# A wideband two-layer microwave measurement method for the electrical characterization of thin materials

код для вставкиСкачатьINFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of th e copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A Wideband Two-Layer Microwave Measurement Method for the Electrical Characterization o f Thin Materials by Trevor Cameron Williams B.Eng., University of Victoria, 2000 A Thesis Submitted in Partial Fulfillment o f the Requirements for the Degree of MASTER OF APPLIED SCIENCE in the Department o f Electrical and Computer Engineering We accept this thesis as conforming to the required standard Dr. M A. StjJ/chly, Supervisor (Dept, of Electrical and Computer Engineering) /—a Ll ^ Dr. J. Bomemanp, Department Member (Dept, o f Electrical and Computer Engineering) Dr. J. Provan, Outside Member (Dept. o< f Mechanical Engineering) Dr. G. Beer, External Examiner (Dept, of Physics) © Trevor Cameron Williams, 2002 UNIVERSITY OF VICTORIA All rights reserved. This thesis may not be reproduced in whole or in part, by any means, without the permission o f the author Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1*1 National Library of Canada Bibtiothdque nationale du Canada Acquisitions and Bibliographic Services Acquisitions et services bibfiographiques 395 WcWnglon Strwt Ottawa ON K1A0N4 Canada 395, rua WaWnglon Ottawa ON K1A0N4 Your I Our a The author has granted a non* exclusive licence allowing the National Library o f Canada to reproduce, loan, distribute or sell copies o f this thesis in microform, paper or electronic formats. L’auteur a accorde une licence non exclusive pennettant a la Bibliotheque nationale du Canada de reproduire, preter, distribuer ou vendre des copies de cede these sous la forme de microfiche/film, de reproduction sur papier ou sur format dlectronique. The author retains ownership o f die copyright in this thesis. Neither die thesis nor substantial extracts from it may be printed or otherwise reproduced without the author’s permission. L’auteur conserve la propridte du droit d'auteur qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. 0-612-74977-0 CanadS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Supervisor: M.A. Stuchly Abstract A novel measurement method was developed to characterize the electrical properties of specific materials within strict requirements. The main measurement parameters needed include the ability to measure thin, lossy materials over a broad frequency range in a relatively expedient manner. The method developed uses a two-layer structure, consisting of a layer o f thin, flexible unknown material supported by a thicker, rigid known material that enables accurate placement inside a waveguide for measurement. A non-linear leastsquares optimization algorithm is used to converge on the complex permittivity and permeability o f the material. An uncertainty analysis is performed to investigate optimal thicknesses o f both the sample layer and the supporting layer. Simulations in FDTD were constructed to explore the effects o f sample layers that are not o f exact dimensions. Results from many different materials show the repeatability, and accuracy of the data set convergences, as well as several limitations o f the procedure. Examiners z /lu.c. Cwk -J Dr. M.A./Stuchly, Supervisor (Dept, of Electrical and Computer Engineering) Dr. J. Bomemann, Department Member (Dept, o f Electrical and Computer Engineering) Dr. J. Provah, Outside Member Membt (Dept, o f Mechanical Engineering) Dr. G. Beer, External Examiner (Dept, of Physics) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents 1 Introduction.............................................................................................................1 Background................................................................................................................... 1 Research Objectives..................................................................................................... 2 Thesis Contributions....................................................................................................2 Thesis Outline...............................................................................................................3 2 Definitions and Background Literature Review................................................. 5 2.1 Definitions.....................................................................................................................5 2.2 Measurement Methods o f Electrical Properties.........................................................5 2.2.1 Time-Domain and Frequency-Domain Measurements...................................... 5 2.2.2 Resonant and Wideband Measurements............................................................. 6 2.2.3 Measurement Method Selected............................................................................ 7 2.3 Guided Wave Measurement Techniques.................................................................... 8 2.3.1 Classical Nicholson-Ross-Weir (NRW) Method [4][6].................................... 8 2.3.2 Multi Sample / Multi Sample-Position Methods...............................................16 2.3.3 Short-Circuited Waveguide Method...................................................................17 2.3.4 Non-Linear Optimization.................................................................................... 18 2.4 Conclusions................................................................................................................. 20 3 Method o f Analysis and Measurements............................................................. 22 3.1 Two-layer Measurements Method........................................................................... 22 3.1.1 Governing Equations in a Waveguide............................................................... 22 3.1.2 Advantages and Disadvantages......................................................................... 25 3.1.3 Sources o f Uncertainty........................................................................................26 3.2 Iterative Solution........................................................................................................ 27 3.2.1 Single Value Optimization in Narrow Frequency Bands.................................27 3.2.2 Polynomial Whole-Band Optimization.............................................................29 3.2.3 Causality of Solution..........................................................................................30 3.3 Main Sources o f Uncertainty..................................................................................... 31 3.3.1 Sample Thickness............................................................................................... 31 3.3.2 Uncertainty in Vector Network Analyzer D ata................................................ 32 3.3.3 Small Gaps in Unknown Sample.......................................................................36 3.4 Modeling of Sample Imperfections.......................................................................... 37 3.4.1 VNA Simulation..................................................................................................43 3.4.2 FDTD Evaluation o f Gaps and Facial Inconsistencies.....................................47 4 Results o f Uncertainty Analysis and Measurements........................................48 4.1 Uncertainty Analysis................................................................................................. 48 4.1.1 Material 1, Dielectric Constant = 80, Loss Factor = 10.................................. 51 4.1.2 Material 2, Dielectric Constant = 20, Loss Factor = 40.................................. 58 4.1.3 Material 3, Dielectric Constant = 100, Loss Factor = 200.............................. 61 4.1.4 Material A, Dielectric Constant = 3, Loss Factor = 0.0001.............................64 4.1.5 Summary Uncertainty Plots............................................................................... 67 4.2 Measurement R esults................................................................................................74 4.2.1 Sample Thickness Optimization........................................................................ 78 1.1 1.2 1.3 1.4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Modeling Results and their Evaluation..............................................................84 5.1 Comparison o f FDTD Results with Measurements...............................................84 5.2 Waveguide gap effect............................................................................................... 88 5.2.1 Full Gaps.............................................................................................................. 88 5.2.2 Dents in Sample...................................................................................................94 5.3 Summary o f FDTD Facial Inconsistency Simulations......................................... 100 6 Conclusions........................................................................................................ 101 6 .1 Measurement Method............................................................................................. 101 6.2 Non-Linear Optimization....................................................................................... 102 6.3 Equipment Uncertainty Analysis...........................................................................102 6.4 FDTD Modeling of Sample Facial Inconsistencies..............................................103 6.5 Results..................................................................................................................... 103 6 .6 Future W ork............................................................................................................104 Bibliography..........................................................................................................................B1 7 Appendix............................................................................................................. A1 7.1 Definitions................................................................................................................ A1 7.2 Summary Uncertainty Plots (Acrylic thickness of 3mm)..................................... A4 7.3 VNA HP8720 Uncertainty Specification Sheet.....................................................A7 V ita........................................................................................................................................ VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V List o f Figures Figure 2.1 Instability problems of the classic NRW explicit solutions [8] 9 Figure 2.2 Sample in rectangular waveguide with sample holder. Represented as three two-port devices 10 Figure 2.3 Sample in rectangular waveguide with different length sampleholder 11 Figure 2.4 Sample in rectangular waveguide with no sample holder 12 Figure 2.S Simulated Sn parameter for a lossless non-magnetic 14 Figure 2.6 Simulated S21 parameter for a lossless non-magnetic 15 Figure 2.7 A.) er and \i, derived from S-Parameters of simulated sample. 16 Figure 2.8 Reproduced from [NIST] to portray smooth curve fit o f least squares solution in contrast to explicitly solved single-frequency point solutions. 19 Figure 2.9 Known substance with unknown coating on its front face that make 21 Figure 3.1 Cross section of waveguide with sample and acrylic inserted. 22 Figure 3.2 The very narrow frequency sections o f constant complex permittivity approximate the ideal solution well while each section remains unrelated. 29 Figure 3.3 A two-layer structure with the sample not perfectly matching the backing acrylic (the waveguide height). 36 Figure 3.4 A two-layer structure with a small dent in the sample. 37 Figure 3.S Graded mesh computational domain constructed for simulations of waveguide. 39 Figure 3.6 Five sofr sources used to excite the fundamental mode, TE|0. in the waveguide. Amplitudes have a sinusoidal amplitude from 0 to 1. 40 Figure 3.7 A cross section of the waveguide at z = 30mm throughout the entire simulation. The fundamental mode is clearly visible, as well as the Gaussian Excitation. 40 Figure 3.8 Plot of three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental fields (Ex, Ez, Hy) collected at z = 15mm, which is 6mm away from excitation and corresponds to the Su collection point. 41 Figure 3.9 Plot of all three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental fields (Ex, Ez, Hy) collected at z = 480mm, which is 6mm away from excitation and corresponds to the S2 1 collection point. 42 Figure 3.10 Plot A shows the Cosine waveform that is multiplied to Plot B, which is the pure Gaussian pulse. Plot C, shows complete time domain Cosine Gaussian Pulse generated over 4000 time steps, Plot D displays the narrow band frequency pulse created with 2 18 points (mostly zero-padded) which places 8.191 - 12.4 GHz between discrete points 480 and 727 witha 17.1 MHz frequency step (df). 43 Figure 3.11 Facial inconsistencies modeled; Gaps are on the left, dents on the right. 47 Figure 4.1 Uncertainty o f the dielectric constant (£’r) due to errors in measurements of S u and S2 1 51 Figure 4.2 Uncertainty of the loss factor (£’’r) due to errors in measurements o f S 11 and S2l 52 Figure 4.3 Uncertainty of the dielectric constant (£’r) due to errors in measurements of Sn and S21 - Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 52 Figure 4.4 Uncertainty of the loss factor (e” ) due to errors in measurements o f Si 1 and S2 1 . Acrylic = I mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 53 Figure 4.5 Uncertainty o f the dielectric constant (£’r) due to errors in measurements o f S2t 53 Figure 4.6 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S2i 54 Figure 4.7 Uncertainty o f the dielectric constant (£’r) due to errors in measurements o f S2 1 - Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 54 Figure4.8 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S: i- Acrylic= lmm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 55 Figure 4.9 S-parameter plot of two-layer structure at the single frequency o f 10GHz. Sample thickness varies from 0.01 mm to 3mm 57 Figure 4.10 The magnitude o f both S-parameters of the two-layer structure at 10GHz, with sample thickness varying from 0.0lmm to 3mm. 57 Figure 4.11 Uncertainty o f the dielectric constant (E’r) due to errors in measurements of S21 59 Figure 4.12 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S2 1 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.13 Uncertainty o f the dielectric constant (£’r) due to errors in measurements of S2| . Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 60 Figure 4.14 Uncertainty of the loss factor (E”r) due to errors in measurements o f S2|. Acrylic = lmm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 60 Figure 4.15 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2, 62 Figure 4.16 Uncertainty o f the loss factor (£”r) due to errors in measurements o f S2| 62 Figure 4.17 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2|.Acrylic = lmm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 63 Figure 4.18 Uncertainty of the loss factor (E"r) due to errors in measurements o f S2i. Acrylic = lmm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 63 Figure 4.19 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S21 64 Figure 4.20 Uncertainty o f the loss factor (£”r) due to errors in measurements o f S2i 65 Figure 4.21 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2|. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 65 Figure 4.22 Uncertainty of the loss factor (£”r) due to errors in measurements o f S2|. Acrylic = lmm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). 66 Figure 4.23 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios o f material at 10GHz, a sample thickness of 0.1 mm and an acrylic thickness of 1mm. 68 Figure 4.24 Uncertainty of the imaginary permittivity due to errors in measurements of S2). Five different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic thickness o f lmm. The erp= 5*erpp has uncertainty too high to be recorded in this plot. 68 Figure 4.25 Uncertainty of the dielectric constant due to errors in measurements o f S2i. Five different ratios o f material at 10GHz, a sample thickness of lmm and an acrylic thickness o f lmm. 69 Figure 4.26 Uncertainty of the imaginary permittivity due to errors in measurements o f S2|. Five different ratios o f material at 10GHz, a sample thickness o f 1mm and an acrylic thickness of 1mm. 69 Figure 4.27 Uncertainty o f the dielectric constant due to errors in measurements of S2i. Five different ratios o f material at 10GHz, a sample thickness of 2mm and an acrylic thickness of 1mm. 70 Figure 4.28 Uncertainty o f the imaginary permittivity due to errors in measurements of S2|. Five different ratios of material at 10GHz, a sample thickness o f 2mm and an acrylic thickness of lmm. 71 Figure 4.29 Uncertainty o f both the real and imaginary components of permittivity at a dielectric constant o f 100 and a frequency of 10GHz. Acrylic thickness for all plots is lmm. e,p = £’r.e,pp = E”r 72 Figure 4.30 An extreme close-up of both S-Parameters and how data smoothing occurs with gating. 75 Figure 4.31 Results from Batch#5 Sample#4, six separate measurements. Arrows indicate increasing time. Sample thickness = 0.15mm, Acrylic thickness = 1 .1 8mm 76 Figure 4.32 Results from Batch#7 Sample#4, four separate measurements. Arrows indicate increasing time. Sample thickness = 0.15mm, Acrylic thickness = 1 .1 8mm 77 Figure 4.33 Results from Batch#9 Sample#5, four separate measurements. Sample thickness = 0.15mm, Acrylic thickness = 1.18mm 78 Figure 4.34 Entire Batch #4. Thickness constraint envelopes were recorded. Sample numbers correspond to Table 4.6 79 Figure 4.35 The polynomial solution and the small-section solution. Sample thickness = 0.14mm, Acrylic thickness = 1.18mm 81 Figure 4.36 Very low-loss sample with very high uncertainty. Also shown is one-layer convergence o f acrylic only. Sample thickness = 0.16mm, Acrylic thickness = 2.07mm 82 Figure 5.1 S-Parameters showing both VNA-collected results and FDTD-simuiated results for sample B21S2 85 Figure 5.2 Sample B21S2, (Left) Dashed line indicates measurements taken with VNA, solid shows FDTD simulated results. (Middle, Right) Zoomed in results showing 10GHz comparison. Blues lines show £r’ , red lines show e ,” 86 Figure 5.3 Comparison o f measured and simulated results for B26S4. Focus is 10GHz results. Green uncertainty enveloped centered on VNA-measured values. 87 Figure 5.4 The five full gap configurations simulated. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.5 FDTD simulated Su parameter for gaps 1-5 for sample B21S2. Blue left: perfect, Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency data points of S-parameters. 89 Figure 5.6 FDTD simulated S2| parameter for gaps 1-5 for sample B21S2. Blue left: perfect. Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency data points of S-parameters. 90 Figure 5.7 FDTD simulated results for sample B 21S2 or dielectric constants and loss factors for one perfectly homogeneous layer and five samples with gap inconsistencies. 90 Figure 5.8 FDTD simulated S| t and S2| parameters for gaps 2-4 for sample B26S4. Blue: perfect. Magenta: gap #2, Cyan: gap #3, Yellow: gap #4,. Red tips indicate low frequency data points of S-parameters. 92 Figure 5.9 FDTD simulated results for sample B26S4 of dielectric constants and loss factors for one perfectly homogeneous layer and three samples with gap inconsistencies. 93 Figure 5.10 The five dent configurations simulated. 95 Figure 5.11 FDTD simulated Sn and S2t parameters for sample B21S2, gaps 1-5. Blue left: perfect. Green: dent #1, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4, Blue right: dent #5. Red tips indicate the low frequency data points o f the S-parameters. 96 Figure 5.12 FDTD simulated results of dielectric constants and loss factors for one perfectly homogeneous layer and five samples with dent inconsistencies. Sample B21S2. On enlarged region plot, E 'r on left, £”r on right. Dent #5 omitted from £’r 10GHz enlarged region plot. 97 Figure 5.13 FDTD simulated Sn and S2| parameters for sample B26S4, dents 2-4. Blue left: perfect. Magenta: dent #2, Cyan: dent #3, Yellow: dent #4. Red tips indicate low frequency data points o f Sparameters. 98 Figure 5.14 Results of dielectric constants and loss factors for one perfectly homogeneous layer and three samples with dent inconsistencies. Sample B26S4. 99 Figure 7.1 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios of material at 10GHz, a sample thickness o f 0.1 mm and an acrylic thickness of 3mm. A4 Figure 7.2 Uncertainty of the loss factor due to errors in measurements of S2|. Five different ratios o f material at 10GHz, a sample thickness o f 0.1 mm and an acrylic thickness of 3mm. A4 Figure 7.3 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios of material at 10GHz, a sample thickness o f lmm and an acrylic thickness of 3mm. A5 Figure 7.4 Uncertainty of the loss factor due to errors in measurements of S2i. Five different ratios o f material at 10GHz, a sample thickness o f lmm and an acrylic thickness of 3mm. AS Figure 7.5 Uncertainty of the dielectric constant due to errors in measurements o f S2I. Five different ratios of material at 10GHz, a sample thickness o f 2mm and an acrylic thickness of 3mm. A6 Figure 7.6 Uncertainty of the loss factor due to errors in measurements of S2|. Five different ratios o f material at 10GHz, a sample thickness of 0. lmm and an acrylic thickness of 3mm. A6 Figure 7.7 HP8720 Equipment Uncertainty Specification Sheet A7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 4.1 The dielectric constant and loss factor of the four materials analyzed.................................................49 Table 4.2 Displaying cut-off frequency, and guided half-wavelength, and guided quarter-wavelength distances for the X-band............................................................................................................................................58 Table 4.3 Sample thickness = 0.1mm, acrylic thickness = lmm, frequency = 10GHz...................................... 73 Table 4.4 Sample thickness = lmm, acrylic thickness = lmm, frequency = 10GHz..........................................73 Table 4.5 Sample thickness = 2mm, acrylic thickness = lmm, frequency = 10GHz..........................................73 Table 4.6 Data corresponding to Figure 4.34 showing variable thickness with S% constraint........................... 80 Table 4.7 Data Set displaying difficulties with optimization convergence. All optimized solutions of thickness lay on the constraint edge o f the envelope, except for the two shown in Red Text.............................81 Table S. 1 Electrical characterization of two representative samples at 10GHz................................................... 84 Table 5.2 Comparison o f measured and simulated complex permittivity values at 10GHz................................ 87 Table 5.3 Summary table showing differences in simulated dielectric constant and loss factor for different configurations o f facial inconsistencies. All values at 10GHz...............................................................93 Table 5.4 Summary table showing differences in simulated dielectric constant and loss factor with different configurations o f facial inconsistencies. All values at 10GHz.............................................................100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments Thank you Dr. Stuchly for your patience and understanding to let me accomplish many other goals simultaneously with this Masters degree. I have bettered m yself in many aspects during the time spent on this research and I feel I owe a great portion o f it to you. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 1 Introduction A measurement method was required for the electrical characterization of thin, flexible materials intended to perform as radar absorbers at microwave frequencies. After a review o f many measurement techniques previously developed, it was determined that an extension to the available techniques was needed for this application. A two-layer waveguide measurement technique has been selected and is described in this thesis. To evaluate the technique, an uncertainty analysis is carried out, several measurement results are analyzed, and a Finite Difference Time Domain (FDTD) simulation is performed that evaluates sample imperfections. 1.1 Background The motivation for this research was provided by an interest and a need to evaluate novel materials for radar cross section (RCS) reduction. Researchers at the Canadian Forces Base (CFB) Esquimalt, Canada, have been developing radar absorbing materials (RAM) based on the specific organic polymers, Polypyrrole (PPy) and Polyaniline (PAni). These materials exhibit unique microwave behaviour if manufactured properly. RAM needs to have relatively high dc and ac-conductivities and a relatively low dielectric constant. Both real and imaginary parts of the permittivity have to be controlled reliably in material manufacture. Design of structures reducing RCS depends on available suitable materials. New materials with unique, controllable microwave properties would dramatically increase the degree of freedom of design. In the process of developing new RAM, measurements of its permittivity and permeability are critical. Since initially, the available samples were made relatively small, it was decided that measurements in the Xband (8.2-12.4 GHz) would suffice. Also, initially only very few materials were expected to have relative permeability greater than 1. Furthermore, it proved difficult to create homogeneously thick sample coatings o f PPy and PAni on the known substrate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1.2 Research Objectives The objectives o f this research and thesis are: • To establish a specific measurement method for the microwave characterization of thin flexible lossy materials. • To develop iterative (optimization) procedures for obtaining complex permittivity (er) and permeability (jir) from measured complex scattering parameters (Su and S21). • To ensure that measurements are repeatable and relatively fast. • To examine the uncertainty associated with the apparatus used during the measurements. • To model the system in FDTD in order to explore how small facial inconsistencies in the sample affect the measurement uncertainty and optimization convergence when solving for complex permittivity. 1.3 Thesis Contributions The main contribution of this thesis is the development of the specific measurement configuration used to characterize the electrical properties of thin flexible electricallyunknown samples. The measurement technique was developed under the constraints of several factors, all of which were met. The measurement process had to be a relatively inexpensive and fast way o f measuring a large number o f samples. A detailed uncertainty analysis was performed for this novel measurement technique for many different types of samples, but the main focus was on lossy non-magnetic materials on the order o f 0 .2 lmm thick. To converge on a solution (permittivity and permeability o f a material) a non linear least-squares optimization procedure was developed. Using the curve-fitting concept devised at NIST [1], a refined program was developed to process the input from a Vector Network Analyzer (VNA), and to calculate the complex permittivity and permeability o f a material. Uncertainty was then calculated based on the VNA accuracy specifications in collecting S-Parameters (magnitude and phase). Although complex Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 permeability is not explored in this thesis, the convergence program maintains the ability to accommodate magnetic materials, though no testing or uncertainty analysis has been performed. The procedures described for the permittivity are directly applicable for permeability. The measurement technique developed is transferable to free-space measurements and will be used with broadband horn antennas once the ability to deposit these new materials over large surfaces is attained. FDTD simulations were performed to illustrate the effect that slight imperfections in the sample layer had on data collection and subsequent permittivity determination. 1.4 Thesis Outline Chapter 2 contains a review o f previously developed measurement methods. Several methods are discussed in detail and their drawbacks are identified with respect to the measurements, which is the goal in this thesis. Relevant waveguide theory is discussed, as well as the electrical characterization o f materials at microwave frequencies. Reasons for selecting the two-layer approach are given. Chapter 3 describes the selected method (the two-layer technique) and procedures needed to obtain electrical characterization of materials. The optimization procedures developed for this work are covered in detail and reasons are given why one method may be better at converging on a solution than another for a specific set o f collected data. Sources o f uncertainty are enumerated. Uncertainty due to the VNA is derived. Uncertainties due to small gaps or dents existing in the sample are noted and FDTD simulation is introduced as a way o f investigating this practical problem. The FDTD simulation of the entire waveguide system is presented along with safeguards used to ensure an accurate solution for the data collected. The procedure to convert data from a simulated waveguide system to a useable set o f values mimicking an actual waveguide set up and calibration is presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Chapter 4 gives the results of the uncertainty analysis developed in Chapter 3 as well as several measurement results of samples delivered to the lab. Four general materials are introduced and their electrical characterizations presented. Each material is subsequently analyzed in terms o f its uncertainty in relation to its thickness as well as the thickness of the supporting structure. Useful uncertainty plots portraying materials with various sample and acrylic thicknesses are then presented, and trends and recommended configurations are explored. Measurement results o f seven materials are then presented with the optimization convergence characteristics analyzed. Limitations of the two-layer optimization approach are shown for several of the presented materials. In Chapter 5, the FDTD simulation results and evaluations are shown. A comparison between actual sample measurements and FDTD simulated measurements is given to validate the computational domain setup. Two lossy materials are used in this comparison. Facial inconsistencies in the form o f gaps and dents in each o f these two samples are then simulated. Five gap and dent configurations are explored for one material, three configurations for the other. The gap and dent results are compared to the results for a perfectly homogeneous sample layer. Chapter 6 contains conclusions comparing the quality o f the solution to the initial problem. General recommendations regarding functionality and limitations of the optimization procedure are given. Directions for future work on this measurement technique are given and possible avenues of further research suggested. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Chapter 2 2 Definitions and Background Literature Review 2.1 Definitions Conventional definitions and notations used in this thesis are given in Appendix 7.1. They include the permittivity, permeability, characteristic impedance, reflection and transmission coefficients, and scattering parameter definitions. 2.2 Measurement Methods of Electrical Properties Electrical properties o f materials have been characterized successfully by many different methods. A comprehensive literature review was performed to determine a well-suited measurement method for the project. After consideration o f the project requirements, a specific measurement method and sample configuration was selected. A outline o f the different approaches toward material measurement is given, a more comprehensive review is given elsewhere [2 ] 2.2.1 Time-Domain and Frequency-Domain Measurements Time-Domain (TD) measurements use a material response to the spectral content o f a fast rise-time (subnanosecond) pulse [3]. The scattering parameters are determined by taking the Fourier transform of this response [4]. This type of measurement has only become possible in the early 1970’s due to the development of fast sampling oscilloscopes and tunnel-diode step generators with picosecond rise times [2]. Advantages o f TD measurements include a potentially lower equipment cost and the ability to take measurements without the need of a calibration procedure [2]. However, resolution and accuracy are often sacrificed in TD techniques, if broadband measurements are required Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 at high frequencies. This, however, is becoming less o f a problem with advancing technology [2 ]. Frequency-Domain (FD) measurements require the use o f an Automatic (or Vector) Network Analyzer (VNA) that collects the scattering parameters (reflection and transmission coefficients) directly. Other methods have been used in the past, but they are cumbersome and less accurate [5]. After a calibration is performed and a measurement plane is established, the scattering parameters can be read at a frequency resolution dependant on the particular analyzer [6 ]. The disadvantages of this method are the cost o f the VNA, and a time-consuming calibration procedure that must be performed before every measurement if accurate results are desired. 2.2.2 Resonant and Wideband Measurements Resonant measurement methods can be performed in either a closed-cavity or an open environment. These methods rely on the change o f the Q-factor and the resonant frequency of a resonator after the test sample has been inserted. In many cases, very specific geometries and exact dimensions o f the test sample are required. The most popular is the cylindrical TMoio cavity, which requires a rod-shaped solid sample, or a liquid in a tube [2]. Other cavity methods can be used that have similar restrictions on sample shape. Open resonators are more convenient than cavity methods, but at relatively low frequencies large homogeneous samples are required. Both types of resonant methods can only measure the properties in a very narrow frequency band requiring a different measuring device or sample size at each frequency range [7]. Mathematical expressions required to solve for the electrical characteristics in one cavity do not convert to different cavity geometries. Only for the perturbation method does the analysis remain unchanged [5]. Wideband measurements can be performed in a closed environment using TEM, TE or TM waves, or in free-space using plane waves. Coaxial lines or waveguides can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 loaded with a test sample provided the correct sample geometry is achievable. A very wide frequency range can be attained, particularly for coaxial line measurements. Usually the fundamental mode is utilized as the only propagating wave. This limits the complexity of mathematical expressions. Again, exact geometries of the test samples are required to limit the uncertainty o f the measurements. Free space measurements require larger samples, as the directivity o f the antenna is the limiting factor on sample size. However, as long as a sufficiently large homogeneously thick sample can be prepared, the width and height of the sample does not have to have exact dimensions. The mathematical expressions required for the determination of the electrical properties o f the test sample do not need to be altered significantly when transferring between fundamental-mode waveguide and free-space measurements. 2.2.3 Measurement Method Selected Because of the requirement to obtain data in a relatively wide frequency range, and the availability of a VNA (HP 8720), further focus has been restricted to frequency domain wideband measurements. Wideband measurements of small samples are possible using coaxial lines, but the test sample geometry is difficult to machine accurately. Neither the manufacturing process nor the facilities to mill samples suitable for coaxial measurement were available at the start o f this research. Rectangular cross-section samples are much easier to manufacture with high tolerances. Because only small thin sheets of sample material were readily available, X-band waveguide was chosen for the measurements. The same method and technique required to process the scattering parameters to determine the electrical properties of the test sample with minimum uncertainty can be directly carried over to other frequency ranges o f the rectangular waveguide, as well as to free-space. Additionally, since only thin flexible samples were available, the proper measurement technique was needed to accommodate this limitation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 2.3 Guided Wave Measurement Techniques A brief outline o f the equations required to solve for the complex permittivity and permeability using the scattering parameters will be given in the following chapters. This method was established in 1970 and is still predominately used today for measurements o f relatively low to medium loss materials. Limitations of this method will be explained and several different solutions shown. 2.3.1 Classical Nicholson-Ross-Weir (NRW) Method [4][6] Nicholson and Ross derived an explicit solution for the permittivity and permeability from the measured scattering parameters. The solution was based on transmission line theory. These explicit solutions were derived before the invention of a practical ANA/VNA, therefore their work was performed in the time domain. After the advent o f the ANA/VNA, Weir [6 ] further developed the technique in the frequency domain. The method is often referred to as the NRW method. Limitations in the NRW method encouraged further research in this area. At certain frequencies close to and including guided half-wavelength (Xg/ 2 ) multiples in the sample, the NRW equations completely breakdown [8 ]. Figure 2.1 illustrates the common problems associated with a sample thickness that is A,g/2 or a multiple at some frequency for a low-loss material [8 ]. This limits the sample to thicknesses less than Xg/2 o f the highest frequency. However, this often results in the scattering coefficient values that are in the range where VNA exhibits large uncertainties [8 ], [1]. An in depth uncertainty analysis o f data collected from the VNA using thin samples relates to this research and is presented later in this chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 2.5 10 0 12 Fr«quMcy(GHz) 10 12 Prequwef(QH< Figure 2.1 Instability problems o f the classic NRW explicit solutions [8 ] Although the NRW method is explicit, it requires a relatively good initial guess o f the electrical properties measured. This can be a limiting factor when the substance is completely unknown. With a poor initial guess, the equations may produce wrong values with no warning o f failure. The importance of the initial guess arises from an infinite number o f roots existing for equation (2.11). An initial guess is required to determine the correct root coefficient for equation (2 . 11 ) and is accomplished by solving for the group delay [6 ]. A brief outline is now given detailing the NRW method o f explicit determination o f the electrical properties of materials. Parameters Er and |ir o f a homogeneous sample in a waveguide (or, in general any transmission line) can be derived by modeling the sample under test as three two-port devices. These are two planes at which reflections between the two materials are considered (Di and D3), and a section of the waveguide uniformly filled with the material (Dj). This is shown in Figure 2.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Port 2 Plano 03 Plane D1 I • Length of Sample and Sample Holder Port 1 Figure 2.2 Sample in rectangular waveguide with sample holder. Represented as three two-port devices The matrices defining these sections are: r z>,= ' 0 i - r ' D, = t + r - r -T T D, ‘ - r = 0 i-r i + r (2. 1) r where T and T are complex as defined in the Appendix 7.1. When cascaded, the three matrices form: D 4 = Dxo D 2 ° Di T r ( i - T 2)‘ ’ T ( i - r 2 )' i - r :T 2 i - t 2t 2 [ T ( i - r : )‘ ’ r ( i - T 2 )' ■^11 = Al S\2 (2.2) S 22 [ l - r 2T 2 where, for isotropic, reciprocal (e.g. non-ferrite) materials: S„ = r(i-T2)' t - r 2T 2 Sl2 = 5 ,, = T (i-r2) i - r 2T2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.3) 11 Modifications must be made to these equations when the sample holder is not o f the exact thickness of the sample. In this case, a linear transformation translates the S-Parameter matrix to the calibration planes, shown in Figure 2.3. Port 2 Plan* 03 12 - Langth batwean Plana 2 and Sampla I - Length of Sampla and Sampla Holder Plana D1 - Length between Plana 1 and Sampla Port 1 Figure 2.3 Sample in rectangular waveguide with different length sample holder Translation can be performed as: dJ (2.4) =0£>ae where: -r»L, ©= 0 (2.5) Therefore: 5,, —e ' r ( i - T 2)' |_i—r 2x 2J 5,, = e -2i^ [' ri (- i r- T2T2)‘2 _ s ,, = S p =e-r>^*L'-' * T ( i - r 2)' [_ 1—r 2x 2J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.6) 12 In the experimental procedure used, the sample was placed flush with the calibration plane, as shown in Figure 2.4. Port 2 Sample 02 Plane 01 I - Length of Sample Port 1 Figure 2.4 Sample in rectangular waveguide with no sample holder For the setup o f Figure 2.4, Li = 0, and L2 = -L. The equations for the S-parameters reduce to: r(i-T2)' [ i - r 2T2J ’r(i-T2)' S„ = e 2r°L |_i-r2x2J S,, = S r 'T(i-r2)' [ t - r 2T2 = e r°L To solve for the permittivity o f permeability explicitly, only two S-Parameters are needed. Therefore, the system (three complex equations) is over-determined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Using S21, T is solved for and replaced in the equation for S| 1. Solving for T then results in: r=K±J{tc2-i) = " 2 ( 2 .8 ) 2s„ and: T = s n +s2i- r i - ( s lt+5:,)r (2.9) Using previous and appendix definitions for T, T, and y, explicit equations can be obtained for complex p.r and Er. These are: ro+iD vA , , 0 - r ) r f 1) \ 'is !' e. = (2. 10) fl. 1 VC / With Xq and Xc being the free-space wavelength and waveguide cut-off wavelength respectively and: -U fl' ( 2 . 11) T A2 The expression ln(l/T) contains multiple roots. Solving for the correct root requires a guess o f the permittivity, or an analysis o f the group delay [6 ]. For a low loss, non-magnetic sample, when L = Xg/2, Si 1 -> 0, and S21 C 2 K = _C 1, it leads to - 4.1 (2. 12) 25, as the magnitude o f T < 1 : 0 c o T= ^ tl(-» 0 ) +C ^ ( " ^ I I I —. 0 ) + I) 5 _r * ( — 0) l ( — I ) ^ '( - . 0 ] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.13) 14 As T 1: (2.14) The equations dictate that either |ir or er is equal to zero. This is not a physical solution, therefore |ir and Er are no longer separable. Theoretically, the equations are only non-separable at one frequency. However, since computers have a finite numerical accuracy, the result is a small frequency band within the frequency range of interest that is affected. To illustrate the problem, a sample with er = 2.6+j0.0001 and |ir = I is evaluated, and the results are shown in Figure 2.5 and Figure 2 .6 . S11 - P olar plot without error of non-magnetic sa m p le with ei= 2 .6 * f0 001 270 Figure 2.5 Simulated Si i parameter for a lossless non-magnetic sample of £r = 2.6 and thickness 2.256cm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 S21 - P o la r plot w ithout error o f n o n - m a g n e tic s a m p le w ith er=2.6-*j*0.001 120 0.8 0.6 150 02 180 210 330 240 300 270 Figure 2.6 Simulated S21 parameter for a lossless non-magnetic sample of er = 2.6 and length 2.256cm It can be noted that on one side of the frequency range the Sn parameter approaches zero and the S21 parameter approaches one. In both cases, the errors in measurements tend to infinity. The material properties derived from the S-Parameters using equation (2.12) - (2.14) are shown in Figure 2.7. Figure 2.7a illustrates the half-wavelength problem, Figure 2.7b displays the narrow band where results cannot be explicitly derived. The half-wavelength multiple in the frequency range of the X-band waveguide for this sample occurs at 9 194 720 620 Hz. The frequency resolution for these graphs has been increased to highlight the half-wavelength problem, and does not represent true VNA measurements. It would however, if the half-wavelength multiple occurred near a standard VNA 1.05 MHz step. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 R oil Com ponents of Permiltnaty tntf Perm eoB44r 1 3 't 25 B««l Com ponents o f Pofm4tiwtf <5 25 2 1 2 i P«>mo«feibtY * a i 15 H 3 1 15 * 1 05 !■ * 2 a 0 O SS 09 0 95 i 1 05 '< 1 15 F’o o u o n c f. x-8«nd W noGuiO* Bongo 9 194/ 12 ( , g 14 9 194/ 9 t9 4 / 9 194/ 9 194/ Froquoncv. X-B*n4 Wi»«Ogt00 P tn p o A 9 194/ ( , q* B Figure 2.7 A.) er and |ir derived from S-Parameters o f simulated sample. B.) Narrow frequency band illustrating breakdown o f explicit equations. Apart from the problems related to the half-wavelength ‘transparency of the sample’, and non-separability of er and |ir, uncertainty in broadband measurements of lossless non magnetic materials results from the VNA having the largest error in phase when the Sn parameter approaches zero, and in magnitude when the S| i parameter approaches 1. This, in practical measurements, means that a wide frequency band around the half-wavelength multiple is unusable due to an unacceptably large error. To overcome these limitations of the NRW approach, other methods have been proposed. Namely, multi sample/multi sample-position methods, and short-circuited waveguide methods. 2.3.2 Multi Sample / Multi Sample-Position Methods The main goal of these methods focuses on taking multiple measurements o f one sample in different positions inside the transmission line, or one measurement o f samples o f varying thickness. This method is mostly used for low-loss materials, where the NRW equations exhibit high uncertainty. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 These methods rely on the sample to be either: accurately positioned within the transmission line in two or more locations [9], [10] symmetric and of the same width, length, and thickness [9], [10], [11], [ 12] manufactured at specified thicknesses, e.g. double the thickness of another sample [ 11 ] In all cases, the sample has to be self-supported within a waveguide, or to be machined to fit inside a coaxial line. This was not possible with the samples of interest in this work. The samples could not be positioned within the waveguide without collapsing and accuracy o f placement was very poor. Also, different thicknesses of sample could not be produced without changing the substrate on which the sample was grown, therefore, changing its electrical properties. These methods, thus, were not suitable. 2.3.3 Short-Circuited Waveguide Method Two main methods have dominated this measurement technique, von Hippel’s use o f a fixed short-circuit to determine the electrical characteristics in a relatively narrow frequency band [13], and the sliding-short method [14]. One difficulty with utilizing a short-circuit, or a sliding-short, is the reliance on the Si i parameter alone, which typically have the higher uncertainty associated with the measurements. In the case of the fixed short-circuit method [I], an explicit solution can be obtained for the permittivity. On the other hand, Maze [14], using 48 or more positions in an actuator-controlled sliding-short had to solve for permittivity and permeability iteratively. To use Maze’s method, the sample must be self supporting and placed very accurately while measurements are taken; this was not possible with the samples of interest. Also, a sliding short was not available. Because o f the desire to take wideband measurements with maximum accuracy, von Hippel’s method did not seem practical. Very thin samples Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 do not work well when placed flush with a short circuit, as the electric field is approximately zero at this location. 2.3.4 Non-Linear Optimization All methods in chapters 2.3.1 - 2.3.3, solve for the permittivity and, if applicable, the permeability explicitly (with exception o f [14]). A non-linear least squares optimization approach instead of relying on the NRW equations or the modified explicit solutions of the several other methods was proposed by James Baker-Jarvis et al. at the National Institute o f Science and Technology (NIST) [1], [15], [16]. The NIST approach uses least-squares to solve for the coefficients o f polynomials that describe the complex permittivity (and permeability) over the entire frequency range of interest. An explanation of nonlinear optimization techniques is covered in Chapter 4 and is not reviewed here with a complete derivation of the non-linear optimization technique, the Levenberg-Marquardt Method, in [17]. The advantage of the optimization technique stems from the fact that no explicit equations are necessary for the material properties. Therefore, no initial guess is required as the multiple solutions arising from equation (2.11) are not an issue. Also, sections in the frequency band that are known to be within high equipment uncertainty can be weighted differently from the rest of the collected data. Because optimization is used, convergence on only select parameters o f the collected data is an option, e.g. only converging on S21 magnitude and phase data. Using the optimization technique, a theoretical relationship between the dielectric constant and the loss factor, as frequency is varied, can be represented. This relationship is known as the Kramers-Kronig (KK) relation. The KK relationship can be used as an optimization constraint in combination with the selection of the measured data, to further facilitate the optimization procedure. For the data where the Su parameter is close to 1 (a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 high uncertainty area for the VNA), more weight in the optimization algorithm can be placed on the KK relationship. Where data is in the low uncertainty area, the KK relationship can be set up to only play a role in detecting intermittent erroneous data. As least-squares is a curve-fitting routine, a very smooth curve is generated that is not drastically affected by single-frequency erroneous points, unlike explicit solutions. Figure 2.8 [1] shows a result o f the permittivity obtained in a very wide frequency band. The large variations in the non-optimized solution come from data collection that corresponds to points of very high uncertainty in the VNA. Cor a teadad glaaa over 0.043-11 <JHs tor the ) aad poirt-by-potat Hrh «iipi > ( ------- ) C Figure 2.8 Reproduced from [ 1] to portray smooth curve fit o f least squares solution in contrast to explicitly solved single-frequency point solutions. With this optimization approach, it is possible to converge to a solution when determining both permittivity and permeability even when using the short-circuit approach. This, however, requires at least two positions or two sample thicknesses. A very robust optimization algorithm must be used for characterization o f high-dielectric constant materials, since several local minima may be converged to before finding the global minimum. If the algorithm does not recognize local minima, the results will not be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 correct. An approximate initial guess could be used to speed up convergence, but is not needed as critically as for explicit solutions, as long as computing time is not a factor. The optimization technique seems to be the most promising solution method, as it has several advantages. The use of the KK relation is important and is a definite advantage. However, even without this relationship, the optimization technique seems superior, due to the curve-fitting nature o f least squares. In the single point-by-point explicit methods, many points have very high accuracy while many points have unacceptable accuracy. With the curve fit, the high accuracy o f some points may be sacrificed to a very small degree but the low accuracy points are brought within error tolerances. This makes the overall data set acceptable rather than only select points. A more detailed description of this method is given in Chapter 3.2 as this method is used for permittivity determination in this thesis. 2.4 Conclusions With the specific restraints placed on measurement arrangements, none of the methods previously described were directly suitable for the test samples o f interest. Therefore an alternative solution was required. Support for the material sample and accurate placement within the sample holder were needed. A two-layer approach with a material o f known electrical characteristics as a supporting structure was chosen and investigated. Figure 2.9 shows the sample configuration, which is to be placed inside a waveguide. This relatively large (to the unknown sample) and rigid structure has the ability to be placed very accurately before and during data collection. This setup can also aid in thickness measurement of the unknown sample, as the known sample can be measured very accurately before and after the unknown sample has been set on the structure, and the subtraction o f these two values equals the thickness o f the unknown sample. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 Knowi Substance Unknown Substance Figure 2.9 Known substance with unknown coating on its front face that make up the sample to be measured. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Chapter 3 3 Method of Analysis and Measurements To successfully measure materials o f interest and satisfy several constraints imposed, a modification to the measurement arrangement from previous work was required. The main challenge is the small thickness and flexibility of the samples, coupled with the requirement to measure the electrical characteristics in a broader frequency range than cavity methods allow for. Additional conditions are imposed by the need for measurements of large numbers o f different materials over a relatively brief time-span. The two-layer approach has been selected as the direction to focus the measurement effort. The equations governing the two-layer method are presented in this chapter, followed by an outline of advantages and limitations of this method. An iterative solution o f the equations is discussed next, and an analysis of sources of uncertainty is also presented. An FDTD program has been used for determination of errors that cannot be evaluated analytically, and the FDTD modeling of the problem is briefly described. 3.1 3.1.1 Two-layer Measurements Method Governing Equations in a Waveguide The two-layer configuration is shown in Figure 3.1 Zo 2 Zo Figure 3.1 Cross section o f waveguide with sample and acrylic inserted. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 where Zo is the characteristic impedance o f the empty waveguide, and Z*j and Z *2 are the characteristic impedances o f the respective material inside the waveguide. A (*) indicates a complex number. Reflection coefficients, r\, T*2, and T*3, are at each respective medium interface, as shown in Figure 3.1. The characteristic impedance are defined in the three different media as: where eo is the permittivity in free space (8.85 x 10‘12), po is the permeability in free space (4tc x 10‘7), e \i and e *r2 are the complex relative permittivities o f the respective materials shown in Figure 3.1. The relative permittivities are defined as e \i = e*i / eo and e *r2 = e *2 / eo, fc is the cut-off frequency o f the waveguide and is equal to 6.65GHz for Xband waveguide used in this work, f is the frequency (typically 401 equally spaced data points between 8.2GHz — 12.4GHz are collected). Impedance, Z%, in Equation (3.1) is a general representation for any material, although in this thesis it is a real number, as the loss factor o f acrylic is negligibly small. Parameters e *r2 and p *r2 from now on will be denoted as er2 and p r2 for simplicity, although they are complex. The reflection coefficients at the three interfaces, indicated by arrows, are: p _ ^ 2 ~ Zo 1 z 2+z0 p* _ ~ ^2 2 z; +z, p. _ ZQ—Z\ 3 z0+z; (3.2) Rearranging equations (3.2) and expressing T*2 as a combination o f H and T*3 gives: z;= -z 04 —!1 °r;+ i z ,= -z 0H±! 2 °r,-i r: = -(r 3+r,) - r/^+ i (33) Transmission coefficients within the two materials are defined as: T* = t ' =e~jrzdl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-4) 24 where the propagation constants are: r Iwt \ z co Y\ = ■ c CO Yi = I------------- (3.5) — V ^ r2 « r2 C Ka J where ‘a’ is the width of the waveguide (2.286cm for the X-band). The reflection and transmission coefficients are used to calculate S*u and S*2i at the outside boundary of the interfaces of the two-layer structure i.e. S*n on the material 1 side, S*2i on the material 2 side. For simplicity, stars are suppressed in all expressions that follow. 5 =^ v, S 2, = u2 (3.6) V, where: ", = r, (l - t ,! )•(r,: - t,: )+ r, (1- r/x ,3)•(1- t,’) =t,t,(i—r,’ )•(i—r32) v, = v : =r,rJ(i-T!:)(i-T 13)+(i-rJ!T!1)-(i-r,!Tli) (3.7) (3.8) (3.9) Equation (3.6) for S21 does not accurately reflect the S21 collected from the VNA. For Equation (3.6) to correctly represent actual collected data, a sample holder of the exact thickness o f the two layer structure would have to be inserted between the two calibration planes established during calibration. This is not practical for a large number of samples that have varying thicknesses. To correct for this, the calibration plane must be moved toward the load the distance o f the combined thicknesses o f the two materials. This is done in Equation (3.10), and the corrected S21 is labeled S210 The scattering parameters measured by the VNA are transferred to the plane of reference, thus: (3.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 where, for a non-magnetic medium: V 7 with fc being the cut-off frequency, denoted: with d| and d2 is the thickness o f the sample and the acrylic backing, respectively. Equations (3.6) and (3.10) now relate Sn and Sjic to the variables er), £ri, (iri, pr2, d|, and d 2 for a given waveguide operating in the fundamental mode. If material two has known dielectric constant and thickness, then Si i and S2ic are functions o f only e*ri, p.\i, and dj. 3.1.2 Advantages and Disadvantages The primary advantage of the two-layer method is that it facilitates measurements o f a thin flexible material in the waveguide. Without the second layer, the unknown material cannot be placed accurately. Because o f the material flexibility, even if one edge of the material were placed accurately, there would be no guarantee that the thin sheet was not concave in the center or on a slant. Another serious concern is related to the manipulation o f very thin, flexible and fragile samples, as their integrity may be compromised. Without a supporting structure, the unknown material can easily be damaged during the first measurement preventing repeated measurements, which are often required for comparison. One o f the major problems with the one-layer approach, as discussed in Chapter 2, was the ‘breakdown’ o f the explicit solution o f the NRW equations for guided half wavelength multiples. Furthermore, when approaching the equation breakdown, the VNA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 measurements have extremely high uncertainty, as the |Su| parameter approaches zero and IS21I approaches one. As discussed in chapter 2, and documented in the literature [I], a way to combat this is to use sufficiently thin samples of unknown material. However, this leads to a very small phase change in S21, and consequently to high uncertainty. By using the two-layer method, it is possible to use different backing materials (material 2 ) or backing material with different thickness to bypass conditions o f high uncertainty (|S| i| close to 0 ). Disadvantages of having the supporting block include added uncertainty due to the uncertainty in material thickness, and reliance on consistent material characteristics o f the known support dielectric. A minor disadvantage includes the added time of sample preparation. However, this can be alleviated with the mass production of material 2 blocks to very high tolerances, all from the same batch o f acrylic. As will be explored in further detail, the uncertainty in the solution for thin lossy samples is slightly increased by the presence of material 2 , which is obviously a disadvantage. 3.1.3 Sources of Uncertainty Many sources of uncertainty related to the mechanics of the measurement process have been reduced over the course of this project. These include: movement of the coaxial cables, repeated disconnection of the waveguide, and jostling o f the coax-coax and coaxwaveguide junctions, during and afrer calibration and measurements. These sources have been reduced by two vice-grips secured to the sample table holding the waveguide on each side o f the calibration plane. This way, very minimal movement o f the whole system is achieved during and afrer calibration. During the calibration, matched-load and short circuit attachments, as well as subsequent sample material insertions, only a slight twist of one waveguide is required. This drastically reduces movements o f the coaxial lines and torque on the junctions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 More important sources of uncertainty include thickness measurement o f both the unknown sample and the known material, uncertainty in the collected data from the VNA, and uncertainty due to small gaps in the unknown and known materials. The latter will be discussed in Chapter 3.3. 3.2 Iterative Solution Since the equations associated with the two-layer method are much more complicated than for the single layer solution, an explicit solution for e \ and ji*r is not feasible. Thus, optimization is used. Several optimization algorithms can be used, as many excellent routines for solving non-linear equations exist in commercial mathematical programs. Two ways o f solving for e*r and |i*r were explored, both for specific reasons. Ideally, causality should be used in the optimization, as discussed in chapters 3.2.1 and 3.2.2. However for reasons outlined in chapter 3.2.3 the two optimization methods used are non-causal. 3.2.1 Single Value Optimization in Narrow Frequency Bands This routine was created on the assumption that within a very narrow frequency range, on the order of 2-10MHz, e \ and fi\ are constants. The X-band is divided into a userdetermined number o f small sections (typically ~40) that are each treated completely separately from the others. Each section contains at least 6 data points collected from the VNA. Because the data points collected are complex, this gives 12 known convergence points in the system. Typically, only e \ is solved for, as magnetic materials have not been explored in as much detail, but when both e \ and |i*r are needed, this corresponds to 4 unknowns to be converged to. Optimization does not require an over-determined system or even an equally determined system, but the more points provided, the more accurate the convergence for the narrow frequency band. The optimization routine uses the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 MatLab ‘Isqnonlin’ function, which uses either the Gauss-Newton or LevenburgMarquart methods (user-determinable). The splitting of the frequency band into small sections was done initially because of an advantage over solving for one e \ and | i \ for the entire frequency range. This allowed both the permittivity and permeability to vary over the 8.2-12.4GHz frequency band. Although the X-band is not at all considered to be a wide-frequency band, it is known that the permittivity can vary inside this range. To dictate that the permittivity and/or the permeability must only be a single value would incur large errors. Figure 3.2 displays the ideal parameters of salt water, and the converged upon values for the narrow band optimization routine solution. It is clearly seen that while the narrow band single value optimization is not error free, it produces incomparably more accurate results than a complete X-band single value solution. Figure 3.2 only displays a portion o f the X-band (9.5 - 10.5GHz) frequency range to better show the piecewise (staircased) convergence of the algorithm. Better convergence would be attained with more sub-bands. A total of 30 (for entire X-band) are used in Figure 3.2, while 50 is the maximum. In Figure 3.2 and in subsequent figures throughout, e’r is equal to ep and e,p, and e”r is equal to epP and e^p. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Narrow FrcQjcncy Optimization Solution for an Ideal Salt Water Samnle. Frequency (GHz) Figure 3.2 The very narrow frequency sections o f constant complex permittivity approximate the ideal solution well while each section remains unrelated. Another advantage of this method pertains to erroneous data readings. Any completely erroneous S-parameter readings are contained inside a small portion o f results and are quite apparent. If a single value for permittivity were converged upon with several erroneous data points in the set, the value would be skewed in the direction of the error. Results on these look ‘staircased’. It may appear tempting to fit a straight line into this piecewise solution; however, this is not likely to be a correct solution for most samples that are measured. 3.2.2 Polynomial Whole-Band Optimization While the method o f Chapter 3.2.1 is quite robust, a more accurate solution can be attained by optimizing the coefficients o f a polynomial to represent each part o f the complex permittivity. Using a 4th order polynomial, this routine converges to the coefficients required to minimize the squared error (distance) from each individually solved frequency point. A ten-coefficient polynomial was found to be a good balance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 between processing time required and reliability of the algorithm. If more coefficients were used, the algorithm becomes less reliable in terms o f convergence. The whole-band optimization uses the piecewise solution of chapter 3.2.1 to provide an initial guess. Once these coefficients are supplied, the optimization refines the solution. This method works well and gives smooth full X-band results. It is the method predominately employed to obtain the results o f chapter 4. Under some circumstances, this polynomial method has difficulty converging to a reasonable result. In this case, the method of chapter 3.2.1 is used. An example of the erroneous solution is shown in chapter 4.2.1. 3.2.3 Causality of Solution The polynomial convergence requires introducing constraints regarding how e’r and e” r are inter-related. The method outlined in chapter 3.2.2 discussed polynomial convergence without this inter-relation. Using constraining equations developed by Debye, two separate polynomials for epsilon prime and double prime should be solved for in the ideal situation. Unfortunately, when these constraints were applied, it was found that the convergence highly depended upon the initial guess. According to the equations (3.13), the variables to converge to include £«, £r», and a time constant, x [18]. (3.13) For a single relaxation process, £ „ is the real part of the complex permittivity at very low frequency, £ ^ is the real part o f the complex permittivity at a high frequency limit, and x is the relaxation time constant [18]. Therefore, if e re, £r», and x are known, then the complex permittivity at any frequency is determined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 With these constraints, the optimization never converged to a consistent solution. The problem is due to the too narrow frequency range o f the X-band to give a unique relationship between e’r and e” r. Also, a material may exhibit more than one Debye relaxation [18]. Multiple Debye relaxations, represented by Equation (3.14), can be set up in the optimization algorithm, but this adds more variables and still does not guarantee that a unique, or even close to accurate solution, can be attained. With the multiple Debye relaxations, once again solutions were highly dependant upon the initial guess. Therefore it was decided not to use this convergence method until measurements in a much broader frequency range were available. However, multi-waveguide, or free-space, measurements are beyond the scope of this thesis. A material exhibiting more than one Debye term can be described as in equation (3.14). e ' M = e „ + e" e~ - + £" - £~ +... (3.14) l +(awj2 3.3 l + (awr i )‘ Main Sources of Uncertainty The sources o f error listed in Chapter 3.1 constitute only a part o f the uncertainty associated with measuring materials in a waveguide. With due care, these uncertainties can be minimized. Three main sources o f uncertainty are: sample thickness measurements, uncertainty in the collected data from the VNA, and small gaps or facial inconsistencies in the sample material. 3.3.1 Sample Thickness While the acrylic backing layer is very rigid and milled to very high tolerances, the unknown sample is often either grown directly on the acrylic, or prepared separately and placed on the backing acrylic. When the sample is grown directly on the acrylic, the total Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 material thickness is measured and the acrylic thickness subtracted to give the unknown sample thickness. When the sample is placed on the acrylic, it can be measured only when it is not so thin and flexible that the measurement process would damage it. In many cases, even when the sample is grown without the acrylic, it is first only measured once on the protective backing. In addition to being flexible, the samples often have a pliable, rubber-like consistency, which makes measuring the material with calipers quite challenging, as no pressure must be exerted on the sample. In many cases, two people measuring the same sample arrive at slightly different results, differing up to 0 . 1mm. With the high tolerance o f the backing acrylic, slight force was needed to place the block inside the waveguide. Also, because of the waveguide arrangement, the structure was placed in the waveguide acrylic side first. The two-layer structure was then slid even with the calibration plane. This slight compression of the unknown material was distributed uniformly by using a waveguide short-circuit to apply pressure to the structure until it was precisely even with the calibration plane. Both factors in sample handling give rise to an uncertainty in its thickness. To rectify this situation, the sample thickness can be added as a variable in the optimization algorithm with its initially measured thickness as the initial guess. A constraint is placed on the envelope of possible thicknesses the optimization program may converge to; often set at 5 or 10%. Comparing the originally measured thickness with the converged-to thickness is a good indication o f how well the optimization program performed. This, will be discussed briefly in chapter S 3.3.2 Uncertainty in Vector Network Analyzer Data Equations (3.15) and (3.16) represent general expressions for the uncertainty o f both real and imaginary permittivity and permeability introduced by the measured scattering parameters [I]. The brief derivation that follows evaluates complex permittivity dependent on Sit. The real and imaginary uncertainty components can be evaluated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 separately by separating the complex term o f permittivity into its real and imaginary part. To evaluate the uncertainty o f complex permittivity depending on both Sn and S21, the square root o f the addition of the squared terms is taken. The same is applied to the uncertainty o f the permeability. v rI _ ** (3.15) AIS. d0„ -r 1 _ 1 M' ri V ty'rl ' P » * . (3.16) Mrt where a = 11 or 21, superscript * stands for prime (‘) or double prime (“), A0 is the uncertainty in the phase o f the scattering parameters, A|Sa| is the uncertainty of the magnitude o f the scattering parameter. Both A6 and A|Sa| are provided in the System Performance Specifications supplied with the VNA 8720C used Figure 7.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 Uncertainty of complex permittivity depending on S2 1 - A£r, _ 1 a*;, AS,, a|s2,| d 02l (3.17) w V 1 a*;, as,. as,, as,, _ as,, ar, as,, azj a^'“"af'ai^+"a^"ai^ 1 2 3 (3.18) 4 Equation (3.18) has 4 unrelated components that have been labeled 1-4 and will be evaluated individually; only the significant derivation steps will be shown. (3.19) ^ = -2 r,r,r 2(i-r32) ar, |^ - =r 3(i- r,2)-(i- r22)- 2r,r,2(1- r32r22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.20) 35 2r,r,r3(1- r,2)- i r rr, (1- r 32r,2) (3 .2 2 ) where m, U2,vi, and v2 can be found in equations (3.7), (3.8), and (3.9), respectively. Su and S2i are found in equation (3.6). Ti, r 2, H , T| and T2 are stated in equations (3.2) and (3.4). where Ti and Z| are stated in equations (3.2) and (3.3) Therefore: » L -----------------(z, +Z 0)2TJo2Mr d£r‘, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.23) where Ti and yi are given by equations (3.4) and (3.5) Therefore: (3.24) 3.3.3 Small Gaps in Unknown Sample On occasion, test samples have been delivered with small facial inconsistencies, either appearing as dents on the face of the sample or gaps straight through to the backing acrylic. Depending on the material, preparation of a homogeneously thick sample may not have been possible. Also, when the sample is made separately and placed on the acrylic, it may not have exact dimensions. The sample may be up to 1mm too short in height (see Figure 3.3), but more typically is about 0.1mm shorter than the acrylic. 1mm Figure 3.3 A two-layer structure with the sample not perfectly matching the backing acrylic (the waveguide height). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Figure 3.3 illustrates the ‘worst-case* sample imperfection. Typically, as mentioned previously, if any inconsistencies are present, they occur as small gaps between the sample and the waveguide or dents located on the external face. Figure 3.4 shows a typical dent configuration. 0.5mm Figure 3.4 A two-layer structure with a small dent in the sample. It is impossible to analytically determine the uncertainty produced by these unsymmetrical gaps and dents in the sample surface. A Finite Difference Time Domain (FDTD) simulation package has been used to model these inconsistencies in test samples and the results are shown in chapter S. A complete list of simulated sample facial defects is found in Chapter 3.4.2 3.4 Modeling of Sample Imperfections The FDTD modeling method was chosen to perform the simulations for two main reasons. The time domain approach enabled the gathering o f information required at all frequencies of interest in one simulation. This was a significant advantage, as the measurements entailed collecting data over the entire X-band. The second reason was the availability of tested FDTD software that had been developed at the UVic Electromagnetics Lab. As the structures simulated (waveguide, rectangular blocks o f dielectric material) and the use o f Perfectly Matched Layers (PMLs) were previously tested components o f this FDTD package, no additional test procedures were needed to complete the modeling. The FDTD program enables two ways of constructing the finite computational domain. These include using a standard cubic / parallelepiped Yee-cell grid [19], [20] of constant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 volumes, or the graded mesh [19], [20]. The graded mesh allows for higher resolution inside the computational domain in specific areas and lower resolution in the areas of lesser interest. This enables the computing time of simulations, e.g. those performed for this thesis, to remain reasonable. A two-dimensional FDTD approach was not feasible in this case as transient and fundamental mode analysis is needed to ensure the propagating waves interacted with the sample and acrylic in the same manner as in a real X-band waveguide. Also, asymmetrical gap and facial inconsistencies are introduced in chapter S. To simulate the waveguide, termination o f four of the six computational domain walls was done with infinitely conductive electric walls. These four walls simulated the waveguide walls along the x and y axis. To terminate the positive and negative z-axis, 9layer Berenger's PML with approximately -60dB reflection for normal incidence were used. This amount was adequate as the VNA noise floor is approximately at -40dB, and there was no need to sacrifice computational time to further reduce the reflection. Graded mesh was used in two directions, namely the y and z-axis. It was needed in the y-axis to increase the resolution near one edge o f the waveguide to simulate thin gaps. The cell size for this direction varied from 0.5mm to 0.1mm with a maximum side-by-side cell difference o f 11.56%. The graded mesh in the z-axis was needed as the samples were in the range o f 0 . 1mm thick, and this resolution was not needed in the rest of the free-space waveguide. The maximum side-by-side cell difference in the z direction was 2.5%. The x-axis (modeled at 23mm in width) and the y-axis (modeled at 10mm in height) corresponded well to the actual X-band waveguide dimensions of 22.86mm and 10.16mm, respectively. The computational domain in the z direction needed to be relatively long for several reasons. The first reason was higher order non-propagating evanescent waves generated by the excitation. The distance between the excitation and the material had to be sufficient to ensure that only the TEoi was present. The second reason was related to data collection. From the excitation plane, the wave travels along the positive z-axis o f the waveguide, it first encounters the Su data collection point that is recording the Ey and Hx fields at every time step. The full excitation pulse must completely travel through this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 data collection point before reaching the sample and reflecting back in the negative z direction. Both incident and reflected waves must be completely distinguishable to enable splitting the time sequence o f the simulation, in order to determine the Sn parameter. 500 mm 968 voxels 23 mm 23 voxels Figure 3.5 Graded mesh computational domain constructed for simulations of waveguide. Smallest voxel size = (1 x 0.1 x 0.1) mm Largest voxel size = (1 x 0.5 x 1) mm Number of voxels = 756976 The excitation used simulated the Ey field that exists in the waveguide for the TEio mode. As shown in Figure 3.6, the Ey was excited with a sinusoidal amplitude pattern to stimulate the TEio mode. The sinusoid was approximated with five equally space voltage spikes across the waveguide, at 4mm, 8 mm, 12mm, 15mm, and 19mm. The field inside the waveguide was then sampled at points further along the z axis to ensure that TEio (the fundamental mode) was the only significant mode. The source was modeled as a ‘Soft Source’. This is opposed to the more common ‘Hard Source’. Both types of sources dictate an electric or magnetic field o f a certain amplitude in the computational domain at a series of time steps. A hard source did not fit the specific criteria for the simulations, as the wave needed to travel back through the excitation point o f origin in the negative z direction once the excitation was zero (or turned off). A hard source is and remains a short circuit, much like a physical antenna, even when turned off. This is not acceptable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 in a waveguide where metal wires have a significant impact on the propagation o f the wave. A soft source, once turned off, becomes free space, as required. 0.608 0.924 1 0.924 0.609 Figure 3.6 Five soft sources used to excite the fundamental mode, TEio. in the waveguide. Amplitudes have a sinusoidal amplitude from 0 to 1. E at X . 2mm intervals inside waveguide Pointe collected at S ,, Data Collection Point Time Steps X Component in Waveguide (mn Figure 3.7 A cross section of the waveguide at z = 30mm throughout the entire simulation. The fundamental mode is clearly visible, as well as the Gaussian Excitation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 Comparison of fundamental Ey, Hx. and H . wilh non-fundamental Ex, E;, and Hy a t S ,, collection point ' 0 J 'Jl ' ■' - - T- ■ 1T11 I 1 .lif lj 1lU . - ,Jli , jp 1 1 1 ' ! 1 uTui'uT 111 1 2000 4000 6000 8000 10000 Time Steps 12000 14000 16000 16000 2000 4000 6000 10000 6000 Time Steps 12000 14000 16000 16000 Figure 3.8 Plot of three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental fields (Ex, Ez, Hy) collected at z = 15mm, which is 6mm away from excitation and corresponds to the Su collection point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Comparison of fundamental Ey Hx, and H, wilh non-fundamental E(. E2, and Hy a t S 2, collection point 2000 4000 5000 8000 10000 Time Steps 12000 14000 16000 18000 .*10 ^ 2 u. * H. ,ii I -«*r o I-1 1-2 -3 o_ 2000 4000 6000 8000 10000 Time Steps 12000 14000 16000 18000 Figure 3.9 Plot o f all three fundamental fields (Ey, Hz, Hx) in relation to non fundamental fields (Ex, Ez, Hy) collected at z = 480mm, which is 6mm away from excitation and corresponds to the S21 collection point. Excitation is accomplished with a Frequency Shifted Gaussian Pulse (FSGP) centered at 10GHz and approximately 4GHz wide. The pulse is created over 4000 time steps with a discrete time difference (dt) o f 2.2354*1013 s. Figure 3.10c is the time domain plot of the pulse, while Figure 3.10d displays the frequency domain plot. It is noted that the time domain pulse starts at time step = I with an amplitude o f 0.0138, which is a deviation from zero. This may produce higher frequency transients, however, they are very low in magnitude. The frequency domain plot of Figure 3.10d shows the narrow band pulse generated, the bars on the plot indicating the frequency band o f interest (8.2 - 12.4 GHz). The time step numbers that correspond to the frequency band bars are explained in chapter 3.4.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.10 Plot A shows the Cosine waveform that is multiplied to Plot B, which is the pure Gaussian pulse. Plot C, shows complete time domain Cosine Gaussian Pulse generated over 4000 time steps, Plot D displays the narrow band frequency pulse created with 2 18 points (mostly zero-padded) which places 8.191 - 12.4 GHz between discrete points 480 and 727 with a 17.1 MHz frequency step (df). A total o f 18000 time steps were used which corresponded to a total simulation time of 40.24ns with a dt o f 2.2354xl0'13 s. 3.4.1 VNA Simulation In order to manipulate the data to simulate data collection from the VNA, calibration o f the FDTD S-parameters is essential. Standard calibration equations are used after four specific FDTD simulations are completed; three simulations are for Sii and one for S21. After determining where the calibration plane had to be placed inside the modeled waveguide, the first simulation was a short circuit. The calibration plane was located on the z-axis value where all the subsequent materials to be measured would be placed. For all simulations, this value was set at z = 250mm with the total z-axis length o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 waveguide being 500mm. Also, before calibration, all observation and excitation points had to be determined and remain unchanged, so as not to change the phase angle of the collected Su and S21 parameters. The calibration process for the Su parameter involves three specific simulations [21]. These three simulations are required to create a determined system of three equations with three unknowns. The three vector unknowns in the system are E d f, E sf, E r f which relate to three specific calibration requirements for a VNA. The Directivity Error, Edf, compensates for the power that is reflected back to the VNA, by devices such as couplers and transitions, before it reaches the sample. The Source Match Error, E sf, compensates for the imperfect transitions from the system back to the source. A small portion of the signal is reflected back into the system and interacts with the sample for repeated times. The Tracking Error, E rf, is caused by variations in magnitude and phase flatness versus frequency between the test and reference signal paths. Equation (3.25) shows the three main uncertainty parameters in a system [21], where Sum is measured S u , and Sua is actual. Equation (3.27) shows the system equation with three unknowns. Three circumstances exist where the S ua is known and they are outlined next. The first, a perfect electric conductor (PEC) is used for calibration plane termination. The information gathered from this simulation is given the label Susc, where the Sua is known to be -1 (or magnitude 1, phase 180°) and corresponds to equation (3.26)a. The second simulation is an open circuit. This is not possible in practice, but can be taken advantage o f while modeling. A perfect magnetic conductor (PMC) is placed at the same location as the short circuit and maintains an Sua of 1 (or magnitude 1, phase 0°) throughout the entire frequency band. This corresponds to equation (3.26)b. A small compensation is added to the gathered data, as the PMC specified at a certain interface in FDTD, actually exists half way between the voxels. Therefore, to compensate for the slight shift in phase that would occur, 0.05mm o f phase shift is added to the collected Su data (for the grid size o f 0.1mm). The data collected from this calibration run was labeled Suoc- The third and final calibration procedure is the matched load. This was accomplished by placing the PML discussed in chapter 3.4 exactly at the calibration plane. The third procedure, labeled S u mi, is not very critical in FDTD simulation, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 especially with the simple setup being discussed, and the result is very close to zero, this corresponding to equation (3.26)c. s ,, u = e „ + 1 t.SF • 01I/( < General Form o f Calibration Equation S C =E i ( 1) ' e kf ^ l - £ s f (-l) (a) _ C _ —/T s _£ + ^ l- £ * .( l) (b) 3-25> (3.26) ^ x 0 • E rf *** 1 - £ JF o (c) Once the three vectors are established, S ua is solved for to produce equation (3.27). Any measured Si i is now applied to this equation before analysis. S UA = ----E s f C^i t Af (3.27) ^ df ) "F E rf Calibration o f S21 only involves one simulation, namely a full waveguide with no sample inside and no discontinuities at the calibration plane. Once data for calibration and data for an unknown material have been collected, relevant data are selected from the observation record. After the truncation is completed, a Fast Fourier Transform (FFT) is performed on both data arrays. Zero Padding is utilized to increase frequency resolution (df decreases). The equally sized arrays are then divided. This gives the magnitude and phase difference data, as both arrays are complex. An Inverse Fast Fourier Transform (IFFT) is applied to the array to obtain time domain data. FFT(Sl m ) N S Ma = IFFT FFT(Sl m ) / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.28) 46 Since the purpose o f simulating the measurement procedures in the FDTD is to mimic the actual collected data from the VNA, the data collected from the FDTD will have to be manipulated to be as similar as possible to the VNA output. Two programs were written to condition the modeled data, one for Su and one for S21. The FDTD output produces a spectral data set o f 4096 magnitude and phase output points, while the VNA produces a set of 401 data points. Upon inspection, approximately 3600 points of the data set from the FDTD program were within the X-Band frequency. For Su after calibration, out o f these -3600 points, every ninth was recorded into a text file in the same format as the HP VNA output. This is a three-column set o f data containing the frequency (from 8.212.4GHz), magnitude and phase arrays. To condition the S21 output, a more complicated procedure had to be constructed. As the current state o f the data contained the frequency response from 0 to 2236.8GHz (a result from a 218 zero padding and a dt = 2.23535034e-l3), only a very small portion of this array was needed. Based on the highest frequency of 2236.8GHz and the array size, this corresponded to a discrete frequency step of df = 17.1 MHz. It was determined that array steps 480 and 727 were the lower and upper points of the X-Band frequency range of interest. Once a frequency array was established that contained the S21 data in (727480=) 247 points, and the associated calibrated magnitude and phase S21 data were linked, a polynomial is fit to this data. A very high degree polynomial (20) was used, as computation time or memory limitations were not an issue. Two high degree polynomials now existed, both in the form y = C + C|X + C2X2 + ... one o f S21 magnitude the other of phase, both with frequency corresponding to x. From the polynomial, the SI 1 frequency array was systematically inserted as the values for x in both polynomials, and the magnitude and phase of S21 recorded. Two files with exact frequency arrays and corresponding S-Parameters now exist, simulating the VNA. Once all other relevant data is inserted into the files, the data is ready to be processed with the same procedure as standard VNA data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 3.4.2 FDTD Evaluation of Gaps and Facial Inconsistencies Simulations were conducted with the l st-layer (the unknown material) having symmetric and asymmetric inconsistencies. The configurations are illustrated in Figure 3.11. Results o f simulations are in Chapter S.3. 0.5mm 0.1mm 9.9mm □ O 0.5mm □ .9 mm O 0.5mm □ Figure 3.11 Facial inconsistencies modeled; Gaps are on the left, dents on the right. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Chapter 4 4 Results of Uncertainty Analysis and Measurements The electrical properties o f the test samples varied with e r ranging from close to 1 to several hundred, while e r ranged from ~0 to several hundred. With materials having also different electrical characteristics within each batch, an intelligent initial guess for the optimization algorithm was not possible. Not all results from the measurements are presented, for several reasons. Many o f the samples received were too thin and lossless, too inhomogeneous, or o f a thickness resulting in high uncertainty. Results shown are for samples that have acceptable ‘perceived’ uncertainty. ‘Perceived’ uncertainty is determined based on consultation of uncertainty plots of thickness o f both layers versus complex permittivity, perceived thickness measurement accuracy and acceptable deviation in terms of sample thickness and facial inconsistencies (dents and gaps). Static conductivities have been measured for several o f the materials at their point of manufacture and is plotted on several graphs (as converted to the DC conductivity component of the imaginary permittivity). Equation (4.1)[18] portrays this relationship and its definition is explained in detail in chapter 7.1. €' ^d c _ + ^ac_ (4J) coe0 (oeQ 4.1 Uncertainty Analysis The thicknesses of the sample and backing acrylic both have a profound effect on the uncertainty of the measurements. The value o f permittivity also has a large effect on the uncertainty. While the thickness is determined before any measurements are made, optimizing the thickness for the particular complex permittivity of an uncharacterized material is impossible. Therefore, minimizing the potential uncertainty is the goal when Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 choosing a sample and acrylic thickness for the initial round o f measurements of a new sample. In the following sections, only realistic (practical) sample thickness and complex permittivity ranges are analyzed. Typical sample thickness has been in the order of 0.1 1.0mm, although several successful manufacturing attempts at a thicker sample have been made. The thickest sample analyzed for uncertainty is 3.0mm. Acrylic thickness is treated similarly. The minimum thickness available is 1.2mm, while acrylic thicknesses over 5.2mm become too difficult (awkward) to insert into the waveguide. All results of the uncertainty analysis presented in chapters 4.1.1 - 4.1.3 assume ideal samples with no facial inconsistencies. Four combinations of property characteristics have been chosen as a representative of the samples tested. The characteristics of these four materials are summarized in Table 4.1. Table 4.1 The dielectric constant and loss factor of the four materials analyzed. Material 1 2 3 A Real Permittivity Imaginary Permittivity ( E ’r) ( E ” r) 80 20 100 3 -10 -40 -200 -0.0001 Data in Table 4.1 do not show separately static (DC) and AC conductivities, only the imaginary permittivity (loss factor). The static conductivity is measured directly after the sample is manufactured and can be subtracted from the effective conductivity if desired, in post-processing. All of these samples and subsequent plots assume non-magnetic materials (|i’r = 1, M-’V = 0). Uncertainty o f real and imaginary parts of the permittivity determined from both S| i and S21 will be displayed for Material 1. After the analysis, it became apparent that for non magnetic materials is was unnecessary to include the uncertainty plots due to the combined uncertainty of the S-Parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Chapters 4.1.1. - 4.1.4 analyze the materials 1,2,3, and A, listed in Table 4.1, respectively. Three different uncertainty plots are presented, the unique format o f which are beneficial for the analysis. All contain the uncertainty of either e’ or e” due to Si i and S21 (or due solely to S21) on the vertical axis. The plots are as follows. Plot Type I: Three-dimensional plots with sample thickness and acrylic thickness as variables on the opposing axes (e.g. Figure 4.1, Figure 4.2). These plots are for a specific material (e.g. £’ = 80, e” = 10) at a single frequency (e.g. f = 10GHz). The thickness of the sample material varies from 0.1mm-3mm, while the acrylic varies from 1mm to 5mm. Although the plots include sample thicknesses greater than 1mm, these are very rarely manufactured in practice. Measurement uncertainties of many materials are reduced when their thickness is increased beyond 1mm (as will be shown in Chapter 4.1.1. 4.1.4.), therefore including these sample thicknesses that may possibly be attainable in the future is useful. Plot Type 2: Pseudo three-dimensional plots of uncertainty vs. sample thickness with view oriented directly down the frequency axis (8.2-12.4GHz) (e.g. Figure 4.3, Figure 4.4). These are given in a series of four plots of increasing acrylic thickness (1mm, 2mm, 4mm, 5mm) These plots are very useful as the uncertainty for many materials varies substantially over the X-band. Each plot shows the entire envelope of uncertainty over the frequency range of interest. Plot Type 3: Two-dimensional plots containing five X = e’/ e” ratios. In these plots, e’r increases from 1 - 1000, while frequency remains at 10GHz, and sample and acrylic thickness remain constant. Values for X were chosen to be 0.2, 0.5, 1, 2, and 5, as these values best represent many measured values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 4.1.1 Material 1, Dielectric Constant = 80, Loss Factor = 10 eP • #0 ePO „ • 10. sample thickness 0.1-3mm. mstensi thickness ^ freq. 10GHz 0.9 . 0.4. 0.1 1 Thickness of matenal (mm) Thickness of Acetic (mm) Figure 4.1 Uncertainty o f the dielectric constant (e\) due to errors in measurements o f Sii and S21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Cp — 8 0 6pp — 10, sample thickness 0.1-3mm. material thickness 1-Smm. freq. 10GHz Thickness of matenal (mm) Thiekness of Acrylic (mm) Figure 4.2 Uncertainty of the loss factor (e” r) due to errors in measurements of Su and S21 crQ• 00 er„Q• 10 Freq • 9 - 12 GHz Sample thickness • 0.1 - 3mm 0.3 r 2 1 Thickness of Sample xIO J Thiekness of Somple Thickness of Sample x 10 ' Thickness of Sample 2 1 0 x 0 s iq ° Figure 4.3 Uncertainty of the dielectric constant (e’r) due to errors in measurements o f Su and S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 e r • ?? er • ?? Freq > 9 - 1 2 GHz Sample thickness • 0.5 - 3mm Thickness of Sample 110~3 Thickness of Sample -J 3 Thickness of Sample x 2 1 Thiekness of Sample 0 * ro 3 Figure 4.4 Uncertainty o f the loss factor (e” ) due to errors in measurements o f Sn and S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). ep • 90 c9Q • 10. sample thickness 0.1-3mm. material thickness 1-5mrn. freq. IOGHi 0.1 1 Thickness of material (mm) Thickness of Acrylic (mm) Figure 4.5 Uncertainty o f the dielectric constant (e’r) due to errors in measurements of S21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 eQ■ 80 • 10. sample thickness 0.1-3m m material thickness 1-5mm. freq. 10GHz Thickness of matenal (mm| Thickness of Acrylic (mm) Figure 4.6 Uncertainty of the loss factor (e” r) due to errors in measurements of S21 erp • 60 er_Qa 10 Freq • 9 - 12 GHz Sample thickness ■ 0.1 - 3mm 0-1 r 2 Thickness of Sample % 1 Thickness of Sample 0 * io*'J 0.1 r € 0.05 r ! 0.0S 2 Thickness of Sample x 1Q-3 1 Thickness of Sample 0 x iq ' 3 Figure 4.7 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 erp ■ BOCTpp • 10 Freq • 9 - 12 GHz Sample thickness • 0.1 - 3mm ■ 2 0.9 r m 1 Thickness of Sample 0 0 IM 3 Thickness of Sample , io '3 3 TNckness of Sample Thickness of Sample Figure 4.8 Uncertainty of the loss factor (e” r) due to errors in measurements o f S 21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Both Figure 4.1 and Figure 4.2 show relative uncertainties due to errors in measurement of Su and S21 - Although the uncertainty decreases down to between 5% and 15% when the thickness of the sample is 1.6mm, for other thicknesses, it is unacceptably large. For this sample, a very thin sample coupled with a thin acrylic is beneficial, as the uncertainty rapidly decreases for real and imaginary permittivities near the smallest values. The uncertainty is unacceptably high mostly because of the Si 1 uncertainty term. After analysis o f several other plots of various materials, it was determined that the inclusion o f Su uncertainty into the optimization algorithm contributed more uncertainty to the convergence than was beneficial from the extra data points. The Su uncertainty cannot be eliminated when the material is expected to have magnetic properties, but for materials with |i.’r = 1, only Su or S21 data suffices. Uncertainties are presented for material 1 that include Su and S21, but for the remaining materials, only the uncertainty o f S21 is given. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Although acceptable uncertainty can be achieved using the parameters from Figure 4.1 and Figure 4.2 because prior knowledge o f the material properties is not available, very high uncertainty would likely result. Figure 4.3 and Figure 4.4 again show the uncertainties o f both Sn and S21. If the main priority for this sample were to determine only the dielectric constant, then maintaining a sample thickness greater than 1.6mm ensures an uncertainty o f less than 17%. Unfortunately, the uncertainty for the imaginary permittivity is very large in this case for a large sample thickness. There is an abrupt drop in uncertainty when the sample is approximately 1.6mm but the slopes on either side o f this value are very steep and achieve a value o f 100% very quickly. Figure 4.5 and Figure 4.6 show the uncertainty of the same material when only S21 data are used. The uncertainty is much lower when compared to Figure 4.1 and Figure 4.2, respectively. Several combinations of thicknesses o f the sample or acrylic are acceptable for the dielectric constant. The maximum and minimum errors occur for various sample and acrylic thicknesses. Their locations, however, cannot be easily explained by the relative (to the guided wavelength) sample and acrylic thickness. The relevant thicknesses are given in Table 4.2. The wave undergoes multiple reflections at the three interfaces, thus a quarter and half-wave transformer analogy cannot be used. However, the minima and maxima o f the uncertainty can be easily explained on the polar plots shown in Figure 4.9. With reference to Figure 7.7 in the appendix, it is clear that the Si 1 uncertainty is very large with approaching a magnitude value of 1. Both Figure 4.9 and Figure 4.10 show Su approaching 1 for specific thicknesses of sample. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 S „ and Sg, at 10GHz, for a matarial with a rp = 80 and a rpp = 10 120 0.8 0.6 \3 0 150 Blue Green Purple Cyan n ■»' acryUc - 1mm - acrylic - 2mm • acrylic * 4mm - acryliq * 5mm 180 330 210 240 270 Figure 4.9 S-parameter plot o f two-layer structure at the single frequency o f 10GHz. Sample thickness varies from 0.01mm to 3mm Magnitude of both S11 and S21 at 10GHz for a matarial with a r_ = 80 an d e rnn = 10 osj0.8 is 0.7 i Blue Green Purple Cyan - acrylic * 1mm acrylic*2m m acrylic*4m m acrylic • 5mm 0.3 0.1 I 1.5 2 Thickness ot A crylic in mm 2.5 Figure 4.10 The magnitude o f both S-parameters o f the two-layer structure at 10GHz, with sample thickness varying from 0.01mm to 3mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Table 4.2 Displaying cut-off frequency, and guided half-wavelength, and guided quarter-wavelength distances for the X-band. Free-Space (Inside Waveguide) Material 1 (e’r = 80) (Inside Waveguide) Acrylic (e’r = 2.65) (Inside Waveguide) fc (GHz) 6.56 X,. (mm) 40 0.5A* (mm) 20 0.25A* (nun) 10 0.734 3.4 1.7 0.85 4.03 20 10 5 Figure 4.7 and Figure 4.8 show the uncertainties due to only S21. It can be seen that the majority o f the values are much lower than when Su is included, as noted previously. The high uncertainty region is now visible and exists (Figure 4.7) for a sample thickness o f between 1 - 1.4mm. However, the uncertainty does not exceed 8% for any sample or acrylic thickness at any frequency. Very low uncertainties are obtained for samples thicker than about 1.5mm. Also, as desired for this work, low uncertainty is obtained for thicknesses of 0.1 -1 mm with 1-2mm acrylic. 4.1.2 Material 2, Dielectric Constant = 20, Loss Factor = 40 Figure 4.11 to Figure 4.14 present the results o f the uncertainty analysis based on S21 data only. Material 2 is very consistent in terms o f trends in uncertainty values for both the dielectric constant and the loss factor. This makes the optimum thickness obvious. As seen in Figure 4.11, the uncertainty o f the dielectric constant steadily decreases with increasing sample thickness, and increases slightly with acrylic thickness. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 ep • 20 • 40. sample thickness 0.1-3mm, material thickness 1-5mm, freq. 10GHz TNckness of material (mm) TNckness of Acrylic (mm) Figure 4.11 Uncertainty of the dielectric constant (e’r) due to errors in measurements o f S21 e • 20 c D 0.1 3C • 40. sample thickness 0.1-3mm, matenal thickness 1-5mm» freq 10GHz j I i Thickness at material (mmt Thickness at Acrylic (mm) Figure 4.12 Uncertainty of the loss factor (e’\ ) due to errors in measurements of S21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 er„ • 20 er^ • 40 Freq * 9 - 1 2 GHz Sample thickness ■ 0.1 - 3mm 0.1 O.OS I 0.05 3 Thiekness of Sample x io ’3 TNckness of Sample * iq ' j TNckness of Sample x io*j 0.1 I.0.0S 0.09 Thiekness of Sample Figure 4.13 Uncertainty of the dielectric constant (e’r) due to errors in measurements of S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). erp ■ 20 erCO <• 40 Freq • 9 -12 GHz Sample thehneas • 0.1 - 3mm f fgO.OS 3 | 0.05 0 Thickness of Sample TNckness of Sample x io '2 * iq - 0.1 £ | 0.05 I 0.05 3K 0 TNckness of Sample % TNckness of Sample Figure 4.14 Uncertainty of the loss factor (e”r) due to errors in measurements o f S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 The envelope o f uncertainty over the entire frequency range (Figure 4.13 and Figure 4.14) remains nearly constant with changing acrylic thickness. If only very thin samples are considered, a thin acrylic thickness is preferred. Thus, an acrylic thickness of approximately 1.Omm is optimal for samples thinner than 0.5mm. This advantage is very small. For material 2, measurements benefit from a sample thickness in excess of 0.7mm, and placed on the thinnest acrylic available, e.g. 3mm or 1.2mm. Samples o f the permittivity similar to Material 2 were quite common and usually had thicknesses o f approximately 0.2 - 0.5mm. A concerted effort was made to increase this thickness once these plots were analyzed. 4.1.3 Material 3, Dielectric Constant = 100, Loss Factor = 200 This material is similar to the material of chapter 4.1.2 and was also very common. Once again, a thick sample with a thin acrylic is the optimal configuration, as apparent from Figure 4.17 and Figure 4.18. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 ep *100 ■ 200. sample thickness 0.1-3mm. matenal thickness 1-Smm. freq. 10GHz TNckness ol matenal (mm) TNckness of Acrylic (mm) Figure 4.15 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f S21 e„C> 100 e 9 _C• 200. sample thickness 0.1-3mm. material thickness 1-5mm, freq 10GHz 0.1 - I 0.0S 4 3 TNckness of material (mm) TNckness of Acrylic (mm) Figure 4.16 Uncertainty o f the loss factor (e”r) due to errors in measurements o f S 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 erp • 100 er^ • 200 Freq • 9 -1 2 GHz Sample tfvckness • 0.1 - 3mm 0.1 1 0.05 I 0.05 Thidinejs of Sample x 10 0 Thiekness of Sample x 1fl-- Thickness of Sample t 0.1 £c■ 1 O.OS a £ | 0.05 3K TNckness of Sample x Figure 4.17 Uncertainty of the dielectric constant (e’r) due to errors in measurements of S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). erp • 100 erx • 200 Freq ■ 9 - 12 GHz Sample thickness « 0.1 - 3mm 0.1 9 I 0.05 O.OS Thiekness of Sample TNckness of Sample x i q° Thickness of Sample x lfl-i *c i 0.05 1 0.05 TNckness of Sample xIO' Figure 4.18 Uncertainty o f the loss factor (e”r) due to errors in measurements of S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 A thick sample of such a high-dielectric constant material may produce smaller measurement uncertainty. These uncertainty equations do not consider over-moding within the sample. Other modes that may be excited upon transition between propagation mediums and which are evanescent everywhere [18], but inside the sample, may contribute additional convergence error. 4.1.4 Material A, Dielectric Constant = 3, Loss Factor = 0.0001 Figure 4.19 to Figure 4.22 present the results o f the uncertainty analysis for this material. The uncertainty of the loss factor of this material is very large unless the sample is sufficiently thick. This is not unexpected, as any deviation from 0.0001 (considering an accuracy o f 4 decimal places collected from the VNA) will result in over 100%. Therefore, for this material, plots o f the loss factor can be used as a guide and cannot be considered accurate. e* • 3 • 0. sample Vtiekness 0.1-3mm. material thickness l-Smm. frcq. 10GHz 0.6 . 0.2. 2.5 1.5 0.5 Thickness of material (mm) 0.1 Thickness of Acrylic (mm) Figure 4.19 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f S21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 e • 3 e • 0. sample OiicKness 0.1-3mm, material thickness 1-Smm, freq. 10GHz Thickness of material (mm) Thickness of Acrylic (mm) Figure 4.20 Uncertainty of the loss factor (e”r) due to errors in measurements o f S21 er9 • 3 ercc • 0 Freq > 9 - 1 2 GHz Sample thickness • 0.1 - 3mm 0.1 i ! 0.0s loos 2 1 Thickness of Sample x10 Thickness of Sample Thickness of Sample x Thickness of Sample •3 0 0.1 f°« x Figure 4.21 Uncertainty of the dielectric constant (e’r) due to errors in measurements of S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 erp • 3 er^ ■ 0 Freq * 9 - 1 2 GHz Sample thickness • 0.5 - 3mm 0.5 0.5 0.2 0.2 0.1 0.T Thickness of Sample 2 x 1 Thickness of Sample Thickness of Sample 0 x 2 1 Thickness of Sample *io*3 x 0 Figure 4.22 Uncertainty o f the loss factor (e”r) due to errors in measurements o f S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right). Samples o f this material with thickness greater than 1mm can be measured with uncertainty o f 5% or less, if acrylic thickness is approximately 1mm. Figure 4.21 and Figure 4.22 clearly show increasing uncertainty of measurement with decreasing sample thickness. In both cases, the envelope also ‘thickens’ with increased acrylic thickness. For this material, a sample o f over 1.5mm backed by the thinnest available acrylic (1.2mm) would be optimal and produce a measurement with uncertainty o f less than 5%. Unfortunately, most very low-loss samples measured had a sample thickness o f less than 0.3mm resulting in very large uncertainty. The uncertainty analysis presented so far indicates that for the non-magnetic materials having the dielectric constant above 10 and loss factor typically greater than 20, the optimal acrylic thickness is roughly 1mm. Therefore, the analysis that follows is limited to that thickness. Data for another acrylic thickness is given in Appendix 0. It should be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 noted, that most o f the measurements o f actual materials were made with acrylic thickness equal to 1.2mm. Only some measurements made before the uncertainty analysis had been completed were made with thicker acrylic. 4.1.5 Summary Uncertainty Plots The plots of this section are intended to give an overall picture of a large collection o f sample measurement uncertainties depending on thickness and acrylic thickness. These plots proved very useful to quickly ascertain an approximate uncertainty associated with any material measured. Each plot shows single frequency data, with a constant sample and acrylic thickness, with varying dielectric constant, which is related by a specific constant ratio to the loss factor. All figures contained in this section are at a frequency o f 10GHz, and an acrylic thickness o f 1mm (data for an acrylic thickness o f 3mm can be found in Appendix 0). The frequency o f 10GHz was chosen, as it is approximately central in the X-band frequency range, and gives a good approximation for the whole range. Sample thicknesses are 0.1mm, 1mm, and 2mm. Figure 4.23 and Figure 4.24 show the uncertainty for a very thin (0.1mm) sample. It is seen clearly, that for any dielectric constant of less than 20, the uncertainty is very large. The uncertainty has an inverse relationship to the ratio e’r / e”r. For e’ = 0.2 x e” the uncertainty in e ’r is large, while it is small for e”r. This trend is consistent throughout the results, such that when a large difference exists between e’ and e”, and e’ is between I and ~50, then there are large differences in the corresponding uncertainties. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 frequency - lOOfz. sample ttvekneas • 0.1mm. material thickness ■ 1mm 0.1 er0 • 2crBC er • 5*er 0.0S 100 1000 e."o Figure 4.23 Uncertainty o f the dielectric constant due to errors in measurements o f S 21. Five different ratios of material at 10GHz, a sample thickness of 0 .1mm and an acrylic thickness of 1mm. frequency - 10GHz. sample tK k n ess ■ 0 1mm. m atenal thickness • 1mm • 2erM er • 5*er f I IV aosi-I I ! 100 1000 "3 Figure 4.24 Uncertainty o f the imaginary permittivity due to errors in measurements of S21. Five different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic thickness o f 1mm. The erp= 5 x erpp has uncertainty too high to be recorded in this plot. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 0.1 frequency - 10GHz. am ple thickness • 1mm, metenal thickness ■ 1mm * 2Cf« 0 90 er • 5*er f 0.06 3 100 1000 e.s Figure 4.25 Uncertainty of the dielectric constant due to errors in measurements of S21. Five different ratios of material at 10GHz, a sample thickness o f 1mm and an acrylic thickness o f 1mm. frequency - 10GHz. sample thickness • 1mm. materiel thickness • 1mm 0 er . 2er ..80 er m5*er f i 0.05 (- c. ’3 Figure 4.26 Uncertainty o f the imaginary permittivity due to errors in measurements of S21. Five different ratios of material at 10GHz, a sample thickness o f 1mm and an acrylic thickness o f 1mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 With an increasing sample thickness, materials o f lower dielectric constant can be measured with a greater degree o f certainty. Figure 4.25 and Figure 4.26 both show that an uncertainty of less than 5% is realizable for some materials even with a dielectric constant as low as 10. The red line on both plots indicates a material with the same real and imaginary permittivity components at 10GHz. With a dielectric constant and loss factor o f 7, this material measurement uncertainty is an acceptable 5%. frequency - 10GHz. sample thickness ■ 2mm. matents! thickness ■ 1mm era • -2cr .s o er.. fier o.osr 100 e.3 Figure 4.27 Uncertainty o f the dielectric constant due to errors in measurements of S21. Five different ratios o f material at 10GHz, a sample thickness of 2mm and an acrylic thickness of 1mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 frequency - 10GHz. sample thickness • 2mm. material thickness ■ 1mm 0.1 er • 2er er mO'er 0L 1 100 10 e3 Figure 4.28 Uncertainty o f the imaginary permittivity due to errors in measurements o f S21. Five different ratios of material at 10GHz, a sample thickness of 2mm and an acrylic thickness o f I mm. When the sample thickness is 2mm, as in Figure 4.27 and Figure 4.28, the ratios of E’r = 0.2 x e”r and e \ = 0.5 x e”r start to show a similar trend in both plots, once the dielectric constant reaches 100 (the imaginary permittivity is 500 and 200, respectively). The high uncertainty is now due to too much wave attenuation by the sample for the VNA data to be accurate. This is in contrast to the results in Figure 4.23 and Figure 4.24 where the attenuation was not large enough to alter the S21 magnitude or phase data from an empty ‘through’ connection. For the cases of Figure 4.23 and Figure 4.24, the reason the uncertainty is kept below 100% when a sample is very thin and lossless is the backing acrylic, which provides additional attenuation. Figure 4.29 shows close-up views of the results of Figure 4.23 through Figure 4.28 for the dielectric constant = 100. Here, a sample thickness o f 1mm is a good compromise for any loss factor except e”r = 20 (as it is out o f range for the e”r uncertainty plot for this thickness). For very thin samples of 0.1mm, uncertainties below 5% for both parts o f the permittivity are only obtained for e’r = e”r. All thin samples (0.1mm) however, can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 measured with uncertainty of less than 10% for 50 < e”r < 200 (likely 300). Thicker samples (2mm) show very low uncertainty for all e”r. Uncertainly of 0.1mm 1mm 2mm Uncertainty of #rpp 0.1mm 1mm 2mm •rp = 100 0.05 •rpp =500 •rpp = 200 •rpp = 100 •rpp = 50 erpp = 20 \ 0.05 I \\ X ou J / \ .. i1 100 . t . 100 100 Figure 4.29 Uncertainty of both the real and imaginary components o f permittivity at a dielectric constant o f 100 and a frequency of 10GHz. Acrylic thickness for all plots is 1m m . e,p = e ’r, e^ p = e ” r Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 73 Summary o f Material Uncertainty Table 4.3 Sample thickness = 0.1mm, acrylic thickness = 1mm, frequency = 10GHz Ratio Coeff. X (e’r = Xe”r) 0.2 0.5 1 2 5 Uncertainty o f e”r Uncertainty o f e ’r < 5% N/A e’r >200 e’r >54 e ’r >38 e ’r >32 < 10% N/A e’’r >400 e”r >54 e”r >19 e”r>6.4 N/A e ’r >22 e ’r >29 e ’r >28 e ’r >28 N/A e”r >44 e”r >29 e” >14 e”r>5.6 < 10% < 5% e’r >9.0 e’r >22 e’r >54 e’r >300 N/A e”r >45 e”r >44 e”r >54 e”r >150 N/A e ’r >4.0 e ’r >8.5 e ’, >29 e ’r >51 N/A e"r >20 e”r >17 e”r >29 e”r >26 N/A Table 4.4 Sample thickness = 1mm, acrylic thickness = I mm, frequency = 10GHz Ratio Coeff. X (e’r = Xe”r) 0.2 0.5 1 2 5 Uncertainty o f e”r Uncertainty o f e ’r <5% e ’r >52 e 'r >18 e ’r >6.3 e’r >5 e ’r >4.4 <5% < 10% e”r >260 e”r >36 e”r >6.3 e”r >2.5 e”, >.88 e ’r >10 e ’r >3.4 e ’r >3 e ’r >2.1 e ’r >1.4 e”r >50 e”r >6.8 e”r >3 e”r > l.l e”r >.08 e’r >l e’r >3.4 e ’r >6.3 e’r >120 e’r >190 < 10% e”r >5 e”r >6.8 e”r >6.3 e”r >60 e”r >38 e ’r> l e ’r >l e ’r >3 e ’r >4.4 e ’r >180 e”r >5 e’’r >2 e”r >3 e”r >2.2 e”r >36 Table 4.5 Sample thickness = 2mm, acrylic thickness = 1mm, frequency = 10GHz Uncertainty o f e’V Uncertainty o f e ’r <5% e ’r >12 e ’r >5.9 e ’r >3.2 e’r >1.8 eV >1.7 e”r >60 e”r >10 e”r >3.2 e”r>0.9 e”r >.34 < 10% e ’r >3.1 e ’r > l.l e ’r >1 e ’r >1 e ’r >1 e”r>15 e”r>2.2 e”r> l e”r >0.5 e”r>.2 <5% e’r >1 e’r>l e’r >3.2 e’r >22 eV >49 < 10% e”r >5 e”r >2 e’V>3.2 e”r> U e”r>10 e ’r> l e ’r >l e ’r > l e ’r >2 e ’r >35 e’V>5 e”r >2 e”r> l e”r> l A k. w Ratio Coeff. X (e’r = Xe”r) 0.2 0.5 1 2 5 Summary data are given in Table 4.3 - Table 4.5 as the sample thickness increases from 0.1mm to 1mm to 2mm, while the acrylic thickness and frequency remain unchanged. The ideal pair o f values, e’r > 1 and e”r > 0, would mean that any material would be below the specific uncertainty. It is clearly seen that as the thickness o f the sample increases, the required pairs o f values for £’r and e”r decreases, with the best result for a sample thickness of 2mm and ratio coefficient of 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 4.2 Measurement Results The first set of plots (Figure 4.31 through Figure 4.34) shows measurement results for several materials for which consistent results have been attained. Throughout the entire set of figures in this section, a dark blue line indicates the dielectric constant (e’r), while the dotted red line shows the imaginary permittivity (e”r). Several measurements o f each material were taken to show repeatability and to observe trends in the characteristics (expected decrease o f conductivity) o f the sample. For many materials degradation occurred with time resulting in a decrease of effective conductivity. A green uncertainty envelope is shown for the ‘Original’ measurement but is left off subsequent measurements. All results are close (in uncertainty) to the original, so there is no need to show uncertainty limits for all measurements, as this would needlessly clutter the plots. Dates are given for each measurement, as this is important for the developers o f these materials. At the time of writing this thesis, batches of materials were still periodically delivered to the lab for characterization. To categorize results, a batch number was assigned to each delivery. A total o f 33 batches o f 3 to 30 samples each had been delivered. This collection o f results shows representative material batches, as well as substrates on which the ppy, or similar conducting polymer, was deposited. Among the different substrates used were silicon rubber, cotton fabric, hardening liquid (epoxy, glue), carbon sheeting, and plastic brills. Depending on the substrate used for the polymer, different effective permittivities were expected, even if the concentration o f polymer was constant. This is understandable, as different substrates have differing thicknesses, electrical characteristics, and densities. The chemist preparing the materials decided which substrate the polymer would be bonded too, and therefore only the overall electrical characterization is in the scope o f this thesis. This information was added to give the reader an idea that differing substrates were used to either increase the thickness of samples, or to enable the increase in concentration o f a specific polymer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Most o f the repeatedly measured samples were originally measured with gating turned o ff on the VNA. The measurements were then taken again after another calibration was performed (labeled ‘redo’ on the figures). For the third measurement, a gating filter was placed on the S-Parameters (labeled ‘gated’ on the figures). Gating essentially tries to filter out additional noise and reflections that were not calibrated out, or were introduced after calibration (e.g. movement o f coaxial cables). A bandpass filter is imposed on the SParameters before they are displayed on the VNA and downloaded for data processing. This bandpass filter is user definable and is set to bracket the main pulse o f energy reflected off (or transmitted through) the two-layer sample. Although gating makes a smoother curve, it also appears that there is a phase shift. An example o f non-gated and gated Sii and S21 parameters is shown in Figure 4.30. This is the raw data used to determine e’r and e”r shown in Figure 4.31. The bandpass filter was set sufficiently broad for all measurements, as the main focus was to measure these samples in bulk and not be required to alter the filter characteristics for each sample, only for each batch. Gated and Non-Gated S-Parameters, Blue is Nan-Gated Red is Gated Figure 4.30 An extreme close-up of both S-Parameters and how data smoothing occurs with gating. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Batch #5 sample 4, original, redo, gated, redo gated Gated Aug09 results, and Gated Aug22 120 100 80 H 60 at a 2 40 OngmeJ measurement taken July 24 Redo measurement taken Juty 30 Gated measurement taken July 31 Redo Gated measurement taken Aug 02 Gated measuremenb taken Aug09 Gated measurements taken Aug22 Static Conductivity ■ 0.960 S/on 20 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Frequency (GHz) Figure 4.31 Results from Batch#5 Sample#4, six separate measurements. Arrows indicate increasing time. Sample thickness = 0.1 Smm, Acrylic thickness = 1.18mm The complex permittivities of the sample in Figure 4.31 have properties opposite that of ‘Material 2’ analyzed with the uncertainty plots of Chapter 4.1.1. Six separate measurements with six individual calibrations were performed for this sample over a one month time period. It is seen that while the dielectric constant changes only within the uncertainty limits, the imaginary permittivity decreases quite significantly over the time period. The dielectric constant is seen to have four overlapping results from measurements taken on July 31st, August 2nd, 9th, and 22nd. Taking the approximate properties of this material at 10GHz to be e’r = 10 and e”r = 80, it is possible to examine the uncertainty plots and determine whether a different sample thickness would give lower uncertainty. Referring to Figure 7.1 through Figure 7.6, from uncertainty plots 3a in Appendix 0, with a sample ratio of e’r = 0.125 x e”r, the closest analyzed is a ratio equal to 0.5. The upper left subplot of each figure is the closest match to this material. In this case, an increase in sample thickness would decrease the uncertainty from ~30% for e’r to below 5%, if a sample thickness o f at least 1.5mm was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 available. The uncertainty difference would not be that drastic in the case of e”r, as it already has an acceptable uncertainty of 3.3%, but it would also decrease. Batch #7 (PS240701), Sample *4, Original, Redo, Gated, Redo pated-fMutta- 200 r 180 160 9 'e S’ 2 140 Original meostffemcnt taken Aug06 Redo measurement taken Aug 09 Gated measurement taken Aug 13 Redo Gated measurement taken Aug 16 120 Static Conductwity • 1.40 S/cm * 10080 60- ♦ 40h 201 8.5 9 9.5 10 10.5 11 Frequency (GHz) 11.5 12 12.5 Figure 4.32 Results from Batch#7 Sample#4, four separate measurements. Arrows indicate increasing time. Sample thickness = 0.15mm, Acrylic thickness = 1.18mm The measurement results in Figure 4.32 closely resemble the permittivity o f ‘Material 4’ considered during the uncertainty analysis. At 10GHz, this sample shows e’r = 78 and e”r = 178. Once again it can be seen that the imaginary permittivity gets smaller with time, while the dielectric constant is within the uncertainty envelope and only changes slightly with the changing conductivity. The results o f the uncertainty analysis o f Figure 4.15 and Figure 4.16 clearly show that the uncertainty for this material is acceptable, being 5.6% and 2.2%, respectively for 8’r and e”r. It would be possible to decrease this further if the sample thickness were increased. From Figure 4.15 and Figure 4.16 is can be noted that the sample thickness of 0.15mm in one o f the largest uncertainty regions plotted. By increasing the thickness to I mm, the uncertainty would drop to 3% and 1.7%, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 B atch 19 (PSQ10801), S am ple #5, O riginal, Redo, G a te d , a n d i-teqo aatea neauita ... % 8.5 9 9.5 10 10.5 11 F re q u e n cy (GHz) 11.5 12 12.5 Figure 4.33 Results from Batch#9 Sample#5, four separate measurements. Sample thickness = 0.15mm, Acrylic thickness = 1 .1 8mm The low loss sample o f Figure 4.33 shows an uncertainty o f 32% and 8.4% respectively for e’r and e”r. All measurements are contained within the uncertainty limits except for the degrading of conductivity, which happened over a period o f two weeks. In the case of these low loss samples, a dramatic decrease in uncertainty occurs with even slight increases in sample thickness. This is very apparent from Figure 4.23 and Figure 4.24 for this sample, namely e’r = 0.3 le’Vat 10GHz. 4.2.1 Sample Thickness Optimization As mentioned in Chapter 3, the thickness of the sample was often included in the optimization. The caliper measurement was given as an initial guess and a userdeterminable constraint was placed on this parameter, usually 5-10%. Examination o f the convergence of the thickness gave an indication o f how well the optimization program fared with the data. If the converged thickness o f the sample was on the edges of the constraint envelope, it showed the algorithm needed to compensate somewhat for poor data. If the optimized thickness changed by only a fraction o f a percent, then the results Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 were usually quite reliable and repeatable. Figure 4.34 shows the results for all samples from the same batch, where the thickness o f the sample was questionable because of the substrate the polymer was deposited in. Original thickness measurements and optimized thickness were recorded, as summarized in Table 4.6. The substrate used for this batch was a plastic brill (resembling a plastic SOS pad) with the polymer grown around every fiber. It was expected that the difference in sample thickness, between measured and optimized, would be uncharacteristically large because o f the nature of the material, but this was not the case. Batch #4, TH3 White and Green, TH5, TH6, TH7, TH8, July 17 and 18 2i--------------*----------------------------------- 1------------- ; Sample 9 Sample 10 -1 2 Sample 1 - 8 Sample 7. 8 .sam ple H ........— Sample 9 ' ,\y„ P*8 5 ^ S a m p l e 3. 4 " 8.5 9 9.S '' -•: 10 10.S 11 Frequency (GHz) V Sample 10.12 - 11.5 ..... 12 ■ 12.5 Figure 4.34 Entire Batch #4. Thickness constraint envelopes were recorded. Sample numbers correspond to Table 4.6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Table 4.6 Data corresponding to Figure 4.34 showing variable thickness with 5% constraint. 1 2 Optimized Thickness (mm) 2.55 2.54 3.26 3.27 TH6 TH7 TH7 2.57 2.57 3.27 3.27 3.35 3.35 3.62 3.62 2.47 2.47 3 4 5 6 7 8 9 10 3.32 3.34 3.61 3.62 2.4 2.47 0.90 0.30 0.28 0 2.92 0 2.59 <— > 2.35 TH8 TH8 2.77 2.77 2.76 2.77 0.36 0 2.90 <— > 12.63 TH3 White TH3 White TH3 Green TH3 Green TH5 TH5 TH6 11 12 Percent Difference (%) 0.78 1.18 0.31 0 5% Thickness Constraint (mm) Caliper Measurement (mm) Sample 2.70 <— > 2.44 3.43 <— >3.11 3.52 <— >3.18 3.80 <— > 3.44 As shown in Table 4.6, the optimization program did not change the thickness of the sample significantly (although within the constraint imposed, it was able to change it as much as 5%). Figure 4.35 shows a data set o f a sample (from Batch #7) for which difficulties were experienced with polynomial convergence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 Batch *7 (PS240701), Sample #1, Original, Redo, Gated, and Redo, Gated remits ,------------’Original an d R edo Original a n d R edo F requency (GHz) Figure 4.35 The polynomial solution and the small-section solution. Sample thickness = 0.14mm, Acrylic thickness = 1 .1 8mm Table 4.7 Data Set displaying difficulties with optimization convergence. All optimized solutions o f thickness lay on the constraint edge of the envelope, except for the two shown in Red Text. Measurement Original Redo Gated Redo Gated Sample Thickness o f 0.14mm 5% Thickness Constraint = 0.147m m «— »0.133mm 10% Thickness Constraint = 0.154mm <— >0.126mm 20% Thickness Constraint = 0.168mm <— >0.112mm 10% Solution 20% Solution 5% Solution 0.154 0.147 0.153 0.154 0.147 0.156 0.154 0.168 0.147 0.154 0.168 0.147 The Gated and Redo Gated results o f Figure 4.35 in polynomial form are obviously erroneous. This material is a perfect example o f the polynomial convergence algorithm having difficulty as previously mentioned. This type o f erroneous result happened very rarely, (approximately 1 time out of 30) and shows how the ‘small-section’ method can be used as a back up. If the sample was again delivered on a different thickness o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 acrylic, or itself was a different thickness, this erroneous convergence difficulty would most likely not be repeated. Even though realistic and repeatable results were obtained for the sample of Figure 4.35, because o f the constraint-edge convergence o f the thickness, the results were not considered to be accurately determined by the uncertainty limits. For this particular sample, facial inconsistencies were believed responsible for the convergence errors. Only when the thickness constraints were relaxed to 20% did the optimization converge to a central location inside the envelope, and only for two samples (shown in red text on Table 4.7). This is not considered acceptable. Batch #10, TH280601, Sample #6 Original, Redo, 5 Gated Results and Gated Blank Sapiple with High Uncart; Original 4.5 (- 3.5 r 8 3‘ 3 S2.5F ? 2 2 Blank Ongmai meaa*ement taken Sep 17 Redo measurement taken Sep 19 Gated measurement taken Sep 10 S t a l e C o n d u c tiv ity ■ 0 .0 0 0 3 0 3 S t a n 1.51- 0.5 Blank 8.5 9.5 10 10.5 11 Frequency (GHz) 11.5 12 12.5 Figure 4.36 Very low-loss sample with very high uncertainty. Also shown is one-layer convergence o f acrylic only. Sample thickness = 0 .16mm, Acrylic thickness = 2.07mm Figure 4.36 is included to explicitly illustrate one o f the limitations of this measurement method, namely very thin, lossless materials. From Figure 4.19, the dielectric constant uncertainty for this material is approximately 55% at 10GHz. Considering the Redo measurement, this would place the uncertainty limits at 6.2 and 2.8%, which the other two measurements fall within. Lossless thick samples, e.g. pure acrylic indicated with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 ‘Blank’ on the figure, show low uncertainty in measurements. At 10GHz, the Blank dielectric constant solution is 2.71, while many text references have this material classified as having an e \ between 2.50 and 2.75, depending on the manufacturer [13]. Since e”r = 0, the measurement data (mean e”r = 0.0044) clearly show the small erroneous results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 Chapter 5 5 Modeling Results and their Evaluation Two central ideas are explored in this chapter concerning the FDTD simulations. The first, a comparison of the simulated results obtained with results from measurements taken, and the second, the analysis of specific facial inconsistencies that occur in the sample layer in practice. 5.1 Comparison of FDTD Results with Measurements The FDTD-collected data was compared to the VNA-collected data as part o f the FDTD analysis, before facial inconsistencies were explored. Two samples were chosen from the unknown materials supplied by the DND, namely, Batch #21 Sample 2 (B21S2), and Batch #26 Sample 4 (B26S4). Both samples were 0.14mm thick and adhered to a 1.18mm block of acrylic with er’ = 2.65. Electrical characteristics o f B21S2 and B26S4 are given in Table 5.1. Table 5.1 Electrical characterization o f two representative samples at 10GHz Sample VNA measured e. Static Conductivity (<u,) B21S2 7.18 - i 3 1.8 17.7 B26S4 156.5- i 174 96.9 The procedure is as follows: The VNA is calibrated and S-Parameter data collected after the insertion of B21S2. The optimization routine is used on the VNA collected SParameters and an array o f frequency dependant complex permittivities is converged upon. The complex permittivity from the VNA at 10GHz is used in the material characterization in the FDTD code. The FDTD simulation is performed and a second array o f S-Parameters is collected. The optimization routine is used on the FDTDcollected S-Parameters and a second array o f complex permittivity values is converged upon. There now exists an array of VNA S-Parameters and complex permittivities, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 another array of FDTD S-Parameters and complex permittivities. These two sets of two arrays are now compared. It should be noted that the FDTD data could only legitimately be compared to the VNA data at 10GHz and not in the whole X-band frequency range. On the other hand, the blue curves in Figure 5.1 show the FDTD data over this entire range, compared to the red curves showing the VNA S-Parameters. It is possible for a material to be set up to be frequency dependant using the FDTD software. However, the FDTD algorithm accounts for the frequency dispersion represented by the Debye equation. Development o f a polynomially frequency-dependent dispersion in the FDTD code was beyond the scope o f this thesis. S-Parameters. Sj j and S 2 1 of VNA Results and FDTD Results 130 90 oa 06 tso 0.4 02 180 1 330 300 270 Red - VNA Blue - FDTD Figure 5.1 S-Parameters showing both VNA-collected results and FDTD-simulated results for sample B21S2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 Comparison batwean Batch #21 Sample 2 and FDTD Simulation 40 3f 10 35 33.5 9.5 1 9 33 8.5 32.5 8 c I 32 7 .5 1 [ 31.5 7j 31 j 6.5 j 6| 6.51 8.5 9.5 10.5 Frequency (GHz) 12.5 30.5 i 10 30 ^ 10 Frequency (GHz) Figure 5.2 Sample B21S2, (Left) Dashed line indicates measurements taken with VNA, solid shows FDTD simulated results. (Middle, Right) Zoomed in results showing 10GHz comparison. Blues lines show er\ red lines show er” Figure 5.2 shows measured and simulated data for the complex permittivities for B21S2. Data at 10GHz are highlighted and measurement error uncertainty envelope is placed around VNA-measured data. Firstly, it can be seen that the simulated and measured parameters are both well within the uncertainty limits. Secondly, the values obtained from the simulation of the dielectric constant and loss factor at 10GHz corresponding to the measured S21 are er’ = 7.81 and er” = 31.6, respectively. Thirdly, the difference in er” values is in this case very small, less than 1%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 B atch 26 S am ple #4. C om parison of M e a su re d a n d Sim ulated R e s u lts 190 180 Met LfttsFacfar 170 160 Mm 150 140 9.75 10.25 10.5 F requency (GHz) Figure 5.3 Comparison o f measured and simulated results for B26S4. Focus is 10GHz results. Green uncertainty enveloped centered on VNA-measured values. Uncertainties in measurements of er’ for sample B26S4 are much lower than for B21S2 and approximately equal for er”, as illustrated in Figure 5.3. Table 5.2 summarizes the results of the comparison. Table 5.2 Comparison of measured and simulated complex permittivity values at 10GHz Sample B21S2 B26S4 Measured 7.18 ±1.59 156.5 ±6.50 Er’ Simulation Results 7.81 155.9 % Difference 8.8 0.38 Measured 31.8 ±1.55 174.2 ±6.00 Er” Simulation Results 31.6 175.7 % Difference 0.63 0.86 The small differences between measured and simulated results (primarily in sr’ of B21S2) are at an acceptable level, assuming that it has been sufficiently well documented that the measurement results presented are within the computed limits o f uncertainty o f the VNA use. For the FDTD simulations performed, specific care has been taken to limit to negligible errors due to: reflections from the graded mesh (maximum voxel dimension Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 difference = 11%), evanescent modes (sufficient distance between probes for Su and S 21 and sample interfaces), and PMLs. The stability factor was at the conservative level of 0.95, and the source excitation was given sufficient amount of time steps for the Gaussian Pulse to start and end very close to zero (0.0138 with a normalized maximum of 1). A very small permutation of the S-Parameters between the VNA and FDTD collected values gives rise to differing permittivity values, in the case of B21S2, as large as 8.8%. It would be a very worthwhile endeavor to explore the effect on converged permittivity from slight variations in the S-Parameters from facial inconsistencies routinely seen in practice. To elucidate errors caused by gaps in the sample and other imperfections, selected representative cases are simulated with the FDTD and analyzed. 5.2 Waveguide gap effect Well defined facial inconsistencies (gaps and dents) in the test sample have been placed in various spots and the FDTD convergence results analyzed. This simulates the problem o f completely covering the acrylic supporting structure with a perfectly homogeneous layer free o f small irregularities. Simulations of two types of the facial inconsistency were performed, namely full gaps in the unknown samples, and partial gaps (dents). Select configurations have been drawn in Chapter 3, while a full collection of simulated models can be found in Figure 5.4 and Figure 5.10. Gap configuration #2 in both o f these figures is the most probable facial inconsistency, with Gap configuration #4 being the next most likely. 5.2.1 Full Gaps Gaps were placed in areas close to the edge o f the sample where they were most likely to occur, not where they would have the biggest theoretical impact on the TE 10 electric field. Gap #5 is an extreme inconsistency but is not completely unfounded for simulation, as similar samples have been delivered for measurement. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 o.Mmm 18 mm j t.ism m 0.1mm - l ^ ^ 9.9mm- I i D Gap *2 Gap *1 0 Mmm I 1 18 mm l i 9mm 1mm9mm- - O j 1ismm I I □ Gap *4 Gap *3 o u14mm mr 18mm V Gap *5 Figure 5.4 The five full gap configurations simulated. S-Parameter Sf j of Five Gapped Samples, a Perfect Sample, and no sample Figure 5.5 FDTD simulated Su parameter for gaps 1-5 for sample B21S2. Blue left: perfect, Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency data points of S-parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 S-Parameter Sj) of Five Gapped Samples, a Perfect Sample, and no sample Figure 5.6 FDTD simulated S21 parameter for gaps 1-5 for sample B21S2. Blue left: perfect, Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency data points of S-parameters. FDTD Simulation Results Comparison lap #1 'erfect Gap #3 Gap *2 iap #4 Gap #5 G a p *1 lap #2 /G a p #3/G a p §4 /Perfect / 8.5 10.5 Frequency (6Hz) 9.5 — 11.5 Figure 5.7 FDTD simulated results for sample B21S2 or dielectric constants and loss factors for one perfectly homogeneous layer and five samples with gap inconsistencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 O f the five configurations of gaps and dents, #2 and #4 were the most common. An inconsistency centering on the middle of the x-axis like #3 and #5 is expected to make a larger impact than an inconsistency on the edges o f the x-axis like #4. This is because of the half-period sinusoidal magnitude configuration o f the TEio mode in the waveguide. In Figure 5.5 and Figure 5.6, one set of S-Parameters is displayed of a simulation run without any sample at all, only acrylic. The acrylic placement was still 0.14mm recessed from the calibration plane to be consistent with the other simulation runs. Displaying this result alongside a result from a very large inconsistency like Gap #5 shows that even when the entire 4mm center (x-axis) of the sample is missing, the S-Parameters, and finally the permittivity, are still reasonable close to the homogeneous layer. The sample with higher conductivity (B26S4) was also simulated. Gap #1 and Gap #5 configurations are not shown as the S-Parameters and consequent permittivities were too far removed from the ‘Perfect’ layer (Figure 5.8 and Figure 5.9). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 S-Param eters o f Three Gapped Samples, a Perfect Sample, and no sample. , < ii\ \ 120 0.6 150 0.4 330 210 300 240 270 Figure 5.8 FDTD simulated S| i and S21 parameters for gaps 2-4 for sample B26S4. Blue: perfect, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4,. Red tips indicate low frequency data points o f S-parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 FDTD Simulation Results Comparison 200 Perfect«' 180 G ap #3 ip#4 e" 8) ■o a c O ) CD Z 160 P e rfe c t £' lap #2 e' 140 Gap #2 e' 120 8.5 9.5 10 10.5 11.5 Frequency (GHz) Figure 5.9 FDTD simulated results for sample B26S4 of dielectric constants and loss factors for one perfectly homogeneous layer and three samples with gap inconsistencies. As shown in Figure 5.8 and Figure 5.9, certain gap inconsistencies in this sample have a much larger effect on the data collected and consequently the permittivity convergence. Table 5.3 displays a comparison and summary o f permittivity results at 10GHz. Table 5.3 Summary table showing differences in simulated dielectric constant and loss factor for different configurations of facial inconsistencies. All values at 10GHz. Sample B21S2 B26S4 B21S2 B21S2 B26S4 B21S2 B26S4 B21S2 B26S4 B21S2 Condition Perfect Gap #1 Gap #2 Gap #3 Gap #4 Gap #5 e ’r 7.81 155.94 9.63 9.01 144.4 7.99 140 7.84 153.6 5.22 AE’r / e ’r 0.19 0.13 0.08 0 .0 2 0.11 0.004 0.015 0.50 tf ” r 32.0 175.72 26.63 30.76 116.59 30.60 171.21 31.76 175.34 24.08 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ae’V /e’V 0 .2 0 0.04 0.50 0.05 0.03 0.008 0 .0 0 2 0.33 94 The values in bold in Table 5.3 display the solution o f the perfect (no facial inconsistency) sample, to which all other samples are compared. O f particular interest are values contained in the heavy outline, namely the Ae’V / e”r values o f 0.04 and 0.50, for B21S2 and B26S4, respectively. The increase in error associated with this particular gap configuration with an increase in conductivity indicates that even a relatively small and often encountered facial inconsistency can significantly alter the convergence solution. Again, referring to Table 5.3, it is apparent that each gap configuration must be considered separately in terms o f expected solution deviation from a perfect sample. This is illustrated by loss factor deviation of gap configuration #3 and #4, which is smaller with greater conductivity while gap #2 deviation is significantly larger. The opposite is true for the comparison o f the dielectric constant. 5.2.2 Dents in Sample In the effort to model even more realistic defects, the following simulations were carried out. Many samples delivered to the lab had similar problems to those described in this chapter. The chemists experienced difficulties getting completely homogeneously thick samples on the acrylic backing. It would have been unfeasible to measure most o f the materials without a solid supporting structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 0.06 mm 006 8 rr mm 18 mm 006m m 0oarr 8mm UT n _______ *IlfBmm 4 9.9m m -* M Dent *i 0 06 mm 0 0 8 mm 1.18mm 1m m —♦ 9mm 1 4 □ Dent *2 I 0 06mm 0.08 m m M 8 mm U4 L 9mm—* □ Dent *4 Dent #3 0 06 mm loosmm 1 1 18mm 1. 4 p = D Dent *5 Figure 5.10 The five dent configurations simulated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 S-P aram eters of Five Dented Sam ples, a Perfect Sam ple, and no sam ple 120 0.8 0.6 ISO 0.4 0.2 180 No Sam ple 'N o S am ple 330 300 Figure 5.11 FDTD simulated Su and S21 parameters for sample B21S2, gaps 1-5. Blue left: perfect, Green: dent #1, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4, Blue right: dent #5. Red tips indicate the low frequency data points o f the S-parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 FDTD Simulation Results Comparison 40 30 E 20 8.5 9.5 10.5 Frequency (GHz) 11.5 10GHz Region Enlarged 7.85- Dent (1 7.84 32 /Perfect < « « L /D e n t9 4 31.5 — 31 ' D e n t 8 3 ^ ^ 7.83- Perfect 30.5 jU ^ D e n tJII^ ^ 7.82- Dent 84 29.5 7.81 Dent 12 7.8 Dent 83 29 24-5 J e n t 85 28 7.79- —■ 10 10 Figure S. 12 FDTD simulated results o f dielectric constants and loss factors for one perfectly homogeneous layer and five samples with dent inconsistencies. Sample B21S2. On enlarged region plot, e’r on left, e”r on right. Dent #5 omitted from e’r 10GHz enlarged region plot. Simulation results of the dents resemble the results for the gaps although with less dramatic changes. Figure 5.12 shows that small facial inconsistencies (e.g. Dent #2, #3, and #4) in some areas only slightly affect the results o f the loss factor and dielectric constant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 Sample B26S4 was simulated in the same way, and, once again only configurations #2, #3, and #4 were included (Figure 5.13 and Figure 5.14). S-Parameters o f Three Dented Samples, a Perfect Sample, and no sample i—ra-----------------120 0.9 0.6 150 N« Sm fir* 330 !10l 300 240 270 Figure 5.13 FDTD simulated Si i and S21 parameters for sample B26S4, dents 2-4. Blue left: perfect, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4. Red tips indicate low frequency data points o f S-parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 FDTD Simulation Results Comparison 200 180 160 O) 140 120 8.5 9.5 10.5 11.5 Frequency (GHz) 10GHz Region Enlarged Perfect s' 174 ^^^erfect i* Dent f 4 c' 15s[ Dent >4 e*N 173 Dent #2 s' V^Dent # 2 e* 172 I54[ \ D e n t f 3 e* Dent #3 c' 1 152: 171 170t 10 10 Figure 5.14 Results of dielectric constants and loss factors for one perfectly homogeneous layer and three samples with dent inconsistencies. Sample B26S4. Table 5.4 shows a comparison o f errors between each sample and the respective ‘Perfect’ layer. It also gives a comparison between the same dents on low and high conductivity samples. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Table 5.4 Summary table showing differences in simulated dielectric constant and loss factor with different configurations of facial inconsistencies. All values at 10GHz. Sample B21S2 B26S4 B21S2 B21S2 B26S4 B21S2 B26S4 B21S2 B26S4 B21S2 Condition Perfect Dent #1 Dent #2 Dent #3 Dent #4 Dent #5 E’r 7.81 155.94 7.85 7.80 154.4 7.80 153.0 7.81 155.4 6.95 Ae’r / e ’r 0.005 0.001 0.010 0.001 0.019 0 0.004 0.12 e’V 32.0 175.72 30.93 31.75 174.59 31.56 173.0 31.92 175.31 28.53 Ae”r / e”r 0.035 0.008 0.007 0.014 0.016 0.002 0.002 0.12 The values in bold in Table 5.4 display the solution o f the perfect (no facial inconsistency) sample, to which all other samples are compared. Contrary to the data shown in Table 5.3, there are no significant differences in errors between similar gaps. Many o f the errors incurred with the dents are insignificant. 5.3 Summary of FDTD Facial Inconsistency Simulations A comparison between VNA and FDTD generated data was performed on two samples o f differing dielectric and conductive properties. The differences between measured and simulated dielectric constant and loss factor were very small with 8.8% being the largest difference between two results. FDTD simulation o f small facial inconsistencies on the two samples were generated and compared to the ‘perfect’ solution o f the completely homogeneous sample. Very realistic gaps and dents proved to permute the final solution by as much as 50% and 2% respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 Chapter 6 6 Conclusions A novel microwave measurement method was introduced for the characterization o f permittivity and permeability o f materials. The method was developed to conform to a set o f specific criteria. The measurement method utilized a supporting structure of known electrical properties inserted with the unknown material into the waveguide. An iterative algorithm was used to converge on the material characteristics. An explicit equation was not possible given the increased complexity of dependence on sought parameters on measured data compared to the more traditional one-layer measurement methods. An equipment uncertainty analysis was performed to find the optimum thickness o f the sample and supporting acrylic second layer as a function o f the properties of the sample. General thickness guidelines were developed for both layers. The FDTD method was used to quantify the effect o f small gaps and dents on the surface of the sample on the final solution. Several results were shown, displaying many properties o f the convergence o f the algorithm. Limitations o f the method were also illustrated by the results that were obviously erroneous. 6.1 Measurement Method The two-layer approach was developed primarily to satisfy the following conditions for a measurement procedure: broadband relatively expedient limited initial startup costs accommodate very thin material without the ability o f self-support accommodate materials only available on thin sheets (essentially unmachinable) achieve minimum measurement uncertainty Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 The final measurement configuration required the sample to be placed or grown on one face o f a known supporting structure that conformed to the height and width dimensions o f the waveguide. 6.2 Non-Linear Optimization Two optimization approaches were used, both utilizing the same Levenburg-Marquardt algorithm but formulated differently. The first approach (and was later used as an initial guess for the second approach) solved for a single complex permittivity (or both permittivity and permeability) in approximately 40 separate frequency bands inside the X-band frequency range. The second optimization solved for the coefficients o f a polynomial that described the permittivity characteristics over the entire X-band frequency range. An option was given to include the thickness of the sample in the optimization o f the second approach. This proved to be a very quick and useful way to determine whether the optimization procedure had had difficulty converging on a solution. This optimization routine proved to be very reliable and quite robust, its limitations were explored in the Results o f Chapter 5. 6.3 Equipment Uncertainty Analysis The VNA used to obtain the S-parameters introduces uncertainty in the measurement data. An uncertainty analysis was performed to determine what combinations of sample and acrylic thicknesses provided the lowest uncertainty for a wide range o f lossy materials. Several specific plots were generated that illustrated the effects of change in sample thickness with constant acrylic thickness and frequency, while general plots were also shown that encompassed a broader range of materials and thicknesses that could be used for reference. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 6.4 FDTD Modeling of Sample Facial Inconsistencies Samples measured did not always have perfect dimensions. Often small gaps or dents existed at the edges o f the sample, sometimes completely exposing the acrylic. A waveguide measurement model was constructed in FDTD and several gap or dent configurations seen in practice were simulated. The results were then compared to the ‘perfect’ sample layer and differences analyzed. This was done for both a low-loss and a medium-loss sample, finding that higher loss in a sample did not directly correspond to a larger percentage difference between the converged solution o f the perfect sample and the non-homogeneous sample. 6.5 Results Example plots were given o f differing materials displaying many different characteristics. Results were chosen to highlight either strengths or weaknesses associated with the method. Very consistent measurement repeatability was observed, and consistent tracking o f material properties as a function o f time (displaying decreasing conductivity). The results were invariably contained within uncertainty limits. Results were also given that included data where the sample thickness was allowed to vary in the optimization. Circumstances where this method worked very well, and when it did not, were shown and discussed. The benefits o f the ability to retain the optimization data of the non polynomial solution (the stair-cased solution) were illustrated. Very low-loss thin sample results were given with very poor convergence, displaying a limitation of the optimization program. Finally, a one layer solution of the acrylic being used as the supporting structure was given. In summary, the method selected together with the uncertainty analysis and the FDTD simulations proved to be fully satisfactory for measurement o f the permittivity o f lossy, thin samples. For most o f the samples, the permittivity could be evaluated with uncertainty better than 5%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 6.6 Future Work The measurement program in Matlab is currently being ported to LabView. With its GUI and communication with the VNA, this will upgrade the usability of the program, if not the function. Adding uncertainty in the thickness of the sample into the theoretical calculations to observe how much error is introduced with small measurement discrepancies would be very beneficial and may prove to warrant more accurate measurements o f thickness. An investigation o f over-moding o f relatively thick medium-to-high loss samples would be interesting and could provide the software program with alternatives if an acceptable solution cannot be found. Transferring this method to free space, once the samples become available, will offer a challenge as signal processing may have to become involved to decrease the interference by outside factors such as wave scatter and diffraction on the sample edges. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B- 1 Bibliography [1] Baker-Jarvis, J., Janezic, M. D., Grosvenor, J. H. Jr., Geyer, R. G., ‘Transmission/Reflection and Short-Circuit Line Methods for Measuring Permittivity and Permeability’ NIST Technical Note 1355 (revised), Dec. 1993. [2] Afsar, M. N., ‘The Measurement of the Properties of Materials’ Proceedings of the IEEE, Vol. 74, No. 1, pp. 183-199, Jan. 1986. [3] Courtenay, C. C., ‘Time Domain Measurement o f the Electromagnetic Properties o f Materials’ IEEE Transactions on Microwave Theory and Techniques, Vol. 46, No. 5, pp. 517-522, May 1998. [4] Nicholson, A.M., Ross, G.F. ‘Measurement o f the Intrinsic Properties o f Materials by Time-Domain Techniques’ IEEE Transactions On Instrument and Measurement, Vol. IM-19, No. 4, pp. 377-382, Nov. 1970. [5] Stuchly, M. A., Stuchly, S. S., ‘Industrial, scientific, medical and domestic applications o f microwaves’ IEE Proceedings, Vol. 130, Pt. A., No. 8, pp. 467503, Nov. 1983. [6] Weir, W.B. ‘Automatic Measurement o f Complex Dielectric Constant and Permeability at Microwave’ Proceedings o f the IEEE, Vol. 62, No. 1, pp. 33-36, Jan. 1974. [7] Stuchly, S. S., Matuszewski, M., ‘A Combined Total Reflection-Transmission Method in Application to Dielectric Spectroscopy’ IEEE Transactions on Instrumentation and Measurement’ Vol. IM-27, No. 3, pp. 285-288, Sept. 1978. [8] Boughriet, A.H., Legrand, C., Chapoton, A. ‘Noniterative Stable Transmission / Reflection Method for Low-Loss Material Complex Permittivity Determination’ IEEE Transactions on Microwave Theory and Techniques, Vol. 45, No. 1, pp. 5257, Jan. 1997. [9] Wan, C., Nauwelaers, B., De Raedt, W., Van Rossum, M. ‘Complex Permittivity Measurement Method Based on Asymmetry o f Reciprocal Two-Ports’ Electronic Letters, May 1996. [10] Back, K.H., Sung, H.Y., Park, W.S. “A 3-Position Transmission/Reflection Method for Measuring the Permittivity of Low Loss Materials’ IEEE Microwave and Guided Wave Letters, Vol. 5, No. 1, pp. 3-5, Jan. 1995. [11] Wan, C., Nauwelars, B., De Raedt, W., Van Rossum, M. ‘Two New Measurement Methods for Explicit Determination o f Complex Permittivity’ Determination’ IEEE Transactions on Microwave Theory and Techniques, Vol. 46, No. 11, pp. 1614-1619, Nov. 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B-2 [12] Janezic, M.D., Jargon, J.A., ‘Complex Permittivity Determination from Propagation Constant Measurements’ IEEE Microwave and Guided Wave Letters, Vol. 9, No. 2, pp. 76-78, Feb. 1999. [13] Von Hippel, A. R., ‘Dielectric Materials and Applications’ The M. I. T. Press, Mass., 1966. [14] Maze, G., Bonnefoy, J. L., Kamarei, M., ‘Microwave Measurement of the Dielectric Constant Using a Sliding Short-Circuited Waveguide Method’ Technical Feature, Microwave Journal, pp. 77-88, Oct. 1990. [15] Jarvis, J., Vanzura, E.J., Kissick, W.A. ‘Improved Technique for Determining Complex Permittivity with the Transmission / Reflection Method’ IEEE Transactions on Microwave Theory and Techniques, Vol. 38, No. 8, pp. 10961103, Aug. 1990. [16] Jarvis, J., Geyer, R.G., Domich, P.D., ‘A Nonlinear Least-Squares Solution with Causality Constraints Applied to Transmission Line Permittivity and Permeability Determination’ IEEE Transactions on Instrument and Measurement, Vol. 41, No. 5, pp. 646-652, Oct. 1992. [17] Williams, T. ELEC540 Project, ‘Use o f a Non-Linear Least-Squares Strategy in the Evaluation o f Parameters Concerning the Design of a Multi-Layered Radar Absorber’. University o f Victoria, 2000 [18] Balanis, C. A., ‘Advanced Engineering Electromagnetics’ Wiley, Toronto, 1989. [19] Taflove, A., Hagness, S. C., ‘Computational Electrodynamics The FDTD Method Second Edition’ Artech house, Boston, 2000. [20] Booton, R. C. Sr. ‘Computational Methods for Electromagnetics and Microwaves’ Wiley, Toronto, 1992. [21] Vector Network Analyzer Calibration, http://morph.demon.co.uk/electronics/new.htm [22] Pozar, D.M. ‘Microwave Engineering’ Second Edition, Wiley and Sons, U.S.A., 1998. [23] Ligthart, L.P. ‘A Fast Computational Technique for Accurate Permittivity Determination Using Transmission Line Methods’ IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-31, No. 3, pp. 249-254, Mar. 1983. [24] Rzepecka, M. A., Hamid, M. A. K. ‘Automatic Digital Method for Measuring the Permittivity o f Thin Dielectric Films’ IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-20, No. 1, pp. 30-37, Jan. 1972. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B-3 [25] Cheng, D. K., ‘Field and Wave Electromagnetics’ Second Edition, Addison-Wesley, U. S. A., 1989. [26] Iskander, M. F., ‘Electromagnetic Fields and Waves’ Prentice Hall, New Jersey, 1992. [27] Hayt, W. H. Jr., Buck, J. A., ‘Engineering Electromagnetics’ Sixth Edition, McGraw-Hill, Toronto, 2001. [28] Knott, E. F., Shaeffer, J. F., Tuley, M. T., ‘Radar Cross Section Second Edition’ Artech House, Boston, 1993. [29] Sadiku, M. N. O., ‘Numerical Techniques in Electromagnetics’ CRC Press, Ann Arbor, 1992. [30] Pasalic, D. ELEC629 ‘Measurements of Complex Permittivity and Permeability of Various Materials Using Transmission / Reflection Method’, University of Victoria, 2000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A- 1 7 7.1 Appendix Definitions Complex Permittivity and Permeability For non-magnetic materials, the complex relative permittivity defines the material electrical characteristics. The complex permittivity in Ampere’s Law expressed in phasor form, includes both conduction and displacement current. =J + Vx MrMo ) (7.1) & J -a E (7-2) If the current density (7.2) is represented in terms o f the electric field and conductivity, Ampere’s Law is represented as in (7.3). V x 77 = J + ja xE = o E + j a x E (7 3) Replacing conductivity with expression (7.4), (T = ere0a) (7-4) and substituting into (7.3) results in (7.5), V x /7 = OJ£"r£0E + jOJ£r£QE = jO £0E(£r - j £ r ) (7-5) The relative complex permittivity is defined as er = (e r - j e r), and shown in (7.6), V x H = jeae0£rE (7-6) A non-magnetic material has a relative permeability (|ir) equal to 1. Magnetic materials possess a complex permeability represented by pr = |i r - jp r. The constants eo and |io are 8.854xl0'l2and l/(c2eo), respectively, where c is the speed of light. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A-2 Characteristic Impedance In free space, the characteristic impedance for a TEM wave is equal to: Za = . f & (7-7) In the thesis, characteristic impedance has a general form for a TEio propagating wave in rectangular waveguide, shown in (7.8). 1 Z= (7.8) l- i J , where f is the frequency, fc is the waveguide cut-off frequency, and both er and |ir may be real or complex depending on the material through which the TEio wave is propagating. Reflection and Transmission Coefficient The voltage reflection coefficient, shown in (7.9), represents the amplitude o f the reflected voltage wave normalized to the amplitude of the incident voltage wave [22] at a reference plane. The subscripts ‘0 ’ and ‘ 1’ refer to labels o f the specific material the wave is incident from (Zo) and the material it is reflected from ( Z |) , respectively. Z -Z r = =!—— z, +z The voltage transmission coefficient, shown in (7.9) (7.10), represents the normalized transmitted voltage througha specific medium. Here y is the propagation constant, shown in (7.11), and is complex for lossy materials. •p-g-yUc) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 7 . 10) A-3 030 J I.l ~ Y = J•~ r— (7.11) For the dominant mode TEio and a waveguide filled with the material having er and ^ r, the cut-off frequency is (7.12), c 2 a ^er (7.12) where ‘a’ is the greater o f the two waveguide cross-sectional dimensions. Scattering Parameters The scattering parameters, Sn and S21, as measured by the VNA, represent the normalized voltage level change reflected by, and transmitted through a two-port, respectively. The scattering parameters are calibrated in a plane of reference, usually to the junction immediately preceding the sample holder. This way, not only amplitude but also phase-change (with the insertion of a post-calibration object) can be measured. Equation (7.13) shows the full representation of the S-parameters collected by the VNA. When = 0, the system is reduced to the equation (7.14). Therefore, Sn represents the voltage reflected back from the system, normalized by the original transmitted voltage. S21 represents the voltage transmitted through the system normalized by the original voltage o f the incident wave [22]. V y~.. ~su ~K S22..V2. Sa ~ V 'V - V 2 . -V K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.13) (7.14) A-4 7.2 Summary Uncertainty Plots (Acrylic thickness of 3mm) frequency - 10GHz. sample tidiness • 0.1mm. material tidiness • 3mm 0.1 er0 • 2er00 er • 8"er 100 1000 *3 Figure 7.1 Uncertainty of the dielectric constant due to errors in measurements of S21. Five different ratios of material at 10GHz, a sample thickness of 0.1mm and an acrylic thickness of 3mm. frequency - 10GHz. sample tidiness • 0.1mm material tidiness • 3mm er3• .. 2erM er ■ 3*er 1000 e.’3 Figure 7.2 Uncertainty of the loss factor due to errors in measurements o f S 21. Five different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic thickness o f 3mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A-5 frequency - 10GHz. sample tfsckness • Irrnv material h d u c u • 3mm 0.1 er ■ O.S*er ? Iu 3K 100 1000 e.3 Figure 7.3 Uncertainty of the dielectric constant due to errors in measurements of Si i. Five different ratios of material at 10GHz, a sample thickness o f I mm and an acrylic thickness o f 3mm. frequency - 10GHz. sample thickness ■ 1mm, material thickness ■ 3mm e r ■ O.Far e r • 0.5*er e r • erPG 8 “ 80 c r ! . 8 #eT 100 1000 e.3 Figure 7.4 Uncertainty of the loss factor due to errors in measurements o f Si i. Five different ratios o f material at 10GHz, a sample thickness o f 1mm and an acrylic thickness of 3mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A-6 frequency - 10GHz. sample thickness • 2mm. material thickness • 3mm 0.1 0.05 3 100 '3 e. Figure 7.5 Uncertainty of the dielectric constant due to errors in measurements of S21. Five different ratios of material at 10GHz, a sample thickness o f 2mm and an acrylic thickness of 3mm. frequency - 10GHz. sample ttvdmess • 2mm. material inclines* • 3mm f •§ 0 .0 s 3g 100 1000 e ’3 Figure 7.6 Uncertainty of the loss factor due to errors in measurements o f S21. Five different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic thickness o f 3mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A-7 7.3 VNA HP8720 Uncertainty Specification Sheet M easurement uncertainty Reflection m easurem ents 1 > <®tal • M • I MI •1 5 .•I a, Magnitude t «• « .9 «• » .1 .4 St I M l w t i o n I . 9 %• I 1 1 C M tltfiitn t S it N af> aetian C a a fftc ta n t Phase Transm ision m easurem ents 9« S « I <• .9 .t I t« M ! . m %m . 9 Magnitude 5 a Phase Figure 7.7 HP8720 Equipment Uncertainty Specification Sheet Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I la I .9 * • a Vita Surname: Williams Given Names: Trevor Cameron Place o f Birth: Cranbrook, British Columbia, Canada Educational Institutions Attended: East Kootenay Community College University o f Victoria 1994 to 1995 1995 to 2002 Degrees Awarded: B.Eng. University o f Victoria 2000 Top Electrical Engineering Graduating Grade Point Average of 2000 Honours and Awards: BCHydro Scholarship 1998 Nordal Scholarship 1999 URSI Student Competition 1999 NSERC Undergraduate Scholarship 1999 APEG BC Highest GPA Electrical Engineering Graduate of 2000 Advanced Systems Institute Graduate Bursary 2000 NSERC Post Graduate Scholarship A 2000 to 2002 President’s Research Scholarship 2000 and 2001 NSERC Post Graduate Scholarship B 2002 to 2004 Publications: Williams T., Rahman M., Stuchly M.A., “Dual-Band Meander Antenna for Wireless Telephones”. Microwave and Optical Technology Letters, vol 24 (2), pp. 81-85, Jan 2000 Articles in Preparation to Refereed Journals: Williams, T. M.A. Stuchly, “Measurements of Permittivity and Permeability of Thin Lossy Sheets at Microwave Frequencies”. Other Refereed Contributions: Williams, T., M.A. Stuchly, and P. Saville, “Measurement of Thin Radar Absorbing Materials”, 2001 USNC/URSI National Radio Science meeting, URSI DIGEST, p. 27, Boston, Mass., July 8-13, 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bajwa, A., Williams, T., and Stuchly, M.A., “Design o f Broadband Radar Absorbers with Genetic Algorithms”, 2001 IEEE Antennas and Propagation Society International Symposium, Vol. IV, pp. 672-675 Neville, S., Williams, T., Bajwa, A., and Stuchly, M.A., “Measurements and Optimization of Radar Absorbing Materials: Preliminary Considerations”, Department o f National Defense Cansmart Workshop, Kingston, Ontario, Sept. 2000 Stuchly, M.A., M. Rahman, M. Potter, and T. Williams, “Modeling Antenna Performance in Complex Environments”, IEEE 2001 Aerospace Conference, Big Sky, Montana, USA, March 18-25 Non-Refereed Contributions: Williams, T., Bajwa, A., and Stuchly, M.A., Poster presentation, Advanced Systems Institute (AS I) Exchange Day, April 2001 Williams, T., Stuchly, M.A., 1999 URSI General Assembly, Presentation, “Design of a DualFrequency, Wideband, Dual-Meander Antenna for Cellphone Applications”, Undergraduate Student Finalist, August, 1999 Potter, M., Fear, E., Rahman, M., Williams, T., Poster presentation, Advanced Systems Institute (ASI) Exchange Day, April 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. University of Victoria Partial Copyright License I hereby grant the right to lend my thesis to users of the University of Victoria Library, and to make single copies only for such users or in response to a request from the Library of any other university, or similar institution, on it behalf or for one o f its users. I further agree that permission for extensive copying o f this thesis for scholarly purposes may be granted by me or a member o f the University designated by me. It is understood that copying or publication o f this thesis for financial gain by the University of Victoria shall not be allowed without my written permission. Title of Thesis: A Wideband Two-Layer Microwave Measurement Method for the Electrical Characterization o f Thin Materials Author Trevor Cameron Williams Aug. 30, 2002 Note: This license is separate and distinct from the non-exclusive license for the National Library o f Canada. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1/--страниц