1 Eu? ('2’) 2 (”TJE‘FQBMQQ y2) cosh (pd) F2 (7'2) 2 — ([52+— d A 2 . ' . (27r) ps111h(p(1) . [jg sin (150.13) + k0 cos (from?) — 130(9ng (440) £2 - A78 Substituting equations (4.35), (4.38) and (4.40) into equation (4.37) and combin- ing the complementary sohitions as well as the similar integrands over .92, leads to the MFIE at BS2 E r'/ E r :7, / Mamwm’yisi— / M00122 (wilfé') {-75% SI JW‘IU'U 52 = 62 (1)2 0)cos(k012) + D2( shame-01:2) (4.41) where GCV (7"2I7‘2') + GPPN (”zit—2') #0 H GCP2 (IEI'IE') = (4.42) 60 and the new parallel-plate Green’s function is given as 0° 0° (P2 P2) ,h d 2 —k2) G = (122er cos 0’ J“ 4.4 PPN (WIT?) {1C [0 (271) psmh (pd) (52 kg) ( 3) 4.5 MoM Technique In this section, a MOM technique is applied to solve the MFIES developed in the last section. The steps taken to apply the the MOM technique along with a summary of the solution is discussed herein. 4.5.1 Expansion of Unknown Slot Voltages The expansion of the slot electric-field in terms of the unknown slot voltages mirrors Chapter 2 (reference Section 2.5.1). 4.5.1.1 Expanded MFIE at BSl Applying the slot electric-field expansion in equation (2.48) to equation (4.33) leads to the expanded version of the MFIE at 881 L1 L2 / (tr/1V1 (1’1) KWC (.171, 31111711) — / (1.142 V2 (LE/2) KCV‘z (.171, y1|.1’2) —L1 —L2 2C1 (y1) cos (110.11) + Dl (y1)sin(k011) + Aloe—JkyIOyl cos (7121/0) (4.44) where the kernels at 881 are defined as W1 f1(y1- _. KWC('J311’y1l$’1) = [143/ '1 1].W0 y1)GWC (r 1|7‘1') (4-45) —l’l’1 and W2 ) . f2 y s .. ACV2 (11.311136) 2 / dy I Z—gfiGCV (I'llr‘gl) (4.46) -112 Iwito 61 4.5.1.2 Expanded MFIE at B82 Similarly, substituting the same slot electric-field expansion from equation (2.48) into equation (4.41) leads to the expanded version of the MFIE at BS2 L1 ' 12 / (1.1/1 V1 (.13) KCVQ (.12, y2l1'i) — / (1.112% (1%) KCpg (.12, yQII’Q) —11 —12 = 62 (.12, 0) (10.12) + D2 (12, 0) (11112) where the kernels at BS2 are defined as ”"1 f1(yi) 2 -1 K 11.1 ,l = /d,——G "‘ CV2 (F2yzll‘1) 1J1 Jame CV (72l7‘1) —W1 and w.2 f ( I) . 2 yr 2 2 ACP2(-T2~.l/2|-I"§)= / (ll/2 N2 GCP2(7‘2|7‘2’) 4122 4.5.2 Application of the MOM Technique Applying the following poii1t—Inatching testing operator W1 Ll / / dxldyld (I1 — 11ml) 6 (yl) for m, = 1,2, ..., N) —W —L) 62 (4.47) (4.48) (4.49) (4.50) 3*: (i; ( = 1,2 depending on which interface the testing is taking place at) to equations (4.44) and (4.47) respectively, leads to L1 L2 I ,r / dill/1 (110111.1er (11ml,OI1II) — / das’2V2 (1’2) KCV2 (3717n1,0|:r’2) —L1 412 C1 (0) cos (190.117,,1) + D1 (0) sin (I‘ll-171ml) + A10 cos (mum/1 /a) for 711.1=I,2,...,N1 (4.51) for B81 and L1 L2 I I r I I I I / dl‘lhfl (I1) I‘CV‘Z (:FQTnQ’OLTI) — / dLIIQVQ (IE2) KCP2 (11727712,0l$2) _L1 —L2 = (:72 (0, 0) COS (kol‘gm2) + D2 (010) Sin (1503:2171?) for 7112 = 1,2, ...,N2 (4.52) for BS2. These equations are then expanded in terms of the unknown slot voltages, using the pulse-function expansion given as where the individual segment length and location are respectively A171 = 2L1/N) ; .1th 2 —L1 + (n) — 1/2) Ar) (4.54) and the pulse function is defined as ‘ A11 1 Ar) 1 ($11,711 - T) < .Ll < (117” + T) an(I0== ’ 0 othe'rwzse 63 Substituting the expression for the pulse-function expansion into equations (4.51) and (4.52) respectively, gives A11 A11 . $1711 + I I Z V1711 / dleWC($1m1’0l$1) ”1:1 A17 551111—71 A12 + N2 $2712 I I _ Z V2712 / dw2KCV2 ($17n110lx2) "2:1 AI $2112- 2 = ,1 (0) cos (11011qu + D1 (0) sin (kUl'lml) + A10 cos (”1'1qu /a) for 1111=1,2,...,N1 (4.56) for B81 and ‘ AI NI Iln1+ I I Z Vln1 / (1:171ch? ($2m2,0l1'1) 711:1 ‘ All? 11711— A1? 1V2 1‘2”? + 72 r ..I " - 1 I — l“2112 _/ (ill-12A CP2 ($27772 ’ 0'1?) 71-221 /_\..r. T2112 _ 72 : C2 (0, 0) cos (“321712) + D2 (01 0) sin (AU-172,712) for 7712 = 1,2, ..., N2 (4.57) for BS2. These equations represent the MOM solution, for the unknown slot volt- ages (V1, V2), to the MFIEs of section 4.4. However, four constants (C1, D1, C72, D2) still remain unsolved. These constants are evaluated by invoking the appropriate 64 boundary conditions VILNZZO. for l=1,2 (4.58) to muations (4.56) and (4.57). which leads to the final einn'essions for the MOM solution AI , +71 Nl—l $1111 I . I Z V1711 / dIlKl’VC (£11111 1 Ol‘rl) 111:2 A1 $111.1— A12 + A2_1 2712 J , ,1 _ Z ‘61)? / d1‘2ACV2 ($177113()l12) 712:2 AI. r2122- '2 = 21 (0) cos (#0111711) + D1 (0) sin (AIN'lml) + A10 cos (mqml /a) for 'r'n1=1,2,...,.’V1 at 881 and AI . +7—1- N1—1 11711 I r I Z V1,,1 / dzrlhcvg (.rgm2,0|;r1) "1:2 Ar Jll‘nl- 2 A1? 1‘ +72 N2 - 1 I 2'”? /' ~ 7., ' . .I _ 2 L2”? / (1.121XC‘P2 (12")? . 0'12) 112:2 A”, ‘1‘2112_“2_2 _ ~ . . / . . ‘ - . ,. _ C2 (0, 0) cos (140.12,,12 + D2 (0, 0) 3111 (LO-12mg for 1112 =1,2,...,N2 at BS2. (4.60) 4.5.3 MOM Solution in Matrix Form Equations (4.59) and (4.60) is summarized in matrix form as N Z 01771.71C7n : b777, f0?" m : 1, 2, 3, ..., N Where N = N1 "+— N2 (4.61) 12.21 where N = N1 + N2 and. A cos 71.1 . a. 07' m = 1,2,...,N1 m: m ( m/) f mm) 0 for m=N1+1,N1+2,...,N is the forcing vector due to the incident x-directed component of the magnetic field. The unknown slot voltages and constants are defined as C1(0,t) for m. = 1 V1", for m. = 2, 3, ..., N1 — I. D1 (0,1‘) for m : N1 Cm Z 4 (4.63) C2 (0,0) for m. = NI + 1 f0?" 7n=Nl+‘2,N1+3,...,N—1 (QWfl) firmzN 66 and finally, a summary of the moment method admittance matrix elements is given bv U 0711,12. 2 i — cos (1.7011771) — sin “(111m) — COS ([60172711— N1 ) — sm (k0.12771_N1) [H’C’ 171,11 _lCV1 171,11—N1 C V2 111—N1 ,n _ CPQ 'm—Nl ,n—fVl m 77?, 71- m TL m n 171 TH 11 4.5.3.1 Matrix Element Definitions are given here. 67 :12 .,N1 :1 =1,2,...,N1 :le =N1+1,N1+2,... =M+1 =N1+1,N1+2,... =N :12 .,N1 :N1+1,N =N1+1,Nl+2,... :1,N1 =1,2,...,N1 =2,3,...,N1—1 =1,2,...,N1 =N1+2,N1+3,... =N1+1,N1+2,... =2,3,...,N1—1 : A71+131Vr1+ 2, _—. N1+ 2, N1 + 3, ,N (4.64) ,N—l ,N—l The solutions for the various admittance matrix elements (details in Appendix C) For the first quadrant of matrix elements, the combination of the waveguide and cavity self terms at BSl are [WC :lwc +lCV11 (4.65) 7111 ,n 1 m 1 ,nl 1711 .11 1 where we _ -2J' 6) ‘ . .- 171114711 — Layoublvl ; 133776712”, 5m [11:17 (117711 - a/2)] .8111 [kmy (5131711 — 0/2)] O:=1,2,3,... -sin (krfiAg171/2) [e—jkyV'Wl — 1] for (4.66) 13 = 0, 1, 2, and ,. J W [C111 : —— — "11 ~"l wlLOLt/v'g ; [€137 [Cy]? kZI‘ .1205? (1:,F W) sin (13,1. W) cot (Iczrt) (4.67) sin (£71,),A331/2) For the second quadrant of the matrix elements, the coupled cavity terms at BSl are .7 . V12 J 61“ [1(711 n 2 [Si '11. 2 v 1 Z l 2 l 2 1.1.1110th l’LQ F k1,”, kyF kzl‘ sin (ert) -sin (A‘I7A1‘2/2) sin [£11.14 (117,71 —— LN sin [kl’I‘ (127,2 — LN .cosQ (kyFW) sin (1,1. W2) (4.68) 68 For the third quadrant of the matrix elements, the coupled cavity terms at BS2 are C 2 CV ‘21 _ 1m? 711 =lm2 "1 —w—__110LW2 Z Fkkayf‘ [CIT sin I20¢th) sin (kIA/A11/2) sin [k ”IF (2327,12— L)] sin [kIF (1‘1”1 — LN ~c032 (kyrw) sin (kyrw) (4.69) Finally, for the fourth quadrant of matrix elements, the combination of the cavity and parallel—plate self terms at BS2 are C' H C' l/ 22 PP . 17712. 712 :17n2.112+l7712 "2 (4°70) where [C V22 _ J' (T COS (kth) "l 7? r 2 2 w/xcoLW W2 I 1cm7 kyrkzr sin (art) .sin (kIAfAzrg/2) sin [kIr (1:21”? — LN sin [kIF (1172,”? — L)] -C'os 2 n (21")1 ,- .- F1 I Z [e’J‘EUA + 816L313] fOT' in), : n [—ej€v‘4 + ej‘sl’B] for m < n and [e—jkOA — e‘jkOB] for m > n er)2 '. F11 = [e‘Jk0A+eJkOB] for m=n [—ejk0A + ejk’OB] for m < n. where the terms A, B are. given as A : Ax (m — n + 1/2) B = A1:(m~—n— 1/2) and the various spectral constants are 2 jm . . 2 2 WW . 7/3 =1» + 16-2 —€ ; pu = -—d ; m = -J - (773) Egsz—kg ; 2?: k2—k-3 70 (4.76) (4.77) (4.78) (4.79) (4.80) (4.81) with the N eurnann number for the residues given as 1 for z/ = 0 61/ :- (4.82) 2 for 1/ 75 0 4.6 Rectangular Waveguide Scattering Parameters The objective of this section is to obtain expressions for the rectangular waveguide scattering parameters Siliy and Sylly. This is accomplished by comparing the ratio of scattered to incident electric field intensities at specific interfaces in the rectangular waveguide [58]. The effect of certain slot dimensions, parallel-plate dimensions and EM materials are discussed utilizing the formulation. 4.6.1 Reflection and Transmission Coeflicients » _ g . g ‘ - - _ ’ thy , 1' 7th}! - The rectangular waveguide scattering parameters S11 and 521 were developed in Section 2.6, and are repeated here for convenience. The scattering parameters are given in terms of the reflection and transmission coefficients as r = si’fy ; T = sgqv (4.83) The reflection coefficient is given as r (:8, y) = aka/102W r (:8) (4.84) Where the (Ir-dependence with N1 as the number of slot partitions at. B81 is AT All sin (kg W1) 1 F(.r) = — , 10 V1 (11, )cos (k 1‘1 ) (4.85) l’l/labwuokgm n21 "‘1 I10 "'1 71 and the transmission coefficient is T (I, y) = €—)A:,,,0(yT—y,) [1 — r (1)] (4.86) 4.6.2 Analysis of Material Properities The purpose of this section is to analyze how certain material properties (permittiv- ity, permeability and material thickness) affect the rectangular waveguide scattering parameters. Only the magnitude data is considered since the phase data, which essen- tially represents only a phase shift in the waveguide, adds only consistent. information. 4.6.2.1 Relative Permittivity and Permeability The permittivity and permeability are the parameters of most interest, since they are the ultimately the desired result. from the extraction process in the next Chapter. Therefore, an understanding of how each specific component of the complex variables affects the waveguide scattering parameters is of significant importance. For this analysis the slot dimensions are kept the same as those in Chapter 2, a material thickness of 5 mm is used, and the material parameters not under analysis are those of free-space. Figure 4.2 shows the magnitude, in dB, of the reflection and transmission coeffi- cients for various real components of relative permittivity versus a frequency range of 8.2 - 12.4 GHz. As the real component of the relative permittivity increases, the resonance is seen to shift lower in frequency and the strength of signal coupled to the material is seen to decrease. Figure 4.3 shows the magnitude, in dB, of the reflection and transmission coeffi- cients for various imaginary components of relative permittivity versus a frequency range of 8.2 - 12.4 GHz. As the imaginary component of the relative permittivity is increased, the resonance of the signal is seen to shift higher in frequency and the strength of signal coupled to the material decreases. 72 Figure 4.4 shows the magnitude, in dB, of the reflection and transmission coefli— cients for various real components of relative permeability versus a frequency range of 8.2 - 12.4 GHz. As the real component of relative permeability increases, the reso- nance is seen to shift lower in frequency, but not as quickly as with the real component of relative permittivity. Also, there is no decrease in strength of signal coupled to the material. Figure 4.5 shows the magnitude, in dB, of the reflection and transmission coeffi- cients for various imaginary components of relative permeability versus a frequency range of 8.2 - 12.4 GHZ. As the imaginary component of relative permeability in- creases, the resonance is seen to shift lower in frequency and a general trend of the strength of signal coupled to the material decreasing holds till the last value. At which point the strength of signal is seen to jump back up to a higher value. This might be due to the material thickness and the possibility of a higher order mode propagation. 4.6.2.2 Material Thickness The last material parameter of interest for analysis is the thickness of the material. As the material thickness changes, the distance the signal travels before it reflects off the PEC boundary is changed. This leads to higher order modes being excited in the parallel-plate region if the thickness is greater than half a. wavelength in the material. For this analysis the slot dimensions are kept the same as those in Chapter 2. Figure 4.6 shows the magnitude, in dB, of the reflection and transmission coeffi- cients for various thicknesses of a loaded material (ep 2 2 - j1.d-3, mu 2 1 - j1.d-3) versus a frequency range of 8.2 - 12.4 GHZ. As the thickness of the loaded material increases, the resonance shifts lower in frequency and the strength of signal coupled to the material increases as the material nears a half wavelength (12 mm). When the thickness increases above a half wavelength the resonance then shifts higher in frequency and the strength of signal coupled to the material ('lecrcases. The effect of 73 higher order modes beginning to propagate is seen on the thicknesses above a half wavelength. Figure 4.7 shows the magnitude, in dB, of the reflection and transmission coeffi- cients for various thicknesses of a loaded material (ep 2 2 - j1.dO, mu = 1 - j1.d-3) versus a frequency range of 8.2 - 12.4 GHz. As the thickness of the loaded material in- creases, the resonance shifts lower in frequency and the signal coupled to the material is seen to decrease. As the imaginary component of relative permittivity is increased, the signal is attenuated faster and thus does not affect the results as significantly. Figure 4.8 shows the magnitude, in dB, of the reflection and transmission coeffi— cients for various thicknesses of a loaded material (ep 2 1 - j1.d-3, mu 2 2 - j1.d-3) versus a frequency range of 8.2 - 12.4 GHz. Figure 4.9 shows the magnitude, in dB, of the reflection and transmission coefficients for various thicknesses of a loaded material (ep 2 1 - j1.d-3, mu 2 2 - jldO) versus a frequency range of 8.2 - 12.4 GHz. As the thickness increases for the magnetically loaded materials increases, similar properties to the electrically loaded materials with a slightly tighter dispersion of the resonance curves for the magnetically loaded cases. 4.6.3 Analysis of Slot Dimensions The purpose of this section is to analyze how certain slot dimensions (thickness, length and width) affect the rectangular waveguide scattering parameters. This analysis is performed with the material properties mirroring those of FGMI25, the available sample for validation in the next Chapter. Only the magnitude data is considered since the phase data, which essentially represents only a phase shift in the waveguide, adds only consistent information for the analysis. 4.6.3.1 Thickness of Slot Figure 4.10 shows the magnitude, in dB, of the reflection and transmission coefficients for various slot wall thicknesses versus a frequency range of 8.2 - 12.4 GHZ. As the 74 thickness of the slot increases the resonance is seen to shift slightly higher in frequency and have a slightly larger coupling. However, at non-resonant “frequencies the coupling is seen to drastically decrease as the thickness increases. 4.6.3.2 Length of Slot Figure 4.11 shows the magnitude, in dB, of the reflection and transmission coefficients for various slot lengths versus a frequency range of 8.2 - 12.4 GHz. As the length of the slot. increases, the resonance is seen to shift lower in frequency. This matches the results of Chapter 2, except that a smaller slot length is needed to match the resonant length of the wavelength in the material. 4.6.3.3 Width of Slot Figure 4.12 shows the magnitude, in dB, of the reflection and transmission coefficients for various slot widths versus a frequency range of 8.2 - 12.4 GHz. As the width of the slot increases, the coupling to the material is seen to increase. The most significant change is seen at non—resonant frequencies, although at the largest width the resonant frequencies are also seen to have increased coupling. Magnitude of $11 & $21 for Various Values of the Real Component of Relative Permittivity vs. Frequency -S '10 ._ — ‘ ‘_—” * ‘ —-—511-1.5 8 +521 - 1.5 -15 4 3 j. --—-sl1 - 2.0 3 _20 _g ----521 - 2.0 93 ‘ ---c--s11 - 2.5 §_25 521 -2.5 2 -----511 -3.0 _30 -----521 - 3.0 '40 l —“T " ‘ ' ' """—_ 'r r ' fi—r 'T' — " r “1 '_fi'—'T_T——‘l 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GI-l2) Figure 4.2. Effect of the real component of the relative permittivity 011 the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 76 Magnitude of 511 & 521 for Various Values of the Imaginary Component of Relative Permittivity vs. Frequency -15 -20 -25 Magnitude (dB) -30 . -35 as .. ,, - _ as , 2H,, , ‘40 T - r r , r l r 7 f r T ' r v T 1 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GI-I2) —-—sll - 1.d-2 +521 - 1.d-2 --‘--511 - 1.d~1 ----521 - 1.d-1 ----- $11 - 1.d0 ----- 521 - 1.d0 -----511 - 1.d1 -----521 — 1.d1 Figure 4.3. Effect of the inmginary compom-ént of the relative permittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 77 Magnitude of 511 a 521 for Various Values of the Real Component of Relative Permeability vs. Frequency —-—s11 - 1.5 g _15 1 +521 - 1.5 V ----511 - 2.0 g _20 -«—-521 - 2.0 .t‘ """511-2.5 §_25 521 - 2.5 I . -"'"511 - 3.0 -30 1_-.. ,,_,,,, H7 __-_ 7 ,_1, ,, __ """SZI - 3.0 -35 -._____—__. '40 h—j—T‘ T_ ._ _ r—rfi" T i R 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 i i T'fi’—Y Frequency (GHz) Figure 4.4. Effect. of the real component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 78 Magnitude of 511 & 521 for Various Values of the Imaginary Component of Relative Permeability vs. Frequency —~—511 - 1.d-2 8 ——-—521 - 1.d-2 3 —«—-511 - 1.d-1 3 ----521 - 1.d-1 93 511 - 1.d0 g. ....--521-1.d0 g —----511 - 1.d1 -+--521 - 1.d1 I ‘40 %'* 1 ‘ r l r *f’ r "f“ r‘——r r r—T’ r r—vr" T l 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.9 12.40 Frequency (GHz) Figure 4.5. Effect of the imaginary component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 79 Magnitude of 511 & 521 for Various Thicknesses of a Loaded (ep=(2.0-j1.d-3), mu=(1-j1.d-3)) Material vs. Frequency —-—511 - 5 mm —~—s21 - 5 mm -.._.511 — 10 mm --'--le - 10 mm ----- $11 - 15 mm ----- 521 - 15 mm -----511 - 20 mm -----521 - 20 mm Magnitude (dB) -40 , . 4 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 T 7" l ‘" l—f—fi‘ T - Frequency (GHz) Figure 4.6. Effect of the material thickness of the real component of the relative per- mittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 80 Magnitude of 511 a 521 for Various Thicknesses of a Loaded (ep=(2.0-j1.d0), mu=(1-j1.d-3)) Material vs. Frequency 0 r :‘T‘=+_—_——:——_-—:——rr-il+- - -fifi - -5 -1011 F i i if F T “ —-——511 - 5 mm E -15 (t ..._ f_ , +521 " 5 mm v ! -«—-511-1o mm §_20 _ - -«—-521 - 10mm .2 ------511-15 mm §_25, ---°--521-15 mm 2 ‘ —----511-2o mm -30 “""SZI - 20mm ~35 -40 . .4 T 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GHz) Figure 4.7. Effect of the material thickness of the imaginary component of the rela- tive permittivity on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 81 Magnitude of 511 & 521 for Various Thicknesses of a Loaded (ep=(1-j1.d-3), mu=(2.0-j1.d-3)) Material vs. Frequency —--511 - 5 mm --—521 - 5 mm ----511 - 10 mm ----521 - 10 mm ---- 511 - 15 mm ----¢ s21 - 15 mm -'---511 - 20 mm -~---521 - 20 mm Magnitude (dB) -3s 7- 8-~~~~—e 7 , ,if- 1 figs 1 -1- ’40 +' ' T _ ‘T—fi—F_' r—V'T—m'fi— T T — T i 7 T Y 7—1 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GI-i2) Figure 4.8. Effect of the material thickness of the real component of the relative permeability on the magnitude (dB) of the waveguide scattering parameters vs. fre- quency (GHz). 82 Magnitude of 511 a 521 for Various Thicknesses of a Loaded (ep=(1-j1.d-3), mu=(2.0-j1.d0)) Material v5. Frequency —-—511 - 5 mm —-—521 - 5 mm ----511 - 10 mm ----s21 - 10 mm ----- $11 - 15 mm -----n 521 - 15 mm -15 -25 Magnitude (dB) ---~511 - 20 mm '30? -----s21-20 mm -35 f_ -40 v ’T "T i if ' 1‘ l' ‘T Y 'fi' T i ’71" 'l i iT—W ' Y 1’ fl 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GHz) Figure 4.9. Effect of the material thickness of the imaginary component of the rela- tive permeability on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 83 Magnitude of 511 81 521 for Various Slot Thicknesses (FGM125) vs. Frequency —-—511 - 0.5 mm _15 J - 7 , 7 -—521 - 0.5 mm -*-'511 - 1 mm ----521 - 1 mm .-... 511 - 2 mm ----- 521 - 2 mm ----'511 - 3 mm ~le - 3 mm I N U1 1 Magnitude (dB) I i l l l | DJ 0 l f i DJ U1 ‘40 Ff— i 7 T r ' ' Y f— " ' ”‘é—" "' Yifi—Tw T '—r—r~ T' '—1 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GI-i2) Figure 4.10. Effect of the slot wall thickness, for a parallel-plate region loaded with FGMI25, on the magnitude (dB) of the waveguide scattering parameters vs. fre- quency (GHz). 84 Magnitude of 511 & 521 for Various Slot Lengths (FGM125) vs. Frequency O _— r r — - - i _ f - - - 2 I 2 2 t : 4 : -5 .__ _ _4 1,7 '10 l" _ 1'“ “if +511-1cm §_15_ _ __ fl +521-1cm V -.._.511- 1.1 cm "£720 ----521 - 1.1 cm :2 ------s11-1.2 cm §_25 -1 77 7 77 7 7 7 f 7 77 ---°--521- 1.2 cm 2 -----511-1.3 cm -30 5, """521 - 1.3 cm -35 l+ ‘40 __i ’C—‘j' 'T " 7 ’ T‘fi ‘7" i 1 ”fi i ‘T” i ’ i V "T" i "'_—‘ 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GHz) Figure 4.11. Effect of the slot length, for a parallel-plate region loaded with FGMI25, on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). Magnitude of 511 a 521 for Various Slot Length to Width Ratios (FGM125) vs. Frequency - fT if * fir TTTT—fi—TTffTfi I H O 'l— l +511 - 1/25 g _15 7_ _. . _ 77i_3 +521 - 1/25 V l —---511 - 1/20 13 -20 l -~--521 - 1/20 ~13 1 ~----s11 - 1/15 éqs +__- _ "'°"521-1/15 Z —-~~511 - 1/10 —+--521 - 1/10 ‘40 I f r . f T ‘r T f i T Y T T i l 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 Frequency (GI-I2) Figure 4.12. Effect. of the slot width (using slot length to width ratios), for a. parallel— plate region loaded with FGMI25, on the magnitude (dB) of the waveguide scattering parameters vs. frequency (GHz). 86 CHAPTER 5 RESULTS To verify the MF IE techniques developed in both Chapter 2 and Chapter 4, several measurements were taken in the X-band frequency range, 8.2-12.4 GHz. While this technique is valid over any desired frequency range, the dimensions and availability of X—band waveguide sections made this the easiest range for a proof of concept demonstration. In this Chapter, the experimental setup is explained and verified, the extraction process is investigated and verified, and experimental results are shown for MagR AM. 5. 1 Experimental Setup The experin‘iental setup used for this resonant antenna material characterization tech- nique is shown in Figure 5.1 and Figure 5.2. The physical experiment gets connected to a HP8510C Vector Network Analyzer (VNA) via coaxial cables, coax to WRQO waveguide (where the W R stands for “waveguide rectangular” and 90 refers to the inner waveguide width a = 0.90 inches) transitions, and WR90 waveguide sections. The connection between the physical experiment and the WRQO waveguide sections uses precision alignment pins and screws to help minimize the disctmtimiities across the interfaces. Great care is also taken to ensure that the coaxial cables are spa- tially stabilized while calibrating the experimental setup and measuring samples. The two-port experimental setup is calibrated using the thru—reflect-line (TRL) method [65H67]. The full two-port TRL calibration method, including the development of the calibration kit for the HP 8510C, is discussed in [68]. The physical experiment. is built out. of a single sheet of brass (Length = 5.5 inches, Width 2 5 inches, and Height 2 1.625 inches) to help insure that there are no electrical discontinuities, except at the flanges and the transverse slot. To build the device, the 87 machine shop first conductively etched a VVRQO waveguide segment (Length = 5.5 inches, Width 2 0.9 inches. and Height 2 0.4 inches) into the length of the brass sheet. Next, they milled away the excess material above the WR90 waveguide segment to achieve the thinest waveguide wall thickness (0.04 inches) that was structurally durable. Then the precision alignment pin and screw holes were drilled for the WR90 waveguide flanges. Finally, the transverse slot (Length = 0.45 inches and Width = 0.27 inches) is etched into the center of the upper WR90 waveguide wall. 5.1.1 Validation Before measuring any test samples, a standard baseline is checked to ir‘litially validate the experimental setup. After performing the calibration, the two waveguide sections are placed together and the magnitude and phase of the S-parameters is checked to ensure that. ngplg = 140" and Sifggl = 0. Next, the physical experiment, with the slot covered by conductive tape, is connected to the VNA and the magnitude and phase of the S-parameters is again checked to ensure that 3.31712 E 1100 and exp S11,22 E 0. The magnitude and phase are approximate in this case due to the slot discontinuity in the waveguide wall. 5.1.1.1 Radiation into a Half-Space To provide further confidence and validation of the experimental setup, a compari- son of measured and formulated data. for radiation into a. lu—ilf—space. is considered. Figure 5.3 shows the comparison of formulation results and experimental data for the magnitude (in decibels (dB)) of the rectangular waveguide reflection and transmis- sion coefficients versus a frequency range of 8.2 - 12.4 GHz. Figure 5.4 shows the comparison of formulation results and experimental data for the phase (in degrees) of the rectangular waveguide reflection and transmission coefficients versus a frequency range of 8.2 - 12.4 GHz. The curves for both the magnitude and phase of the reflection and transmission coefficients line up very well. 88 The resonance of the slot antenna is seen at just slightly less than 13 GHz, which is expected since the slot is the complement of a strip dipole antenna [55]. Also, the dispersion of the formulated resonance is slightly tighter than the experimental resonance. This is expected since the formulation assumes the brass waveguide is a PEC, when in actuality it has a small amount of ohmic loss. 5.1.1.2 Signal Attenuation In observing Figure 5.2 it is noticed that the experimental setup has a finite parallel- plate region, whereas the formulation assumed and infinite parallel—plate region. The assumption of an infinite parallel-plate region was implemented for two reasons: it greatly reduces the complexity of the formulation; the motivation of this problem was the characterization of lossy EM materials. Since the signal in a lossy media attenuates, no contribution is seen from fringing fields due to the finite boundaries of the physical experiment. Thus making the assumption viable and allowing the formulation to correctly model the experiment. To understand how lossy the EM materials must be to ensure the assumption of an infinite parallel-plate region is maintained, an analysis of the attenuation of signals in the parallel-plate region is given. Figure 5.5 and Figure 5.6 show the attenuation of a wave traveling 6 cm in a parallel-plate waveguide for various real components of relative permittivity versus the imaginary component of relative permittivity for 8.2 and 12.4 GHz respectively. The signals are seen to decay rapidly as the imaginary component increases. As the real component of permittivity increases, a higher imag- inary component is required to achieve the same level of signal attenuation. There is a tighter dispersion between the curves for the real component of permittivity as the frequency increases. The MagRAM sample that is used to validate the formulation is FGMI25 from Cuming Microwave. Using its general material properties [41], a knowledge of the general attenuation properties of the signal in the MagRAM is obtained. Figure 5.7 89 shows the attenuation of a wave traveling in a parallel—plate waveguide loaded with FGMI25 for various frequencies versus distance. The signal is seen to attenuate to -40 dB in less than 1 cm over the desired X-band bandwidth. Thus, showing that no fringing effects are. seen when measuring the experimental data with the experimental setup that has been built. 5.2 Complex Constitutive Parameter Extraction The overall objective of this material measurement technique is to experimentally obtain sample scattering parameters using a network analyzer and compare them with their theoretical expressions. That is, th, Slly (112,641.) — Siip (w) = 0 t} ?' S2113! (w, 6, )1.) — 531110011) = 0 This pair of nonlinear equations with two unknowns has a solution which is seen to decompose into two parts. F irst, analytical theory is needed to relate S [(1111 (w, e, ,u.) and 53]” (5.36.11) to the complex constitutive parameters (see. MFIE analysis in Chapter 4). Second, a technique is needed to accurately measure Sam (15) and 5;?) (w) (see previous section). With the two necessary parts for the solution, equation 5.1 is iteratively solved using a complex two-dirnensional Newton root search method, giving the desired results of the complex constitutive parameters [7]. 5.2.1 Extraction Validation The ultimate goal for this material measurement technique is to sirmiltaneously ex- tract both permittivity and permeability. The intention is to perform the extraction using both the reflection and transmission coefficients from the rectangular waveguide, while measuring only a single layer of the material. Unfortunately, simple validation tests of the extraction process showed that this is not possible under the current for- 90 mulation. Essentially, the two scattering parameters do not provide enough sensitivity to perform the extraction. In order to achieve the desired goal of simultaneous extraction of permittivity and permeability, a second independent experimental measurement. is needed. The simplest method to achieve this goal is to measure a second thickness of the desired material. Then using one of the scattering parameters from each of the two different material thickness measurements, solve the pair of nonlinear equations in 5.1. To decipher how different the two material thicknesses need be to perform the com- plex constitutive parameter extraction, tests were performed using generated acrylic data. Acrylic is used due "to its consistent material properties across the entire waveg- uide bandwidth. F igurc 5.8 shows the extracted permittivity and permeability values for an acrylic material with a material thickness difference equal to 15 percent of the material wavelength. The data is seen to be very good across most. of the bandwidth, however, certain points are still not converging correctly. Figure 5.9 shows the extracted permittivity and permeability values for an acrylic material with a material thickness difference equal to 20 percent of the material wavelength. The data is seen to be very good across the entire bandwidth, leading to the conclusion that a 20 percent material wavelength thickness difference is sufficient to properly extract the desired material characteristics. 5.2.2 Tested Samples The general concept of this resonant antenna material characterization technique is to non-destructively measure simple, lossy medias. Thus, the ultimate verification of this technique is to measure a sample with those properties. MagRAM, in the form of FGMI25 from Cuming Microwave, is a convenient solution in that it exhibits the desired properties, it is readily available, and comparison data is available using a partially-filled waveguide technique [68]. 91 5.2.2.1 MagRAM The results for permeability of the l\»’IagRAM using the resonant slot-antenna tech— nique are compared to the results using a partially-filled rectangular waveguide tech- nique in Figure 5.10. Using the results of the partially-filled rectangular waveguide technique. as the baseline for the expected results for the resonant slot-antenna tech- nique, the values of the permeability are seen to line up very well. The results for permittivity of the MagRAM using the resonant. slot-antenna tech- nique are compared to the results using a partially—filled rectangular waveguide tech- nique in Figure 5.11. Again, using the results of the partially-filled rectangular waveg- uide technique as the baseline for the expected results for the resonant slot—antenna technique, the values of the permittivity are seen to not line up very well. The real component. of the permittivity seems to have the general concept, but oscillates wildly about the desired value. Whereas the imaginary component of the permittivity also oscillates wildly, but not near the desired value. These type of results have been noticed before, mostly in coaxial probe tech- niques. and are due to air-gaps between the probe. flange and the material surface [22],]40].[69]—[70]. Even a small air-gap between the sample and probe causes a signif- icant. discontinuity in the strong, normal electric-fields leading to large errors on the predicted permittivity values. The permeability results however remain largely un- changed as there EM boundary conditions are not significantly affected by the small discontinuity. However, for a couple of reason the issue is believed not to be due to an air—gap. First, the real component of permittivity would be lower across the entire bandwidth than the predicted value. which is not the. case. Second, the electric field is actually tangential and therefore continuous, which should not create such a large eflect on results. Finally, an exceptionally heavy weight was placed 011 the experimental setup ensuring a good contact. between the material and slot antenna. The alternative 92 explanation offered is a poor electric field interrogation of the material. 93 Figure 5.1. Experimental Setup: Rectangular waveguide radiating through a trans- verse slot into a half space. 94 Figure 5.2. Experimental Setup: Rectangular waveguide radiating through a trans- verse slot into a finite parallel—plate region loaded with MagRAM. 95 Comparison of Formulation and Experiment for the Magnitude of 511 a 521 vs. Frequency -10 -15 . $11 - Exp ~**“~521 -Exp ------- 511 - Form 521 - Form -20 .. Magnitude (dB) -25 -30 ,. -35 __ '40 I i 7* 1 T u T r i r Y 8.20 8.60 9.00 9.40 9.80 10.20 10.60 11 Frequency (GHz) r .00 11.40 11 i 1 .80 12.20 Figure 5.3. Comparison of the formulation results and experimental data, using the magnitude (dB) of the waveguide scattering parameters, for the radiation of the signal into a half-space vs. frecpiency (GHz). 96 Comparison of Formulation and Experiment for the Phase vs. Frequency 180 120 60 Phase (Degrees) O \\ l|\ -120 W I f \t -180 8.20 8.60 9.00 9.40 9.80 10.20 10.60 11.00 11.40 11.80 12.20 Frequency (GHz) 511 - Exp 521 - Exp ------- $11 - Form 521 - Form Figure 5.4. Comparison of the formulation results and experimental data, using the phase (degrees) of the waveguide scattering parameters, for the radiation of the signal into a half-space vs. frequency (GHz). 97 Attenuation of a Wave Traveling 6 cm in Parallel-Plate Waveguide at 8.2 GHz vs. Imaginary Component of Relative Permittivity I H U1 _ 5. _ _ l ——-EPR - 1.0 ----EPR-2.S ------- spa-5.0 -----EPR- 10.0 -20 1_u_11__ _ "11__ Attenuation of Wave (dB) 1.5-3 1.5-2 1.5-1 1.E+0 1.E+1 Imaginary Component of Relative Permittivity 1.E-S 1.E-4 Figure 5.5. The attenuation (dB) of a wave traveling 6 cm in a parallel-plate waveg- uide, for various values of the real component of relative permittivity, at 8.2 GHz vs. the imaginary conmonent of the relative permittivity. 98 Attenuation of a Wave Traveling 6 cm in Parallel-Plate Waveguide at 12.4 GHz vs. Imaginary Component of Relative permittivity 0 ~‘ ___ ___—___ “k". -5 __ 4 7 -‘ L a : V- t -10 . ( ‘\ v [ \ -, . g \ \\ ...\ -15 5 ' S ‘ T1 '- \ —-—EPR - 1.0 \ '6 _20 7 1,1. ----EPR- 2.5 8 \ \ \_ ------- EPR - 5.0 .. \ '. \ _._.. _ .5 _25 , _ , '. EPR 10.0 :I ‘ i 5 . \ 2‘ .‘ 3: -30 *— —— l ‘. ', < i i '. - l . '-. l -35 .14___7 1 11‘ ,_ "u l ‘ '2 ‘. ’40 ' fl 7 i r T’ ' 1 i—LY‘ Y #—1” | 1.E-5 1.E-4 1.E-3 1.E-2 1.E-1 1.E+0 1.E+1 Imaginary Component Relative Permittivity Figure 5.6. The attenuation (dB) of a wave traveling 6 cm in a parallel-plate waveg- uide, for various values of the real component of relative permittivity, at 12.4 GHz vs. the imaginary component of the relative permittivity. 99 Attenuation of a Wave Traveling in a Parallel-Plate Waveguide Loaded with FGM125 vs. Distance 6‘ E _ 0 > a f 3 3 7 —8.2 GHz : --—- 12.4 GHz .9. U W II 3 C 0 3: < \ -35 x n L \ \ -40 -)—-—-—--s————— 4 ——- 1 — -- r 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Distance (cm) Figure 5.7. The attenuation (dB) of a wave traveling in a parallel-plate waveguide, at 8.2 and 12.4 GHz. loaded with FGMI25 vs. distance (cm). 100 Relative permittivity & permeability extraction of generated acrylic data (15% material thickness difference) vs. frequency 2.5, .E '5 2 f i! 0 E '0': 15« n . all —EP Re 4; 1 ...................................................................... ——--EP1m '32: ---------- MU Re ._ —-—--MUIm E 0 0.54 n. 0 .2 E 0"]iiw—w—w—1—1—1—v—w—T—T—w—w—w—chifi—v—T—v—fi", g 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 -0.5 Frequency (GHz) Figure 5.8. Relative permittivity and permeability extracted using generated acrylic data that has a. 15 percent wavelength difference between the two material thicknesses. 101 Relative permittivity 81 permeability extraction of generated acrylic data (20% material thickness difference) vs frequency 2.5 E '5 2 — a 0 E 3 15‘I Q ' ] cl ] ——EP Re E 1' ................................................................................. ----EPIm .3 ....... MU Re " -----MU Im E o 0.5~ D. 0 > '5 g 8.20 8.62 9.04 9.46 9.88 10.30 10.72 11.14 11.56 11.98 12.40 1 Frequency (GHz) Figure 5.9. Relative permittivity and permeability extracted using generated acrylic data that has a 20 percent wavelmigth difference between the two material thicknesses. 102 Comparison of Relative Permeability for FGM125 vs. Frequency , o PP-MU-Re ] - PP-MU-Im — PF-MU-Re 2 ____.___. _ ___ ,7 7 , 7, 7 _ Vifl PF-MU-Im Relative Permeability b) l I l -1 '2 r Y fir r . 1 8.00 8.40 8.80 9.20 9.60 10.00 10.40 10.80 11.20 11.60 12.00 12.40 Frequency (GHz) Figure 5.10. Comparison of the relative permeability, for the MagRAM FGMI25, using a partially-filled rectangular waveguide method and the resonant antenna tech- nique vs. frequency (GHz). 103 Relative Permittivity Comparison of Relative Permittivity for FGM 125 vs. Frequency O... if ..~ . .V~.0d> 5 ew— — ° PP-EP-Re ' PP-EP-Im — PF-EP-Re 2 ., ____ , . __ __ _ _ __ _ __ '— PF-EP-Im _ _____ f -_____ 51L.- " _-____ 1 w—‘A‘S’T ‘2 1 1 1 ‘r 1 1 1 1 v 8.00 8.40 8.80 9.20 9.60 10.00 10.40 10.80 11.20 11.60 12.00 12.40 Frequency (GHz) Figure 5.11. Comparison of the relative permittivity, for the MagRAM FGMI25, using a partially—filled rectangular waveguide method and the resonant antenna tech- nique vs. frequency (GHz). 104 CHAPTER 6 CONCLUSIONS This dissertation has provided a waveguide slot-antenna technique for the non- destructive e 'aluation of a PEC backed lossy, simple media. The sample permittivity and permeability were found by using an iterative complex two—dimensional Newton’s root-searching algoritlnn to compare the theoretical S-parameters obtained using the MFIE technique with the experimentally measured S-parameters obtained form the network analyzer. The initial goal of this research was to extract both permittivity and permeability sinuiltaneously with only a single experimental setup. This was not possible under the current formulation. thus a. second material thickness was needed to perform the simultaneous extraction. A couple of special case tests were done to show that the extraction process, using the two separate thicknesses, was converging to the correct complex constitutive parameters. Then the xx'aveguide slot-antenna material characterization technique was experimentally verified, through the comparison of a MagR.-‘-\I\l sample. with a partially-filled rectangular waveguide measurement. The accuracy of the permittivity data. suffered, likely due to a poor electric field interro- gation of the material. This comparison demonstrated a proof of concept, and thus the validity of the technique. 6.1 Suggestions for Future Work As this is initial research into the use of slot antennas for material characterization, many areas of future research are note worthy. First, looking into why the single layer of material was unable to simultaneously perform a full charactcrization of the material. This should include looking into various extraction methods to see if the 2—D Newton’s method used was not the appropriate choice. Next, a study into 105 increasing the coupling though the slot. The most practical way to achieve this is to relax the assumption of a long an narrow slot, and increase the size of the aperture. Another area of interest is using multiple slots. This should also increase coupling to the material, but could also help achieve the initial goal of using only a single material thickness to fully characterize the material. Finally, looking into the ability to extract the complex constitutive parameters for two—layers, which could be done using a two-layer parallel-plate Green’s function. 106 APPENDICES 107 APPENDIX A MAXWELL’S EQUATIONS AND HERTZIAN POTENTIALS A. 1 Introduction Appendix A provides an overview of l\"Iaxwell"s Equations and Hertzian potentials due to a magnetic source. A development of the wave equations, for both EM fields and Hertzian potentials, is also included. A.2 Maxwell’s Equations and the Wave Equation for E and If l\-‘Iaxwell‘s equations, due to a magnetic source, for a simple medium (linear, homo- geneous and isotropic) and the magnetic source continuity equation in the spectral- domain point. form are v X 13:70?) = —j;,. (F) — ijuI-T (m (A.1) v x H (1:) = 31.181505) (4.2) v - 815(5) = (A3) V ’ H}? (F) = Pm (7:) (A4) V ' jfn (7—) = “J's/’07?! (7—) (A5) —O where J,,, is a. volume magnetic current density, pm is the density of the magnetic charge. and 6 : € (1 — ja / 016‘ ) is the effective complex permittivity. The wave equation for E is found by taking the curl of (A.1), substituting (A2) into the resulting relation, then applying the vector identity V x V x E = V (V - E) — VQE and using equation (A3). The result is v21? + 12E" = v x .17., (A6) 108 where 1172 = 022146. Similarly, the wave equation for If is determined by taking the curl of (A2), substituting (A.1) into the resulting equation and then using the vector identity V X V x E = V (V - E) —— VQE and also equations (A4) and (A5). The result is _. _. 5 1 .. VQH + NH = jweJm + .—v (v.1...) (A.7) 301,11, A.3 MT Hertzian Potential The primary use of Hertzian potentials is as a simplifying intermediate step to deter- mining electric and magnetic fields. The MT Hertzian potential may be identified by observing that (A.3) implies that E is written as E = —jw1LV X 7?}, (A.8) since V - V x 7?}, = 0 by vector identity. The magnetic field is then determined by substituting (A8) into (A2) and using the vector identity V X V = 0. This leads to H = 1.271), + 174 (A.9) The wave equation for 7?}, is identified by substituting (A8) and (A.9) into (A.1), applying the identity V x V x 7?), = V (V ' 7Tb) — V271), and then using the Lorentz gauge condition (I) = V ~ 70,. The resulting Helmholtz wave equation is J’In jwu v25), + 1627?}, : — (A.10) Equation (A.10) is decomposed into three scalar equations in Cartesian coordinates (reducing mathematical complexity) as follows —-0 J. V2711“) + (£27028. = * -ma (All) Mt 109 where a = :r, y, 2. Substituting (1) = V - 7?}, into (A.9) leads to H = 127?), + v (V717,) (A12) Since (A.10) shows that 7?), is maintained by a magnetic current, it is called a MT Hertzian potential (ET Hertzian potentials are also possible, but since they do not occur in this problem they are not discussed here). A comparison of (All) with (A6) or (A.7) demonstrates why Hertzian potentials are introduced in mathematical analysis of EM problems. In equation (A.11), each component of 7?), is directly related to each component of .12". The relationship between E, j in (A6) or If, f in (A.7) is more complicated and thus a solution is generally strongly singular and more difficult to obtain. Thus. it. is easier to solve for 7?), first then determine E and If using (A8) and (A.12). 110 APPENDIX B PROPERTIES OF RECTANGULAR WAVEGUIDES AND CAVITIES B. 1 Introduction An analysis of the guided wave structures used in Chapters 2 and 4 is needed to ensure proper application in the formulations developed. Appendix B develops the characteristic eigenmode fields of the rectangular waveguide, for both the y and z- axial directions. These modal fields are then extended to Green’s functions for both the rectangular waveguide and cavity respectively. Followed by the development of the rectangular waveguide scattering parameters for the y—axial direction modal fields. B.2 Waveguide Modes In the interior of a rectangular waveguide, Maxwell’s equations (A.1-A.4) can be divided into two basic sets of solutions or modes. For one set of modes no longitudinal or axial magnetic field component exists, these modes are called transverse magnetic (TM) modes. The other basic set of modes have an axial magnetic field but no axial electric field component, this set is referred to as the transverse electric (TE) modes [58]. The TE modes are used in the rectangular waveguides (with a > 5, see Figure B.1), because the TEM) mode is the dominant. mode due to the fact. is has the lowest cutoff frequency. The cutoff frequency of the TEmn mode is given by Ckc C\/m2b2 + 712a? fc,mn : ET— : 20.0 (13.1) where c is the speed of light in. free space, a. is the width of rectangular waveguide, b is the height, and m and n are the modal values [58]. The cutoff frequency helps define the bandwidth of the waveguide, where the first higher order mode to propagate 111 defines the upper bound of the bandwidth, and the cutoff frequency defines the lower bound of the bandwidth. B.2.1 Geometrical Configuration Consider the cross-sectional view of a rectangular waveguide shown in Figure B1. The origin is located in the center of the bottom plate and the waveguide has dimensions of width —(1/2 3 :r S a/2 and height 3) = b. The width (a) will in general be twice the height (h). IN) N II 0 Va , x=—-—— x=0 x:— Figure B.1. Geometrical Configuration: Rectangular waveguide (y—axially direction) B.2.2 Modal Analysis (TEQC Modes) Generally when describing a set of TE modes, the direction of propagation has no electric-field component. However, sometimes the geometry of the problem lends itself to an alternative set of modes. In the case of the rectangular waveguide used in the formulations of Cl’iapters 2 and 4, where the direction of propagation is the y—direction, the TEJ- modes are the most complete set of modes. This is because the transverse slot can only have an equivalent :r—directed magnetic current. This can 112 alternatively be reasoned by noting that the slot causes higher order modes to be scattered, thus a superposition of T Ey and T My modes would be needed to have a complete set of modes. This superposition of TE, and T My modes is seen to equal the set of TE! modes, for propagation in the y—direction [60], thus the latter set of modes is developed here. 8.2.2.1 y—axial direction To develop the desired set of modes, the appropriate choice of the MT Hertzian potential is necessary. To ensure that the x-directed component of the electric-field is equal zero and the ar-directed component of the magnetic field is not equal to zero, the appropriate choice of the MT Hertzian potential is 7?}, = inf), (B.2) Then, substituting (B.2) into (A8) and (A12) and carrying out the vector operations gives respectively for the EM fields E 231011.02: x Vanna}, (B.3) and 4 a . 2 a? HZV‘Q‘gWh—l—SEQICO—l—gfi) 71'}, (13.4) where the transverse Laplacin operator with respect to the .r-direction is (9 8 V, =’f—-——+2— B5 1.1: Jay 02 ( ) To solve the homogeneous Helmholtz equation v2 .2 __ B 7fh + 1‘07”). _ 0 ( 6) 113 in terms of longitudinal and transverse components requires writing the MT Hertzian potential as 7th = 510,30}, (1:, :5) 63¢” (B.7) Then utilizing (B7) in (B6) along with the separation of variables technique leads to 7th 2 240,3 [A cos (1811') + B sin (101.10] [C cos (kzz) + D sin (1023)] 64:17:; (B8) where the cutoff wavenumber and propagation constant are respectively 13 = 1:3 + r2 = 13, +1.3 (13.9) The boundary conditions on the PEC waveguide walls are Ell/~Zl:r=—a/2,a/2 : 0 _’ 7Tb.l;‘l?=-(l/2,a/2 : 0 (13'1” 00h. 8:: Eylz=t,bl : 0 _) 221,01 : 0 (B12) where the partial derivative required for enforcing the boundary conditions is given HS .8579"; 2 140.,ng [A cos (151.23) + Bsin (kfl:)] [—C sin (1022) + Bees (1523)] (EI‘y (13.13) 2 . leading to the generating eigenfunction for the 7th mode as Why 2 A4,. sin [1014, (.1: — a/2)] cos [1:37. (:5 — 1%)] eTjkyV’y (RM) 114 with the wavemunbers defined as k1... = 91.021.23... I (1 k3,, = 4,718 = 0, 1, 2, where b =(01—t) (8.15) .2 _ .2 __ .2 L ...2 .2 AC, _ 1,, 1y, _ 1,, + 1,, Substituting (B14) into (B.3) and (B4) and applying the mathematical operations, the modal fields are sunnnarized as follows $70 1,1 201: [224 (2)-244.321.8121] J W 03.16) Hi: (F) = [’12, (F) i 9495,1132, ("9] 82112033; (317) e~7 (F) = sin [1513.]. (.1: — a./2)] cos [133,7 (2 — t)] ; A317 : wlL0A7y7A7 (318) €311 (F) = sin [km 7 (r — a./2)] sin [10,37 (2 — t)] ; A97 2 jwnokngV (B19) 17,, (F) 2 27.41.7111,, (a + 2A.,,h,., (72) (13.20) 11,, (F) = sin [11,, (.L- — a./2)] cos [1:2, (2 — 1)] ; .41., = (1:3 — 13.) A, (B21) h3,7 (F) = cos [:3 (I — (1/2) ] sin [k3,], (2 — 1)] ; :1, = —k, Is A (B22) h,, (a = cos [1,, (.1.- — (1/2)] cos [1,, (z — 2)] ; Ag, = — )1. 1a,, .,4 (B23) B.2.2.2 z-axial direction Consider the cross-sectional view of a rectangular waveguide shown in Figure B.2. The origin is located in the center of the waveguide having dimensions of width —(1.C/2 S :1: S (LC/2 and height —-b(-/2 S y g lip/2. Using the same steps as the previous section, the TEI, modes with propagation 115 7%) b / z=—C 2 z=O — >; x 2- b6 2 a I a x=——C x=O x=—C 2 2 Figure B.2. Geometrical Configuration: Rectangular waveguide (Is-axially direction) in the z-direction may be summarized as follows —+ A . A .163 2 E12 (2) = [24.31222F (.2, y) :2 24112.13 (21)] ,2) 1“ (B24) —o —o A ~ 7k? ... 1112(2) = [1% (Ly) 2 243111,,T (211)] e$J ~rZ (13.22)) (23F (.1', y) = sin [luff (.17 — (lg/2)] sin [kl/1“ (y — b(_-/2)] ; A31“ 2 jw11.()kyFAp (B26) (“l/F (.1‘. y) = sin [’1er (.1' — 113/2)] cos [AF/1‘ (y — 113/2)] ; Al/F = 11211013111412 (B27) Hip (it: y) : iA$Fh1F($ay) _ Q/ayphyp (3:? y) (828) [13,312 (41,331) = sin [113-F (.1: — (1.0/2)] cos [kg/F (y — bC/2)] ; {21311 = (1:8 — 13.) AF (B29) h3F (:1:,y) = cos [11,312 (:r. — 113/2)] cos [kyp (y — b(./2)] ; AZI‘ = jkl’I‘kZFAF (B30) 116 11,141.)» = cos [131. (.1: — (182)] sin [1yF (y — bc/2)[ ; 4,,r = kIFkyF/ip (3.31) ’6in : %,a 21,2,3,... 1yF = €313 = 0,1,2,... (B.3?) 2 _ 2 2 _ 2 2 13F _ 10 — 12F _ 1,1. +1yF B.3 Green’s Functions In this section the Green’s functions for the rectangular waveguide and cavity are developed. The rectangular waveguide Green’s function is determined by using the Lorentz reciprocity theorem, while the. rectangular cavity Green’s function is deter- mined by using the method of scattering superposition [61]. B.3.l Rectangular Waveguide Consider the waveguide configuration given in Figure B.3. The slot is assumed to be cut in the bottom waveguide wall between cross-sectional planes 051 at y = y) and C Sg at y = 312. It also supports a field is that is excited by the incident TE;0 mode. Figure B.3. Rectangular waveguide with slot discontinuity 117 B.3.1.1 Slot Excited Modal Fields The first step in the determination of the desired rectangular waveguide Green’s function, is to express the scattered waveguide fields, excited by the slot, using a modal expansion. These electric and magnetic field expressions are given as 12': Z822“ 7(2): ZB2[— 24 3.,,..<-,2)+J42322~,( 22)] “11*" (B23) 1" "kg. 131— ZB,H , (F): ZB,[1,,(., 2—) 2.43,,1y,(x,z)[21 W (13.34) and E Z 2 C775: (7—) Z 2 CV [3342783, (51323) + QAyveyv (33) 2)] e—jkyvy (B35) '7 '7 ... _. _. A ~ —"k H = ZCW’H: (Fl 2 EC? [hm (i172 3) + yAyvh'm (1": 3)] 6 J my (B36) '7' 7 for y < .111 and y > if; respectively. 8.3.1.2 Determination of Modal Expansion Coefficients The next step is to determine the modal expansion coefficients by using the Lorentz reciprocity theorem on a source-free, bounded region S. The closed boundary of this source-free region 5 consists of the cross-sectional surfaces CS) and C82, and the conducting waveguide surface SC, or SZSC+081+CSQ (3.37) The Lorentz reciprocity theorem applied to this source-free region is written as )6 13» (5“ x Eb — E” X H”) (15 = 0 (B38) 8 118 where the fields II to) En (13.39) H :21 Ha represent the EM fields (described in section B221) radiated into the bounded waveguide region by the slot aperture field E; and Eb : EH: 7 (8.40) "b “:t H H, are the waveguide modal field (described in equations (B33-B36)) traveling in the iy-direction. Substituting the fields in (B39) and (B40) into (B38) gives the spe- cialized form of the Lorentz reciprocity theorem - r 72 *2 7' én-(Ex ,,.— 7xH)dS=O (B41) where the supm‘position of the three boundaries set equal to zero determines the modal expansion coefficients. Solving for the conducting surface SC by first using a vector identity [:37] gives [562.(22222- 7,22 “)115=/SC[(2XE).122_(222,2).12]ds (13.42) Application of the PEG boundary condition [58] results in the expression / f1-(Exfif—E?xfi)d$=/ fi-(Esxfif)d8 (B43) 50. where the only remaining contril‘mtion is due to the slot electric-field. Solving the cross-sectional surface CS) next (where the outward normal is defined 119 as —;1)) leads to CS 2/ —;“- Ex "i—“ixfi as 8.44 1 C81y( ,7 'y ) ( ) with the substitution of the scattered and modal field expressions from (B39) and (B40) leading to A A 4 ' k 4:}; _ C51: Z,y B.y [C51 _y. _(2A37637(;1:,z) x I137 (2,2)) ej( In 317)y1 0 (B45) IF (2.3/13.11337. (:r, z) x I237 (:13, 2)) e] ($k.y7+ky7)y1] Then by defining the mode orthogonality relationship as 3476471 = / (189 ' (67%,, (I, Z) X EtL (13.2)) (8.46) CS equation (B45) is simplified to 031 = B,S,ej<1:tl>kyvyl (1 3: 1) (13.47) where the $ is associated with forward and backward traveling waves respectively. Similar steps are taken to solve the second cross-sectional surface 032 (where the outward normal is defined as y) giving 033 2/ g. (E x if? — E3? x H) as (8.48) (:52 Substitution of the scattered and modal field expressions from (B39) and (B40) along with simplifying the expression using (B46), leads to CS3 = C7quej(—1T1)k-’/7y2(lq: 1) (13.49) 120 where the 2F is again associated with the forward and backward traveling waves respectively. Superposing the terms in (B43), (B47), and (B49) and setting them equal to zero generates the expression for all values of 7 as 13787811ITllkyj’yl (1 :l: 1) + C,S,€j(_ITIM‘yTK/Q (1 $ 1) (B50) 2 F “:1: _ +f5a71-(ES X H,.)dS-—0 Solving this equation using either the upper or lower Sign and then rearranging the terms leads to the modal expansion coefficients as _1 . _. _.+ 2 B, = E s. ’11- (ES x H, ) as (13.01) and . —1 _- a 2- c, _ 287/511- (Eb x H, ) as (13.52) res1_)e(.:tively. Substituting the appropriate fields into the mode orthogonality relation- ship, the mode expansion constant is defined as A314 ab , 1 or =0 =_L$7_ . 3,:- f B (8.53) S, .. 267 2 for [3%0 Finally, the modal expansion coefficients are determined as 6‘1 1 2' n , — ». . ... ,_jk' 23/, r B, : _.43,,a.b [90 dSEy (F) sin [1.1.7 (.13, — (#2)] cos [123, (2., — t)]c, 9? (B54) (w , 2 ,, ,, ,‘I ‘ 3 J jk..y’ r... C', : —A3,a.b /Sa (ZSEy (F) sin [1.12, (.1. — (#2)] cos [123, (.2 — 15)]e 31/ (Boo) 121 B.3.1.3 Green’s Function due to a Transverse Slot The Green’s function is now found by simply substituting the modal expansion coef— ficients into equations (B34) and (B36) (repeated here for convenience) =2 324 2 41,12 22) 1 yr” y < 2’ (8.56) 1 ~ F _k . r H; =Zc,.4,,1,,(a,z)e J 2121" y > y’ (13.87) and then manipulating the subsequent expressions into the form of equations (2.9) and (2.10). The resulting .1:—directed component of the magnetic field for all y and y’ is 01)) E2( ) 4 H S A: + (IS—_— G 1' 7' B58 =<2 1) (913233 jwuo V) G (1 ) ( ) where the MT Hertzian potential is the dot product of the slot electric field and the resulting rectangular waveguide Greens function, due. to a transverse slot cut in the l(_)wer broad wall, which is given as GWG (FIF)——— “HE :7 TL sin]A;137 (:1: — a/2)] sin [11:3 ,3, (:1:’— a/2)] ~cos [A3, (2- t)] COS [A337 ~ ,::( — t)]e—jky7 y—y I B.3.2 Rectangular Cavity The Green’s function for the rectangular cavity derived in this appendix follows the development of Tai [61]. This appr1'1ach involves starting with the functions already available for a. rectangular waveguide of the same cross-sectional dimensions and then applies the method of scattering superposition to find the desired function. The method of scattering superposition is applied in two separate steps. The first step is to generate the Green’s function for the semi-infinite waveguide, with a PEC plate placed at the z = CC interface. The second step is to generate the Greens function 122 for the rectangular cavity with a second PEC plate placed at the z 2 0 interface. To solve for the Green’s functions by placing PEC plates at the interfaces 2 2 0,130, the PEC boundary condition (tangential electric—fields equals zero) is applied to the fields at these locations. Since only the y-directed component of the electric-field is tangential at both interfaces, it is the only component of the electric-field that is represented here. To develop the expression for the y—directed component of the electric-field, the modal fields of section B.2.2.2 are used to derive a rectangular waveguide Green’s function using the derivation in section B.3.1. The y-directed component of the electric-field, that is valid for all z and z', of an infinite rectangular waveguide is thus represented as 8 Ey (7:!) E = —w 10—/ 613 G ’ 7" 7—! —,— 13.60) y J 1 5.2 5., we“ ) 3W0 ( where the Green’s function is represented as GWG (FIF) 2 5:17; 2 7g? sin [1:31. (:1: — tic/2)] sin [19:31“ (:13’ — arc/2)] " 1' F ~ (B61) HI) — '11: -cos [Ar-yr (y — bp/2)] cos [k'JI‘ (y’ — bc/2)]€ J 21“ Substituting this Green’s function into the field expression and rearranging terms so only the z-(glependcnce is considered, leads to the following I I 8 :ij~ 2' Ziljkz Z, Ey 2 Ey(:1:y|:1,y) 07 e “T e P (B62) Now that the necessary field is developed, the method of scattering superposition may be applied to generate the first Green’s function for a semi-infinite rectangular waveguide. This is accomplished by added a scattering term to the field expression in (B62) that accounts for the waves scattered off the PEC interface at z 2 cc. The 123 resulting field is | Q.) -. I -..., I}, I 13.21 = Ey (4.1/1411’) eilkzrzei’W +Aslej"*rze’k~rz (8.63) Q.) (I where applying the appropriate boundary condition determines the constant A31 as Asl 2 e_]k3‘/F2CC (B64) Substitution of this constant back into (B63) and then applying mathematical ma- nipulation to the field for the z > 2’ and z < z’ cases respectively, leads to gr (2: — c(-)] 611,3, (5‘06) ejkzflz—cc) cos [kzF (z, _ Cc)] - k EU 2 EU (1:, yl.r’,yl) —0—2 COSA 82 (B60) e.g. the representation of the y-directed component of the electric-field for a semi- infinite rectangular waveguide. The second step is to generate the desired Green’s function for the rectangular cavity. Just as before, this is accomplished by adding a scattering term to the field expression in (B65) that accounts for the waves scattered off the PEC interface at 2 2 0. The resulting field for the 2: > 2’ and z < 3’ cases respectively is .I_ ., cos [ALT (z — 00)] 61/1214 (" 6") ejkzl“(z_CC) cos [kZI‘ (z’ — 6(3)] (B66) +As2 COS [kzr (z — Cc)] COS [kzr (2’ _ 60]] Ey 2 By (:17, y|;1:’,y’) 58—2 2 where applying the appropriate boundary condition determines the constant A32 as ,' _ijI‘CC A32 2 _L__ (13.67) sin ((321462) 124 Then, substitution of this constant. back into (B66) and then applying mathematical manipulation to the field for thez M> s and z < z’ cases respectively, leads to a 2 cos [A311 (2. — ed] cos (Arzrz’> 3] $111 [kfl :Cr (] COS (A3112) COS [k7r ('l "‘ Cc)] Ey— — E3, (1:, yll‘ ,y)— (B68) the representation of the y—directed component of the electric field for the rectangular cavity. Expanding this field, which is valid for all z and 2’, to include the :1: and y- dependencies and manipulating the expression to look like equation (B60) gives Ey (72’) Julie , 8 E!) 2 —]w,110$/S dS GCV (11F) (B69) a where the rectangular cavity Green’s function for the z > z’ and z < 2' cases respec- tivelv is 0,, .,-.¢:—2 (L [H 2)] [AJT M C1 (fl! ) m A? kzr sin(k~r ) sin TF( —ac/) sin a(,/ ) -cos [kg/Ff; —b /2 cos [1‘91“(y (—bL/2) cos [A‘zr (Z — 06)] cos (A P2,) 1 cos (113.51. 2:) cos [A331, (2' — 06)] (B70) 125 APPENDIX C SOLUTIONS TO ADMITTANCE MATRIX ELEMENTS Solutions to admittance matrix elements developed in Chapters 2 and 4 are detailed in Appendix C. C.1 Waveguide Matrix Elements The waveguide matrix elements only interact with one of the equivalent currents defined in the formulations. Therefore, they have only self terms defined as ALT ~T-1n,+ 1"“; — dr’K , T Olr’ (c 1) 1111.111— '-'1 WC “1771.11 "1 ' A.‘ ‘T1711——2I—l where the kernel is represented as 111 , f1 (31 ) Itwg(;r1ml,0|.1"1) 2 / (lg/1 jT—lll Grvg<171m1,0 f|r1,y1 t) (C2) 41/1 and the rectangular waveguide Green’s function, due to a transverse slot cut in the bottom wall, from Appendix B is specialized as 1 f , _ GWG (11",1 0 t|311.y1, t): (—1() E jA': sin [AI7 (.L'lml — (M2)] .7 .sin [AM (.111 —— (re/2)] Jim/"l yli (CS) 126 Substitution of (C2) and (C3) into (C1) and rearranging the terms leads to w, 110_ ._ _ I, g, —J'I.-y.,—y’1 z: 1.11... 012)] 111.111.11.16 , l _I/VI A; $1111 +—2El / dzr’l sin [Am7 (2'1 — (#2)] (C4) A. $111.1 ___;l Where solving the spatial integrals gives the solution to the waveguide matrix elements. Solving the I’l-dependent integral first, the addition theorem for sinusoidal functions is used to split the single integral into two closed form integrals. This is given as W1 we —1 ‘7 .- _ I 1 ~jky~ 1771.71 I , I) Z A' 5111 [111137 (Ilml — 61/12)] / dylfl (311) e I u} [1.0 (1. , m. 7 ——W1 417., +523 ' COS (Afraid/2) / dill Sin (AC-17,1 $1) A1: $1711 — Tl mfié? — sin (AI7 (1/2) / (1.1"1 cos (Aha/1) (C5) . Ar $171.1 — Tl I _,,II where substitution of the solution for these closed form integrals leads to WC 1 6’7 (,1 I I ’jky (*3/1) 1’._—§:_in[k (a; ‘02)]/d' , e '7 771.11 (Ultoab kg,7 S I’Y 17711 / ‘4, yl f1 (gl) '1 2 . . .cos (Ana/2) E; 8111 (“7131711) sm (k1,...fol/2) _sin(Al- a/2) :rcos(A1,Ir1nl) (MAMAIl/2) (C6) 127 Combining the similar sinusoidal terms and again using the addition theorem for sine functions, the matrix elements are written in terms of the only the y'l-dependency as WC —2 67 . . nun. 2 m A), W blIl [kl‘ey (1317711 — (#2)] 5111 [15,177 (11111 '— (1/2>] VVI . _-1 . — ' -sm (k5,.A;L~1/2) / (1y; f1 (y’1)e J 311' ”II ((3.7) _Wl Now, to solve the yi-dependent integral, the yl-directed distribution of the slot electric field from equation (2.49) is substituted, giving 11 C_ . ,. .' .. [m n — 112/111111)” 1 :71. A.“ Sill [AW ($14,721 _ (IA/2)] hm [A17 (11"1 — (#2)] VVl . —"'A', A, --- I .sm (Ang131/2) / dy’le J 3” yli (C8) ‘Wl Analysis of the exponential integral (making sure to handle the absolute value cor- rectly) leads to the final solution for the waveguide matrix elements as 11G__ —2j €~ , 1. , 1,. ~ - rm: :1. 2.5, 1‘14 (1..., - (#2)] 1‘47 ($1., - WM . —jA' u' a =1,2,3,... -s1n (A',,.,,A.1'1/2) [e .154, 1 — 1] for (C9) 5’3 2 0, 1, 2, C.2 Cavity Matrix Elements The cavity matrix elements interact with two of the equivalent currents developed in the formulations of Chapters 2 and 4, thus having both self and coupled terms. The matrix elements are solved (observed) at either BSl or BS2, where the self terms are excited by the equivalent current at the same interface, and the coupled terms are 128 excited by the equivalent current at the opposite interface. C.2.1 BSl - Self Terms The self term cavity matrix elements at BSl are defined as $1711+AI1/2 CV11 CV11 ,1 1,! lmJi : ["11le : / (1.111(CV11 (LEImIsOll‘l) (C'IO) 1.1711 —A1‘1/2 where the kernel is given as l’lr’rl ( I) , / Ifl y] ( / / ) r .I. [I —— ' — I.’ ‘ , , .t 011 1\Ci11(~11,,.1»0l.11). / 111311 J. #0 Gov Timla01|$1 311. (C ) —l'l"1 and the rectangular cavity Green’s function is specialized as 2 2 : 6 J I F GCV (1117711 ’ 0’ till-.9151) — 7 CC C I" kzr Sin (kzFCC) -sin [Li-"1‘ (£11111 —— (LC/2)] sin [1'er (17,1 — (l.(./2) -cos (k’JF (hr/2) cos [19511. (1/1 — (157/2) ~cos [kzI‘ (t — 00)] cos (11‘:th (C.12) 129 Substitution of (CH) and (C.12) into (C10) and rearranging the solution gives (‘V11_ j? (P cos (kart) . ‘ .. [111.11 _ _7 cos AyFbC/2 uJ/LOU'C C F kzl‘ Sin (kzpcc) .sin [lamp (1:17,,1 — ac/2)] cos [kzr (t — Cc):l W1 - / dyifl (310608 [My (111 - 115/2)] __.[171 Tlnl+AT1/2 / (11"1 sin [kl-,1. (:r’l — (LC/2)] (C.13) .rlnl—Arrl/2 ‘fi . I . y o . . . . Solving the .rl-(lependent integral first. 1t is noticed that the waveguide matrlx ele- ‘ . ‘. . 1', . , . ’ ‘ a _. . 4 .‘ ments had a similar .1.. -(.le.pendency, thus Similar steps are taken to solve tlns mtegral. Therefore, the self cavity matrix elements at BSl are written in terms of the only the y'l-dependency as 6 cos k.» t) [CV11 _ _14_ F ( “F ””‘n __w/zoabc — ‘ ' ' ' (, - F [$7.37.]?in Sln (kZI‘CC ~sin (A'MAJ‘l/Q) sin [If-"1‘ (271ml — tic/2)] sin [If-7'1“ ($1711 — (1.0/2)] 11’} / 111/1h (9i) cos [A.3/F (y; — b(./2)] (C14) —-l«1'1 ) cos [RT (1‘. —- (36)] cos (A.3/P bc/2) Again, similar to the waveguide matrix elements, solving the y'l-dependent integral involves the substitution of the yl-directed distribution of the slot electric field from 130 equation (2.49) giving . ros k~ t lC‘Vll _ __L2_ GFC ( HF ) 711,71. _ 7. - WHOO'CbCWl l" [€137 kzr Sln (kZI‘CC ~sin ((1717Aél‘1/2) sin [kII‘ (171ml — Clo/2)] sin [lg/1.1. ($1711 — (LC/2)] ) cos V? (t — (4)] cos (kyFbp/2) W1 . / dy'l cos [kyl‘ (y-Il — bc/2)](C.15) _W'l Then to solve the yi-dependent integral, the addition theorem for cosine is used to split the single integral into two closed form integrals. This is written as ,. 6 CO‘ k7~ 1) lC‘Vll _ ___Z_2____ F b( if 7”,". _ ,1 ‘ j I f c WI1()(11.b(.Wl r 15...,7115F 5111(kzrcc .sin (1:,r7,A.1r1/2) sin [kTI‘ (11ml — (lg/2)] sin [11:151. (17172.1 — (LC/2)] W1 .cos (kl/I“ bC/2) / dy’l cos (kyr 91) ‘W1 W1 . — sin (AryFbC/2) / dy'l sin (kyry'l)(C.16) —W1 ) cos [1131. (t — 1:55)] cos (kl/I“ (JG/2) where it is seen by inspection that using the concept of even/odd integrals over symmetric limits. leads to 6 cos k t) [CV11 _ __fl— F ( 2F 771,71 _ . 7 ‘ WHOQCbCW 1 I" kl") A3311 Sln (kZI‘CC .sin (1111.57 Axl/2) sin [kl'I‘ ($17711 — (lo/2)] sin [kl’l‘ (111,1 — (LC/2)] W71 . / (1.1/1 cos (kg/F;y’1>C.17) 0 ) cos VT (15 — Cd] cos2 (kyrbC/Q) 131 Solving this cosine integral leads to the final form of the self cavity matrix elements at. 881, with unspecified dimensions [ox/1.124;: . F (151) 771,11. wltoachWI k k k . k I‘ 1.7 9F ZI‘ Sln zFCC .sin (k137Axl/2) sin [11:11. (11317711 — aC/2)] sin [kip ($1711 —— aC/2)] -c082 (arm/2) sin (15,. W1) (C.18) ) cos [11721, (t — 0.)] Applying the specific cavity dimensions used in the formulations of Chapters 2 and 4, given as a... = 2L ; be = 2%” ; cc 2 t ; 1V1 = W (019) the self cavity matrix elements at BSl are finally written as 7 f i 6 , / [(111 = ——J— ———I:—— sm km Aug/2) "2’1"” 11,111.0sz 21‘: kkayrkzr K 7 -sin [kIF (331ml -- L)] sin [kip ($1711 — L)] -cos2 (kyI‘ W) sin (kl/1“ W) cot (kzrt) (C20) C.2.2 BSl - Coupled Terms The coupled term cavity matrix elements at BSl are defined as 1‘12"? +/_\..r2/2 (ll/’12 CV12 J ’ .’ 1111,11. : [1111,1112 : / (LL2ACV12 ($11711 a 0'12) (C21) 12722 —A.'r.2/2 where the kernel is given as 1V2 ( I) l, f .1 [f2 y2 , , I [I Acm (171112110112) 2 / (1312 Mo GCV(.11m1,0,t|.1:2,y2,0) ((3.22) 4112 132 and the rectangular cavity Green’s function is specialized as —2 6F (chc 1" er Sill (kZI‘CC) -sin [11151. (371ml — (LC/2)] sin [19551. (17,2 — aC/2)] -cos (keyring/2) cos [kl/F (yé — bc/2)] cos [ls/Zr. (t — 60)] (C23) I I GCV (-T1,,,1.0,t|;r2,y2,0) : Substitution of (C22) and (C23) into (C21) and rearranging the solution gives r 2 [$117112 2 _J___ 61‘ cos (kyI‘ (Dc/2) , 11211011ch I‘ 1931‘ sin (kzrcc) .sin [lg-Ir (171ml — aC/2)] cos [11131. (t — 612)] ”"2 . / (lyéfg (11,2) cos [kl/1‘ (y'z — bC/2)] _W2 272,712 +A112/2 / das’2 sin [er (:c’z — Claw/2)] (C24) 12712 —A;1‘2/2 Solving the spatial integrals in a similar manner as the self terms at B81 and then applying the following cavity dimensions a..=2L ; b...—_2w ; 136:1 (C25) the final solution for the coupled terms of the cavity matrix elements at 881 is F k3,»), [Cy-F kzF Sin .sin (km Amy/2) sin [11th (111ml — LN sin [kit-F (:1an2 —- LN .cos? (kl/1‘ 1V) sin (kyI‘ 1V2) (C26) [CV12 = __‘2—7— CF 772.,” WHOLW W2 Z (kth) 133 C.2.3 B32 - Self Terms Similarly, the self term cavity matrix elements at BS2 are defined as 12722 +A1‘2/2 CV22 CV 22 . J r .1 1111,11 :11112112: / (112ACV22 (1721",210l-l’2) (C27) $2712—A12/2 where the kernel is given as 1'12 . .3/f2( '2) [\(w(,'22(1-) 1311121 0|12)= /d y2 jam CC V (33211121 0 ()[12, 112. 0) (C28) —ll"'2 and the rectangular cavity Green’s function is specialized as —2 61“ a'CbC I‘ kzr Sin (kzFCC) vsin [111T (.1‘2m2— (10/2)] sin [11' II‘ (12 —— (LC/2)] .cos (k grin/2) cos [k ”I“ (y 2- b /2)] cos (kzrcc) (C29) GCV (2:2,,,2,0,11|I’2.yg,o) = Substitution of (C28) and (C29) into (C27) and rearranging the solution gives .- creos k2 cc C122 12 ( F ) . .. 1,... ZW . “bf W) . C C F kZP Sln (kgr Cc) .sin [k‘T’F (172",2 — (LC/2)] W2 - / 11.11312 (ya) [15. (y; — 1121] _WQ 127,2 +AJ‘2/2 / (1.172 sin [17,. F (.12 (1.5/2)] (C30) .12”? —A.r2 /2 134 where solving the spatial integrals in a similar manner as the self terms at 881 and then applying the following cavity dimensions 11.5 : 2L ; be = 21V ; c5. :1 (031) the final solution for the self terms of the cavity matrix elements at. BS2 is [($22 _ j 61‘ cos (krgrt) "'~” _ .11 11111 E - ~sin (k5,.7A12/2) sin [11151, (12,772 — L)] sin [MP (1172722 — L)] .cos2 (kyFl/V) sin (kyf 1V2) (C32) C.2.4 BSZ - Coupled Terms Finally, the coupled term cavity matrix elements at BS2 are defined as $1111 +AI1/2 CV21 CV21 (I r ‘ , I l111.11 : [1112.111 2 f (1‘1 IRCV21 (3 27,12 2 0'11) ((133) 11111 ~A1‘1/2 where the kernel is given as W1 , f1 (31') KCl/Ql (1.27722101111) = / dyi‘mGCV ($2771.2101013731yllat) (0'34) _WI and the rectangular cavity Green’s function is specialized as —2 6p (1ch P kZI‘ sin (kl/TC“) -sin [1.ng (172,712 — (1.5/2)] sin [kw (.1'1 — (1.5/2)] .cos (kl/Pb“/2) cos [2'91“ (Ll/,1 — 115/2)] cos [ksr (t — c5.)] (C35) I I chr (1.2-"1.250701121HI/1ét) : 135 Substitution of (C34) and (C35) into (C33) and rearranging the solution gives CV21 32 6F ,. — fl: . (11112) w/l'Oac C F kzF Sln (kZFCC) .‘sin [Amp (12",? - (LC/2)] cos [kZI‘ (t — (5)] W1 11711+A11/2 . / dy’lfl (y'l) cos [Ayr (y'l — 115/2)] / 111'18i11 [A71 F (1'1 —a5/2)] (C36) —11'1 $1111 —A.1?1/2 Again, solving the spatial integrals in a similar manner as the self terms at B81 and then applying the following cavity dimensions (153:2L ; bC=2W ; cczt ; W1=W (037) the final solution for the coupled terms of the cavity matrix elements at BS2 is [C 1 21__ m 11 —WA10L-W 2 Zr: 16:12 AyrAz:F sin (k Ft) .sin (A'557A11/2) sin [ACIF (:rgm2 - L)] sin [“1“ (117,1 —- LN -cos2 (113,1, W) sin (11,1. W) (0.38) C.3 Halfspace Matrix Elements The halfspace matrix elements only interact with the equivalent current defined at BS2 in the formulation of Chapter 2. Therefore, they have only self terms defined as . A1? $25,112 + 17%;”? : / (1.172KHS (121112101312) (C39) A. 112112 — ‘22 136 where the kernel is given as W2 f2 11 KH5(;1-o 2m2 01.12) = / .122 ’ juE11—_O—)GHS(1"2’"2 0 01142 .22 0) ((3.40) 4’12 . and the halfspace Green's function is specialized as 2 2 _jk0\/(‘r21112 —I,2> + (_312) e 2 . I I 2 27r\/<12m.2 - 1‘2) + ("Z/2) Substitution of (C40) and (C41) into (C39) and rearranging the solution gives (C41) I I G115 (12mg ., 0. 01172, 1112, 0) = A1 2 2 +72 if 0 112112112 —Jk0 “(1.27712 _12) + (112) 8 [HS_ _ [A (1.1., d" I m 11 7,1104%” 2 2 .92 2 2 433-1 ) “”2 \/(‘T21112 — (1:2) + (3’2) where (_‘lepending on whether or not 1112 equals 112, the solution of the spatial integrals ((3.42) T2112“? are ap1'11‘oxi111at.ed differently, leading to the final solution for the halfspace matrix elements. C.3.1 1112 74 112 For the 1112 75 112 case, the source points are approximated as follows .12 ——> 127,2; y2 —> 0 (043) which is essentially a point excitation at the center of each partition, leading to the distance between source and observer as 2 Rmyén : \/(-T27112 — 172-112) : $21112 — $132712 (CA4) 137 This approximation is valid since the dimensions of the slot are long and narrow relative to the wavelength, where by the distance between separate partitions will vary insignificantly for any point in either partition. Substitution of (C44) into (C42) and then rearranging the solution gives A11? 12712 + —21 W2 / .11'2 / 1125 ((3.45) f A1? —W 2 T2112 — 2 . —J"**0 [HS ___ _J 8 "”1 (12110471145 127712 _1‘2712 $27712 _ 33211.2 which, after solving the spatial integrals, leads to the final form of the halfspace matrix elements for 7712 7é 712 as —j1€.0 1‘27” -—:l?2 ,' _. .2 71.2 HS film-26 , [111,71 : 7 (C46) 11211021 |$2m2 — $2712 (3.3.2 mg 2 71.2 For the T712 2 712 case, the source points are approximated as follows . / .. . I . . _ . . I2 _) 41211.2? y2 _) 0’ 121112 "” 12112 (CA7) where, since the source and observer locations are at essentially the same point, the distance lmtween thorn is approximately R1112=n2 EV: 0 ((3.48) 138 Substitution of (C48) into (C42) leads to the following , AI I2112+ 1V2 HS ‘1 I I 1 [771.71 = W / d1)? / (1112 2 2 ((3.49) A1 —W \/(.1‘2 2 —- 1') + y’ 1‘2712—72. 2 ”2 2 ( 2) where the 1?,,22=,,2 in the denominator is not substituted, since this would lead to an indeterminate form of the equation, and thus a bad apmoximation. Instead, a change of variables given as 121,2 -— 17/2 : 11; (1.171; = —(111 (C50) is applied to (C49) to simplify the analysis of the spatial integrals, leading to AJ‘ W2 , _ u (1 C51 111,11 w;1(-_)—_4711V2 / / 312—— W ( ) _A12 412 +312 Now the use of a change of variables (11. = 1') along with the concept of an even function integrated over synnnetric limits gives the solution as A1? W2 _:_1 1115— _ 111’ / d’ 052 "1"" 1.2—___,11027r1V2 y2—-——— ,2 ( ) £2.12 0 +312 The integral over the yé-dependent function was found in an integral table [71] as {—1.1‘ 2(21' + b ) 2 sinh —— ; R = a. + 11.1? + Cl? C53 VB =\/(_‘ 1< v4ac — bQ ( ) where I I2 rzyz, (1:12; (1:0; c=1 (C54) 139 specializes (C53) to the integral in (C52). Substituting the solution to (C53) into (C52) leads to A1 HS_ - -1 W2 ,. [m n —— u,‘1_1—_—"—027r112 (1;1-’25111l1 (E) (CO5) —%2 where the simple trigonometric relationship 1 sinh—1(1) = csc h_1 (—) (C56) IL‘ allows (C55) to be written as 1” ‘9 _ ___—L_ ’" " 5110211112 ,I (1.112 csc h—l 12 (C57) 1172 5133\1? The integral over the Ié—(lependent function was also found in an integral table [71] [(1.1 (sch—1 (§)— - :ccsch 1 (Z) + asinh 1(—:—) (C58) as where :1: 2 13,2; a 2 W2 (C59) specializes (C58) to the. integral in (C57). Solving this integral, substituting the solution back into (C57), and then using simple mathematical 111anipulations, the final solution for the halfspace matrix elements when 1112 = 112 is HS_ —j A172 . _1 2W2 A132 _ h W2 1 —— . 0 l'" 7" wMOWW2 [ 2 sm (A532) + Sin] (2%)] (C6 ) 140 C.4 Parallel-plate Matrix Elements The parallel-plate matrix elements only interact with the equivalent current defined at BS2 in the formulation of Chapter 4. Therefore, they have only self terms defined as I2"? +AI2/‘2 ) 3 7’ , 1,1,5"? = / 111/21113133 (121122» 011’?) (C61) 121,12 —A.‘1.'2/2 where the kernel is given as yl) V12 Km (.11-2,,.,.01r’2) =/d .11”); (“2 app 3(xz,,,,.o 0115 112.0) (0.62) 4'12 and the parallel-mate Green’s function from Chapter 3 is specialized as 00 00 It: ($21112 I2) —j7l3/, 2 2 I I 2 e e 2cosh(pd) (E — k) GPP3(17271122010|41’221J2a0) = d A 2 - 2 2 _OC DO (271) psmh (pd) (If — k0) (C63) Substitution of (C62) and (C63) into (C61) and then rearranging the solution leads to PP3__ 00 00 2 ejgrz’”? cosh (pd) (£2 — k2) 1111.11 _ 2 d A 3 2 2 w114j7r psmh (pd) (g — k0) —00 —OO ”[2 1132712 +AJT2/2 I . . I _. , - / «11512 (1’2) 6W2 / das’ge 3“? (0.64) —l'l"'2 $2712 —A.1‘2/2 141 where the solution is in terms of both spatial and spectral integrals. Solving the 1.5-dependent spatial integral first leads to oo 00 j£_( .12 <12 ‘ ) pp3 2 ( m? "2 sin (5A132/2) cosh (pd) (£2 — k2) 1111,11: ,_J22 d/\ . 2-2 an 71 Epsmh (pd) (5 k0) —OO —OO ”’2 . 1 - / c1612 61) e‘3"y2(c.65> 41/2 Next, the yé-dependent spatial integral is solved after substituting the yg-directed distribution of the slot. electric field from equation 2.49, yielding [PPB— _ _j "1"7’66'112712—W2 J'E (I2 ‘ -4l“ ) . (12A sin 11112)c "22 212.2 sin “Alp/2) cosh (pd) (£2 — k2) nfp sinh (pd) (£2 — 13(2)) (C66) Z the solution in terms of only spectra1 integrals. 8\8 C.4.1 Spectral Integral Analysis The use of the Fourier transform domain method in Chapter 3, to solve for the parallel— plate Green’s function, lead to the spectral integrals in equation (C63). Solving the spatial integrals in equation (C64) has then given the solution to the parallel- plate matrix elements in terms of only the spectral integrals in equation (C66). Solving these spectral integrals will produce the desired solution for the parallel-plate admittance matrix elements in Chapter 4. Cauchy’s integral theorem, integral formula and residue theorem, are used to solve the spectral integrals in equation (C66). Cauchy’s integral theorem states that if a function f (:3) is analytic at all points interior to and on a simple closed contour C 142 [64], then fflzwz = 0 (C67) C Cauchy’s integral formula states that if f (z) is analytic everywhere within and on a simple closed contour C, then if 2:0 is any point interior to C z—zo f “3) dz = 21:271ij(20) (C68) C depending on whether C is oriented in the positive or negative direction. Cauchy’s residue theorem states that if f (z) is analytic on the closed contour C, then the value of the integral of f (3) around C is f f(z) ..e: = i2er' 2 Resz=zkf(z) ((3.69) C k=1 C.4.1.1 17 Analysis Rearranging equation (C(56) so that all the 17 dependent terms are separated and then re-writing the siimsoidal 1] component in exponential form leads to the following form of the solution ( ) 1Pp3_ -1 7 56’ 2’”? 2”? enema/2162412) 111,11. —w1.l.471’2l’V2 €(g2—k8) 00 (eJ'UW2 — {inn/2) 008110901) .70 d" npsinh (M) (C ) 143 Since only the 17 spectral integral is of interest in this section, a separate g-dependent function is defined as follows 00 (ej111V2 _ e_jnW2) COSh (pd) 17(6): /d1) 17psinh (pd) (C71) —00 The parameter p z p (1}, g) is a function of 17, thus the solution written in terms of 17 is ac (cjull’g _ e_j1}l’lv'2) COSll ( In? _ "8(1) F (g) 2 / .11, (0.72) where 77(2) ___ p2 + k2 _ £2 ((3.73) Equation (C71) now reveals the singularities in the complex n—plane, where at first glance there appears to be both simple poles (ie. 17 = 0) and branch cuts (ie. p. sinh (pd) = 0). However, upon further evaluation the apparent branch cuts actu- ally reduce to simple poles, the key to this important reduction is the small argument approximation for the sine function (sin (I) E .1: for :1: << 1). This is because no matter what value of p makes sinh (pd) : 0, there is always ends up being a p2 in the denominator which gives simple poles at 1] 2 i110. By allowing 1] to be complex (ie. 17 = 17”). + jmm), the integrand in equation (C71) becomes analytic in the entire complex n—plane, except at the simple pole singularities stated earlier. This allows Cauchy’s integral theorem to be invoked by deforming the integration contour off the real axis, where the specific closed contour C is chosen based on an examination of the integrand. More specifically, the exponential terms eij77w2 are considered since proper choice of the half-plane closure provides GXponential decay and thus convergence of the integral. Splitting the integrand into 144 two terms as follows oo jnl’VQ oo jnW ‘ I — 2 ‘ F(€) / d' e cosh (pd) _ / d e cosh (pd) .74 111) sinh (Pd) np sinh (pd) (C ) _‘30 —oo allows the f—dependent function to be defined as F“) = FUHPK) +FLHP(€) (C15) where U H P and LH P describe the half-plane closure used to ensure convergence of the integrand. Figure C .1 shows the closed contour for the integrand of F UH P (5), where applying Canchy’s integral theorem leads to FUHP (5) "if CFUHP(€,11)dn=/ FUHP(€.11)d11 CR+CJ +082 2 211j 2 Res,- (FUHP (5.17)) (C76) The infinite contour integral is examined, via inspection, by letting 1} tend to infinity. The hyperbolic sinusoidal terms are seen to cancel each other and the p term tends to 1], giving 81' '71W2 FUHP 5,1 d =/ d1; _0 (3.11 A; ( 1)11 a; 112 ( ) where since 1) is complex and proper half-plane closure was chosen, the exponential term decays, driving the value of the integral to zero. The contour integral around the pole at 17 = 0 is examined using Cauchys residue them‘em. The semi-circle arc has a negative orientation to ensure the desired integral has the proper sign. Since the arc in only half a circle, the residue is only multiplied 145 A 1m {17} x11; Figure C.1. Evaluation contour for UHP closure in the complex n—plane by —71 j instead of —21rj, leading to L6. FUHP<£J1>d11= -7rj [Res (FUHP (6,0))] (C18) where the residue is Res (FUHP(§,0)) ___ ngPW = 0) _ cosh (pod) _ , C19 .1311me posmh(pod) ( ) Substituting the results for (C77) and (C79) into (C76) and rearranging the function to achieve the desired result in terms of the remaining residues, gives UHP _ . CO811019060 ] fie F (517])(177 _ 711 i190 sinh (pod) Res (FUHP(§,—110)) + i Res (FUHP (g, 41111)] (C80) 1/=1 +271j where the rmnaining residues have full circle arcs orientated in the positive direction, 146 thus they are multiplied by 2117' and are found as UHP . -. . . p- (71 = ~10) e‘J’IOH'Q Res (FD HP (5» —770)> = [0313p : —2,_ (C81) 1 -1,” (71 = —no) ’70“ and , UHP , _ _ —" W1 ,. p_ — 1 J 1721/ 2 q _1,,, (11 = -71:») "M1 respectively. Substituting (C81) and (C82) into (C80) yields the final desired result as , _ . ..‘l 7 ,— 3170115 00 '—j17VW'2 Fl” Hp (15) = 1r" [ cos1(p()() e 26 (C83) posmhad) 113d ,2, 11.1 where the various spectral wavenumbers are given as . . 71/11 ./ 715:1)12/‘11I62—52 i [91/27 ; 778=I€2-€2 ; p()=] 132—f2 (0.84) Figure C2 shows the closed contour for the integrand of F “I P (g), where applying Cauchy’s integral theorem leads to FLHPe) = f FLHP(€.11)d11=/ _ _ FLHP (6,11) d-n C CR+C0 +000 2 —27rj 2 Res, (FLHP (817)) (C85) The infinite contour integral is evaluated in the same manner as the UHP, where 17 is allowed to tend to infinity. The hyperbolic sinusoidal terms are then seen to cancel each other and the 7) term tends to 17, giving LHP e—jnwg / F (6,1)d11=/ (in—=0 (0.86) 00:, C— 2 OO 17 where again since 17 is complex and the proper half-plane closure was chosen, the 147 Figure C2. Evaluation contour for LHP closure in the complex 17-plane exponential term decays, driving the value of the integral to zero. The contour integral around the pole at 17 = 0 is again examined using Cauchy’s residue theorem. The semi-circle arc has a positive orientation for the LHP to ensure the desired integral has the proper sign. Since this are is also only half a circle, the residue is only multiplied by 117 instead of 2717', leading to fr F L” P (5.01110 = 11' [Res (FLHP (€,0))] (C81) wl'iere the residue is . LHP( .—_ 0) cosh ( (I) P». (FLHP 0 = p0 ’7 = — I L m C88 163 k (6» )) (1,611}? (77 = 0) p0 sinh (pod) ( ) Sul.)stituting the results for (C85) and (C87) into (C84) and rearranging the 148 function to achieve the desired result in terms of the remaining residues, gives p0 sinh (pod) f‘ FLHP (517)6111 Z 117 [ cosh (pod) ] C , Res (FLHP (t. 00)) + 2 Res (FLHP (5. 72.11)] —271j (C89) V21 where the remaining residues have full circle arcs orientated in the negative direction, thus they are multiplied by —21rj and are found as LHP _ _,-. w p17 (71— 770) 6 J00 2 R98 (FLHPQJKD) = ,BHP . _. = ——§‘21— (C90) (11,0 (11 110) 710 ‘ and LHP , _. —«' 1V . I — 1771/ 2 He.9(FLHP({,17U)) = P’nLVHPU 771/) = _LW— (091) q TIV (TI : 771/) UV respectively. Substituting (C90) and (C91) into (C89) yields the final desired result for the LHP as FLHP (g) , I: cosh (pod) e‘j’IOWQ 7"" _— p0 sinh (de) 178 d 00 _ . W 2 JUV 2 —e-————] (C92) 2 u=1 17,/d Substituting (C84) and (C92) into (C75) and combining like terms gives the following form of the E-dependent function P0 sinh (190(1) —— C93 178 d V 173d ( ) , cosh (pod) e-j’70w2 00 2e—j7IVW2 F (a) = 2m i —— =1 A more compact form of the {-dependent function is found by noticing that the middle term of (C93) is the 1/ = 0 term of the series in the last term of (C93). The compact version of the {-dependent function is thus written as 710 sinh (pod) ..‘l '7 '00 —j17ul/V2J F(§) = 211j[ COMP”) + -e—————“—] (C94) K) /=0 77;; d 149 where the Neumann number is defined as 1 f 01 l/ = 0 61/ Z ((3.95) 2 for 1/ # 0 The f-dependcnt function in (C94) is then substituted back into (C70) giving the parallel-plate admittance matrix elements in terms of only a 5 spectral integral. Splitting the matrix elements into two separate terms gives 00 1f (1“ ., -I . ) pp3 _ —j / e 2 n? 211.2 sin (€Amg/2) (£2 -— k2) cosh (pad) 1 1. _ 1 ' (If i i I 97.21112 6 (£2 - 1'13) 1908111110909 -—2‘ 0° °° ej g <22 ”2712) sin (EAx2/2) (£2 — 1.2) e‘j’W2 ._-_,ZEV / dg 2 2 2 (C96) quTrl/l/g 11:0 _ g (g — 11:0)an Solving the 5 integrals in closed form would be the most advantageous computa- tionally. However, if that is unfeasible to attain, then solving the integral numerically is the only option. In the case of (C96) only the first term is able to be solved in closed form, because the second term has a branch cut due to the multi—valued 17,, term. The parallel-plate admittance matrix elements are thus redefined in terms of two functions as . —j 1 2 00 lP‘P‘g : —— —F — E F C97 where F1 corresponds to the first term in (C96) and is evaluated in the next sec- tion. After expanding the sine function in terms of exponentials and simplifying the function, F1 is given as 00 (18ng — €3.53) p0 cosh (pod) a=fe . g (£2 — k8) smh (pod) (C98) 150 F2 corresponds to the second term in (C96) and is solved numerically, where noticing that the function is even in g and is written as (C99) °C cos [t (12..., — 2:2,,2)] Sin (gags/2) (g2 — 12) e-inz/W2 1112:7015 1(52—k3)03 0 thus simplifying the necessary compuatations. Other definitions include 71/11 . 113=p3+k2-€2 ; 121): d ; p0=J\/k2-€2 (C1100) and A B = (272m2 — 11:2"2 — Axg/2) 2 A12 (111 —- 11 —1/2) (C101) C.4.1.2 5 Analysis The 6 spectral integral analysis of (C98) is carried out in almost exactly the same 111aI'1ner as the 17 spectral integral analysis of the last section. Splitting the integrand into two separate terms gives Flz—Z [E (Impocosh pod _Z1€ 6158110cosh(p()d) (0102) £(52 — kg) sinh( (pod) {(52 — k8) sinh (pod) where depe1'1ding on matrix element being filled, the upper or lower half-plane closure is defined for the first term as A = A1011. — n +1/2) > 0 V m 2 11 UHP Closure (0103) < O \7’ 111. < 11. LH P Closure 151 and for the second term as B :2 Acr(m—n— 1/2) > O V m > n UHP Closure ((3104) < 0 V m S n LHP Closure This leads to three different closure cases for the function F1 given as F1 (m > n) = {{qu — FIUBHP ((3.105) 171(711271) = F941“) — Ffif" ((3.106) F1 (m <71) = FILAHP — F112;”) (C.107) Both the first and second terms have the need for upper and lower half-plane closure. Since they have the same integrand, except for the terms A and B, new functions in terms of a dummy variable (DV) are defined as OO ejilDV) cosh d F913: fdg 2 p20 , (1)“) (0.108) D _ {(5 -ls:0) smh(p0d) and LHP 0C €j€(_DV)p0608h(P0d) 0v me —ko)smh for the upper and lower half-plane closure respectively. This is done so that the complex plane analysis only has to be calculated once instead of multiple times. The desired F1 functions are then created by substituting in the appropriate term for the dummy variable. Figure C.3 shows the closed contour for the integrand of F 1UDIl/ P , where applying 152 Cauchy’s integral theorem leads to UHP _ UHP _ UHP F119,, — lodge)”, (a -— (magma; FlDV (M = 27rj ZResi (Ff/DIX}D (0) (0.110) (~ “{5} C01; ; x a x _ EEO “{5} El Figure C.3. Evaluation contour for UHP closure in the complex f-plane The infinite contour integral is examined, through inspection, by letting { tend to infinity. The hyperbolic sinusoidal terms are seen to cancel each other and the po term tends to 5, giving 81am) 2 [ch FFD$P(g)dg=/C+ dg 5 =0 (0.111) where since «S is complex and proper half—plane closure has been chosen, the exponen- tial term (:lecays, driving the value of the integral to zero. 153 The contour integral around the pole at g = 0 is examined using Cauchy’s residue theorem. The semi—circle arc has a negative orientation to ensure the desired integral has the proper sign. Since the arc in only half a circle, the residue is only multiplied by —7rj instead of —27rj, leading to (<11 Fifi? (a) 11 = —m’ [12... (pg? (09] (0.112) where the residue is Res (FUHP(0)) _ ”SHIP“ = 0) _ kcos(kd) _ _ ______ (0.113) 1DV q,(()]HP (é ___ 0) 15(2) sm (ch) Substituting the results for (C.111) and (C.113) into (C.110) and rearranging the function to achieve the desired result in terms of the remaining residues, gives f k . . . U HP - , (Ob (1.1.1) . [ _ ( UHP F — —7r] ———-———— + 27? Res F —k()) W l 13 sin (M) J IDV ( ) + 2 Res (FIUDIi/P (—§l.))] (C114) where the remaining residues have full circle arcs orientated in the positive direction, thus they are multiplied by 27r j and are found as R (FUHP( 1‘ )) - png (6 _k0) _ e—jk°(DV)EC°S(%d) (C 115) (8 10V ,0 — (1,9ng = —ko) " 2% sin (Ed) . and UHP __ 1) ‘ (6 = #61,) _e-Wpra, Q’UHP (5— _ £1)—€l (53—113)d respectively. Substituting (C.115) and ((3.116) into (C.114) yields the final desired 11152.9( FlUfo/P(_ g,.,.-_-.)) (C.116) 154 ail result as UHP_ . ICCOS (kd) Bujk0(Dv)7C-COS (Ed)+ e 'j5U(DV)p0U [.0 sm (kd) k0 sm (kd) v: 1 £3 (EU- (€861) where E = k2 — 13 (0.118) 2 2 2 6U=pov+k ; 170,):— Figure C.4 shows the closed contour for the integrand of F LH P ,where applying Cauchy’s integral theorem leads to LHP FLHP _ LHP F10, — #10 th 1m (51— /CR+C()‘+C_ FDV (odg = —27erResz(F1LgVP (5)) (C.119) _ a: 111mg} X _ + '5' E-ko x ... x ; CR Figure (3.4. Evaluation contour for LHP closure in the complex {-plane 155 The infinite contour integral is examined, through inspection, by letting i tend to infinity. The hyperbolic sinusoidal terms are seen to cancel each other and the po term tends to 6, giving LHP _ €j€(—DV) _ /CO_O 15‘le (§)d§ _ [6‘30 (lg—ET— _ 0 (0.120) where since 6 is complex and proper half-plane closure has been chosen, the exponen- tial term decays, driving the value of the integral to zero. The contour integral around the pole at g = O is examined using Cauchy’s residue tl'ieorem. The semi-circle are a positive orientation to ensure the desired integral has the proper sign. Since the arc in only half a circle, the residue is only multiplied by NJ instead of 27rj, leading to LHP __ . LHP /00‘ F1 DV (5) dg _ m [Res(F1DV (0))] (0.121) where the residue is LHP , = 0) 1c cos (kd) R2.) FLHf) 0 2 po (6 = —— C.122 (S( 1DV (l) qlng(€: ) kgsinUcd) ( ) Substituting the results for (C.120) and (C.122) into ((3.119) and rearranging the function to achieve the desired result in terms of the remaining residues, gives , kcos(kd) , LHP _ _ , _ __ LHP ., F _ 7r] l k3sin(kd)l 27f] lReS(F10V (W) + 2 Res (#51; (5.0)] (0.123) 11:1 where the remaining residues have full circle arcs orientated in the negative direction, 156 thus they are multiplied by —27r j and are found as ., LHP 21719;”) (E : k0) _ ejk0(_DV)lccos (Ed) R68 F1 (k0) LHP - 2 . — ((3.124) DV —kq’ k0 (6:130) 2k0 sm (kd) and LHP ' —DV 2 p _ €12 J€11( ) . RCS< F1L1§IVP(§“=)) szvHPG ) = 82 2 501’ (0125) q gv (6:511) €12 (512 _ k5) d respectively. Substituting (C.124) and (C.125) into (C.123) yields the final desired result as 101 + 2 A8 sin (kid) k3 sin (Ed) “___1 {.3 (£5, — A8) d (C.126) FLHP— _ __W, [_ 1.. (020.11) dill—Dvltcos (Ed) 00 26.15.1—0mpgv Substituting the results from (CH7) and (C.126), with the appropriate A, B term replacing the dummy variable, into equations (C.105-C.107) respectively, leads to the following set of three distinct groups of F1 functions P121.) [e—jévA _ e-jaB] 21r' 00 F1071 >71.) = 7J2: 11:1 612' (£21 _' 1‘8) Fees (Ed) [e‘jkOA — 13‘3"”03] +1rj 2 _ _ (C.127) k0 sin (ktd) 51 A 361 :2—7—Lj p31, [6- J ‘l' 6 B] F1 (111— - 11) . 77,2. 22( (£3 2) ’Fcos (Ed) [e‘jkOA + ejkOB] kcos (kd) +7TJ — 27Tj --—— (C.128) k0 sin (kd) JkO sin (kd) 157 A) F (m < n) 27Tj i pg” [_ejEUA + ejngl 1 I ‘ = _— d ,,=, 23 (£3 — k8) Ecos (Ed) [-ejk0A + ejkOB] k0 sin (kd) +721 (0.129) Substituting (C.127-C.129) into (C97) and rearranging the terms leads to the final representation of the parallel-plate admittance matrix elements, given as 15.5.3: 21222121) +12%. +13...” (0130) where the first and second terms given as ll _] i2 2pOUFC'lp1 + kcos ()lcd)Fle‘Bp2 (C 131) 22‘pp Tut/2 ”:15; (5,2, -13).) 1,212. (2.1) ' and ”k ‘ kl 1? — ’ “M (l for 111.211 (0.132) m "PP 1211211211,) sin (ltd) respectively, are associated with the F1 function and the third term, given as 11311 11 Z 61/ ’ PP: wwngdV 7 cos [g (1'2",2 — 172,,2)] sin (€Arg/2) (£2 — k2) e—j’qu'? - (l5 0 2 . (C.133) 5(12 — 12M? is associated with the F2 function. Other definitions include [e‘jgl’A — e’jg‘B] for 111 > 11 Flax-[)1 : [€_jf’vA + ejguB] f0]. 7” 2 77. (C134) {-63.61% + ejEL’B] for 111 < n 158 A] and [e—jkOA — e—ijB] for 111 > 11 Ferp2 __ 1 [e’jkOA + ejkOB] for m = 11 [—ejk0A+ejkOB] for m<11 where the terms A, B are given as A=A17(m—11+1/2) B = A1011 — 11 —1/2) and the various spectral constants are 2 jmr . 6U : p31, + A32 1 p01) : T 1 6U : —] — ( jmr , 713:1?12/‘l‘k2’52 i 1912:?“ ; 77u=—] —(7712/) 12:12 —k0 , k: 122—kg with the Neumann number given as 1 for 1/ = 0 61/ = 2 for 1/ 72$ 0 159 (0.135) (0.136) (0.131) (0.138) (0.139) (0.140) A] BIBLIOGRAPHY 160 [ll BIBLIOGRAPHY W. Williams, and K. 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