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 2/05 p:/ClRC/DateDue.indd-p.1 NUMERICAL S-PARAMETER EXTRACTION AND CHARACTERIZATION OF INHOMOGENEOUSLY FILLED WAVEGUIDES By Pedro Barba A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Electrical and Computer Engineering 2006 ABSTRACT NUMERICAL S-PARAMETER EXTRACTION AND CHARACTERIZATION OF INHOMOGENEOUSLY FILLED WAVEGUIDES By Pedro Barba A numerical tool based on the finite element method (FEM) is developed in order to assess the parameter uncertainty vulnerability in a novel inversion algorithm to extract the electromagnetic constitutive parameters from a material sample. This inversion algorithm relies heavily on the assumption, when having the cross-section of the testing waveguide partially filled, that the material sample has to be perfectly centered. In the present work, the effect of having the material sample displaced from the center is measured by comparing its extracted constitutive parameters (6, u) with the values corresponding to the perfectly centered case. The finite element method formulation presented here, can also be used to provide the theoretical data (that otherwise would have to be obtained via traditional mode matching techniques or the hybrid mode decomposition), required for the inversion algorithms corresponding to non-regular samples. The results of this work identify some of the cases in which errors are originated from the sample preparation or from the measurement technique utilized. This information is used to identify the band of frequencies in which the error in the inversion algorithm can be minimized. The numerical method is further extended to investigate the behavior of waveguides loaded with layered as well as anisotropic materials. To Edith, my lovely girlfriend iii ACKNOWLEDGMENTS I would like to extend my appreciation and gratitude to my academic advisor, Dr. Leo Kempel, for providing me with the opportunity to work under his guidance and making me part of his research team. To Dr. Shanker Balasubramaniam, for helping me write my computer codes faster and more efficiently. To Dr. Edward Rothwell, for always having the time and willingness to answer all my questions. Also, Dr. Gregory Kobidze, for his friendship and help during our time at the Computational Electromagnetics Lab at MSU. A very special Thank You to Dr. Barbara O’Kelly and Dr. Percy Pierre for providing me with the opportunity to come to this wonderful institution where I have spent the happiest years of my life. My eternal gratitude for my parents Irma and Sergio, for always being there unconditionally for me. My grandfather, Isauro Medina Hinojos, for always being an inspiration in all my endeavors. To Edith, my lovely girlfriend, for spending all this time with me and be willing to work with me no matter a what time, no matter for how long. I love you! This work was partially supported by the National Science Foundation under grant ECS—Ol34236 and the Air Force Office of Scientific Research under grant FA9550-06- 1-0023. I would also like to gratefully acknowledge the Michigan State University High Performance Computing Center (HPCC) for providing computational resources for this project. iv “Wenn die Tugend geschlafen hat, wird sz'e fr'ischer aufstehen.” Menschlz'ches, Allzumenschlz'ches Friedrich Nietzsche TABLE OF CONTENTS LIST OF TABLES ................................. viii LIST OF FIGURES ................................ ix KEY TO SYMBOLS AND ABBREVIATIONS ................. x111 CHAPTER 1 Introduction and Background ........................... 1 CHAPTER 2 Preliminary Work .................................. 3 2.1 The Thm-Reflect-Line Calibration Technique ............. 3 2.1.1 Introduction ............................ 3 2.1.2 Scattering and Transmission Parameters ............ 3 2.1.3 Derivation of Equations ..................... 5 2.1.3.1 The Thru Measurement ................ 7 2.1.3.2 The Line Measurement ................ 8 2.1.3.3 The Reflect Measurement ............... 11 2.1.3.4 Postprocessing of Measured Standards ........ 13 2.2 Derivation of the Reflection and 'fiansmission Coefficients for a Fully- Filled Rectangular Waveguide. TElO Mode .............. 18 2.3 The Inversion Algorithm ......................... 28 2.4 Hybrid Modes and the 'IIansverse Resonance Method ......... 32 2.4.1 Hybrid Modes ........................... 32 2.4.2 The Transverse Resonance Method ............... 39 2.4.3 Conclusion ............................. 39 CHAPTER 3 The Finite Element Method Formulation for Inhomogeneous Waveguides . . . 41 3.1 Formulation ................................ 41 3.1.1 Domain Discretization ...................... 41 3.1.2 Interpolation Basis Emotions .................. 42 3.1.3 Formulation of the System of Equations Using The Ritz Method 50 3.1.3.1 Solution of Integrals .................. 56 3.1.4 Solution of the System of Equations ............... 70 3.1.5 Numerical S-Parameter Computation .............. 71 3.2 Validation ................................. 74 vi CHAPTER 4 Results ....................................... 79 4.1 Error Generated by Cross-Sections Shifted from Center ........ 79 4.1.1 Low-Contrast Material ...................... 81 4.1.2 High-Contrast Material ...................... 81 4.1.3 Magneto-Dielectric Materials .................. 82 4.2 Layered Materials ............................. 83 4.2.1 Perpendicular in the Direction of Propagation ......... 83 4.2.2 Parallel to the Direction of Propagation. Horizontal and Verti- cal Layering ............................ 85 4.3 Anisotropic Formulation: A Ferrite ................... 85 CHAPTER 5 Conclusions and Future Work ........................... 124 BIBLIOGRAPHY ................................. 126 vii Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 4.1 LIST OF TABLES Definition for each volumecfunction (5 within each tetrahedron. Definition for each edge on a tetrahedral element. ......... Definition for each edge on a triangular element with its constitu- tive nodes ................................ Definition for each area-function 1,12: within each triangular element. Parameters for the four-point triangular surface Gaussian integra- tion rule. ............................... Dimensions for a rectangular waveguide for the frequency bands used to conduct the numerical simulations and inversion operations [27]. .................................. viii 46 47 60 63 67 8O Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 LIST OF FIGURES A two-port linear network with input and output signals ...... A two-port network with connectors. ................ The Thru standard connection. The Line standard connection. The Reflect standard connection. .................. Signal flow graph for the Reflect standard .............. Rectangular homogeneous source-free waveguide. Sideview of a fully-filled cross-section waveguide propagating the TEIO mode ............................... Extracted relative permittivity and permeability for an acrylic sam- ple using the algorithm in [10] .................... Extracted relative permittivity and permeability for an alumina sample using the algorithm in [10] .................. Vertically loaded waveguide to illustrate the LSE and LSM mode decomposition. Vertically loaded waveguide ...................... OOOOOOOOOOOOOOOOOOOOOOOOOOOO Horizontally loaded waveguide. Suspended sample rod loaded waveguide ............... Tetrahedron element for waveguide mesh discretization ....... Mesh for the waveguide cross-section with material sample inside. Tetrahedron element showing its vertices and the interior point p. Definition for a tetrahedron. Showing its nodes, edges and edge directions ................................ Rectangular waveguide with obstacle ................ 2-D Element for ‘Ilt. ......................... Comparison between the FEM and theoretical closed-form solution in a fully-filled d/a = 1 rectangular waveguide. Acrylic (e = 2.5, p = 1), sample length 8 = 5mm. 000000000000000000 Comparison between the FEM and mode-matching solution in a partially-filled d/a = 0.5 rectangular waveguide. Acrylic (e = 2.5, p = 1), sample length l = 7.5mm. Comparison between the FEM and mode-matching solution in a partially-filled d/a = 0.25 rectangular waveguide. Acrylic (e = 2.5, p = 1), sample length 3 = 7.5 mm. ix macs». 12 13 18 22 30 31 33 36 37 38 41 42 45 47 49 60 75 76 Figure 3.10 Comparison between the FEM and theoretical closed-form solution in a fully-filled d/a = 1 rectangular waveguide. Alumina (e = 9.0 — 30.0027, p = 1), sample length 6 = 3mm. .......... 78 Figure 4.1 Vertically loaded waveguide with material sample shifted from the center by a distance 6. ........................ 80 Figure 4.2 Comparison of the transmission and reflection coefficients (magni- tude) for acrylic when the parameter 6 is varied from 0 to 5 mm in ten steps. (6 = 2.5,}; = 1). .................... 87 Figure 4.3 Comparison of the transmission and reflection coefficients (phase) for acrylic when the parameter 6 is varied from 0 to 5 mm in ten steps. (6 = 2.5,;1 = 1). ........................ 88 Figure 4.4 Percent error on the S-parameter magnitude resulting from shifting the center of the acrylic material sample from 6 = 0 to 6 = 5mm. 89 Figure 4.5 Percent error on the S-parameter phase resulting from shifting the center of the acrylic material sample from 6 = 0 to 6 = 5 mm. . . 90 Figure 4.6 (a)Extracted relative permittivity e for an acrylic sample when the parameter 6 is increased from 6 = 0 to 6 = 5 mm, (b) error. . . . 91 Figure 4.7 Comparison of the transmission and reflection coefficients (mag- nitude) for alumina when the parameter 6 is varied from 0 to 5 mm. .................................. 92 Figure 4.8 Comparison of the transmission and reflection coefficients (phase) for alumina when the parameter 6 is varied from 0 to 5 mm. . . . 93 Figure 4.9 Percent error on the S-parameter magnitude resulting from shifting the center of the alumina material sample from 6 = 0 to 6 = 5 mm. 94 Figure 4.10 Percent error on the S-parameter phase resulting from shifting the center of the alumina material sample from 6 = 0 to 6 = 5 mm. . 95 Figure 4.11 (a) Extracted relative permittivity e for an alumina sample when the parameter 6 is increased from 6 = 0 to 6 = 3.5 mm, (b) error. 96 Figure 4.12 S-parameters for a lossy-magneto—dielectric material (magRAM) sample (a)magnitude and (b) phase. ................ 97 Figure 4.13 Waveguide with a layered material in the direction of propagation of the incident field. ......................... 98 Figure 4.14 S-Parameters (magnitude) for a perpendicularly layered material. Material A: (e = 9 -— 30.0027, p = 1), Material B: (e = 1, u = l). . 99 Figure 4.15 S-Parameters (phase) for a perpendicularly layered material. Ma- terial A: (e = 9 —j0.0027, It = 1), Material B: (6 =1, u =1). . . 100 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 4.27 Figure 4.28 Figure 4.29 Figure 4.30 Extracted relative permittivity for a layered material perpendicular in the direction of propagation. Material A: (e = 9 — 30.0027, p = 1), Material B: (e = 1, p =1). ................. S-Parameters (magnitude) for a perpendicularly layered material. Material A: (e = 1, p = 1), Material B: (e = 9 — 30.0027, [1. = 1). . S-Parameters (phase) for a perpendicularly layered material. Ma— terial A: (e = 1, p = 1), Material B: (e = 9 — 30.0027, u = 1). Extracted relative permittivity for a layered material perpendicu- lar in the direction of propagation. Material A: (e = 1, u = 1), Material B: (e = 9 - 30.0027, [1. = 1). ................ Extracted relative permittivities for a layered material perpendicu- lar to the direction of propagation and the asymptotic permittivity for a homogenized material. The plot on top shows the result when Material A has a higher permittivity, the plot on the bottom M8:- terial A with a lower permittivity: (e = 1, p = 1), (e = 9 -30.0027, 11 = 1) .................................. Waveguide with a layered material parallel to the direction of prop- agation. Horizontal layering ...................... Waveguide with a layered material parallel to the direction of prop- agation. Vertical layering. ...................... S-Parameters (magnitude) for a horizontally layered material. Ma- terial A: (e = 9 — 30.0027, p = 1), Material B: (e = 2.5, u = 1). S-Parameters (phase) for a horizontally layered material. Material A: (e = 9 — 30.0027, u = 1), Material B: (e = 2.5, u = 1) ...... Extracted relative permittivity for a layered material parallel to the direction of propagation. Horizontal layering. Material A: (e = 9 — 30.0027, [.1 = 1), Material B: (e = 2.5, p = 1). ...... S-Parameters (magnitude) for a horizontally layered material. Ma- terial A: (e = 2.5, u = 1), Material B: (e = 9 —— 30.0027, )1 = 1). S-Parameters (phase) for a horizontally layered material. Material A: (e = 2.5, u =--- 1), Material B: (e z 9 -— 30.0027, [1 = 1) ...... Extracted relative permittivity for a layered material parallel to the direction of propagation. Horizontal layering. Material A:(e = 2.5, u = 1), Material B:(e = 9 — 30.0027, u = 1). ............ S—Parameters (magnitude) for a vertically layered material. Mate- rial A: (e = 9 —3'0.0027, ,u = 1), Material B: (e = 2.5, p = 1). . . . S-Parameters (phase) for a vertically layered material. Material A: (e = 9 - 30.0027, [1 = 1), Material B: (e = 2.5, u = 1). ...... 101 102 103 104 105 106 107 108 109 110 111 112 113 114 Figure 4.31 Figure 4.32 Figure 4.33 Figure 4.34 Figure 4.35 Figure 4.36 Figure 4.37 Figure 4.38 Extracted relative permittivity for a layered material parallel to the direction of propagation. Vertical layering. Material A: (e = 9 — 30.0027, [1. = 1), Material B: (e = 2.5, u = 1). ......... S—Parameters (magnitude) for a vertically layered material. Mate- rial A: (e = 2.5, u = 1), Material B: (e = 9 - 30.0027, )1 = 1) . . . S-Parameters (phase) for a vertically layered material. Material A: (e = 2.5, u = 1), Material B: (e = 9 — 30.0027, u = 1). ...... Extracted relative permittivity for a layered material parallel to the direction of propagation. Vertical layering. Material A: (e = 2.5, u = 1), Material B: (e = 9 — 30.0027, ,u = 1) ............. Transmission and Reflection Coefficients (magnitude) for a mag- netized ferrite. 47rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, Ho = 100 06. ............................. Transmission and Reflection Coefficients (magnitude) for a mag- netized ferrite. 47rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, H0 = 300 06. ............................. Transmission and Reflection Coefficients (magnitude) for a mag- netized ferrite. 41rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, H0 = 500 06. ............................. Transmission and Reflection Coefficients (magnitude) for a mag- netized ferrite. 47rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, H0 = 800 08. ............................. xii 116 117 119 120 121 122 123 KEY TO SYMBOLS AND ABBREVIATIONS FEM: Finite Element Method LSE: Longitudinal Section Electric LSM: Longitudinal Section Magnetic MagRAM: Magnetic Radar Absorbing Material MUT: Material Under Test NRW: Nicolson—Ross-Weir TE: 'II‘ansverse Electric TM: Ttansverse Magnetic TRL: Thru-Reflect-Line TRM: 'IIansverse Resonance Method VNA: Vector Network Analizer xiii CHAPTER 1 INTRODUCTION AND BACKGROUND Typically, a material may be described by its bulk electromagnetic constitutive param- eters: electric permittivity e and magnetic permeability p. In general these quantities are complex valued (in a time-harmonic scheme) and can either be scalar or tensor functions. The extraction of these parameters by an indirect measurement such as the amount of electromagnetic energy that they reflect and transmit, is known as electromagnetic material characterization. deitional methods for material characterization often use rectangular metallic waveguides because of the simplicity of the geometry to produce suitable material samples and the nearly universality of these components available in microwave labo- ratories. The mathematical models that describe the behavior of the electromagnetic fields in a rectangular waveguide are also simpler that those for many other geome- tries. One of the most popular methods to characterize materials is the Nicholson-Ross- Wier (N RW) technique [2, 3]. The biggest advantage of this technique is that once the scattering parameters of the material sample are known (from experimentation), the permittivity and permeability for the test sample are then provided in closed form. Certain conditions, however, must be met by the material sample to be char- acterized properly. One of these conditions is that the sample has to fill the entire cross-section of the waveguide. If the material is lossy or highly reflective, a poor transmission coefficient will be obtained, yielding to poor results in the extracted constitutive parameters. Other conditions for the material sample is that it must be linear, homogeneous and isotropic. Also, the geometry of the sample must have parallel front and rear faces, perpendicular to the waveguide walls [7]. An alternative method uses a two-dimensional root-search algorithm in lieu of the NRW approach. This alternative method, for example, was used in [10] for solving the case of a partially filled waveguide. When implementing this inversion method, a word of caution is in order. The material sample being tested has to be perfectly centered in the waveguide cross-section. The objective of this practice is to simplify the mathematical analysis by exploiting the symmetry of the problem. In this thesis a simulation tool is developed to assess the robustness of the inversion algorithm mentioned above when perturbations are present on the experimental setup, specially when the test sample is misplaced inside the waveguide (e. g. laterally shifted from the cross-section center.) since this violates a major assumption during the inversion algorithm. Other perturbations to the experimental setup, that need to be numerically simulated, include the frontal and rear faces of the sample not being parallel to each other or not being perpendicular to the waveguide walls. Chipping of the material sample during its manufacturing process or during handling in the laboratory could also be a contribution for error when extracting its constitutive parameters. It will be shown in the present work how different methods, mainly the longitudinal-section electric (LSE) and the longitudinal-section magnetic (LSM) [11] decompositions or the transverse resonance method (TRM) [22] fail to be practical in modeling this inhomogeneous waveguide problem, especially when the material under test consists of an increasingly number of layers or the material exhibits anisotropic properties. Since the finite element method (FEM) treats each element as a homogeneous entity and the discretization of the computational domain can, for all intents and purposes, conform to any shape, the finite element method is a powerful tool for uncertainty analysis. CHAPTER 2 PRELIMINARY WORK 2.1 The Thru-Reflect-Line Calibration Technique 2.1.1 Introduction Errors resulting from imperfections of a measurement system can be classified as either random or systematic. Systematic errors, like the ones resulting from the use of equipment not being properly calibrated, are the repeatable errors that can be measured and then mathematically removed from the measurement via calibration. The T hm-Reflect-Lz’ne (TRL) calibration technique was first introduced in 1979 by Eugen and Hoer [23]. Consider Figure 2.2 in which a two—port network is formed with connectors A and B and a waveguide segment labeled here as “MUT” (mate- rial under test). Then, the TRL calibration technique effectively removes the error introduced into the measurement by connectors A and B when measuring the S- parameters of the network. Also, at the end of the calibration, the reference planes are at the boundaries of the MUT as shown on Figure 2.2, rather than the VNA ports. The technique requires the measurement of three standards in addition to the “total” measurement, which comprises of the S-parameters of the connectors A and B altogether with the MUT. These standards are: 1) the Thru, corresponding to the S-parameter measurement of a zero-length (or thru) connection between connector A and connector B; 2) the Reflect, consisting of the 311 and $22 measurement of a highly reflective one-port device, I‘; and 3) the Line which is the measurement of the S—parameters of an empty transmission line of known length. 2.1.2 Scattering and Transmission Parameters Figure 2.2 shows a two-port linear network. The network can be completely charac- terized by means of the scattering parameters, which relate inward to outward waves from each port, as shown in Equation (2.1). If the [S] matrix is symmetric it means that the network is reciprocal. Also, for a lossless network [S] is unitary [27]. Vlm 2 P rt V31 - 0 Linear 1210'" Network v3)”, Figure 2.1. A two-port linear network with input and output signals. 'UYu’t = $11 512 vi" (21) Ugat 8'21 322 '05” A physical interpretation to the scattering parameters is to think of them as the reflection and transmission coefficients of the network. These coefficients are complex quantities consisting of a magnitude and a phase and are computed as follows: vout = R = [R] 3197‘ (2.2) in_ v2 ——0 511 = - m ”1 out 7.72. m ”1 -_- T = IT] gift (2.3) in- v2 -—0 321 = with similar expressions for $22 and 312. Another way to characterize this network is by relating the waves, ingoing and outgoing, at port 1 to those at port 2. This relationship is known as the transmission parameters (T-matrix). ,UO’Ut t t .0231 1. ____ 11 12 2 t (2.4) 2121"“ t21 t22 '03“ If multiple networks are connected in series, it is possible to obtain one equivalent transmission matrix for the whole array by multiplying all matrices in the same order as their network position in the array [25]. By using the theory described in [34, pp.539-541], two useful relationships to convert T—parameters to S-parameters and vice versa are obtained: t11 t12 312521-511322 511 1 lTl = = 8—. (2-5) 1521 t22 21 -322 1 511 512 1 t12 t11t22 — t12162 [S] = = g— (2-6) 521 322 22 1 -t21 2.1.3 Derivation of Equations The derivation presented here follows that of Matthews and Song [24]. Let a MUT be connected to a vector network analyzer (VNA) by using the connectors A and B as shown in Figure 2.2. By first measuring the total S-parameters of the network, as seen by the VNA, the effects of each network component can be de-embedded from the total measurement by means of the transmission (T) parameters as follows. The conversion from S- to T-parameters is done with the help of the formula (2.6). [Sm] —" lel [Tml = [Tal ' [Tmutl ' [Tbl (2-7) To To VNA % VNA A / // B 2 —’ Reference Reference Plane Plane “A” “B” Figure 2.2. A two-port network with connectors. where [Tm] ____ tm11 tml2 (28) tm21 tm22 [Ta] 2 tall ta12 (2.9) ta21 ta22 tmutll tmut12 [Tmutl = (2-10) tmut21 tmut22 t(>11 t(>12 [Tb] = (2.11) tb21 tb22 [Tm] is the “measured” transmission matrix as seen by the VNA, [Ta] is the con- nector “A” transmission matrix, [Tmutl is the “Material Under Test” transmission matrix and [Tb] is the connector “B” transmission matrix. The strategy is now to accurately characterize the connectors A and B so that their contribution can be re moved from the measured values of the VNA. Then, the MUT transmission matrix can be expressed as [Tmutl = [Tar-1 - [Tm] - [Tb] ‘1 (2.12) 2.1.3.1 The Thru Measurement The Thru standard consists on joining both of the connectors A and B as shown in Figure 2.3 and then measure its S—parameters, lStl- With the aid of Equation (2.6) T0 VNA ToVNA .— A B __. 1 I Figure 2.3. The Thru standard connection. the resulting measured T-matrix for the Thru standard, [Tt], is written as [St] —* thl thl = [Tal' thhrul ° [Tb] = lTal' [Tbl (213) where thl = tm Q” (2.14) tt21 tr22 and the theoretical T-matrix is given by 1 o _ thhrul = (2.10) 2.1.3.2 The Line Measurement The Line standard measurement, shown on Figure 2.4, consists of placing an empty waveguide section of length 3 between connectors A and B. The S-parameters for this configuration are then measured, obtaining [S ll Using Equation (2.6) the conversion To To VNA VNA <— l A Line B —-> A L. Reference Reference Plane Plane “A” II B” Figure 2.4. The Line standard connection. [51] “r [Tll is performed. The transmission matrix for the Line standard is given by [TI] = [Ta] ' [Tlinel ' [Tb] (2'16) where tm 1*112 [Tl] = (2.17) t121 tz22 and its theoretical T-matrix is given by lle'nel = Using equations (2.13) and (2.16) the following is obtained: lTll 2 [Ta] ' [Tlinel ° lTGl—l ‘ thl Now, defining [Tltl with tltll trm tlt21 tlt22 = [Tl] ' thl-1 tltll tlt12 trt21 t1:22 1 7f" (t111tt22 - t112tt21) 1 lT—tl (tll2tt11 — t1111212) 1 fir] (tz21tt22 - tz22tt21) 1 [7ft] (t122tt11 - t121212) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) where |Tt| is the determinant of matrix thl- Then equation (2.19) can be written as [Tltl [Ta] = [Tal- [Tune] 01' tltll tzt12 tlt21 t11:22 tall tal2 ta21 1Q7122 tau ta12 ta21 ta22 (2.25) (2.26) Carrying out the products of each side tltllta11+tlt12ta21 = tame—ll (2-27) tltllta12+tlt12ta22 = tamell (2.28) t112112111 +tlt22ta21 = ta21‘3—7l (2-29) tlt21ta12+tlt22ta22 = ta22ell (2-30) Taking the ratios of the equations above (the ones that share the same sign on the exponential function) the following set of quadratic equations are obtained: 2 all t 11 _ tltll C“ ) + (tlt22 " tltll) (t: ) “tlt12 — 0 (231) a21 21 2 t 3—1—2)+ t —t (3‘11— ——-)—t = o (2.32) lt21(%'_ 22 +(zt22 an) t a22 lt12 These quadratic equations share the same set of solutions, namely t 11 t 12 1 (L) ,(L)=tlt11 — t11:22 i l/( (tlt22 “ tlt11)2 + 4 (tlt21tlt12) ta21 ta22 2trt21 (2.33) The choice of roots will be made later in §2.1.3.4. Now, from Equation (2.13), solving for [Tb] [Tbl = lTal—1 ' thl (2.34) or alternatively written as t(>11 t(>12 = 1 ta22 —ta.12 . tt11 tt12 (2 35) T . tb21 t1222 I a] "ta21 . tan tt21 tt22 10 Carrying out the product, each element of the connector B transmission matrix is obtained as a function of the thm measurement given in Equation (2.13) and the transmission parameters of connector A. tbll = filttllta22—tt2ltal2) tb12 = fila(tt12ta22_tt22tal2) 1(321 = |_T17| (tt21ta11 - ttllta21) t322 = lflu-l(tt22ta11-tt12ta21) (2-36) Taking the ratios t tb—Ql = t2:21 - t311 ($33) (237) tm tt22 - 13:12 (2:3) t 5,112 = tt12 - tt22 (Egg) (2.33) W tt11 - t1:21 ( 022) the following relationship is found t (£2.11) . (fill) = tm — tm (tam) (2.39) ta22 t321 3,22 — tm (1%?) H 2.1.3.3 The Reflect Measurement The measurement of the Reflect standard, Figure 2.5, consists of placing a reflectome— ter of value F at the end of either connector A or B and then measure its reflection coeflicient as seen from the VNA. Using the signal flow graph theory presented in [27] and with the help of Figure 2.6, the reflection coefficient, as seen by the VNA, can be expressed as a function of the reflection coefficient I‘ and the intrinsic S-pararneters 11 To VNA ' Reflect ‘— A Of B Standard F Figure 2.5. The Reflect standard connection. of the connector as follows: 5 S r s = _a1_2__42_1_ r11 Sall + 1—Sa22I‘ fifflt‘fth 1+(§2§-‘21)r The following ratio is obtained t t11 1 5711—1422 a. ta22 F 1‘57"11t all (2.40) (2.41) Following the same procedure on connector B using the same reflection coefficient F, these expressions are found Sb125321F 1 -- Sbllr t t _ 12217322? 1—{2131‘ 5122 = Sb22+ t 5+ £142.33; ”’22 P 1+Sr22t bll 12 (2.42) (2.43) (2.44) taking their product, the necessary expression is obtained $21 A 7 1‘ lSll S24) 1‘ S12 ll Figure 2.6. Signal flow graph for the Reflect standard. t t (£4.11) . (11222)_(S"11 _ 14%) (1 + S’mZgII) ta22 tbll (5 r22 + £2312.) (1.3711211) 2.1.3.4 Postprocessing of Measured Standards (245) The next step is to assign the correct root to Equation (2.33) This will be accom- plished by using the value of the reflection coefficient I‘ from the Reflect standard. By placing a metallic plate during this measurement, and by recognizing that the ratio ta12 /ta22 is the reflection coeflicient as seen by the VNA, then it is clear that the root with the larger magnitude is assigned to the reflection coefficient, the ratio ta12/ta22- All of the three standards have been measured at this point. All what is left now is to combine the results of these measurements to express the S—parameters of the MUT from Equation (2.12). Combining Equation(2.39) with Equation (2.45) yields to tan (tt11 - t421 ($15)) (5 r11 '- $3,129 (1 + 3122 (2)1,» ta” (Q22 -tt12 (211)) (S 7‘22 + $521) (121163?» 13 M y—s (2.46) The sign of the ratio (2.46) is chosen in such a way that when using the same value of I‘ during the measurement of the Reflect standard, (2.41) and (2.46) yield to the same numerical value. Also from Equation (2.39) t £211 _ ttll ‘tt21 (IS-2%) . (t_a_1__1_)——1 (2.47) t022 — tt22 - ttl? (HI-TI) ta22 and by using the relation (2.5), the parameters Sall and Sa22 are constructed. t San = tal2 (248) t 5a22 = —(—“21) ta22 tail t422 From Equation (2.6) the product Sa12 - Sa21 is found to be t 1 t 12 ta21 512-521: aa-]1—(—a—-)(—— (2.50) a a t422 t422 tall For connector B, equations (2.5) and (2.36) are used to construct its 511 and 322 parameters as 5011 = SIM 022 t t = t“? - :1522' (321%) ° (ta22) (2.51) t122'( 422) 4112'( c122) Sb22 = 3,21 . (Egg) - ttll . (:022) (2.52) 722' ( 422) - til? ( (122) Taking the determinant of [Sb] and using the previous two results _tb21t012 _ tbll + t5210212 (2.53) lel =3511'5522-5512'Sb21= 41,223,322 tb22 tbzztb22 tb11 5012 ' 5521 = a); + 5511 ° 3022 (2-54) It is now necessary to separate the off-diagonal elements of [Sa] and [Sb]. With this objective in mind, two assumptions will be made [24]: 1) The determinants of the measured transmission matrices for the Thru and the Line standards are the same 5 12 5512 IT |= T =(——“ ).(_ (2.55) t III 5421 5521 and 2) the determinants of the connectors A and B transmission matrices are also the same. |Ta| = [Tb] (2.56) With these assumptions the ratio 5412 2 _San = 2412 (2.57) Sa21 5521 St21 is found, decoupling both off-diagonal S-parameters on [Sa]. S Sa12 = \J‘Sa12 'Sa21 ' 3%? (2:58) S 'S S ___ a12 (2.21 (2.59) 021 8012 The same process applies for the off-diagonal S-parameters for the B connector. All the information necessary to construct the elements of Tmut as expressed in equations (2.10) and (2.12) are now available. Using equations (2.6) and (2.5), and 15 using the elements of [Sa] and [Sb] as shown above, the T—parameters for the MUT are obtained as: _ t422 (tm11tb22 - tm12tb21) - 1412 (tm21tb22 - tm22tb21) t t11 - m“ 061115622 - t6121621) (tblltb22 - t6121621) t “2 __ t622 (tm12t611 -tm11tb12) — t41207722215611 —tm21tb12) mu _ (tai1ta22 - t012t021) (tblltb22 — tbthb21) t (:21 _ tau (tm21tb22 - tm22t621) - ta21 (11772111622 - t7721213621) mu _ (116111622 - tal2t021) (tbllt022 '— tbl2tb21) t _ tall (tm22tbll - tm21t012) - 1621 (tm12tbll '- tmlltb12) mut22 — (tallt022 - ta12ta21) (tbllthZ '— tb12t021) Finally, the S-parameters for the MUT are given by: [s t]_ Smutll Smut12 "I'll, ‘— 5mut21 Smutzz With each individual element given by: t t12 Smutll = $3.22 1 Smut12 = tmut22 (tmutlltmut22 —tmut12tmut21) 1 t , t2 Smut22 = ’fnfii mu (2.60) (2.61) (2.62) (2.63) (2.64) (2.65) (2.66) (2.67) (2.68) Or alternatively, expressing the results in terms of only: 1) the scattering param- eters of connector A, [83], 2) the scattering parameters of connector B, [Sb], and 3) the total measured scattering parameters, as seen from the VNA, [Sm ]; the final expressions, as presented in [24], are: 16 Smutll = Smut12 = Smut12 = Smut22 = 5611 - (5611 "9711.22 - ISml) + (Sm11 - 5611) ° lsbl (2.69) Sb11'(Sm22'lSal- $322 - Isml) + (Sm113a22 — ISal) - lel " m12 ' Sa21 'Sb21 (2.70) 5611'(5m22° ISal - 5322 ' ISmI) + (33.118622 —— lSal) - lsbl - m21 ' 3612 ' 5612 (2,71) Sbll ' (51,122 ' [Sal _ Sa22 ' lsml) ‘l' (Sm115a22 — lSal) ' lel 5622- (5622517411 - ISal) + Sm22 - lSal - 3622 ' lsml (2 72) 3,1,.(sm22- ISal — 3.22 - lSml) + (83.115622 - lSal) - [Sb] Where the operation | - | represents the determinant of the argument. 17 2.2 Derivation of the Reflection and Transmission Coefficients for a Fully- Filled Rectangular Waveguide. TElO Mode A y A 0 a Figure 2.7. Rectangular homogeneous source-free waveguide. Consider the metallic waveguide cross-section shown in Figure 2.7. The structure has infinite length in the 21:2 directions. Because of its solenoid nature in a source-free region, the electric flux density vector can be written as V-D=0 —> D=—VXF or E: —% VxF (2.73) and obeys the homogeneous vector Helmholtz wave equation v2 F + k2 F = 0 (2.74) 18 The objective is to reproduce TEZ modes, which means Ez=0 ——> F$=Fy=0and Fz7é0 (2.75) Hence, Equation (2.74) is written as the homogeneous scalar Helmholtz wave equation V2 F; + k2 F: = 0 (2.76) The vector potential F 2 can be expressed as the product of three independent func- tions Fz($, y, z) = f($) 9(9) hlz) (2-77) Substituting (2.77) into (2.76) subject to the boundary conditions Edy = 0) = Ex(y = b) = 0 (2-79) and assuming a wave propagating in the +2 direction, the following is obtained. Fz(ar,y, z) = A0 00309;; :5) 00506;, 33) exp(—jkz z) (2.80) where k2 = w2eu=kg + k5 + kg (2.81) kg = \/k2 — (£7702 — (2:102 , m,n = 0, 1,2,3, . .. (2.82) 19 In addition, A0 is a constant. The fields E and H are then constructed by using E __123 H =__j_.. 62F“ I 6 (9y :1: wen (91:82 E _lii . —_-___j__ 82E? y e a: y wen 83/62 2 E2 —-0 H = ~52; (322- + k2) F; (2.83) Horn the previous relations and the solution provided in Equation (2.80), each com- ponent for the electric and magnetic fields are expressed as: k Ex = A0 :y— cos(kx :r) sin(ky y) exp(—3'kz 2) (2.84) E9 = —A0 E} sin(k$ 2:) cos(ky y) exp(—-3'kz 2) (2.85) E2 = 0 (2.86) kxkz . . H3; = A0 wen sm(kx cc) cos(ky y) exp(—3kz 2) (2.87) k k Hy = A0 if: cos(kx 3:) sin(ky y) exp(—3'kz 2) (2.88) k3, + 715 Hz = —jA0 Woosfix :13) cos(ky y) exp(~—3'kz 2) (2.89) Defining the constant E0 = —Aokg;/e, the TEZ dominant mode is obtained by letting m = 1 and n = 0 on Equations (2.82) and (2.84) to (2.89), yielding k2 = w26u=kg + kg (2.90) 1r 2 4,0 = 332...-(3) (2.91) as the dominant wave number. The modal fields for the mode of interest are given 20 Ex = 0 (2.92) By = E0 sin(k;g 17:) exp(-3'kz 2) (2.93) as = 0 (2.94) Hz; = -E0 53:; sin(kg; :r) exp(—jkz 2) (2.95) Hy = 0 (2.96) Hz = 3E0 sic—‘- cos(kx :r) exp(—3°kz 2) (2.97) A side view of a loaded waveguide is shown in Figure 2.8. A dielectric and magnetic discontinuity exists in Region 2, defined in 21 < 2 < 22, filling the entire cross- section in the my plane. Regions 1 and 3 share the same electromagnetic properties. On Figure 2.8, the coefficients A, B, C and D are the constant amplitudes of the incident and reflected electric fields in each of the waveguide regions; and the functions 1111,2(23, :l:2) are given by: 1111(x,—2) = sin(k$x)-exp(—3k12) (2.98) 111104.74) —_- sin(kg;.r)-exp(jk12) (2.99) 1112($,-z) -_— sin(kg;:r)-exp(—jk22) (2.100) 92mg) -_- sin(kg;:c)-exp(3'k22) (2.101) Suppose that an incident field coming from 2 < 21 hits the material discontinuity at 2 = 21, partially transmitting into Region 2 and reflecting back into Region 1. Then, the total electric and magnetic fields in Region 1 can be expressed as the superposition 21 of their incident and reflected quantities in this way: E3“ = sin(k;,;:r) [A exp(——3'k12) + B exp(3'k12)] 3 (2.102) H3“ = (Ll-Si) sin(kg;a:) [—A exp(—-3'k12) + B exp(3'k12)] x k + j (‘61—?) cos(kg;:r:) [A exp(——jk12) + B exp(3°k12)] 2 I (2.103) Following the same procedure as in Region 1, the total electric and magnetic fields in Region 2 are given by A Y L ,. 1 Z 1 Region 1 Region 2 Region 3 (81’ ”1) (822 ”2) (813 “1) A-TIIX, -z) C'-‘F;(x, -z) ELF—17x, -z) B-‘I’l(x, z) D-‘I’2(x, 2) F °‘Pl(x, z) 0 21 22 E Figure 2.8. Sideview of a fully-filled cross-section waveguide propagating the T1310 mode. > 22 E3013 = sin(k;pa:) [C exp(—3'k22) + D exp(jk22)] 3 (2.104) It Hg“ = (62—55) sin(k$:r) [-C exp(—3'k22) + D exp(3'k22)] x + j (5%) cos(kg;:r) [C exp(—jk22) + D 6111107923” 2 (2.105) Assuming that no reflection is produced from 2 > 22 (i ——> 00), then the constant F shown in Figure 2.8 is set equal to zero, producing in this way, only a transmitted wave. The total electric and magnetic fields in Region 3 are then given by E?” = E sin(kg;:r) exp(-—3’k12)y (2.106) HgOt = E (W) [—k1 sin(kg;:r) x + 3' k1; cos(kg;:r) 2] (2.107) 1 Tangential electric and magnetic field continuity must be maintained across each section interface. This means that at the first waveguide discontinuity, 2 = 21, the two conditions that must be satisfied are: E33110: = 21) = Egan(2 = 21) A exr)(—J'k121) + B expljk121) - C exp(-J'k2z1) + D exp(jk221) (2.108) and Htlan(2 = 21) = Hgan(2 = 21) k1) . (k1) . A — ex —— k 2 — B — ex k 2 = (#1 p( 3 1 1) #1 9(3 1 1) C (%)exp(-jk2Z1) - D (S25) expik2z1) (2.109) 23 Similarly, at the second waveguide discontinuity, = 22 838% = 2.2) = 239% = 2.2) C eXp(-jk2Z2) + D eXPUk2Z2) = E expf—jk132) (2.110) and Hgan(2 = 22) = Haawz = 22) k k k C (—2) exp —-3'k 2 — [3(1) ex 3k 2 = E (4) ex —3k 2 “2 ( 2 2) #2 p( 2 2) #1 P( 1 2) (2.111) From (2.110) and (2.111) two of the unknowns are expressed as a function of the transmitted wave amplitude E. I [12(161 k2) C = —-— —+— -ex (2 k —k )oE 2.112 2 k2 #1 #2 1) 2(2 1) ( ) 1 k1 #2) D = —-1——-— -ex —2 k +k -E 2.113 2( “1 k2 p( 2(1 2)) ( ) from (2.108) and (2.109) A and B are expressed as a function of the previous constants 24 found. A = $913611- + %)-exp(21(k1-—k2)-C + % fl—i - (7%- - 5%) ~exp(21(k1 - k2) - D (2.114) B = 1 — éi—i— (E + 525)] texp(—21(k1+k2))-C + [I — éfi - (2% - 2%)] -exp(21(k2 — k1)) - D (2.115) Using Equations (2.112) and (2.113) in (2.114) the following ratio is found )- cl eXpUk (Z -z )) = 1 2 1 2 (2.116) k 2 k L603(12 (22 — 21)) +1“ (:1; S ((5)) sin(k2 (22 - 21)) IbID‘J d similarly using Equations (2.112) and (2.113) in (2.115), the second ratio is expressed (492 — (3)2 -sin(k2(22 — 21)) - eXp(—J'k1(22 + 21)) (2117) From the previous two ratios found, (2.117) and (2.116), taking their product provides 25 the third and last ratio B__'B E A—LE A k 608092 (Z2 - 21)) H ((5: 3 (£2) Sin(1’€2(Z2-21)) #2 . (2.118) Recognizing that the reflection coefficient is related to B /A and that the transmission coefficient is related to E /A, the 311 parameter evaluated at the z = 21 plane is then expressed as B sin(k$:r) exp(jk121) A sin(k;r:c) exp(—jk121) Slllz=zl : = €- exp(j2k1 21) , '( (£1) (5%) (5%) cos(k2 (22 — 21)) +3 ((92 (3’2 " 21)) l. #1 (2.119) 26 while the 521 parameter evaluated at z = 22 is expressed as E sin(k$:z:) exp( —jk122) 521': = 22 A sin(kg;33) exp(-216131) E . = Z eXP(-Jk1(Z2—Z1)) L - 1 cos(k2 (22 - 21)) +j 5111092 (22 " 21)) (2.1‘20) 27 2.3 The Inversion Algorithm The objective is to solve the system of simultaneous equations given by th Slly(w,6,p) — Sffphu) = 0 th $213,041,611) — ngphu) = 0 (2.121) where 5313/ and 531713 for ij = 11,21 are the theoretical and experimental S- parameters of the material sample, assuming a symmetric S-matrix. The theoret- ical S-parameters are obtained by means of the mode matching technique. Equation (2.121) is solved using the iterative complex two—dimensional Newton’s root search algorithm. A successful solution to (2.121) will provide the permittivity and perme- ability of the material sample in question. In short, the method developed in [4, 5, 6, 10] consists in: 1. Expansion of the electric and magnetic fields into orthogonal modes in each region of the waveguide. 2. Application of boundary conditions to the tangential components of the fields at each material/waveguide interface. 3. Application so symmetry properties of the incident TEIO mode and geometry of the waveguide (e.g. centering the material in the cross-section.) 4. Test the resulting equations with orthogonal modes to obtain a linear system of equations. . . thy thy 5. Solve the linear system to obtain 511 (w, 6,#) and 521 (w, 6,11). 6. Solve the system of equations (2.121) iteratively and extract the permittivity e and permeability p. 28 The algorithm is validated in this work by feeding the theoretical values for the S- parameters developed in (2.119) and (2.120) for two different non-magnetic materials. The first case is acrylic (e = 2.5 + 30), Figure 2.9, with a sample length of 7.5mm and the second is alumina (e = 9.0, tand = 0.003), Figure 2.10, with a sample length of 3mm. Both cases fill the entire waveguide cross—section (a = 22.86mm, b = 10.16mm) and are tested in the X-Band (8-12 GHz). The agreement throughout the whole band is consistent with the values of permittivity and permeability used to generate the theoretical S-parameters for both materials. 29 Extracted Relatlve Permlttlvlty for Acryllc 3 2.5 2 - . — epsilon, real 15 _ - - - epsilon, imaginary 1 1 - . 0.5 - . OI-I-I-IIII ..... I... IIIIIIIIII III-IUI1 ’0'58 9 1o 11 12 Frequency (GHz) (a) Relative permittivity. 1 2 Extracted Relative Permeablllty for Acryllc 1 0.8 ~ 0.5 . — mu, real 1 - - - mu, imaginary 0.4 r 0.2) 4 CLO-IIIIICIIIIIIC-.-.----------------. '0'28 9 1o 11 12 Frequency (GHz) (b) Relative permeability. Figure 2.9. Extracted relative permittivity and permeability for an acrylic sample using the algorithm in [10] 30 Extracted Relative Permittivity for Alumina 10 - 1 3 . 6 ’ -- epsilon, real i - - - epsilon, imaginary 4 L 2 b 1 0III-I..-I-----.-----III-I-I-I-I-D-I-Id 8 9 10 11 12 Frequency (GHz) (3) Relative permittivity. 1 2 Extracted Relative Permeability for Alumina 1 0.8 - 1 0.6 - 1 — mu, real 0.4 P - - - mu, Imagnary ‘ 0.2 1 . 0P --------------------------------- --~ "0'28 6 10 1'1 12 Frequency (GI-12) (b) Relative permeability. Figure 2.10. Extracted relative permittivity and permeability for an alumina sample using the algorithm in [10] 31 2.4 Hybrid Modes and the 'h‘ansverse Resonance Method 2.4.1 Hybrid Modes In solving for the eigenfunctions and eigenvalues for any of the waveguide struc- tures shown in Figure 2.12, Figure 2.13 and Figure 2.14 by trying to decompose the electric and magnetic fields in TE3 or TMZ, as it was done for the empty waveg- uide in §2.2, will not lead to the correct imposition of boundary conditions in the air / material interface [21]. Instead, a decomposition of fields known as hybrid modes will be used [11, 21, 22]. The term hybrid arises from the fact that each decompo- sition that is a solution for this problem is a combination of TE2 and TM2 modes. Hybrid is used interchangeably with the terms LSE or LSM (longitudinal section electric/magnetic) decomposition. Modes LSE“: ory mean TE“: cry and similarly LSM“; 0W mean TM“: ory. Structures exhibiting a material discontinuity in the x direction are solved in such a way that either no component of the electric field exists in this direction, meaning it is LSE‘”; or that the magnetic field lacks the a: compo- nent, meaning it is LSM'T. The same principle is applied to material discontinuities in y. Take for instance the geometry of the waveguide shown in Figure 2.11, the same geometry used by the inversion algorithm [10] when the material sample is centered. Assuming the interior of the structure is free of sources V ' D = 0 ——+ D = —jwueV X IIh (2.122) the electric and magnetic fields are written as a function of the magnetic Hertzian potential as E = -—jw,qul'Ih (2.123) H = VxVxl‘Ih (2.124) 32 A -gx 0 a Figure 2.11. Vertically loaded waveguide to illustrate the LSE and LSM mode de- composition. The Hertzian potential obeys the wave equation V2 II}, + k2 11,, = 0 (2.125) The discontinuity of the material exists in the a: direction. In order to obtain LSE“: modes, it is necessary to define the magnetic hertzian potential as traveling waves in the +2 direction with only one component, the m-component, as follows: H}, = May) exp(-jkz 2))? (2-126) The wave equation (2.125) then becomes v9r my) + (k2 — k?) may) = 0 (2.127) where 2 2 V21. = 19— 52— (2.128) 8x2 611,12 The tangential field components E; and Hy are constructed using . , a E: = JWH exp(—szz) 5:694:31?!) (2-129) . ()2 Hy = exp(—szzlaan—y¢($iy) (2130) Applying the separation of variables technique as it was done in §2.2, solutions that satisfy the boundary conditions at the waveguide walls are obtained for (0(2, y). (p1(:r, y) = A1 sin(kxl as) cos(ky y), 0 S :1: S :51 (0201:, y) = [A2 sin(kx2 :c) + 82 cos(k;1;2 23)] cos(ky y), ‘51 g a: 3 2:2 l‘ ‘ Q (a) Along the side wall. 1A; 7 ‘, (b) With off-set. Figure 2.12. Vertically loaded waveguide. Q“ 0“ ---->~<> ---->1<> (8.) Along the bottom wall. nod S (b) With off-set. Figure 2.13. Horizontally loaded waveguide. X> .. ‘> Cl (a) With rectangular cross-section. --,.<, on .> a (b) With arbitrary cross—section. Figure 2.14. Suspended sample rod loaded waveguide. 38 X> X> 2.4.2 The Transverse Resonance Method The cross-sections depicted on Figure 2.12, Figure 2.13 and Figure 2.14 can also be modeled as an equivalent transmission line circuit, formed with different sections, each having different impedance values. In doing so, a transcendental equation in the form of (2.137) is obtained. This approach is referred to as the transverse resonance method (TRM) [22]. For obtaining the propagation constants for the electric and magnetic fields, the TRM approach is a much more straightforward and simple solution when compared to the hybrid mode decomposition method from §2.4.1. But when it comes to recon- struct the distribution of the fields, the TRM lacks in the ability to provide with the information to do so. For this reason this approach will not be further discussed in this work. 2.4.3 Conclusion If any of the geometries shown in Figure 2.12, Figure 2.13 or Figure 2.14 is changed by increasing the number of material discontinuities they possess (e.g. become layered), then the complexity of a solution also increases, since more boundary conditions need to be satisfied at each interface. A hybrid decomposition solution for this problem becomes impractical, specially if the materials in question are anisotropic in nature. Furthermore, if a material sample, like the one depicted in Figure 2.14 is considered, a solution in closed form is not possible, even if the material geometry is rectangular. For this case a variational/perturbational approach is required [11]. With this in mind, it is convenient to develop an alternate tool to assess the robustness of the inversion algorithm described in §1 and §2.3. As a requirement the simulation tool has to be able to handle eigenproblems with material inhomogeneities and dispersive materials having complex geometries. The finite element method (FEM) is particularly suitable for modeling three- 39 dimensional bodies with complex geometry features. It can also incorporate materials of any composition without the need to reformulate the problem [1, 13, 16]. The FEM formulation for three-dimensional inhomogeneous waveguides is the subject of the next chapter. 40 CHAPTER 3 THE FINITE ELEMENT METHOD FORMULATION FOR INHOMOGENEOUS WAVEGUIDES 3. 1 Formulation 3.1.1 Domain Discretization Because of the versatility to conform to many shapes, the element chosen to discretize the waveguide space is a tetrahedron, shown in Figure 3.1. As a first step, the Figure 3.1. Tetrahedron element for waveguide mesh discretization. cross-section of the waveguide is drawn in two dimensions, with the material sample aligned. Then, a mesh consisting of triangles is generated. The number of triangles increases with the electrical density of the material sample. The two-dimensional mesh is extruded into depth with a finite number of layers, producing triangular- prism elements which in turn are partitioned into tetrahedra. In this way the three- dimensional waveguide is generated. 41 71 T—j— ‘ ,T . T ‘v v— “ W I v _ . '1 ‘ .1 l ooooo l 1 «'91 . i 7‘ ‘ 1 ‘ r 0- 1) 4" , l _o _A a— r _n t o I cccccc ] 4 s . t ' “ [ ,K ‘, 1 ~- . 4- r - . , ' 1 ' - - ~ . x . '. fl: 9 v“ a .’ I a 3 r r o . . , t . ‘ s ‘11-" 't ‘s. 1 r—F“ ‘ [ ‘ -1 ‘ ( ‘ V[\ . 1 , . ,V . 1 ‘\ ‘ H ¥~¢+ ‘J‘v- Ar :T 7 fi [ \l ‘ " ;’ 1 1 _ ' - . 1 ' . . ‘ ‘- 1L/ ' . 1 ' 1 [{“rr‘v..¢.o.irwr.t I 14-. 4‘ 3 '4. =’ I l - , _ r 1 g I '_r I _ \ 1 \ \ r [ - 1 .1 ‘ r - r'"_‘"— *‘—$—*- IIIr-Jarovrs > o I t» 3 «3+ ““"*' ‘—1 1. * ' , ‘ 1 ~ _ ' 1 _ ’ i - Q 5 a f OJ 0 o I r i .' ( c f g u '4 g [ I .' . V. . , .‘ . \‘ , l l ' . b 3333 o o - of" . o ¢ 4 r 1 x u .‘ ' i- + _e— k 4 + I If u f . "_f ) o 1.» .f A a ,,,,, “ i (*3; , 3 I“ __ d - . 1 ‘ ‘ ’ . ' 1 I . : 1 1 I ~ .4]. r.‘ _..+_>,( . .‘ , , _. r P 4 A ‘ t 1. I ‘1 , . 1 , . ~ . ‘ 1 . . y , 1 l 4 1 ‘1 -~ ‘_'—. k ’7' e 4 _o v , : (~f it I A? I v [ .\.. 1 J. 7»»-,- 9— A \’- .7-’ ' ~ . '\_ ‘. ~ L~O I 4 a . kr O o a 4 ; ' b“ l». ' _ . . ‘ . , ,7, F > . ' "T7 \. \r t '5 ttttt ' of, ooooooooooo + o 4 q . ' /.- ‘K T .- q +. .‘ a 4 A aaaaa I r o a D I w 1 4 . .\1* ‘Q l‘ . .’ ‘ . [ L MAJ; x x 4; A V x' 1 ' ~ I ' ' ' . ‘ , ( fir ‘ ' '+ v ', . ‘. ’ - A ' . l I _-T." j -;_ ’1‘" . .3. a (,1. 0r 1 ooooo I .. .1 .++ LU , .. 1. ' 1 . 1 . ‘ . 11‘ .1 [ ,i 2: .- _ 1 _ [E r— ? ’1 1 ,” ‘ \., _x I l 1 . ’ ‘ A 1 ' | ‘ 1 L A ‘ 4* M A A M1. 1 A ‘l 1 . \ ‘ . .r vfi v V—Y. r *+* TI ' ' f1 . ~ 1 ‘.' 1,’ 1 . | ' '- ‘ L '3: I + (.7‘ l 1 * r a | 1 0 c v c r a 1 i ' ‘11 d, x ‘— ‘ ‘. . A4 f V 4 A A A ¢ 1 h, 'x f ] ‘1 ‘ Y1 1'- #~..]/ ,1 K 1 ‘1‘ _fi- h l a 4 I - 5 § 0 I o 1 91: .1 i o q .' ., ‘ , \ '. Kin #EL fix 7 _ . ‘.~+1‘ 5 g} 1‘ 1 [ 1 \l 4‘ 1 _{L'l _l A; _L 1.4__' ‘ ’ 1‘ UJL .1 \Ju’ l u .1 Figure 3.2. Mesh for the waveguide cross-section with material sample inside. 3.1.2 Interpolation Basis Functions Within each tetrahedron, the unknown field can be interpolated from each node value by using the the first order polynomial pe(3:,y,z) =06 +bex+cey+d€z (3.1) The value of the field at each vertex (node) of the tetrahedron is therefore pi($1, yl, 21) = a8 + 061131 '1' 081,11 + dezl (3.2) [05(332, yg, 7.2) = ae + berg + ceyg + (162.2 (3.3) p§(x3, ()3, 23) = a8 + beat3 + ceyg + (1623 (3.4) pi(;r4, y4, z4) = a6 + bex4 + cey4 + (1624 (3.5) 42 Each coefficient in Equations (3.2)-(3.5) can be expressed as a function of the coor- dinate values for each vertex. This coefficients are then given by be where I - Ve PS $5 yzi e 23 Pi xi vi e 24 l = __6Ve (aipi + 0393 + 03103 + 03851) (3-6) = 6V—5 (bfipfl3 + bgpg + b§p§ + bipi) (3.7) 1 = —6Ve (Cipi + 0503 + 6383 + 0303) (3-8) 1 = ave— (dipi + 4292 + dgpg + dipi) (3'9) )= % -abs(Ae + B6 + Ce + DB) (3.10) 43 Expanding each determinant on Equations (3.6) through (3.10) and grouping them, each element constant is then obtained in terms of its nodal coordinates (3535 - 353535 + (3535 - 3535>$5 + (3535 - 353535 (3535-3535)$1+(3535 y53i>$5+<353i yi$5>$5 (9535-3535>$5+$ $2 (3535-3153535 (3.535- 3535>$1 + (3535 y535>$2 + (3535 - $53935 (9:5 - 95)3§ + (95 - 95):;5 + (:95 - y§)35 (95 - y§)35 + (95 - 95)35 + (95 -- 95)35 (95 — 95)35 + (95 - 95% + ('95 — 9335 (293" 92)21 + (.711 ($2 — $§)z§ + ($3 ($3 — $5025 + ($2 ($3 — $S)zf + ($513 ($5 —— $92513 + ($3 - $5)z§ + ($513 — $5) <95 -— 35915 + <35 ($2 -— $5”? + ($5 -— $i)y§ + ($3 — $513) ($2 —- $4)y1+ ($4 — $50313 + ($3 - $5)y - $3925 + ($5 -— $§)z — $9.25 + ($5 — $§)z — $2)226 + ($3 — $5) “ (Dali/3 + (332 " $3M y3)ZQ + (112'- yflzg <3 N N Am Em was 1am rum Ace £6!) 8 ($5 - $935 + ($5 - $3315 + ($5 - $595 [($35 - 35):!5 + ($5 [(935 - $§)y5 + ($5 [(15 - I5)y5 + ($5 [($35 - 335% + ($5 -— $§)y3 + ($2 — $5) 3,14]: - $5515 + ($5 - $5) 131513 e 1 e "2 - $5)y§ + ($5 - $§)y5l$’§ — ISWS +(1‘S - $5)y§l‘35 (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) Equation (3.1) can now be rewritten as $803, y, 3) = 2 <5 ($1 11, 3)p5 (3-31) The function (f is a first-order polynomial given by 1 . <\llt}dS (3.58) {13 } = ffsl (—2ij10) {fith}T-{Eincxfi}dS (3.59) [13] = ff/V6{VXN8}T-[z7r]-{VxNe}dV (3.60) [1%] = [M —k%{N8}T.[zr1.{Ne}dv (3.61) Noting that the formal derivative of the functionals (3.54) and (3.55) is given by [1] 6 6F(E)= a {E ___} F(E) (3.62) and using the partial differentiation rules for matrices and vectors as described in [16] a We} 635} ({$}T‘[Al'{$}) = 2[Al-{x} (3.64) (C {3}) = C (3.63) Q3 A then the first variation of the functionals (3.54) and (3.55) after setting them equal to zero, 6 F (E) = O, as the Ritz procedure requires; and enforcing the boundary con- ditions (3.42), (3.43) and (3.45), the following linear system of equations is obtained [1? + 1313-] {E} = {12 } (3.65) [1% +1?3 + 15] . {E} = {13} (3.66) where the system (3.65) solves the isotropic problem depicted in Figure 3.5 while 55 (3.66) solves the anisotropic version of the same geometry. 3.1.3.1 Solution of Integrals The solution of the integrals (3.56 — 3.61) is performed within each element, assuming the homogeneity of the tetrahedra. Using the result from (3.35) and taking the dot product in the integrand of (3.56) (Vfo)o(VxN;) 4Q!- -— J.¢:._.¢.€.€._€.€: “ ((6V8)4) (Czldfg dflClzXCJldD dchJ2) +(d‘3b‘? -—b¢ )(3336. —66 ) Tij Z1 22 21 2'2 31 12 1'1 3’2 6 6.2 _. <2 ¢ 6. e ._ e e + (611622 c21b22)(b]1c]2 c116]2 )) = constant, i,j = 1,2,3, . . . ,6 (3.67) Since the integrand T,- j is always a constant, the value of 3.56 reduces to the volume of the tetrahedron multiplied by a set of constants as follows: 56 45.: w... 8.. fl .((c6d6 _d¢c¢ )(ce. d6 _d606.) 21 22 2122 3132 3132 6 _ 6 . . 6 ._ 6 . +(dflbi2 6,1,2)(d§1bj2 b11d§2l 6 6 ._ 6 6 6 6 .. 6. 6 + (b21c22 czlbz2)(bjlcj2 CJle2 ), z',j=1,2,3,...,6 (3.68) As for the second integrand, it involves the dot product of the vector basis functions, Xij($»y42)- may. z) = (466) (NS) - (N3) = (4636") ° ((53?) (<51 Vszg2 _ qungl) ' ( (3-73) fl +3338 43$< m '(‘633 — 4532 — ”33 + ¢i2l (3-74) \1 bag 8 01% < a: (€532 — $34 — i2 ‘6 2¢i4l (3-75) «I #538 m m< C) CD ' ((634 - (1’33 — ¢i4 + ¢i3l (376) K) (0 O < (D “($33 — ¢i3 + ¢irl (377) (2334 _ 2633 — 2¢i4 + 6%) (3.78) r——1 p—c NCO l__J [\D 03 ll '(633 — 9633 " ¢i3 + 2¢i2l (3'79) (433 — 634 — 6‘62 + 634) (3.80) '(¢i3 — (633 ‘ ”in: + €534) (3'81) ‘ (4534 — ¢i4 + ¢i1l (3-82) 720ve ' ($34 — $34 “ (35:33 + Q5332) (3.83) 58 [e 38 [lg ]35 “ (“kg”) ' (72%39) ' ($34 " (1524 - 2¢€2 + (15%;) (3.84) 38 (e [13 l36 = (463%) - (72:33.3) - (6534 - 6534 — 65‘13'4 + 265333) (3.85) (8 £8 [13144 = (4434.) . (3566(2) - (933 — 633 + 632) (3.89 [e 38 [13 Ms = (49357“) ‘ (7213342) 10653 “‘ 2634 “ €632 + €634) (3-87) Ie _ -k2 . £66126 . e _ e _2e e 388 l 2l46 — ( 067") 720Ve (@534 33 ¢24 + ¢23l (- l (8 (36 [13155 = (4.3...) ‘ (365634) - (432 - «a + 63..) (3.89) 1e _ __k2 . [Egg . e _ 2 e __ e e 390 [ 2l56 — ( 067') 720ve ($24 $23 $44 + $34) ( 0 ) Ie —k2 . €636 . e _ e e 391 l 2l66 — ( 051') 360W W44 ¢34 + ¢33) (- l The function (3% is defined as 6% = 655; + 0:30; + dfdg (3.92) And, of course, [ lg lij = [ 13 ]jz" The integrand of [ lg ] and [12 ] require the tan- gential component of the surface basis function on surfaces 81 and SQ. This tangential component translates directly into the face of each tetrahedron that constitutes either surface. For simplicity, the author chooses to treat each tetrahedra face lying on 81 and 82 as two-dimensional triangular basis functions, whose outward pointing normal fl is always directed to the outside of the mesh (e.g. —2 for 31 and +2 for 82). Following an analogous procedures as the one for the three-dimensional tetrahe- dron, the two-dimensional triangular element is constructed as shown on Figure 3.6 and with the edge definition as given on Table 3.3. 59 Figure 3.6. 2-D Element for \I't. Edgez' Nodez’l Node 2'2 1 1 2 2 2 3 3 3 1 Table 3.3. Definition for each edge on a triangular element with its constitutive nodes. 60 Any quantity, £t(:z:,y), inside the triangle shown in Figure 3.6 can be readily approximated by means of the first order polynomial €t(x. y) = 6t + ftcv + Qty (393) evaluating this polynomial at each node, three different values are obtained for the interpolation function 615631,“) = at + ftatl + gtyl (3.94) $062.32) = J + ft-Tz + 9ty2 (3.95) f§(x3,y3) = at + ftxg + Qty/3 (3.96) then, the value of fit at any point (x, y) inside the triangular element, can be expressed as a linear superposition of its own value at each vertex as 3 Ef—(x. y) -—-— Z wits. y)€f (3.97) i=1 where the function ([2,? is the two-dimensional triangular basis function in area coor- dinates: 1 . 636,31) = '27:? (e; + ffa: + gzt-y) z = 1,2,3 (3.98) To illustrate the meaning of area coordinates, suppose that the point p inside the triangular element depicted in Figure 3.6 has the coordinates (mpg/1;). Then, each 61 t basis function 0’22- is expressed as: Area of triangle p23 _ A 1223 = Area of triangle 123 — A 123 (3.99) Area of triangle p13 _ A p13 — Area of triangle 123 — A 123 (3.100) Area of triangle p12 _ A 1912 1 466....) = 52.666669.) 1 65(6969) = _2At (63 + 1‘51”}? + 953/19) '— , 1 6:369 yp) = {It (83 + f 31729 + 93910) From which it follows naturally t 916.9) + 9504, y) + 9304. y) = 1 = Area of triangle 123 — A 123 (3.101) (3.102) It is also important to note that 212:. = 1 at node 2' and (D: = 0 at the edge opposite to node 2'. Each constant for I/Jzt- (2:, y) in Equation (3.98) is found as a function of the triangular element vertex coordinates, giving 61 = 6563 -w§y§ 6% — 6361-6163 91 = 63—65 95 = 61-63 (3.103) (3.104) (3.105) (3.106) (3.107) (3.103) (3.109) (3.110) (3.111) Function vertex 1 vertex 2 vertex 3 (pg p 2 3 025 p 1 3 495, p 1 2 Table 3.4. Definition for each area-function 0’): within each triangular element. The area of the triangular element is also found to be 1 At = -2~ ffgffg — fégi (3-112) Similarly as it was done in §3.1.2 for the three-dimensional basis function 3.33, the Whitney vector finite element for a two-dimensional case is generated by operating on the function 11%(22, y) with the wronskian operator as follows 11:: = (H $166 yW’fI/J 2(6 y)- w-2(6.y)\7’w,1($ 9)] = 3§[¢§1V¢‘26‘2V¢,] = 5% [(th 2%1 _ 261%)" +(gl'l2wf1 _ 911%)5'] (MB) The basis function ‘11: shares the same prOperties as its three-dimensional counter- part, N9, in the sense that it is divergence-free (see Equation(3.34)) and that it offers a finite, constant and difierent-from—zero curl (see Equation(3.35)). With these prop- erties, the electric field E133}, 212 on each triangle constituting the surfaces SI or SQ, can also be written as the expansion of the two dimensional basis function ‘11: as 63 follows 3 Et(x,y,212)=2:111!§12) (3.114) _— _— The integrands of [19] and {IE } call for the tangential component of the basis function ‘11: on either surface. With this objective in mind, the new basis function, A: is now defined as ,t . ,I,t Z -— 11X 2' 3' t t t t t,t t .t . 2Azt [(962131 — 9211/52)" ’2'(f 2%1 — f2'1"”i2)yl (3'115) The integral (3.58) can now be re-written as as e _ 3 tT. t [I3] — ”31092 (116210) {A } {A } <18 (3.116) or in its discretized version (”32:12) [(gi2912+itf21t'f2")’lt'1¢t1 2A (9:2:51H2ff1)d%12(91t1932+_fit1th2'pt)‘/’2'2u:§1 +(gf19;1+f 1131M 21¢ 122] dS’ ”"3. = 1’2’3 (3.117) Using the formula [16] [5 (ml)? (wg)s(’1’3)t ‘15 = (3+zirisitimA (3.113) 64 all the components of [IS] are found: [I u [I [I [I e 3 e 3 e 3 e 3 e 3 e 3 ]11 " ]12 l13 _ ]22 ‘ l23 _ 133 “ ) ‘ [féffi + 9595 - fffé + gigfi + fifi+fifil [$3+£é-$$+éé 2(ftft+tt) tt tt 13 9193 +f1f2+9192 [fifi + 959i - 2 (féfé + 959:?) fifi+fifi+fifi+fifiJ [fifi+fifi-$fi+dfi fi$+fifil [fifi+éd-$$+éfi 2(ftft+tt) tt tt 21 9291 +f2f3+9293 [fid+dfi-fifi+dfi fifi+fifil 65 (3.119) (3.120) (3.121) (3.122) (3.123) (3.124) The basis function ‘1': is also used for the computation of IS; {1311 = [[1—211110111x111t1T {Eincxnms = [[3: (411310) {At}T-{szin°}ds Z/[gl 0“) [(821952 E2931) (1 11.) 111(3) 3 = a 2 [pi-I41 + qZ-1142] (3.125) i=1 and each of its parameters are defined as . 2k E , a = 1((Tzl3—)§9)-exp(—sz1ozn (3.126) p1 = (61:15:52: ef:gfl)-:i, i=1.2,3 (3.127) 141 = f]: sin( )dxdy, 3:51 (3.129) I42 = [[3 xsin( )dxd'y, 5:31 (3.130) The integrals I41 and 142 are evaluated numerically using the gaussian quadrature rule for triangles with four sampling points. The values for each parameter in this integration rule are given in Table 3.5. 4 . . . ffs F(x,y) dandy m At :WZ-F ((39315) (3.131) i=1 The approximation for I41 is written as 66 1 (i1 C; C}; Wi 1 1 1 1 _m 3 3 3 48 2 3 1 1 m 5 5 5 4'8 3 1 3 1 % 5 5 5 4'8 4 1 1 3 % 5 5 5 48 Table 3.5. Parameters for the four-point triangular surface Gaussian integration rule. In and for I42 as M2:- 27 48 —-21 - At - sin (2 01) 48 - At - 01 - Sin (£- 01) 67 + + + 25 25 48 25 m' 25 25 48 25 m' -sin( -sin( -sin( an» planlanla ~ 62 - sin (:1: 02) -63 - sin (Z- 03) - 04 ' Sin (2 04) 3mm both integrals having the arguments 61 = $331 + £172 + $173 (3.134) 62 = 2:171 + $232 + $173 (3.135) 03 = £231 + 2222 + £233 (3.136) 94 = $271 + gzg + §$3 (3.137) For the anisotropic case, [1?) ] and [18 ], the tensors for [Dr ] and [Er ] are decomposed and the product of the integrands is carried out. For the first anisotropic integral, [1%], the integrand is a constant. Like it was performed with [1?], the integrand is just multiplied by the volume of the tetrahedron element. [ISL-r f/Ve{Vfo}T-[firl-{VxN§}dV 4vee,ej e e e = (W) [(C Ci1df2 _CfiCi2)V$$(CJ1d§2 -CJ'1 CJ'2) +(CZC1CE2—Ce 2'1 C12 ) ny (C5182 11311132) +(C2'1CC3 z'2 C5192 ) ”552 ( C§1C§2 _CJ1C52 ) +le1Cze2 21:2 ) ”593(C51d32 d§1CJ2l +“151%” 2'1 2'2 ”ill/(C1193; C§1d§2l “‘51sz 112-11152 ) ”1'32 ( C§1C§2 —CJ1C§2) +(Cf1 C52 *Ci1bfz)”§$ (“31052 dglCJ'z) (C5142 — CZ'1CZC2 ) ”Cy “131% — C§1C§2 ) + (C5143 C11Cf2 ) 1132 ( C§1C§2 CJ1C;2 )l (3.138) For the integral [1%], four terms involving the product of the basis functions are 68 , 6 obtained. Just like it was done with [IS ], the integrals given by fve ((8 <21” C371) dV for i,j = 1,2,. . . ,6 and m,n = 1,2 are computed using the formula (3.70). [1512-]: ]f/V —k3{N§’}T [e1{N§}dv _ 8 8 . . _ _(figj) {/Ve C11 le dV [biz (633be + €$yC32 + 6:1;sz2 +C'i2 (éyxbj 32 + €yij2 + €yzdj2 +di 22 (fszjz + Ezijz + Gzzdj2 ) ) ) _ [VG gfl €162 dV [bl-2 (emxbjl + 63:3,ch + ezzdjl) +cz-2 (nybjl + eyycjl + eyzdjl) +0122 (eszjl + ezycjl + ezzdjl) _ Leg C262 C331 dV[bz-1 (graph)? + €33ij2 + 633; 61-1-2) +611 (eyxbj J2 + €yij2 + €ygdj2) ) ) ) ) +6121 (szbjz + 6,3ij + fzzd' 2 + Les-e (2'2 Cj2 dV[bz‘1(€x$bj1 'l' 6:1;ij1 + 6132de +Ci1 (nybjl + Eyijl ‘1'" 63/2de (3.139) where the constant 19:?- is defined as: kg (,3 - 19??- = — ——l- (3.140 U ( (6V9 )2 ) ) 69 3.1.4 Solution of the System of Equations The linear system (3.65) above is solved by using the biconjugate gradient method (BiCG) for antisymetric systems [33]. The BiCG method solves the system [A] {x} = {y}. The notation A“ stands for the adjoint of matrix A, the superscript at for the conjugation of a complex quantity and the inner product (1‘, y) is defined as :ry* . $1 7‘1 1111 For 71 an $n+1 Tn+1 9114-1 pn+1 wn+1 0n lhn+1“ llyll until T 0 P1 - y (11 = 11* 1, ,N DO (Tn a (In) (Apr; 1 wn) 1‘71 + an Pn Tn — an A PR 971 — (1:140 Min 7'n+1+c71 1”” qn+1 + c; “’71 (Tn+l ’ qn+1> Tn . (In tol erance 70 3.1.5 Numerical S-Parameter Computation The TRL calibration technique described in detail in §2.1 is applied for the compu- tation of the S-parameters of the material under test (MUT). Refering to Figure 3.5, the region of the waveguide defined between the surface 31 at z = 21 and the frontal face of the MUT would be referred to as, what is called in §2.1, Connector A; While for the region delimited by surface 82 at z = 22, and the rear face of the MUT as Connector B. Using the definition given in §2.1.2 Figure 2.2, Equation (2.2) and using the prop- erty of orthogonality of the waveguide modes, the scattering parameter 511 is derived as vout(z = z ) S11 |z=z1 = lug—TL v1 (2 — 21) ”SI [ (E($,y, 21) - Einc(:z:,y, 21) ) - (sin (% 2:) y) ] dxdy 7 ([51 (Einc(:c,y, 21)) . (sin (% :5) y) dxdy 2 exp (ijIO 21) ff . 7T .. _ ( aon 51 E($,y,21) (Sln (ax) y) dxdy 1 and with Equation (2.3), the scattering parameter 5'21 is derived as S _ ”gut“. = 22) 21 lz=22 - m [[52 (E(x.y.z2) ) . (sin (gm) y) dxdy H31 (EinC($,3/a 31)) - (sin (217: 11:) y) dxdy 2exp(jk.,1021))/'/ , 7r .. = " E , ,z - — d d ( ,, (... 2> (mm) .. (3.142) 71 Using the same principle as the two previous cases, 5'12 is expressed as 812 lz==22 and $22 as S22 lz=22 (z = 21) v§"(z = 22) 2 exp(-jkzlo 22) (1on ,6th = 22) 1131(2: 2 z2) 2 exp(—jkzlo 22) (1on )//31 E(ar.y.zl).(sin (ii-CC) )//.S'2 E($,y,z2)- (sin (3 as) ) dxdy (3.143) )dxdy - 1 (3.144) Using the expansion of the electric field on the surfaces 81 and 82 given by Equation (3.114) and substituting it in all four expressions above, the discrete versions of the S-parameters for the MUT are given by [Pi 3 CM“ 1» H H [m vs II t—i s s M“ Mo. 2.? s H H .8 l’l“ '5 s ll H '141+ (14 '141+qz' '°141+q2' "141+Qz' '142l — 1 '142l '142l '142l - 1 (3.145) (3.146) (3.147) (3.148) with the integrals I41 and 142 as defined in §3.1.3.1 and evaluated at the surface 72 S = 81 for 811 and 812; and S = SQ for 821 and 322. a1 = (zexPUkflO 21)) (3.149) (1on (2At)2 2 k 02 = ( exp( 3 21032)) (3.150) taon —(:tAtt qi (ff 19,-2( ff 2:951)” i=1,2,3 141 = [[5 sin( )dxdy, 3:513 142 = [[5 :csin( )dxdy, 5:512 73 3.2 Validation The FEM code developed in section 3.1 is validated on X—band (8-10GHz) by com- paring its results with the theoretical S-parameters formulated in section 2.2 and the mode-matching technique developed in [10]. The two compared cases are a low— and high-contrast materials. The first example, Figure 3.7 , consists of acrylic (e = 2.5, u = 1) as a material sample with length f = 5mm which fills the waveguide cores-section entirely. The dimensions of the waveguide are a = 22.86 mm, b = 10.16 mm [27]. In the second example, Figure 3.8, the acrylic sample partially fills the waveguide cross-section by 50% (d/ a = 0.5) and has a length 2 = 7.5 mm. Finally, for the low-contrast material case, the acrylic—to—waveguide width is d/ a = 0.25 with a sample length of l = 7.5 mm; the results are depicted in Figure 3.9. For the high-contrast material validation, a sample of alumina (e = 9 — 30.0027) with length I = 5mm and d/a = 1 is simulated. Figure 3.10 shows the magnitude and phase of the reflection and transmission coefficients. 74 S-Paramotors, Magnitude G v _1 . . 821's ____ ________ ..2/ —— ‘ -3t --- S11Theory , m --- 821 Theory 1: — S11 FEM ‘4 — $21 FEM -5; -6- -78 9 1o 11 12 Frequency (GI-l2) (a) Reflection and transmission coefficients, magnitude S-Parametors, Phase 200 ' 1 ' 1 150,, 811': d 100' 321 Theory ‘ .. --- S11 Theory 8 50- -- S11FEM - §a — $21 FEM 0 0b ‘50' 821’: -1mb fl -1508 9 1o 11 12 Frequency (GHz) (b) Reflection and transmission coefficients, phase Figure 3.7. Comparison between the FEM and theoretical closed-form solution in a fullyJilled d/a = 1 rectangular waveguide. Acrylic (e = 2.5, p = 1), sample length 3 = 5 mm. 75 S-Parameters, Magnitude t 0.8 / $21 ’8 d 0.6 0.4- — 821. FEM - - - 821, Mode Matching 0.21 — S11. FEM - - - S11 , Mode Matching 08 9 1o 11 12 Frequency (GHz) (a) Reflection and transmission coefficients, magnitude S—Parametere, Phase 200 . . - 150 . 811’s «1 1001 i a 50 . -— S11, FEM 1 3 - - - S11. Mode Matching a O . — S12, FEM 1 8 - - - S12, Mode Matching -50 . ‘ 400' 821’s ‘ -150 . '2008 9 1o 11 12 Frequency (GHz) (b) Reflection and transmission coefficients, phase Figure 3.8. Comparison between the FEM and mode-matching solution in a partial- ly-filled d/a = 0.5 rectangular waveguide. Acrylic (e = 2.5, p = 1), sample length S-Parameters, Magnltude 1 , r 0.81 321 s 1 (16- . 0.4 — S11. FEM --- S11, Mode Matching 0,2- -— S21, FEM . - - - $21, Mode Matching 93 e 16 1} 12 Frequency (GHz) (a) Reflection and transmission coefficients, magnitude S-Parameters, Phase 200 . . 811’: 150 _ 1 1001 1 — S11, FEM a 50L - - - S11, Mode Matching . O 3 — $21, FEM g’ - - - $21, Mode Matching o 0' 1 -5of _100 _ $21 8 J -1508 9 1o 11 12 Frequency (GHz) (b) Reflection and transmission coefficients, phase Figure 3.9. Comparison between the FEM and mode-matching solution in a partial- ly-filled d/a = 0.25 rectangular waveguide. Acrylic (e = 2.5, u = 1), sample length 12 = 7.5 mm. 77 S-Parameters, Magnitude 0 r -1 - ______________ -2 .- ..3 1- -4 . g -5 1. -6 . - - - S11 Theory ’7 ' - - - 821 Theory — S11 FEM '8‘ — $21 FEM ‘ '98 53 1‘0 1‘1 12 Frequency (GHz) (a) Reflection and transmission coefficients, magnitude S—Parameters, Phase 200 . . 150' 811’s ‘ 100- w - - - S11 Theory § 50 - - - 821 Theory 8’ O — $21 FEM , O — S11 FEM -50 821’s -100 -150 - ‘ Frequency (GHz) -2008 9 1o 11 12 (b) Reflection and transmission coefficients, phase Figure 3.10. Comparison between the FEM and theoretical closed-form solution in a fully-filled d/a = 1 rectangular waveguide. Alumina (e = 9.0 — 30.0027, p = 1), sample length 3 = 3 mm. 78 CHAPTER 4 RESULTS 4.1 Error Generated by Cross-Sections Shifted from Center The error measurement is performed using the waveguide configuration shown in Figure 4.1 and the dimensions provided on Table 4.1 [27]. The length € of the sample (its length in the z direction) as depicted in Figure 3.5, is set to a value between /\ z / 4 and Az/2, where A2 is the guiding wavenumber (27r/k210) [4, 5, 6]. The shifting parameter 6 is varied from 0mm to 5 mm, a distance that can realistically represent variations common in a laboratory environment. For each step the S-parameters of the sample are measured and then inverted using the algorithm in [10] to extract its constitutive parameter(s). From this information, the error of the extracted 6 and )u is quantized using the value entered during the forward simulation as a reference. The error is computed with the formula x - xO error = l—— 1 100% (4.1) x0 11 ” where x is the extracted constitutive parameter and “xo” is the reference value. Three cases will be simulated and then inverted: 1) a low-contrast lossless dielectric material (acrylic), 2) a high-contrast dielectric material with low loss (alumina) and 3) a lossy magneto-dielectric material (magRAM). Each of the previous three cases will be simulated using a constant ratio of d to a, where, as Figure 4.1 shows, (1 is the width of the material sample and a is the width of the waveguide. The hight of the sample is always b. 79 >‘<> N|Q ..4}... ..JL.. 53.. Figure 4.1. Vertically loaded waveguide with material sample shifted from the center by a distance 6. Band finitial (GHz) ffinal (GHz) a (m) b(mm) X 8.0 12.0 22.86 10.11 S 2.6 3.95 72.14 34.04 Table 4.1. Dimensions for a rectangular waveguide for the frequency bands used to conduct the numerical simulations and inversion operations [27]. 80 4.1.1 Low-Contrast Material The numerical simulation is performed in X-band, with the waveguide dimensions specified on Table 4.1. The lossless low-contrast material chosen is acrylic. During the computation of the S-parameters the constitutive parameters for the acrylic material sample where e = 2.5 and u = 1 with d/a = 0.5 and a sample length 8 = 5mm. The acrylic sample was shifted in ten equal steps from the center of the waveguide, 6 = 0 mm, to 6 = 5 mm. The resulting scattering parameters are depicted in Figure 4.2 and Figure 4.3. The error for each parameter is shown in Figure 4.4 and Figure 4.5. The percent error plot for the extracted relative permittivity, Figure 4.6, clearly shows that its error is maximum and minimum when also the error on the magnitudes of the transmission and reflection coefficients are maximum and minimum. The error peaks 14.5 % at both ends of the frequency band at 5 = 5mm. It also shows that for each displacement of 6 there exists a frequency, f0, for which there is no error, no matter what the displacement of the sample is. The frequency f0 increases approximately from 10 GHz to about 10.6 GHz for increasing values of 6. Each frequency f0 possesses a band for which the error is fairly small. Out of this band the error increases rapidly. 4.1.2 High-Contrast Material The slightly lossy high-contrast material for this run of simulations is alumina, par- tially filling the cross-section of the waveguide by 50%. The alumina sample has a relative permittivity of e = 9.0 and a loss tangent (tan 6) of 0.0003. The sample length is l = 3mm. The simulation is also performed on X-band incrementing the shifting parameter 5 in ten steps, from 6 = 0mm to 6 = 5mm. The reflection and transmissions coefficients appear in Figure 4.7 (magnitude) and Figure 4.8 (phase). The percent change for the transmission and reflection coefficients for both, magni- tude and phase, are shown in Figure 4.9 and Figure 4.10 respectively. The error on the extracted permittivity (6) is shown in Figure 4.11. It is important to point out 81 that there exists a limit at which the displacement parameter 6 can be increased for this material. It was found that at displacements equal to 6 = 4mm the inversion algorithm does not converge for a large number of frequency points, and as 6 keeps increasing (6 _>_ 4.5 mm) the algorithm does not converge at all. This is due to the fact that with such a big variation from the assumed pair (6, u) it is not expected that a root for Equation (2.121) can be found. As it was shown in the previous case, acrylic, the extracted relative permittivity error for alumina also exhibits maxima and minima throughout the whole frequency band for all the 6 tested. The error characteristic also follows those of the percent change of the magnitude of the trans- mission and reflection coefficients. But in contrast to acrylic, alumina exhibits two frequencies, f1 % 8.6 GHz and f2 a: 10.5 GHz, at which the error is zero. Each of these frequencies, f1 and f2, have a narrow band about them for which the error is fairly small. Out of these frequency bands, the error increases rapidly. 4.1.3 Magneto-Dielectric Materials As a third and last example a dispersive lossy magneto—dielectric material was simu- lated on S-band (see Table 4.1 for waveguide dimensions.) The material chosen was magnetic radar absorbing material (MagRAM) with sample- to waveguide width of d/a. = 0.09 and its length 2 = 3mm. The sample constitutive parameters used for the S-parameter extraction were obtained from [8]. As it can be seen on Figure 4.2 and Figure 4.3, the effect of shifting the material sample from the center 6 = 0 to 6 = 3.5 mm is negligible. This effect can be attributed to the magnetic field distribu- tion in the cross section of the waveguide. As Equation (2.97) shows, the 2 component of the magnetic field exhibits a null at the center of the cross-section, the position at which the sample is located. As the sample is shifted sideways, this component of the magnetic field increases, but not as rapidly as required for it to have a significant contribution on the measurement of the reflection and transmission coefficients. In order to to appreciate the effect of shifting a material sample, it would have to be 82 placed as an initial position, at either sidewall of the waveguide, where the z compo- nent of the magnetic field is maximum and starts to decay towards the center of the waveguide. 4.2 Layered Materials By creating composite mixtures of different dielectric and magnetic materials in layer- ing structures, the overall effective dielectric permittivity and magnetic permeability can be engineered to obtain a desired value for an specific frequency or throughout a whole band. One advantage of obtaining the effective constitutive parameters of a sample in this way, is to reduce its cost, since using homogenous samples might be a more expensive way to proceed [26]. In this section three cases for layering structures are studied: 1) structures layered in the direction of propagation, 2) structures layered horizontally and parallel to the direction of propagation, and 3) layered vertically and parallel to the direction of propagation. 4.2.1 Perpendicular in the Direction of Propagation The structure is shown in Figure 4.13. The length of the overall length K is kept constant and equal to 3 mm while the number of layers is incremented progressively in odd numbers, in this way the material of the same type (material “A” as shown in Figure 4.13) is always on either side of the waveguide. This practice ensures the reciprocity of the sample, and the algorithm in [10] can be applied. The first case consists of material of type—A being alumina(e = 9, tan6 = 0.0003 and p = 1) and material type-B free-space, as an approximation for foam. The number of layers is incremented from three to nineteen, and then the S-parameters are computed at a fixed frequency f0 = 9 GHz. The reflection and transmission coefficients are depicted in Figure 4.14 (magnitude) and Figure 4.15 (phase). The results are validated using the wave-matrix technique [11, 25, 26]. The transmission parameters for each layer are obtained by using the theoretical reflection and transmission coefficients for a 83 fully-filled rectangular waveguide by using the equations derived in §2.2, Equation (2.119) and Equation (2.120). Horn these two Equations, assuming the reciprocity of the materials involved, the relationship (2.6) is used to convert the previous coeffi- cients to T-parameters. This process is repeated for each layer, then each T-matrix is multiplied in the same order as the layer in the waveguide to produce a “total” trans- mission matrix. From this result and using the relationship (2.5) the S-parameters for the total structure are then obtained. _ tAll tA12 tBll 13312 13311 tBlZ tAll t.1112 thotl— - - tA21 tA22 ItB21 13322 tB21 t322 tA21 tA22 (4.2) Figure 4.16 shows the extracted relative permittivity as a function of the number of layers. As the number of layers increases the material homogenizes to the approxi- mation [26] EeffzeA'VA—l'fB'VB (4.3) where V A is the volume fraction of material A and VB is the volume fraction for material B. When the thickness of layer A is the same as that of layer B, Equation (4.3) is written as Geff=(-2-}N-)-[(N+1)-€A+(N-1)-€B] (4.4) where N is the total number of layers in the structure. Clearly, from Equation (4.4), the effective relative permittivity eeff becomes the mean average of both dielectric permittivities e A and e B when the number of layers is sufficiently large (n —+ 00). 6 + E _ eeff 2: _14__2__B_ (4.0) 84 Figure 4.17, Figure 4.18 and Figure 4.19 show the results when the values of e A and e B are reversed. Figure 4.20 shows the extracted permittivities for both cases and the asymptotic approximation in (4.5). 4.2.2 Parallel to the Direction of Propagation. Horizontal and Vertical Layering The structures are constructed by filling the entire cross-section of the waveguide with horizonal and vertical layers, as shown in Figure 4.21 and Figure 4.22 respectively. The first exercise consists on having a high-contrast material in the outer layers (material A) and then alternating with a low-contrast material (material B). The constitutive parameters for material A are those of alumina and for B acrylic, with the same values for e and p as described in previous sections. The values are then reversed. The number of layers are increased from three to nineteen. The simulations are performed on X-band. For both cases it is seen from the scattering parameters and from the extracted relative permittivity that the material homogenizes as the number of layers increases. Figure 4.23 through Figure 4.34 show the results for the reflection and transmission coeflicients as well as the extracted values for the relative permittivity. 4.3 Anisotropic Formulation: A Ferrite Anisotropic materials, like ferrites, find multiple applications in microwave circuits. These applications include directional devices such as isolators, circulators, gyrators [27], phase shifters, microwave switches, and microstrip antenna applications [28]. The simulation of a ferrite is performed using the Pfilder model described in [27] and with the specifications in [29, 30]. Figure 4.35 through Figure 4.38 show the magnitude of the reflection and transmission coefficients for the same ferrite at different applied magnetic fields. The ferrite specifications are: saturation magnetization 47rMs = 5000 Gauss, anisotropy field Ha = 200 Oe, linewidth AH = 500 Oe. The applied magnetic 85 fields are H0 = 100, 300, 500 and 800 Oe, all in the y direction. The sample width is d = 4 mm and its length 2 = 25 mm. The ferrite is place along the waveguide sidewall at x = 0. 86 Transmission Coefficient, Magnitude 0.95 1 1 5mm 0mm 0.9 ] _ ...... / 0.8 1 . 0.71 0 mm / 0.65 - 5 m'“ 0'68 9 1o 11 12 Frequency (GHz) (9) Reflection Coefficient, Magnitude 0.62 1 . . 0.6 . 058- /° "'m 5 "‘m 0.561 \ . 0.54 - l 0.52 - >1”; ’ ~ 0.5. \ 0.48 - \ 5 mm 0'468 9 10 1 1 12 Frequency (GHz) 0)) Figure 4.2. Comparison of the transmission and reflection coeflicients (magnitude) for acrylic when the parameter 6 is varied from 0 to 5 mm in ten steps. (6 = 25,11 = 1). 87 Transmission Coefficient, Phase 1 ‘ V c» (O h _s _e. ..s N 1 0 Frequency (GI-12) (9) Reflection Coefficient, Phase 1 170 160 150 ..l 140 8 9 11 192 10 Frequency (GHz) 0)) Figure 4.3. Comparison of the transmission and reflection coefficients (phase) for acrylic when the parameter 6 is varied from 0 to 5 mm in ten steps. (6 = 2.5, p = 1). 88 Percent Error on Transmission Coefficient, Magnitude 5 1- ——————.. _ tax 4 1 V 25 20- 5mm % Error A 01 / ,// 10- 1 0%@0mm 5mm 1 8 9 10 if“ 1 2 Frequency (GHz) (8) Percent Error on Reflection Coefficient, Magnitude T 16' 14F / 12‘ 10_ 0%@0mm % Error O N 1? O) on ‘\ \ §\ ) 10 Frequency (GHz) 0)) Figure 4.4. Percent error on the S-parameter magnitude resulting from shifting the center of the acrylic material sample from 6 = 0 to 6 = 5 mm. 89 Percent Error on Transmission Coefficient. Phase 1 V 5mm 10 1 1 3 . 0%@0mm % Error 0) b b 10 Frequency (GHz) (8) Percent Error on Reflection Coefficient, Phase 14 . . . 12 1 5mm 10’ % Error 0%@0mm 08 9 1o 1 3‘ 12 Frequency (GI-ix) (b) Figure 4.5. Percent error on the S-parameter phase resulting from shifting the center of the acrylic material sample from 6 = 0 to 6 = 5 mm. 90 Extracted Relative Permittlvlty for Acrylic 3 . 2.8 1 2.6 ~ 2.4- 1 2.2 - \ 1 5 mm 28 9 10 1 1 12 Frequency (GHz) (8) 25 Error on Extracted Permittlvlty for Acrylic 5 mm x 20 . a 5 mm 8 151 1 15 0% error @ 0 mm 3‘ 10' 5 013 9 10 1‘1 12 Frequency (GHz) 0)) Figure 4.6. (a)Extracted relative permittivity e for an acrylic sample when the pa- rameter 6 is increased from 6 = O to 6 = 5 mm, (b) error. 91 0 8 Transmission Coefficient, Magnitude 0 Frequency (GHz) (9) Reflection Coefficient, Magnitude 1 0 Frequency (GI-12) (b) Figure 4.7. Comparison of the transmission and reflection coefficients (magnitude) for alumina when the parameter 6 is varied from 0 to 5 mm. 92 0 Transmission Coefficient, Phase a 9 1'0 1‘1 12 Frequency (GHz) (8) Reflection Coefficient, Phase 10 Frequency (GI-I2) (b) Figure 4.8. Comparison of the transmission and reflection coefficients (phase) for alumina when the parameter 6 is varied from 0 to 5 mm. 93 0Percent Error on Transmission Coefficient, Magnitude 7 V I e “— 5mm / 5mm 10 1112 Frequency (GI-I2) (8) Percent Error on Reflection Coefficient, Magnitude 11 12 10 Frequency (GHz) (b) Figure 4.9. Percent error on the S-parameter magnitude resulting from shifting the center of the alumina material sample from 6 = 0 to 6 = 5 mm. 94 Percent Error on Transmission Coefficient, Phase 10 Frequency (GHz) (9) 20 Percent Error on Reflection Coefficient, Phase 15 °/. Error A O 8 10 Frequency (6112) (b) Figure 4.10. Percent error on the S-parameter phase resulting from shifting the center of the alumina material sample from 6 = 0 to 6 = 5 mm. 95 Extracted Relative Permitlvity for Alumina 16 Percent Error on Extracted Permitlvity for Alumina 10 Frequency (GHz) (8) N O 0% Error@0mm . 0 Frequency (Gt-l2) (b) Figure 4.11. (a) Extracted relative permittivity e for an alumina sample when the parameter 6 is increased from 6 = 0 to 6 = 3.5mm, (b) error. 96 nTransmission and Reflection Coefficients, Magnitude .U V t r 0.75 - 0.7 0.65 0.6 - 0.55 - 0'26 2.8 3 3.2 3.4 3.6 3.8 Frequency (GHz) (3) Transmission and Reflection Coefficients, Phase 821’s L 183.6 28 s 3.2 3.4 3.6 3.8 Frequency (GHz) 0)) Figure 4.12. S-parameters for a lossy-magneto-dielectric material (magRAM) sample (a)magnitude and (b) phase. 97 (b) Figure 4.13. Waveguide with a layered material in the direction of propagation of the incident field. 98 Transmission Coefficient, Magnitude -4.3 1 1 . e -4.4 -4.5 ~ -4.6 In 1: -4.7 -4.8 - — S21, FEM - - - $21, Wave Matrix -4.9 _5 4 n a 4 n A I n 4 6 8 10 12 14 16 18 Number of Layers (a) 1 m: Reflection Coefficient, Magnitude -1.7 1 -1.75 1 1 — S11, FEM -1.8- --- S11,WaveMatrix 1 8 -1.85 - -1.9I- -1.95 . :1 6 91012141618 NumberofLayers (b) Figure 4.14. S—Parameters (magnitude) for a perpendicularly layered material. Ma— terial A: (e = 9 ~- 30.0027, 11 = 1), Material B: (e = 1, a = 1). 99 Transmission Coefficient, Phase 282 . . . r . . 281 ~ 2801 1 i 27 - 8’ 9 a 278‘ 1 -— $21, FEM 277 _ - -- $21, Wave Matnx ‘ 276 4 6 8 10 12 14 1‘6 118 Number of Layers (8) Reflection Coefficient, Phase 192 . - . . . . 1 191 - 190- 0) 8 '5.189 It a 188[ — S11, FEM 137. --- S11, Wave Matrix 186 L ‘ ‘ 11 6 81012141618 Number of Layers (b) Figure 4.15. S-Parameters (phase) for a perpendicularly layered material. Material A: (e = 9 - 30.0027, )1 = 1), Material B: (e = 1, p = 1). 100 Extracted Effective Relative Permittivity 6 I I T I I i r I I l l 5 4 6 8 10 12 14 16 18 Number of Layers Figure 4.16. Extracted relative permittivity for a layered material perpendicular in the direction of propagation. Material A: (e = 9— 10.0027, )1 = 1), Material B: (e = 1, u = 1). 101 Transmission Coefficient, Magnitude -3.2 a - . r a r -- 821, FEM '3-4 - - - $21, Wave Matrix _3.61- -1 m -o -3.81 _4 . 1 '4'2 4 6 81012141618 Number Of Layers (a) 2 1 Reflection Coefficient, Magnitude -2.Zr ‘ 1 -2.3 1 -2.4 g -2 5. -2.6 — S11, FEM -2.7 - - - Si 1, Wave Matrix -2.8 1 4 6 6 10 1‘2 14 1.6 1.8 NumberofLayers (b) Figure 4.17. S-Parameters (magnitude) for a perpendicularly layered material. Ma- terial A: (E = 1, u = 1), Material B: (e = 9 —j0.0027, p = 1). 102 292 Transmission Coefficient, Phase 29‘ — $21, FEM - - - $21, Wave Matrix 284 4 6 6 140 1‘2 14 1.6 18 Number of Layers (9) Reflection Coefficient, Phase 202 . r . . . r 201 -— S11, FEM 200- --- $11, Wave Matrix 1 199~ 1 a E 5,1981 1 0 a 1971 196 1 195* 1 194 t ‘ J 4 6 8 11) 112 14 16 18 Number of Layers (b) Figure 4.18. S-Parameters (phase) for a perpendicularly layered material. Material A: (e = 1,11 = 1), Material B: (e = 9 — 30.0027, )4 =1). 103 4 9 Extracted Effective Relative Permittivity l l l 4 4 6 8 10 12 14 16 18 Number of Layers Figure 4.19. Extracted relative permittivity for a layered material perpendicular in the direction of propagation. Material A: (e = 1, u = 1), Material B: (6 = 9—j0.0027, u = 1). 104 Extracted Relative Permittivity for the Two Case I I i I i I A: High-Epsilon B : Low-Epsilon Asymptotic Approx. 1 for Eps. 5b-II-I-I-I-...--...-....II-I-I-I-I-I-C i A: Low-Epsilon B : High-Epsilon l L l 8 10 12 14 16 18 Number of Layers .5 a) b Figure 4.20. Extracted relative permittivities for a layered material perpendicular to the direction of propagation and the asymptotic permittivity for a homogenized material. The plot on top shows the result when Material A has a higher permittivity, the plot on the bottom Material A with a lower permittivity: (e = 1, ,u = 1), (e = 9 — j0.0027, p = 1). 105 (b) Figure 4.21. Waveguide with a layered material parallel to the direction of propaga- tion. Horizontal layering. 106 } ' -----> a (b) Figure 4.22. Waveguide with a layered material parallel to the direction of propaga- tion. Vertical layering. 107 Transmission Coefficient, Magnitude 0.75 . . . 13 Layers \ 0.7 - 0.65 06 3 Layers q 0.558 9 Frequeligy (GI-12) 11 12 (8.) Reflection and transmission coefficients, magnitude Reflection Coefficient, Magnitude 0.85 - 0.8 0.75 0-7 ' 13 Layers A 8 e 10 1‘1 12 Frequency (GHz) (b) Reflection and transmission coefficients, phase Figure 4.23. S-Parameters (magnitude) for a horizontally layered material. Material A: (e = 9 - 30.0027, p = 1), Material B: (e = 2.5, p = 1). 108 -1008 9 1o 11 12 Frequency (GHz) (8.) Reflection and transmission coefficients, magnitude Reflection Coefficient, Phase 205 . . . 200 \ 195 1 3 Layers i f ”190 5 £185 / 180'- 3 Layers 5Laye . 175' 170 ‘ ‘ 8 9 1o 11 12 Frequency (GI-I2) (b) Reflection and transmission coefficients, phase Figure 4.24. S—Parameters (phase) for a horizontally layered material. Material A: (e = 9 -—- 30.0027, ,u = 1), Material B: (e = 2.5, p = 1). 109 Extracted Effective Relative Permittivity 5.5 . a - 3 layers 5 - 1 5 layers 4'5 D 7 layers ' 4 . \ 13 layers - 8 9 10 1 1 12 Frequency (GHz) Figure 4.25. Extracted relative permittivity for a layered material parallel to the direction of propagation. Horizontal layering. Material A: (6 = 9 — 30.0027, [.1 = 1), Material B: (e = 2.5, 11 = 1). 110 0 8 Transmission Coefficient, Magnitude 0.78 ' 0.76 ' 0.74 0.72 0.7 - 0.68 L 8 8 1o 11 12 Frequency (6”!) (8) Reflection and transmission coefficients, magnitude Reflection Coefficient, Magnitude 0.74 i V I 0.72 0.7 - 0.68 0.66 5 layers ...—9 0.64 » 3 layers 0.82 - 0'63 e 10 11 12 Frequency (GHz) (b) Reflection and transmission coeflicients, phase Figure 4.26. S-Parameters (magnitude) for a horizontally layered material. Material A: (e = 2.5, p = 1), Material B: (e = 9 — 30.0027, 11 = 1). 111 Transmission Coefficient, Phase ‘— 3 layers 5 layers 1 13 layers -80 u '858 9 1o 1 1 12 Frequency (GHz) (a) Reflection and transmission coefficients, magnitude 210 Reflection Coefficient, Phase 190 p L J 185 10 1 1 12 Frequency (Griz) (b) Reflection and transmission coefficients, phase Figure 4.27. S-Parameters (phase) for a horizontally layered material. Material A: (e = 2.5, p = 1), Material B: (e = 9 - 30.0027, 11 = 1). 112 Extracted Relative Permittivity 13 la ers 3.8k y 3.75L adul— 3.7 - 3.65 ~ 5 layers 3.6 - . l 3.55 - - , 35* 3 layers 3.45 - 3'48 9 10 11 12 Frequency (GHz) Figure 4.28. Extracted relative permittivity for a layered material parallel to the direction of propagation. Horizontal layering. Material A:(e = 2.5, p = 1), Material B:(e = 9 —- 30.0027, ,u =1). 113 Transmission Coefficient, Magnitude f r 1 0.65 0.6 0.55 0.5 - 12 10 Frequency (Gl-lz) (8) Reflection Coefficient, Magnitude 0.9 0.85’ 03* 13layers \\ -\1 .. “\ \ 0758 L A 11 12 10 Frequency (GHz) (b) Figure 4.29. S-Parameters (magnitude) for a vertically layered material. Material A: (e = 9 — 30.0027, 11 = 1), Material B: (e = 2.5, p = 1). 114 Transmission Coefficient, Phase -80 layers 5 -85 r 1 D a _90+ “Yam 7 \ layers 13 ‘ 0 ° \ — L 1 95 layers 3 _100 l. \ 8 8 1'0 11 12 Frequency (GHz) (3) Reflection Coefficient, Phase 1 95 layers 1 3 190 g layers 5 layers 3 a K I 180 _ layers 7 ‘ \ layers 3 | 175i . I 1708 8 1o 11 12 Frequency (GI-12) (b) Figure 4.30. S-Parameters (phase) for a vertically layered material. Material A: (e = 9 — 30.0027, 11 = 1), Material B: (e = 2.5, u = 1). 115 Extracted Relative Permittivi 6.5 I 3 ‘ ' f i r ‘ l 3 ‘ r ‘ I 1‘. 1' _ . - - - Layers: 3 . ~‘ 3 6'4 I Layers: 5 g : ' ,~' 1 ’ 1-1-1 Layers: 7 3 I ~' '- l I : 0’ ll 6'3 ,' — Layers: 9 g i ’g/ f — Layers: 11 i ,3}: 6.217 — Layers: 13 E Jr" Layers: 11 . \ ,g" I 5 6.1 ". \\\\ "'\.1' E : .v"\“' E : 6+ ...—1" i ' ‘ ,..m1""l.ayers: 9 g : 5.9- 2 : 5 1' E I § 5.8 i :5 t 1 : 3 Layers: 13 5'73 9 1o 11 12 Frequency (GHz) Figure 4.31. Extracted relative permittivity for a layered material parallel to the direction of propagation. Vertical layering. Material A: (e = 9 - 30.0027, 11 = 1), Material B: (e = 2.5, p = 1). 116 0 6‘ Transmission Coefficient, Magnitude layers 13 0.6 0.55 0.5 ' 0.45 fly»: 3 /\ 0'48 8 10 1‘1 12 Frequency (GHz) (8) Reflection Coefficient, Magnitude 0.9 . laye 3 0.88» \ /1 0.86 0.84 - 0.82 0.8 0.78 - layers 13 0'768 3 1‘0 1'1 12 Frequency (GI-l2) (b) Figure 4.32. S-Parameters (magnitude) for a vertically layered material. Material A: (e = 2.5, p = 1), Material B: (e = 9 — 30.0027, 11 = 1) 117 Transmission Coefficient, Phase -75 f -85 , layers 13 l -1001 iayeres 5 Degrees «b 3' layers 3 10 Frequency (GI-12) (a) 12 Reflection Coefficient, Phase layers 13 Deg _L ~i O 10 11 Frequency (Gl-iz) (b) 12 Figure 4.33. S—Parameters (phase) for a vertically layered material. Material A: (e = 2.5, p = 1), Material B: (e = 9 -— 30.0027, 11 = 1). 118 9 Extracted Relative Dielectric Permittivity layers 3 7 " .1 | 7 layers 9 6 Av 5.5 layers 13 I 8 9 10 11 12 Frequency (GHz) Figure 4.34. Extracted relative permittivity for a layered material parallel to the direction of propagation. Vertical layering. Material A: (e = 2.5, 11 = 1), Material B: (e = 9 - 30.0027, [.1 =1). 119 Transmission Coefficient, Magnitude -2’ / '3 P A 8 9 1‘0 11 12 Frequency (GHz) (8) Reflection Coefficient, Magnitude l N N h A 8 9 10 1 1 12 Frequency (GI-l2) (b) Figure 4.35. Transmission and Reflection Coefficients (magnitude) for a magnetized ferrite. 47TMS = 5000 Gauss, AH = 500 08, Ha, = 200 06, H0 = 100 06. 120 Transmission Coefficient, Magnitude V r 8 9 10 1‘1 12 Frequency (GHz) (8) Reflection Coefficient, Magnitude Ch 12 10 Frequency (6112) (b) Figure 4.36. Transmission and Reflection Coefficients (magnitude) for a magnetized ferrite. 47rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, Ho = 300 08. 121 Transmission Coefficient, Magnitude -98 9 10 1'1 12 Frequency (GI-l2) (8) Reflection Coefficient, Magnitude I N 01 l 00 C) p 10 1 1 12 Frequency (GHz) (1)) Figure 4.37. Tiansmission and Reflection Coefficients (magnitude) for a magnetized ferrite. 41rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, Ho = 500 08. 122 Transmission Coefficient, Magnitude -9 . . l Frequenfze (GHz) 1 1 12 (3) Reflection Coefficient, Magnitude dB :1: o: I (3 Ci 8 9 10 1'1 12 Frequency (GHz) 0)) Figure 4.38. Transmission and Reflection Coefficients (magnitude) for a magnetized ferrite. 47rMs = 5000 Gauss, AH = 500 Oe, Ha = 200 Oe, Ho = 800 Oe. 123 CHAPTER 5 CONCLUSIONS AND FUTURE WORK In the present work, the FEM was used to assess the error originated from misplacing a material sample within a waveguide and whose constitutive parameters are being extracted by using the algorithm described in [10]. It was found that for low-contrast materials the repercussions of shifting the sample are tolerable if displacements are present. A maximum error of 22% was found at 5 mm. For a high-contrast material, however, errors of nearly 80% were found for displacements of 3.5 mm. The inversion algorithm did not converge when greater displacement were simulated. It was also found that no matter how far the sample is placed from the center of the waveguide, there is always at least a frequency at which the error in non-existent. 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