# Microwave nondestructive testing and evaluation of electricalproperties of lossy materials

код для вставкиСкачатьMICROWAVE NONDESTRUCTIVE TESTING AND EVALUATION OF ELECTRICAL PROPERTIES OF LOSSY MATERIALS BY UGUR CEM HASAR B.Sc., Cukurova University, Adana, Turkey, 2000 M.Sc., Cukurova University, Adana, Turkey, 2002 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the Graduate School of Binghamton University State University of New York 2008 3324358 2008 3324358 © Copyright by Ugur Cem Hasar 2008 All Rights Reserved Accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the Graduate School of Binghamton University State University of New York 2008 Charles R. Westgate, Department of Electrical and Computer Engineering, Binghamton University Mehmet Ertugrul, Department of Electrical and Electronic Engineering, Ataturk University, Turkey James Constable, Department of Electrical and Computer Engineering, Binghamton University Vladimir Nikulin, Department of Electrical and Computer Engineering, Binghamton University iii ABSTRACT The measurements of reflection and transmission complex scattering (S−) parameters are generally utilized for characterization of electrical properties of materials at microwave frequencies. There are three disadvantages of these measurements: a) ambiguities in phase; b) errors caused by shifts in calibration planes; and c) phase uncertainty of reflection scattering parameters of low–loss materials. The connection between the characterization of electrical properties of materials and the application of these measurements in industrial−based applications is simply lacking. The first part of this dissertation focuses on two new measurement methods to fill this gap by eliminating the aforementioned disadvantages. In the literature, microwave nonresonant transmission/reflection methods for materials electrical characterization eliminate either the dependency of calibration−plane on measurements or the dependency on thickness in the measurements. A recently proposed method to simultaneously eliminate these dependencies is inadequate for both complex permittivity and complex permeability measurements. In the second part of this dissertation, we propose a unique technique for calibration−plane invariant and thickness−independent complex permittivity and complex permeability determination of granular and/or liquid materials. Cement−based materials (cement paste, mortar, concrete, etc.) are the most widely used construction materials in the world. Knowledge of mechanical, mixture proportion, chemical properties as well as electrical properties of such materials is iv important for evaluation of their quality and integrity. The methods commonly used in civil engineering for characterizing their mechanical properties are generally destructive. The third part of this dissertation presents microwave nondestructive reflection and transmission properties of young mortar specimens with different w/c ratios over their early curing periods and hardened mortar and concrete specimens with different w/c ratios over their late curing periods. A direct application of available microwave methods in the literature to industrial−based applications may not be appropriate since these methods generally necessitate expensive instruments. Therefore, there is a need for a simple and relatively inexpensive microwave sensor for electrical characterization of materials in industrial−based applications. In the final part of the dissertation, we propose a microcontroller−based microwave free−space sensor for measurements of electrical properties of lossy materials. v DEDICATION I would like to dedicate this dissertation to my parents, Harun Hasar and Ayse Hasar, my brothers, Hasan Alper Hasar and Ahmet Alpay Hasar, and my fiancée, Hafize Yuzgulec. I would not have been able to accomplish this goal without their invaluable and endless supports, patience, and understanding. Bu çalışmayı, aileme, kardeşlerime ve sözlüm Hafize Yüzgüleç’e ithaf ediyorum. Sizlere ne kadar teşekkür etsem azdır. vi ACKNOWLEDGMENTS I wish to express my gratitude to my academic advisors, Dr. Charles R. Westgate, of Electrical and Computer Engineering Department, Binghamton University, and Dr. Mehmet Ertugrul, of Electrical and Electronic Engineering Department, Atatürk University, for all of their knowledge, guidance, patience, and encouragement throughout the initiation, preparation, and completion of this dissertation. Without their great help and supports, I would not have achieved this. I would like to thank Dr. James Constable, Dr. Vladimir Nikulin, and Dr. John Fillo of the Thomas J. Watson School of Engineering and Applied Science for participating in and serving on my committee and for their practical insight regarding my dissertation material. I also would like to take this opportunity to thank my former academic advisors, Dr. Sergey N. Kharkovsky, of Electrical and Computer Engineering Department, Missouri University of Science and Technology (formerly University of Missouri-Rolla) and Dr. Victor Pogrebnyak, of Electrical Engineering Department, University at Buffalo for his inspiration and invaluable support. With this opportunity, I also would like to thank all the faculty members in Electrical and Electronics Engineering Department and my friend Cüneyt Gözü and faculty members in Industrial Engineering Department at Atatürk University for their help and endless support. vii Finally, I would like to thank the Higher Education Council of Turkey (YOK) for providing me with a scholarship to pursue my Ph.D. degree in USA. Also, I would like to express my thanks to the Scientific and Technological Research Council of Turkey (TUBITAK) for supporting me two scholarships during my M.Sc. and earlier Ph.D. studies. viii TABLE OF CONTENTS LIST OF TABLES .......................................................................................................... xi LIST OF FIGURES ....................................................................................................... xii CHAPTER 1 INTRODUCTION ............................................................................... 1 1.1 Background .......................................................................................... 1 1.2 Overview of the Chapters .................................................................... 7 CHAPTER 2 2.1 2.2 2.3 2.4 CHAPTER 3 3.1 3.2 3.3 3.4 REVIEW OF PREVIOUS STUDIES ................................................ 10 Novel Methods for Lossy Materials Characterization ....................... 10 A New Sample Holder for Low-Loss and Lossy Materials Characterization ................................................................................. 13 Electrical Characterization of Cement-Based Materials .................... 16 A Simple Microwave Sensor for Electrical Characterization of Materials in Industrial-Based Applications ....................................... 21 TWO MICROWAVE METHODS FOR COMPLEX PERMITTIVITY DETERMINATION OF LOSSY MATERIALS USING AMPLITUDE-ONLY S-PARAMETER MEASUREMENTS ......... 23 Theoretical Analysis .......................................................................... 24 3.1.1 Complex Permittivity Extraction Using the Classical MultipleReflections Model .................................................................. 27 3.1.2 Validation of the Derived Forward Scattering Parameters and Derivation of Reverse Scattering Parameters ........................ 29 An Amplitude-Only Method for Permittivity Determination of Lossy Materials ............................................................................................ 31 3.2.1 Selection of a Suitable Approach for Permittivity Extraction 32 3.2.2 Single-Pass Technique ........................................................... 33 3.2.3 Derivation of a Simple Objective Function ........................... 35 3.2.4 Functional Analysis of the Derived Objective Function ........ 39 3.2.5 Refining the Computed Complex Permittivity ...................... 40 An Amplitude-Only Method for Thickness-Independent Complex Permittivity Determination of Lossy Materials ................................. 41 3.3.1 Zero-Order (ZO) Approach for Complex Permittivity Determination ........................................................................ 43 3.3.2 Higher-Order (HO) Approach for Complex Permittivity Determination ........................................................................ 46 Measurements .................................................................................... 47 ix CHAPTER 4 4.1 4.2 4.3 CHAPTER 5 5.1 5.2 5.3 5.4 5.5 5.6 CHAPTER 6 6.1 6.2 6.3 6.4 CHAPTER 7 7.1 7.2 A WAVEGUIDE SAMPLE HOLDER FOR AUTOMATED THICKNESS-INDEPENDENT COMPLEX PERMITTIVITY AND COMPLEX PERMEABILITY MEASUREMENTS OF GRANULAR AND LIQUID MATERIALS ............................................................ 56 Theoretical Analysis .......................................................................... 58 The Method ........................................................................................ 60 Measurements .................................................................................... 62 A SIMPLE FREE-WAVE METHOD FOR ELECTRICAL CHARACTERIZATION OF CEMENT-BASED MATERIALS …. 69 Theoretical Background ..................................................................... 71 The Method ........................................................................................ 72 The Measurement Set-up ................................................................... 73 Calibration Procedure ........................................................................ 75 Measurements .................................................................................... 77 5.5.1 Measurements of Young Mortar Specimens .......................... 77 5.5.2 Measurements of Hardened Mortar and Concrete Specimens 90 Measurement Uncertainties ............................................................. 101 A SIMPLE AND RELATIVELY INEXPENSIVE MICROCONTROLLER-BASED MICROWAVE FREE-SPACE MEASUREMENT SYSTEM FOR INDUSTRIAL-BASED APPLICATIONS ............................................................................. 108 The Measurement System ................................................................ 109 Calibration Procedure ...................................................................... 112 Algorithm ......................................................................................... 116 Results .............................................................................................. 121 6.4.1 Validation of Programs and the Microwave Section ........... 121 6.4.2 Validation of Whole Measurement System ......................... 123 CONCLUSION AND FUTURE WORK ........................................ 127 Conclusion ………………............................................................... 127 Future Work ..................................................................................... 131 REFERENCES ........................................................................................................... 133 x LIST OF TABLES Table 4.1 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 6.1 Table 6.2 The parameters ε ∞ , ε s , and τ for distilled water and methanol. The data are estimated by interpolation from [104],[105] for room temperature (20–25 Co) ............................................................................66 Mass percentages of raw materials of cement samples ........................... 78 The absolute temporal change of reflection coefficient of each young mortar specimens ..................................................................................... 81 Mixture proportions of mortar and concrete specimens .......................... 90 Reflection properties from various sides of mortar and concrete specimens with different w/c ratios after 3 months of curing .................. 96 The comparison of the permittivity results ............................................ 121 The comparison of the measured R and T from [66] and computed Rc and Tc by all µC programs .............................................................. 122 xi LIST OF FIGURES Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 5.1 Fig. 5.2 The scattering parameter measurements of a sample sandwiched between two dielectric plugs .................................................................... 25 The diagram for the analysis of the five–layer structure in Fig. 3.1 by multiple reflections approach ................................................................... 27 Conformal transformation between ε 3 and ( χ , ξ ) pair ........................... 35 Dependency of f (ξ ) over ξ for different ε 3 ......................................... 40 The measurement set–up .......................................................................... 47 Comparison of the frequency response (R) with/out aperture smoothing and the TRL calibrations for (a) waveguide short and (b) thru connections ................................................................................................................... 50 Measured ε of an antifreeze solution extracted by the plug–loaded two– port transmission–line method (PLTL) [25] and the proposed method (PM) in section 3.2 without aperture smoothing and with 20 % aperture smoothing ................................................................................................. 52 Measured ε of a binary mixture of ethyl alcohol (75%) and water (25%) by the plug–loaded two–port transmission–line method (PLTL) [25] and the proposed method (PM) in section 3.2 without aperture smoothing and with 20 % aperture smoothing ................................................................. 53 (a) Real part and (b) imaginary part of the measured ε of methanol by the plug–loaded two–port transmission–line method (PLTL) [25] and by the proposed method (PM) in section 3.3 without aperture smoothing and with 10 % and 20 % aperture smoothing ......................................................... 55 The complex S–parameter measurements of granular and liquid samples sandwiched between stable and movable dielectric plugs ....................... 58 (a) Real part and (b) imaginary part of the measured (dashed line) and theoretical (solid line) ε of distilled water ……...................................... 63 (a) Real part and (b) imaginary part of the measured (dashed line) and theoretical (solid line) ε of methanol ….................................................. 64 (a) Real part and (b) imaginary part of the measured ε of an antifreeze solution by the plug–loaded two–port transmission line method (PLTL) [25] and by the proposed method ............................................................. 65 Dependency of the computed length of air–filled section, L0 , inside the proposed sample holder from commercially available antifreeze solution measurements over frequency .................................................................. 68 The configuration for the measurement of reflection and transmission properties by using a free–space method ................................................. 71 The schematic diagram of the measurement set–up ................................ 74 xii Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Hourly reflection properties of young mortar samples with different w/c ratios at 8.5 GHz (a) between 20–30 hours and (b) between 44–54 hours .................................................................................................................. 80 Hourly transmission properties of young mortar samples with different w/c ratios at 8.5 GHz (a) between 20–30 hours and (b) between 44–54 hours .................................................................................................................. 84 (a) Real part and (b) imaginary part of the complex permittivities of young mortar samples with different w/c ratios between 20–30 hours .............. 88 (a) Real part and (b) imaginary part of the complex permittivities of young mortar samples with different w/c ratios between 44–54 hours .............. 89 Results of transmission properties from right sides of mortar and concrete specimens with different w/c ratios .......................................................... 91 Application of the relative difference approach to (a) the same type specimens and (b) different type specimens ............................................ 93 The real part of the ε of mortar and concrete specimens with different w/c ratios and sides ......................................................................................... 99 The imaginary part of the ε of mortar and concrete specimens with different w/c ratios and sides .................................................................. 100 Dependencies of reflection and transmission properties of w/c=0.40 mortar sample with ε = 21 − j 0.69 on sample thickness, L ............................. 102 Dependency of Z 2 versus specimen thickness, L , for ε = 6.95 − j 0.22 and ε = 11.18 − j 0.60 .............................................................................. 105 The schematic diagram of the microcontroller–based microwave free– space measurement system .................................................................... 110 The circuit board of the electronic section ............................................. 111 The schematic diagram of the calibration technique ............................. 113 The simplified algorithm for each program ........................................... 118 The memory footprint, total ROM space allocated, and some information related to the program, which uses Newton’s method for ε , written in BASCOM ............................................................................................... 120 Hourly measurement of r and t at 10 GHz for the cement paste sample with w/c=0.375 ....................................................................................... 124 Determined ε of the cement paste sample using the measurements in Fig. 6.6 ................................................................................................... 125 xiii CHAPTER 1 INTRODUCTION 1.1 Background Material characterization is an important issue in many material production processes and management applications in agriculture, food engineering, medical treatments, bioengineering, and the construction industry [1]. In addition, microwave engineering requires precise knowledge of the electromagnetic properties of materials at microwave frequencies since microwave communications are playing more and more important roles in military, industrial, and civilian life [2]. For these reasons, various microwave techniques, each with its unique advantageous and limitations [2], [3], are introduced to characterize materials’ properties. Materials and manufactured products are often tested before delivery to the user to ensure they will meet expectations and remain reliable during a specified period of service. It is essential that any test made on a product intended for future use in no way impairs its properties and performance. Any technique used to test under these conditions is called a nondestructive testing (NDT) method. The monitoring of any structural changes and properties such as defect initiation and growth in materials and products during the production and service lives by NDT methods is known as nondestructive evaluation (NDE) [4]. 1 Methods in NDT can be roughly divided into five categories. These are: 1) radiological methods: X–rays, gamma rays, and neutron beams; 2) acoustical and vibrational methods: ultrasound and mechanical impedance measurements; 3) electrical and magnetic methods: eddy current, magnetic flux leakage, and microwave testing; 4) visual and optical methods: interferometry, holography, and dye penetrants; and 5) thermal methods: infrared radiation and thermal paints [4]. Two or more NDT methods may often be suitable for NDE; but, this does not necessarily imply they may be regarded as alternative techniques. It often happens that one of the methods is used to complement another or perhaps to verify the findings of the other [4]. Microwave nondestructive testing (MNDT) methods have advantages over other NDT methods (such as radiography, ultrasonics, and eddy current) in terms of low cost, contactless feature of the microwave sensor (antenna), good resolution and superior penetration in nonmetallic materials [5]. They are known to be excellent candidates for aquametry (detection of moisture content or bound– and free–water) in environments in which moisture content determination is sought [6],[7]. In addition, MNDT techniques are insensitive to environmental conditions, such as water vapor and dust (compared to infrared sensors) and high temperatures, safe (compared to radioactive radiation), and fast. However, microwave sensors are expensive (the costs have generally fallen with time), must be adapted to a specific application, sensitive to more than one variable, and must be calibrated separately for different materials [8]. Microwave methods for materials characterization generally fall into resonant or nonresonant methods [2]. Resonant techniques have much higher accuracies and sensitivities than those of nonresonant techniques. A shift in the resonant frequency and a 2 decrease in quality factor after insertion of a material are, respectively, utilized for the determination of the real part (dielectric constant) and the imaginary part (loss factor) of the complex permittivity. However, materials resulting in a large frequency shift from the resonance frequency or a considerable decrease in the quality factor are not accurately measurable by resonator methods which use a perturbation technique [9]. As a result, these methods are generally utilized for electrical characterization of medium– and low– loss materials. In a recent study, it was shown that resonant techniques are also applicable to high–loss materials provided that very small samples are prepared or higher volume cavities are constructed [10]. Nonetheless, resonant methods require meticulous sample preparation before measurement. In addition, for an analysis over a broad frequency band, new measurement set–ups (or cavities) must be made, which are not feasible from a practical point of view. Tunable resonators can be used for a wider frequency band analysis; but, they are expensive and an increase in the frequency bandwidth accompanies a decrease in the accuracy. Examples for some resonant instruments are open (or Fabry–Perot) resonator [11]–[13], cavity resonators [14], whispering gallery modes [15]–[17], etc. Nonresonant methods have relatively higher accuracy over a broad frequency band and necessitate less sample preparation compared to resonant methods. Additionally, they allow frequency– or time–domain analysis, or both. These features make these methods very attractive for industrial–based applications. In these methods, electrical properties of materials are usually derived from amplitude and/or phase measurements of reflection and/or transmission coefficients [2]. Reflection measurements require that the reference plane be well defined, and the higher the frequency the more 3 they are affected by the surface characteristics. However, they are convenient in some instances because the sensor can be placed on one side of the material (for instance, measurements on walls and pavement, back–filled concrete wall, roadway, etc). Additionally, transmission measurements have the advantage of providing more information on the whole volume, because the wave propagates through all the material in its path; however, they require two sensors for their measurements. Because nonresonant methods are applicable and feasible for industrial–based applications, we will restrict our study to focus solely on nonresonant methods. Outcomes of this dissertation are fourfold. Firstly, we will propose two methods for characterization of the electrical properties of lossy materials. In the literature, methods used for materials characterization generally employ complex scattering (S–) parameter measurements. To measure these parameters, one needs to utilize a vector network analyzer (VNA). However, these instruments are expensive, complex in operation, and require expertise. Therefore, there is a need in industrial–based applications for characterization of materials by simple and relatively inexpensive measurement set–ups. Scalar network analyzers (SNAs), for example, can be used as a relatively inexpensive solution for materials property measurements. However, because they are limited to measurements of the amplitudes of S–parameters, it is not obvious how to use amplitude–only measurements to fully characterize materials. To fill this gap, we developed two amplitude–only methods. The first method extracts complex permittivity of lossy materials at one fixed frequency, and the second determines both the complex permittivity and thickness of lossy materials using multiple frequencies. 4 Second, we will present a design of a highly sensitive and accurate waveguide holder for characterization of liquid and granular materials. It is well known that measurements of electrical properties of solid materials are easier than those of liquid and granular materials provided that the solid materials are not very thin. This is because granular and liquid materials produce a meniscus, which requires a different approach than those used for solid materials. Another problem that an experimentalist encounters while performing the measurements on granular and liquid materials is that in most instances he cannot explicitly evaluate the thickness of these materials. The final problem, which is related to the second one, is that one needs to implement the transformation from the calibration plane (the measurement point the system is calibrated) to the measurement plane (the measurement point where the sample is located). In the literature, there is no unique technique which eliminates all these drawbacks. In this dissertation, we propose a method which determines the thickness of materials, calculates the calibration plane factors (used for transforming the calibration plane to measurement plane), and characterizes the complex permittivity and complex permeability of solid (thin), liquid, and granular materials. Third, we will propose a method to measure and monitor the electrical properties of cement–based materials (cement paste, mortar, concrete, etc.) over their long service times using amplitude–only measurements. These materials are the most widely used construction materials in the world. Knowledge of mechanical, mixture proportion, and chemical properties (compressive and tensile strengths, water–to–cement ratio (w/c), sand–to–cement ratio (s/c), coarse aggregate–to–cement ratio (ca/c), porosity, degree of hydration, etc.) of such materials is important for 5 evaluation of their quality and integrity. These mechanical and physical properties are correlated with the electrical properties of materials. Therefore, the characterization of electrical properties of these materials can be utilized to improve the physical and mechanical properties and to detect any integrity problems such as air bubbles, cracks, etc. The general methods used in civil engineering for characterizing the mechanical properties of cement–based materials are either to prepare a sample with the same mix proportions in the lab, or to extract a sample from the structure, and then test in the lab. The sample preparation method does not reflect the actual measurement of interest since the sample in the lab and the materials cast in the field are not cured in the same environment [18]. The sample removal method is clearly harmful. As a result, the test techniques used for both methods are destructive [19] because the sample is damaged and cannot be used in future. In the literature, characterization of these materials has been done extensively using reflection measurements. However, these measurements cannot fully characterize these materials. It was shown that transmission properties are also important for structural inspection of these materials [20]. For these reasons, in this dissertation we present reflection and transmission measurements for inspection and full characterization of these materials. Finally, we will develop a simple microwave sensor which is constructed from relatively inexpensive and readily available components for characterization of materials for industrial–based applications. The sensor uses free–space (free–wave) measurements, which are very suitable for noncontact, nondestructive, noninvasive, and in situ characterization of materials. To reduce the cost of the system, we used a microcontroller to acquire the measurement data, process these data, and calculate the complex 6 permittivity of materials. In addition, because free–space measurements suffer from diffraction from sample edges, we also developed a calibration procedure for highly accurate measurements by the set–up. 1.2 Overview of the Chapters This dissertation is organized as follows: In chapter 2, we will present a review for each of the four aforementioned research studies in this chapter. Then, in chapter 3, we will derive forward and reverse reflection and transmission S–parameters for a five–layer structure using a conventional multiple–reflections model. The derivations are sufficiently general that they can be used in subsequent chapters. Furthermore, we derive a well–defined objective function which depends on only one unknown. This is significant since fast and accurate computations could be achieved by this function. We introduced two amplitude–only methods (methods based on amplitudes of S–parameter measurements) for complex permittivity determination of lossy materials. While the first method extracts the complex permittivity of lossy materials sandwiched between two low–loss dielectric plugs at one frequency, the second uses the complex permittivity of lossy materials without any prior physical thickness measurement using amplitude–only measurements at different frequencies. Both of these methods are validated by a general purpose rectangular waveguide structure using amplitude–only measurements after a simple calibration technique, which only corrects the frequency errors, and a full two– port calibration technique are applied. We verified that the extracted complex permittivity from amplitude–only measurements calibrated by a simple calibration technique is in very good agreement with that achieved by the full two–port calibration. Therefore, the 7 proposed methods allow the measurements to be carried out by scalar network analyzers which are relatively inexpensive and simple in structure compared to vector network analyzers; and they are suitable candidates for industrial–based applications. In chapter 4, a waveguide sample holder for automated thickness–independent complex permittivity and complex permeability determination of low–, medium–, and high–loss materials is introduced. There are three main advantages of the holder as a) it eliminates the dependency of calibration plane on measurements; b) it takes into account the effect of low–loss plugs, which are utilized to hold the liquid or granular samples in place; and c) it determines the complex permittivity and complex permeability of materials without any previous physical thickness measurement of samples. In addition, the holder is very important for measurements of solid thin materials since these materials can result in sagging over the plane that measurements are carried out, and accurately determines their complex permittivity and complex permeability by avoiding the sagging problem using low–loss dielectric plugs. For validation of the sample holder, we conducted measurements of the methanol, distilled water, and a commercially available antifreeze solution. It is observed that the extracted complex permittivities of these samples are in good agreement with those obtained from a theoretical model (Debye model) and with those determined from a robust technique reported in the literature. In chapter 5, we demonstrate the free–space microwave measurements of cement– based materials (concrete, mortar, etc.) using a relatively inexpensive and simple set–up. Because these measurements are noncontact, nondestructive, and noninvasive, the set–up can be used for in situ measurements. We carried out two types of measurements as: a) 8 measurements of reflection and transmission properties of young mortar specimens with different water–to–cement ratios over 20–54 hours after sample preparation, and b) measurements of reflection and transmission properties of hardened mortar and concrete specimens with different water–to–cement ratios over 3–36 months after sample preparation. We measured these properties using relative measurements, and analyzed the possible measurement uncertainties. In chapter 6, we present a microcontroller–based microwave free–space sensor for electrical characterization of lossy materials. We have tried to keep the overall cost of the sensor minimum such that the sensor can easily be adapted for industrial–based applications. We also developed a calibration technique, which is similar to the well– known Thru–Reflect–Line technique, for the sensor to decrease and eliminate undesired reflection from environments, the losses in the measurement set–up, and diffractions from edges of the sample. This is imperative since free–space measurements are prone to these unwanted reflections and diffractions. We refined the domain for computations by using the objective function derived in chapter 3, and coded three different programs for complex permittivity determination of lossy materials. We validated the programs using the measurement data taken from the literature by comparing their results with those of the techniques in the literature. For validation of the sensor, we carried out reflection and transmission measurements of a lossy sample (fresh cement paste) immersed into a wooded container and calculated its complex permittivity. 9 CHAPTER 2 REVIEW OF PREVIOUS STUDIES Because the dissertation can be divided into four main sub–fields, in this chapter, we will present a review of previous studies for each sub–field. 2.1 Novel Methods for Lossy Materials Characterization Due to its relative simplicity, the transmission–reflection (TR) technique is a widely used measurement technique [21] for broadband measurements of materials. The well–known Nicolson–Ross–Weir (NRW) method [22], [23] is a type of TR technique, which determines the complex permittivity, ε , and permeability, µ , of materials noniteratively. Because initially the NRW method was considered as a three–layer structure, Williams et al. modified it to suit for simultaneous ε and µ determination of lossy thin materials backed by a low–loss material [24] and Bois et al. adapted it for simultaneous ε and µ determination of granular and liquid materials plugged (or sandwiched) by two low–loss materials [25]. The complex scattering (S–) parameters are generally measured to achieve this goal by TR techniques [26]. However, there are three main problems associated with these measurements. First, they often yield multiple solutions. The general approach to determine one solution was to choose a thickness for the material less than one–quarter wavelength [27]. However, thin materials not only result in sagging, which alters the 10 theoretical formulations [28], but also decrease the feasibility and repeatability of measurements. Though other test materials can be attached or adhered to the material under test, the inhomogeneity in these materials can notably reduce the accuracy of ε and µ estimation. To circumvent the problems arising from using thin materials for extracting one solution, Weir et al. proposed a technique which compares calculated and measured group delays and determines an initial guess for ε and µ after this comparison [21],[23]. The second problem with complex S–parameter measurements is that TR techniques drastically suffer from the phase uncertainty of reflection measurements of low–loss materials at frequencies which correspond to multiple–half wavelengths inside these materials [21],[29],[30]. At these frequencies, amplitudes of reflection measurements decrease significantly, and measured phases of reflection measurements diverge from their actual values. This results in a sharp ripple in the extracted ε and µ over frequencies. To overcome this problem, Baker–Jarvis et al. used an iterative numerical technique [30], which removes the ripples at those frequencies. This technique uses a parameter whose value depends on the length and electrical properties of the material and can be set zero or infinity. Although the technique is effective, it yields unambiguously extractions of ε only in low–loss materials since that parameter is set to zero. This eliminates the reflection S–parameter measurements, and eventually forces one to use transmission S–parameter measurements in the iteration process. In addition, the technique requires a good initial guess for ε before the iteration. The authors proposed that this initial guess can be determined from the solution of the NRW method at one frequency. However, the extraction of ε for the entire frequency band may result in erroneous results if the initial guess is evaluated at a frequency which corresponds to 11 minimum measured reflection S–parameters. More importantly, this technique does not solve the multiple–solutions problem in ε determination since it uses, as an initial guess, the extracted ε at one frequency from the NRW method, which inherently has multiple solutions for ε and µ . Later, Boughriet et al. proposed a noniterative and stable effective parameter (NSEP) method to eliminate the ripples in the measured ε and µ [31]. The method is superior to the iterative technique [30] in that it explicitly determines ε and µ as does the NRW method. Even if the method effectively suppressed the ripples for ε measurements by removing the ill–behaved term, 1 − Γ 1 + Γ , from the ε equation, it is not suitable for broadband and simultaneous ε and µ measurements since that term still appears in the equation of µ . Furthermore, the method also extracts multiple solutions for ε and µ as do the NRW and iterative technique, and, in this sense, requires a prior knowledge about or initial guess for the electrical properties of material under test. Finally, for low– or high–loss materials, the total uncertainty in ε determined by the NSEP method is greater than that by the iterative technique [30]. Another drawback in using complex S–parameter measurements is that measured ε and µ are adversely affected from a small change in phase measurements that arise from any shift in the calibration plane (the measurement point where the calibration is performed) of materials. Finally, accurate complex S–parameter measurements require using expensive VNAs, which makes microwave techniques unsuitable for industrial– based applications [26],[32]. To eliminate all the aforementioned drawbacks of complex S–parameter measurements, amplitude–only S–parameter measurements can be employed. In this dissertation, we develop two methods which use amplitude–only S– 12 parameter measurements for ε determination of materials and also eliminate the well– known multiple–solutions problem. The first method not only considers a three layer structure (the material under test is located inside its holder) but also takes into account of four– and five–layer structures which are needed for completeness of the analysis and for application purposes (the liquid materials placed between two low–loss dielectric plugs). We derived an equation, which depends solely on only one unknown quantity, for lossy materials. This is very important for both practical reasons (fast and accurate ε determination) and numerical stability (one solution for ε ). The second method uses amplitude–only measurements at different frequencies for one–valued (correct and unique) ε determination of lossy materials. This method is different than the first method in that it also determines the thickness of these materials, which removes the necessity of its measurement or priori knowledge. This is very important and provides immunity to methods for industrial–based applications where thickness measurements are dynamically difficult to achieve. For example, materials continuously moving on a conveyor belt. 2.2 A New Sample Holder for Low-Loss and Lossy Materials Characterization When applying the TR techniques for measurements of granular and liquid materials, some form of approximation in the physical nature of the measurement apparatus and/or forward and inverse problem formulations are made [25], [33] or formulations require that the dielectric specimen be low–loss [34]–[36]. To avoid any meniscus formation on top of liquid samples, generally the measurement set–up is positioned vertically [37]. However, this positioning will not totally eliminate the meniscus. For the case of granular specimens, planar measurement planes may not be easily achieved [25] and this can 13 degrade the measurement performance if the formulae characterizing the problem, which in some instances cannot satisfy the requirement of the plane distortion as a result of its unpredictable nature, are not provided. Another problem for measurements of these materials which are retained between two low–loss dielectric plugs (or windows) is the lack of appropriate calibration which takes into account the materials thickness. It is well known that for high–loss liquid or granular materials, the amplitude level of transmission measurements decreases considerably, and, in some instances, this circumstance drastically affects the measurement accuracy if its level is comparable with the noise floor or threshold of VNAs. Calibrating a sample fixture containing two low–loss dielectric plugs where these plugs are separated from each other by a distance in which the liquid or granular material will be retained may degrade the measurement accuracy if the liquid or granular material attenuates the electromagnetic waves, while passing through it, more than VNAs can accurately measure. Coaxial sample holders can be utilized for broadband measurements [38]; however, they suffer from bad contacts between the sample and coaxial inner and outer conductors [39], [40]. On the other hand, if a coaxial line sample holder is used and dimensioned so that only single mode operation is possible, the dimensions become small and meniscus and other dimensional uncertainties may adversely affect the measurement [41]. Because waveguide measurements are highly sensitive, sample preparation for these measurements is relatively easy, and mechanical connections of waveguide sections are simple. Waveguide sample holders can be employed for precise dielectric characterization of liquid samples [25], [33],[41]. 14 Somlo has proposed a self–checking method for extracting ε and µ of liquid or granular materials in waveguide measurements [41]. This method auto–monitors the accuracy of measurements and also removes the calibration problem for these materials by employing only one low–loss dielectric plug for holding the sample. However, the measurement accuracy deteriorates because of the meniscus formation on top of liquid materials where surfaces are in direct contact directly with the air section inside the waveguide. To circumvent calibration plane unevenness, Bois et al. used a waveguide sample holder for granular and liquid materials which are sandwiched between two dielectric plugs [25]. This method not only solves the calibration problem by considering the effects of the plugs on theoretical analysis but also eliminates the calibration plane irregularities by employing two movable low–loss plugs. However, it may experience liquid leakage passing through the movable plugs while compacting the sample, which eventually can decrease the measurement repeatability. In addition, the method requires precise knowledge of the measurement plane and the sample length. In practical applications, a feasible method needs to be developed to eliminate the requirement for precise knowledge of the thicknesses of samples. In this dissertation, we propose a different waveguide sample holder which combines the advantages of these aforementioned methods. It both eliminates the measurement plane dependency from measurements and includes the effect of plugs on theoretical formulations. Other advantages of the method are that the thickness of the sample does not need to be known either precisely or approximately and that the holder could be horizontally or vertically positioned since any meniscus that could form on top of the liquids is eliminated by compacting samples using the plugs. 15 2.3 Electrical Characterization of Cement-Based Materials Cement–based materials (cement paste, mortar, concrete, etc.) are the most widely used construction materials in the world. Knowledge of mechanical, mixture proportion, and chemical properties (compressive and tensile strengths, water–to–cement ratio (w/c), sand–to–cement ratio (s/c), coarse aggregate–to–cement ratio (ca/c), porosity, degree of hydration, etc.) of such materials is important for evaluation of their quality and integrity. The general methods used in civil engineering for characterizing the mechanical properties of cement–based materials are either to prepare a sample with the same mix proportions in the lab, or to extract a sample from the structure, and then test in the lab. The sample preparation method does not reflect the actual measure of interest since the sample in the lab and the materials cast in the field are not cured in the same conditions [42]. The sample removal method is clearly harmful. The test techniques used for both methods are destructive [19] because the sample is damaged and cannot be used in future. Therefore, several NDE techniques have been proposed and applied for quality assessment, mixture content evaluation, and monitoring the internal integrity (voids, cracks, degradation, etc.) of these materials [43]–[45]. Among the techniques, microwave NDE (MNDE) techniques are not hazardous and low in cost compared to radioactive methods, scatter little compared to acoustic waves, and have a good spatial resolution and superior penetration in nonmetallic materials [5]. The spatial resolution of MNDE techniques, varying roughly from 3 mm to 100 mm, depends upon the wavelength (or frequency) and whether a near– or far–field approach is applied. 16 MNDE techniques used for cement–based materials’ investigation can be divided into two groups: near–field and far–field. Assuming that microwave signal unboundedly radiates into free–space, near–field techniques will have a higher resolution and accuracy than far–field techniques since the footstep of the microwave signal for former techniques is much smaller than that of latter ones [42], [46]–[54]. Dr. Zoughi and his colleagues at the Missouri University of Science and Technology (formerly University of Missouri-Rolla) applied the near–field open–ended waveguide contact with the sample under test (SUT) for cement–based materials’ investigation and demonstrated its potential for cure–state monitoring, mixture content estimation, and early age strength prediction [42], [47]–[49]. However, contacting techniques require close proximity probing with the SUT which limits their applicability and efficacy since, if improper sensors are employed, undesired higher–order modes, which drastically affect the measurements, can appear as a result of any minuscule air gap between the sensor and the SUT [55]. Although the SUT can be placed at a distance (the stand–off distance) from the antenna, open–ended waveguide measurements can be drastically affected by any minute change (a few millimeters) of this distance during measurements [50]–[52]. This can be a problem for investigation of real construction structures since they have some surface roughness associated with them [56]. To eliminate the adverse effects of this distance on measurements, a method which uses orthogonally dual–polarized microwave signals can be employed [57]. However, this method is applicable to only anisotropic materials such as carbon fiber reinforced polymers. Next, it is known that electrical properties of the SUT in the near–field of an antenna are known to influence the antenna beam [58]. Because cement–based materials 17 contain water and some varying mixture contents (coarse and fine aggregates, etc.), their electrical properties can notably alter the antenna beam pattern at proximity. Finally, the numerical modeling of wave–material interaction for near–field techniques is relatively complex [59],[60]. Far–field MNDE techniques have also shown a great potential for the investigation of cement–based materials [59]–[65]. These techniques, unlike near–field techniques, are contact–free methods that deal only with the dominant excitation mode and also allow plane wave assumption for modeling. Although their accuracy and sensitivity is lower than near–field techniques, they enable large area coverage and are very suitable for remote sensing [61]. Knowledge of electrical (dielectric, conductivity, or complex permittivity, etc.) properties of cement–based materials is also important for propagation–related research; for example, microwave propagation modeling to develop indoor wireless communication systems [66]–[70]. This is because reflection and transmission characteristics of buildings, walls, etc. are governed by these electrical properties [20], [64]. In addition, for practical purposes, it is attractive to use the simplest measurement system because using expensive microwave instruments, such as VNA, are not convenient for industrial–based applications [32]. Recent investigations [42], [47] by Dr. Zoughi and his colleagues have demonstrated the capability and potential of microwaves to detect the state and degree of chemical reaction (hydration) in cement–based materials. In addition, a strong correlation between the magnitude of reflection coefficient of microwave signals and the water–to– cement ratio (w/c) of these materials, which eventually can be utilized for strength 18 estimation [54],[71], was shown. Although the results are promising, reflection–only measurements do not disclose any new information about these materials after approximately the 28th day of curing since reflection properties stay constant. Dr. Kharkovksy, Dr. Sato, and their colleagues measured reflection and transmission coefficients of these materials to gain more knowledge about their electrical properties. Transmission coefficients are less dependent on surface properties of cement–based materials and can evaluate the overall volume properties of these materials since the wave passes through the whole volume. Periodic assessment of aging in civil structures a over long period of time is important for structural integrity and public safety [59]. As a result, transmission coefficients are also important for structural integrity of these materials. In the literature, although transmission coefficients of these materials are measured, minimal attention was given to what the effects of different surface properties of these materials would be on transmission measurements and whether monitoring the transmission measurements of these materials over long service times would disclose new insight. In this dissertation, we present the measurement results of free–space reflection and transmission coefficients of some cement–based materials with varying mixture contents over their long service lives. We measured early age reflection and transmission properties of mortar specimens with different w/c ratios and introduced a new approach (relative slope approach for reflection properties) to predict the history of the hydration process of these specimens. It is shown that the difference between the relative slopes of reflection properties of mortar samples with 0.40 and 0.45 w/c ratios is greater than that of other mortar samples whose w/c ratio difference is 0.05. This result shows that a 19 mortar with a lower w/c ratio will gain hydration (and strength) faster than the one with a higher w/c ratio (a non–linear relationship between the degree of hydration and w/c ratios at early ages of curing). We demonstrated that monitoring early age transmission properties can be very useful for the quality enhancement of these samples. We also show free–space microwave transmission (and reflection) properties measured from diverse sides of hardened mortar and concrete specimens with different w/c ratios and the same sand–to–cement ratios during a 3–36 months period after specimens’ preparation. The relative difference approach initially developed for reflection measurements is applied to predict the state and degree of aging (curing) of specimens using transmission properties obtained from various sides of specimens. It is observed that while reflection properties of specimens are constant, their transmission properties and aging changed, except for last few months, during 3–36 months. After approximately 30 months, aging of all specimens became constant. It is shown that, for measurements acquired from the same side of specimens, the mortar specimen with a higher w / c ratio will lose free–water faster than the specimen with a lower w/c over approximately 3–9 months (first stage) of curing. On the other hand, the mortar specimen with a lower w/c ratio will drop free–water sooner than the mortar specimen with higher w/c during approximately 9–30 months (second stage) of curing. Moreover, it is illustrated that concrete specimens will complete the aging sooner than mortar specimens with the same w/c ratio as a result of heavy aggregates. In addition, among the specimens, while the concrete specimen with a higher w/c ratio will exhibit quicker aging during approximately 3–6.1 months, the mortar specimen with a lower w/c ratio will display faster aging during nearly 6.1–30 months. 20 2.4 A Simple Microwave Sensor for Electrical Characterization of Materials in Industrial-Based Applications Lossy dielectric materials contain high content of free–water. Microwaves are known as excellent candidates for aquametry (detection of moisture content or bound– and free– water) measurements [72], [73]. Among microwave techniques, nonresonant methods are capable of characterizing low–loss and lossy materials, allow wide frequency coverage, and are feasible for frequency– and time–domain analyses. These features make them very attractive for industrial–based applications. Among nonresonant methods, free–space methods have been used for characterization of a wide variety of materials because they are nondestructive, noncontact, noninvasive, require little, if any, sample preparation [73], and can be effectively used at high temperatures or in chemically poisonous regions [2], [27], [28]. An amplitude–only microwave free–space technique was analyzed for dynamic ε determination of materials immersed into a container [26]. Using a graphical method, it was shown that if the material under test demonstrates at least 10 dB attenuation, its ε can uniquely be determined by using amplitudes of reflection and transmission coefficients. However, the graphical approach is unsuitable for industrial–based applications and dynamic characterization of materials since it takes a long time to determine ε [74] and requires a two–dimensional numerical technique [75]. An optimization technique will also be helpful for this purpose. However, these techniques are often time consuming due to slow convergence and the existence of spurious solutions [76]. They also necessitate a well–defined domain for solutions because any small deviation from a local optimum point can result in an erroneous solution [75]. 21 In this dissertation, we will present a microcontroller–based microwave free– space measurement system for ε determination of lossy materials since there is a need for a simple and relatively inexpensive measurement method in industrial–based applications. We will utilize the derived objective function (the function which is utilized for ε determination) in chapter 3, which depends on only one unknown quantity, for ε determination [20],[74],[77]–[79]. This function is very important since the implementation of ε determination in microcontrollers necessitate a well–defined domain and a simple objective function for computations. Since free–space measurements are very prone to undesired effects of diffractions from sample edges, we also show a useful calibration technique we recently developed to minimize these effects. We describe the main properties of the measurement system and validate it using free– space measurements. 22 CHAPTER 3 TWO MICROWAVE METHODS FOR COMPLEX PERMITTIVITY DETERMINAITON OF LOSSY MATERIALS USING AMPLITUDE–ONLY S– PARAMETER MEASUREMENTS The complex scattering (S–) parameters are generally measured by nonresonant methods to determine the relative complex permittivity, ε , and the relative complex permeability, µ . However, there are three main problems associated with these measurements. First, they often yield multiple solutions [23]. Second, the phase uncertainty of reflection S– parameter measurements greatly increases when the thickness of materials is of integer multiples of one–half guided wavelength [21],[29]. Finally, they are adversely affected by any small shift from the calibration plane of materials which results in enormous phase errors. To eliminate all these drawbacks simultaneously, amplitude–only S– parameter measurements can be employed. In addition, the use of expensive vector network analyzers (VNAs) for complex S–parameter measurements makes microwave techniques inconvenient for industrial–based applications. Therefore, there is a need for a simple and relatively inexpensive microwave method for dynamic measurements. In this chapter, we will present two amplitude–only methods (methods based on amplitudes of S–parameter measurements). In the first method, we will derive an objective function which depends on only one real quantity for determination of the dielectric 23 constants of materials. The objective function is quite general and allows the use of a composite structure with at most five–layers. Second, we will present an amplitude–only method for thickness–independent complex permittivity determination of lossy materials. 3.1 Theoretical Analysis In this section, the problem at hand is first kept general so that the derivations in this section could be used for the following chapters, and then it is restricted to dielectric (nonmagnetic) materials with µ = 1 . The geometry of the problem is depicted in Fig. 3.1 in which a sample whose dielectric properties are unknown is sandwiched between two dielectric plugs with different lengths and electrical properties. In the following mathematical analysis, a harmonic time variation of the time ( e jωt ) is assumed. We also consider that the waveguide walls behave like perfect conductors and the waveguide is operating in its fundamental TE10 (transverse electric) mode. For the following derivations, we assume that lateral dimensions of media 2, 3, and 4 in Fig. 3.1 are flat and there is no air gap between external surfaces of media 2, 3, and 4 and waveguide inner walls. If any of these assumptions is not satisfied, the electric field vector which is transverse to the direction of electromagnetic propagation in an empty waveguide section will deflect and eventually this produces TMmn (transverse magnetic) waveguide propagation modes inside media 2, 3, and 4 [80],[81]. Here, m and n are used to denote different modes of the TM waveguide propagation where m = 1,2,3,... and n = 1, 2,3,... In addition, it is assumed that all media in Fig. 3.1 are isotropic, homogenous, and reciprocal. 24 Fig. 3.1. The scattering parameter measurements of a sample sandwiched between two dielectric plugs. For a homogenous medium, any solution for the time-harmonic electric and magnetic fields must satisfy the following Maxwell’s equations [81] ∇ × E = − M − jωµ H , ∇ × H = J + jωε E , (3.1) ∇ i E = qev ε , ∇i H = qmv µ . (3.2) In (3.1) and (3.2), the superscript right arrow denotes the vector quantities; E and H electric and magnetic field intensities; J and M are electric and magnetic current densities; qev and qmv volume electric and magnetic charge densities; and ε and µ are frequency-varying complex permittivity and permeability of the medium. The forward and reverse S-parameters for the overall structure in Fig. 3.1 can be obtained by using Maxwell’s equations in (3.1) and (3.2) and applying boundary conditions for tangential electric and magnetic fields at different material boundaries in Fig. 1 and at waveguide walls (tangential components of the electric field must vanish on 25 the walls of the waveguide). We can also electric and magnetic vector potentials, F and A , for a representation of E and H to simplify the boundary solutions problem in Fig. 3.1. However, we prefer the network analysis to the problem in Fig. 3.1 than the field analysis using E and H or F and A . This is because attempting to solve Maxwell’s equations subject to boundary conditions for the entire structure in Fig. 3.1 is rather complicated and takes considerable amount of time. In addition, Maxwell’s equations result in expression for E and H (or F and A ) at all points in space, which is more information than it is needed [14]. Instead, we apply microwave network analysis to our problem. There are basically two approaches to analyze this wave–material interaction using microwave network analysis: transmission–line impedance and multiple reflections. While the former is applicable to only a steady–state analysis, the latter one can be applied to both transient– and steady–state analyses. In this dissertation, only the steady–state analysis is considered since the time required the electromagnetic wave to travel from the five–layer structure in Fig. 3.1 is so small. We will use the conventional multi–reflections model and derive complex S–parameters for the structure in Fig. 3.1. It should be stated that in a steady–state analysis, the transmission–line and multiple–reflections approaches can be assumed to be equivalent and can equally be applied to our problem [14]. However, to gain more knowledge about the wave–material interaction, we will employ the multiple–reflections approach. 26 3.1.1 Complex Permittivity Extraction Using the Classical Multiple–Reflections Model The classical multiple–reflections model takes into account only the delays for multiple reflected and transmitted signals in a medium [82]. To visualize how the complex S– parameters are derived from this model for the five–layer structure in Fig. 3.1, a diagram, which shows internal reflections and transmissions inside each medium, is illustrated in Fig. 3.2. Fig. 3.2. The diagram for the analysis of the five–layer structure in Fig. 3.1 by multiple reflections approach. Using this diagram, the forward complex S–parameters ( S11 and S21 ) for the structure in Fig. 3.1 are obtained as S11 = rA + tA tB rBT32 + tA tB rB2 rC T34 + tA tB rB3rC2T36 + ... (3.3) S21 = tA tC T3 + tA tC rB rC T33 + tA tC rB2 rC2T35 + ... (3.4) where 27 ( r12 + r23T22 ) rA = r12 + t12 r23t21T22 1 + r23r21T22 + ... = ( ) ( ) rB = r34 + t34 r45t43T42 1 + r45 r43T42 + ... = , (3.5) , (3.6) 1 + r12 r23T22 r34 + r45T42 1 + r34 r45T42 rC = r32 + t32 r21t23T22 1 + r21r23T22 + ... = − r23 + r12T22 1 + r12 r23T22 (1 + r12 ) (1 + r23 ) T2 , (3.7) ( ) , (3.8) ( ) (1 − r12 ) (1 − r232 )T2 , (3.9) ( ) (1 + r34 )(1 + r452 )T4 , (3.10) tA = t12t23T2 1 + r23r21T22 + ... = tB = t32t21T2 1 + r21r23T22 + ... = tC = t34t45T4 1 + r45r43T42 + ... = 1 + r12 r23T22 1 + r12 r23T2 1 + r34 r45T4 and Z − Zn , n = 1,2,3,4; m = n + 1; rnm = m Zm + Zn Zr = jωµr γr , γr = j 2π λ0 Tk = exp ( −γ k Lk ) , k = 2,3,4. (3.11) 2 λ ε r µr − 0 , λc r = 1,2,3,4,5. (3.12) Here, rnm and Tk are, respectively, the first reflection coefficients at media boundaries and the first transmission coefficients inside the media; Z m , γ k , ε r , µr , and Lk represent, correspondingly, the impedances, propagation constants, relative complex permittivities, relative complex permeabilities, and lengths of media; and λ0 and λc are the free–space and cut–off wavelengths. For free–space and/or coaxial–line measurements, λc → ∞ . 28 Finally, by substituting (3.5)–(3.10) into (3.3) and (3.4), S11 and S21 for the five– layer structure in Fig. 3.1 can be written in a compact form as r12 + r23T22 )(1 + r34 r45T42 ) + ( r12 r23 + T22 )( r34 + r45T42 ) T32 ( , S11 = (1 + r34r45T42 )(1 + r12r23T22 ) + ( r34 + r45T42 )( r23 + r12T22 )T32 S21 = 3.1.2 (1 + r12 ) (1 + r23 )(1 + r34 )(1 + r45 )T2T3T4 (1 + r34r45T42 )(1 + r12r23T22 ) + ( r34 + r45T42 )( r23 + r12T22 )T32 . (3.13) (3.14) Validation of the Derived Forward Scattering Parameters and Derivation of Reverse Scattering Parameters For validation of the derivations in (3.13) and (3.14), we consider two special cases. In the first case, we assume that the lengths and electrical properties of medium 2 and 4 are the same and media 1 and 5 are air. In these circumstances, r45 = − r12 , r34 = − r23 , and T4 = T2 . Then, (3.13) and (3.14) reduce to (1) S11 r12 + r23T22 )(1 + r12 r23T22 ) − ( r12 r23 + T22 )( r23 + r12T22 ) T32 ( = , 2 2 2 2 2 (1 + r12r23T2 ) − ( r23 + r12T2 ) T3 (3.15) 2 2 1 − r12 1 − r23 T22T3 ( )( ) = , 2 2 2 2 2 (1 + r12r23T2 ) − ( r23 + r12T2 ) T3 (3.16) (1) S21 where the number ‘1’ in parenthesis at the superscript of S11 and S21 in (3.15) and (3.16) denotes the first special case. The complex S–parameters given in (3.15) and (3.16) are the same of those in [25],[26]. 29 In the second case, we assume that media 1, 2, 4, and 5 are air and the lengths of media 2 and 4 are zero. Then, we can write r12 = 0 , r45 = 0 , r34 = − r23 , and T2 = T4 = 1 . In this condition, complex S–parameters in (3.13) and (3.14) will reduce to ), (3.17) 2 1 − r23 ( )T3 , = (3.18) (2) S11 = (2) S 21 ( r23 1 − T32 2 2 1 − r23 T3 2 2 T3 1 − r23 where the number ‘2’ in parenthesis at the superscript of S11 and S21 in (3.17) and (3.18) (2) (2) and S 21 are identical to those signifies the second special case. It is obvious that S11 well–known S–parameters used in the literature [21]–[23],[29],[30]. In the manner we derived the expressions for the forward complex S–parameters, we also derived the reverse complex S–parameters for the five–layer structure in Fig. 3.1 by systematically exchanging the individual reflection and transmission coefficients in the media. To demonstrate this, we first exchanged the individual reflection coefficients in the media as r12 → r54 , r23 → r43 , r34 → r32 , r45 → r21. (3.19) Then, we replaced the individual transmission coefficients in media 2, 3, and 4 as T2 → T4 , T4 → T2 . (3.20) Substituting the replacements in (3.19) and (3.20) into (3.13) and (3.14), we derived the reverse complex S–parameters as r54 + r43T42 )(1 + r32 r21T22 ) + ( r54 r43 + T42 )( r32 + r21T22 ) T32 ( S22 = , (1 + r32r21T22 )(1 + r54r43T42 ) + ( r32 + r21T22 )( r43 + r54T42 )T32 30 (3.21) S12 = (1 + r54 )(1 + r43 )(1 + r32 ) (1 + r21 ) T2T3T4 (1 + r32r21T22 )(1 + r54r43T42 ) + ( r32 + r21T22 )( r43 + r54T42 )T32 . (3.22) Finally, using r54 = − r45 , r43 = − r34 , r32 = − r23 , and r21 = −r12 , the expressions for S 22 and S12 in (3.21) and (3.22) reduce to r45 + r34T42 )(1 + r23r12T22 ) + ( r45r34 + T42 )( r23 + r12T22 ) T32 ( S22 = − , 2 2 2 2 2 + + + + + r r T r r T r r T r r T T 1 1 ( 23 12 2 )( 45 34 4 ) ( 23 12 2 )( 34 45 4 ) 3 S12 = ( (1 − r45 )(1 − r34 )(1 − r23 ) (1 − r12 ) T2T3T4 )( ) ( )( ) 1 + r23r12T22 1 + r45 r34T42 + r23 + r12T22 r34 + r45T42 T32 . (3.23) (3.24) We assumed at the beginning of the derivation of the S-parameter expressions that all media are reciprocal. At first moment, it is seen that the final expressions for S21 and S12 in (3.14) and (3.24) are different from each other. However, a close analysis demonstrates that they are identical. To this end, we use (3.11) and (3.12) and obtain (1 + r12 ) (1 + r23 )(1 + r34 )(1 + r45 ) = (1 − r45 )(1 − r34 )(1 − r23 ) (1 − r12 ) . (3.25) On the other hand, it is obvious that the final expressions for S11 and S22 in (3.13) and (3.23) are different from each other. This is because the five–layer structure in Fig. 3.1 is asymmetric. However, if the lengths and electrical properties of medium 2 and 4 and those of media 1 and 5 are the same, the structure becomes symmetric and the expression for S22 in (3.23) becomes equal to that for S11 in (3.13). 3.2 An Amplitude-Only Method for Permittivity Determination of Lossy Materials In this section, we will derive an objective function, which depends on only one unknown quantity, for accurate, fast, and unique complex permittivity of lossy materials using 31 amplitudes of S–parameters. This function is very important since speed is one of the key factors in industrial–based applications. In the analysis, we will use the expressions for complex S–parameters in (3.13) and (3.14). We assume that media 1 and 5 are air and that complex permittivities and lengths of media 2 and 4 are known. In addition, it is assumed that all layers in Fig. 3.1 are nonmagnetic; i.e., µr = 1 . Our goal is to determine ε 3 for a known sample length and a given frequency, f , using the amplitudes of S11 and S21 in (3.13) and (3.14). 3.2.1 Selection of a Suitable Approach for Permittivity Extraction There are two main techniques for ε 3 extraction of the sample using (3.13) and (3.14): a) the direct method and b) the de–embed method. In the direct method, the theoretical S– parameters of the structure in Fig. 3.1 are directly compared with the measured S– parameters. In the de–embed method, known layers (media 2 and 4 in Fig. 3.1) are removed (i.e., de–embedded) before comparing theoretical and measured S–parameters of the sample [84]. In applying the direct method, firstly an initial guess for ε 3 is provided and then theoretical ABCD (A–) parameters of each layer (media 2, 3, and 4 in Fig. 3.1) are computed. Next, these parameters are used to obtain S–parameters of the structure. Finally, theoretical and measured S–parameters of the structure are compared. The value of ε 3 is iterated by a suitable numerical technique until the desired accuracy is reached. In applying the de–embed method, firstly A–parameters of known layers are computed. Then, measured S–parameters of the structure are converted into A– parameters. Next, the sample is isolated from the structure by using the A–matrices of the 32 known layers. Finally, ε 3 can be extracted by the Nicolson–Ross–Weir technique [22],[23] or any other method using S–parameters of the isolated sample [84]. It was shown that while the direct method detects possible sample inhomogeneities, the de–embed method cannot because an averaging occurs while converting S–parameters to A–parameters (e.g., A22 = ( S21S12 − S11S22 ) S21 ) [84]. This detection is important in industrial–based applications since samples are not always made so perfectly (e.g., formation of air inside samples) [84]. It can also be used for measuring the integral ε 3 of quasi–heterogeneous materials (e.g., concrete specimens composed of cement powder, water, and coarse and fine aggregates – all varying in content and size). Therefore, we adapted the direct method as a solution to our problem. Before applying this method to our problem, it is important to analyze S–parameter expressions in (3.13) and (3.14) for lossy materials. 3.2.2 Single-Pass Technique The single–pass technique in frequency domain measurements can be applied under two conditions: a) the sample in Fig. 3.1 has a thickness that is large enough to allow elimination of the signals due to multiple reflections in it, or b) the sample is a high–loss sample. If the sample in Fig. 3.1 possesses at least 10 dB attenuation, the first reflected power at interface 2-3 will be approximately 10 times greater than that at interface 3-4; that is 10dB ≈ 10log (1) P23 (1) (1) ⇒ P23 ≈ 10 P34 , (1) P34 (3.26) (1) (1) where P23 and P34 are, respectively, the powers of first reflected signals at interfaces 2- 33 3 and 3-4. In the same manner, since the first reflected signal at interface 3-4 consumes the same amount of power to return to the interface 2-3, using (3.26), we can write 10dB ≈ 10log (1) P34 ⇒ (2) P23 (1) (2) P23 ≈ 100 P23 , (3.27) (2) where P23 is the power of the second reflected signal at interface 2-3. As a result, total power of all reflected signals at interface 2-3 will be (1) (2) (3) P23 = P23 + P23 + P23 + ..., (3.28) (v) where P23 for v = 1, 2,3,... is the power of v th reflected signal at interface 2-3. It is obvious from (3.28) that for applications in engineering, we can approximate (3.28) to (1) P23 ≅ P23 , (3.29) because the contribution of each reflected signal for v > 1 at interface 2-3 is equal to or (1) less than one percent of P23 . Therefore, multiple reflections inside the sample can be neglected. Assuming that the sample in Fig. 3.1 results in at least 10 dB attenuation [26] and conductor losses are much smaller than dielectric losses [81], provided that the sample length is small, S11 and S21 in (3.13) and (3.14) reduce to s S11 = s = S21 r12 + r23T22 1 + r12 r23T22 (3.30) , (1 + r12 ) (1 + r23 )(1 + r34 )(1 + r45 ) T2T4T3 ( )( 1 + r34 r45T42 1 + r12 r23T22 34 ) . (3.31) where the letter ‘s’ at the superscript of S11 and S21 in (3.30) and (3.31) denotes simplified expressions. In the simplification, we made use of T32 << 1 . It is clear from (3.30) that for lossy samples there will be no effect of the medium 4 on the S11 , and S21 includes only the first transmission coefficient of the sample while multiple reflections inside media 2 and 4 are present. 3.2.3 Derivation of a Simple Objective Function The direct method requires a numerical technique for ε 3 determination. In industrial– based applications, it is always attractive to evaluate ε 3 fast and accurately [26]. In addition, it is appropriate to have a well–defined domain for numerical computations since any small deviation from a local optimum point can result in incorrect solutions [75]. Because ε 3 is unknown, it is difficult to predict a domain for its computation. Nonetheless, we succeeded in assigning a highly restrictive domain for ε 3 computations using the conformal transformation, as shown in Fig. 3.3, between ε 3 and r23 since r23 ≤ 1 . Fig. 3.3. Conformal transformation between ε 3 and ( χ , ξ ) pair. 35 To elaborate on this, we defined a new set of variables as r23 = χ − jξ , T2 = Λ1 − j Λ 2 , r12 = Λ3 − j Λ 4 , (3.32) r34 = k − jl , T4 = Λ5 − j Λ 6 , r45 = Λ 7 − j Λ8 . The definition of new variables changed the inverse problem from the determination of ε 3 into the determination of χ and ξ (or k and l ) because we can determine ε 3 from (3.11) for a known ε 2 (or ε 4 ). Using (3.32), the amplitudes of simplified S–parameters will be )( ( ) ( ) ( 12 ) 2 2 2 χ 2 + ξ 2 + Λ32 + Λ 24 + 2 Λ12 − Λ 22 ( Λ 3 χ + Λ 4ξ ) Λ1 + Λ 2 +4Λ1Λ 2 ( Λ 4 χ − Λ 3ξ ) s S11 = 2 2 2 2 2 2 2 2 2 1 + Λ1 + Λ 2 Λ3 + Λ 4 χ + ξ + 2 Λ1 − Λ 2 ( Λ 3 χ − Λ 4ξ ) −4Λ1Λ 2 ( Λ 4 χ + Λ3ξ ) )( )( ( ) ( )( ( , ) ) (3.33) 12 (1 + χ )2 + ξ 2 (1 + k )2 + l 2 s = Λ9 S21 H1 ( χ , ξ ) H 2 ( k , l ) e Im{− jγ 3}L3 , )( ) (3.34) where ( Λ9 = (1 + Λ3 ) + Λ 42 2 ( ) ((1 + Λ ) )( 7 2 )( )( 12 + Λ82 Λ12 + Λ 22 Λ 52 + Λ 62 ) ( (3.35) 12 ) 2 2 2 1 Λ + + Λ Λ 32 + Λ 42 χ 2 + ξ 2 + 2 Λ12 − Λ 22 ( χΛ 3 − ξΛ 4 ) 1 2 H1 ( χ , ξ ) = −4Λ1Λ 2 ( χΛ 4 + ξΛ 3 ) ( )( )( ) ( ) (3.36) 12 2 2 2 1 Λ + + Λ Λ 72 + Λ82 k 2 + l 2 + 2 Λ 52 − Λ 62 ( k Λ 7 − l Λ8 ) 5 6 H2 ( k,l ) = −4Λ5 Λ 6 ( k Λ8 + l Λ 7 ) (3.37) and Im {i} in (3.34) means the imaginary part of the complex i and the vertical bar i denotes the amplitude of the complex i . 36 We can readily determine χ from (3.33) in terms of ξ as ( ) s 2 −C2 ∓ C22 − C1 C1ξ 2 + C3ξ + Λ 32 + Λ 42 − S11 χ= , C1 (3.38) where C1 , C2 , and C3 are known quantities and are given as ( C1 = Λ12 + Λ 22 ( ) ( 2 ) s 2 1 S − Λ 32 + Λ 42 , 11 (3.39) ) s 2 s 2 C2 = Λ12 − Λ 22 Λ3 1 − S11 + 2 Λ Λ Λ 1 + S 1 2 4 11 , ( ) s 2 s 2 C3 = 2 Λ12 − Λ 22 Λ 4 1 + S11 − 4Λ1Λ 2 Λ3 1 − S11 . (3.40) (3.41) The correct root for χ in (3.38) is selected by imposing r23 ≤ 1 . Below, we will prove that −1 < χ < 0 and −1 < ξ < 0 , which not only helps the selection of the correct root for χ but also reduces the computation times considerably. The proof is as follows. For a low–loss medium 2 and a lossy sample, we can assume that tan δ 3 > tan δ 2 where tan δ 3 and tan δ 2 are, respectively, the loss tangents of the sample and the medium 2 in Fig. 3.1. Next, using (3.32) we expressed χ and ξ as n12 + n22 ) − ( a 2 + b 2 ) ( , χ= ξ= 2an1 ( n2 n1 − b a ) , (3.42) n1 = Re {− j ( λ0 2π ) γ 2 } , n2 = − Im {− j ( λ0 2π ) γ 2 } , (3.43) ( n1 + a )2 + ( n2 + b )2 ( n1 + a )2 + ( n2 + b )2 where a = Re {− j ( λ0 2π ) γ 3} , b = − Im {− j ( λ0 2π ) γ 3} , and Re{i} means the real part of the complex i . It is clear from (3.43) that a > 0 , b > 0 , n1 > 0 , and n2 > 0 for λ0 < λc . Since a and b , respectively, are mainly dependent on 37 Re {ε } and Im {ε } , then tan δ 3 will be proportional to b a . In the same manner, tan δ 2 will be proportional to n2 n1 . Then, it is obvious that ξ < 0 . Assuming, for the sake of simplicity, a = n1 in (3.42) will prove that χ < 0 . We can substitute χ in (3.38) into (3.34) to compute ξ . However, k , l , and Im {− jγ 3} must be written in terms of χ and ξ for this computation. To this end, using (3.11) and (3.32) we found k and l as (1 − χ ) ( k= l= 2 )( ) ( )( + ξ 2 n12 + n22 − (1 + χ ) + ξ 2 m12 + m22 2 Π1 − Π 2 ( ) 2 χ 2 + ξ 2 − 1 ( m2 n1 − m1n2 ) − 4ξ ( m1n1 + m2 n2 ) Π1 − Π 2 ( 2 )( ) ( ), (3.45) , )( 2 ) Π1 = (1 − χ ) + ξ 2 n12 + n22 + (1 + χ ) + ξ 2 m12 + m22 , ( (3.44) ) (3.46) Π 2 = 2 χ 2 + ξ 2 − 1 ( m1n1 + m2 n2 ) + 4ξ ( m1n2 − m2 n1 ) (3.47) m1 = Re {− j ( λ0 2π ) γ 4 } , m2 = − Im {− j ( λ0 2π ) γ 4 }. (3.48) where In a similar way, Im {− jγ 3} is expressed from (3.12) and (3.32) as ( ) 2 2 2π 2ξ n1 + χ + ξ − 1 n2 Im {− jγ 3} = . 2 λo 1+ χ ) + ξ 2 ( (3.49) Then, the simple objective function that depends on only ξ is derived as s m f (ξ ) = S21 − S21 = 0. (3.50) where the superscript ‘m’ in (3.50) denotes the measured parameter. Because (3.50) contains an exponential term, an explicit solution for ξ (and χ ) is not possible. 38 However, since we derived a simple objective function which depends on only one unknown quantity, the solution can be cast by any simple one–dimensional numerical technique [75]. For example, it is observed that the Newton’s method with ξ = 0 initial guess converges to the actual solution in less than 15 iterations. After computing ξ from (3.50), χ can be determined from (3.38). Finally, ε 3 will be 2 ε3 = 2 Λ10 2 λ 2 λ λ − o Im {− jγ 3} + o + j 2Λ10 o Im {− jγ 3} , 2π 2π λc λ Λ10 = ξ n2 − o Im {− jγ 3} + n1 (1 − χ ) 2π 3.2.4 (1 + χ ) . (3.51) (3.52) Functional Analysis of the Derived Objective Function A function g : H → K (from mapping H to the mapping K ) is called “one–to–one” if and only if its functional dependency is either increasing or decreasing for every possible value of the independent quantity [85]. Because it is difficult to show the “one–to–one” property of f (ξ ) mathematically, we choose a numerical technique instead. In the analysis, we drew f (ξ ) over −1 < ξ < 0 for different ε 2 , ε 4 , ε 3 , λ0 , L2 , L4 , and L3 at s s and S21 for given ε 2 , ε 4 , ε 3 , λ0 , L2 , L4 , and L3 . X–band. Firstly, we computed S11 Then, we compared the computed ξ from (3.50) with the assumed one. For example, Fig. 3.4 shows such a dependency of f (ξ ) for different ε 3 . It is assumed that L2 =10 mm, L4 =15 mm, L3 =25 mm, λ0 =29.979 mm, ε 2 = 2.5 − j 0.0184 , and ε 4 = 1.2 − j 0.012 . It is seen in Fig. 3.4 that there is only one point which satisfies f (ξ ) ≅ 0 for different ε 3 . This shows that f (ξ ) is a “one–to–one” function. 39 A function g : H → K is called “onto” if and only if for every possible variable in H there is a correspondence in K [85]. It is clear from (3.38) that not every ξ will yield s since the value of the terms in the a real χ for given different ε 2 , L2 , λ0 , and S11 square–root in (3.38) might be negative. However, by restricting the computations in (3.38) for acceptable χ quantities will help us determine an acceptable ξ . It is because of that the dependencies of f (ξ ) in Fig. 3.4 are given for −0.45 < ξ < 0 . It is noted that restricting the computations not only shrinks the domain for computations but also significantly decreases computation times. 3.2.5 Refining the Computed Complex Permittivity In the derivation for ε 3 of lossy samples in (3.50), we used simplified expressions in (3.33) and (3.34). Accordingly, the determined ε 3 will be fairly accurate. To compute its precise value, the ε 3 determined from (3.50) can be used an initial guess for the expressions in (3.13) and (3.14) in a search algorithm since we are searching for a solution near to this fairly accurate value [75]. Fig. 3.4. Dependency of f (ξ ) over ξ for different ε 3 . 40 3.3 An Amplitude-Only Method for Thickness-Independent Complex Permittivity Determination of Lossy Materials In the previous section, we proposed a method which uses only the amplitudes of S– parameters and derived an objective function, which depends on only one unknown quantity, for fast and accurate complex permittivity determination of lossy materials. Although this method is very applicable for industrial–based applications, the requirement of precise thickness information of these materials before permittivity measurements can limit its potential. It is always attractive to eliminate this need not only in the sense of improving the accuracy of measurements but also for saving time, especially for measurements of liquid or granular materials, which are positioned between two low–loss dielectric materials with known electrical properties [25]. In addition to the investigations of characterization of the behavior of materials reported in the literature, frequency variation techniques have been used for several purposes in nonresonant measurement methods. They have been applied as a decision maker for eliminating the superfluous solutions of the extracted complex permittivity, ε , and complex permeability, µ , of materials by the well−known Nicolson−Ross−Weir technique [22],[23]. They have also been employed for obtaining an initial guess for the ε determination of materials using transmission scattering (S−) parameter measurements [86]. They are also useful in assigning a correct propagation constant, from which the ε of materials can be extracted, using uncalibrated S−parameter measurements [87] and for determining the ε , µ , and/or thickness of materials [88]−[90]. An additional usage of frequency variation techniques is for optimization of measurements of physical properties (thickness, disbond, delamination, etc.) of materials [50]. The main process for the 41 aforementioned techniques is to obtain independent measurements at different frequencies and then extract or optimize the desired information (permittivity, permeability, thickness, etc.). It is known that phase−measurements are not convenient for dynamic characterization of materials since the measured phase may differ by an integral multiple of 2π from its actual value [26]. To overcome this ambiguity, Weir proposed a method involving a comparison between calculated and measured group delays of the material at different frequencies [23],[29]. In a recently published paper, Varadan and Ro utilized the well−known Kramers−Kronig relations to resolve the same ambiguity [91]. Although these methods are very useful, the use of expensive vector network analyzers (VNAs) for complex S−parameter measurements makes microwave techniques inconvenient for industrial−based applications [32]. Therefore, there is a need for a simple and relatively inexpensive microwave method for dynamic measurements. In this section, our purpose is to elaborate on the usage of frequency variation for ε determination of lossy materials with no a priori knowledge of the thicknesses of the samples using amplitudes of S−parameter measurements at slightly separated frequencies. In the analysis, we will use the expressions for complex S–parameters in (3.17) and (3.18) and are repeated for convenience as ), (3.53) 2 1 − r23 ( )T3 . = (3.54) S11 = S21 ( r23 1 − T32 2 2 1 − r23 T3 2 2 1 − r23 T3 42 Applying the single–pass technique described in the subsection 3.2.2 reduces (3.53) and (3.54) to s S11 = r23 , ( ) s 2 S21 = 1 − r23 T3 , (3.55) where we made use of T32 << 1 and the superscript ‘s’ at the expressions of S–parameters in (3.55) denotes simplified expressions. 3.3.1 Zero-Order (ZO) Approach for Complex Permittivity Determination Because ε changes over f , we have to consider or simulate that change in our ε determination. However, for very small frequency shifts, we can approximate ε ( f ) = ε ( f ∓ ∆f ) where ∆f f [88]. We will name this approximation the “zero–order” for the remaining of this section for the reasons which will be explained in the next subsection. To demonstrate this in our problem, we define new variables to simplify the derivations as χ1 − jξ1 = ε − ( λ0 λc ) , 2 κ1 = 1 − ( λ0 λc ) . 2 (3.56) The definition of new variables changed the inverse problem from ε determination into χ1 and ξ1 determination. Using (3.56), we obtain s S11 = ( χ1 − κ1 )2 + ξ12 , ( χ1 + κ1 )2 + ξ12 s S21 = 4κ1 χ12 + ξ12 e −2πξ1L3 λ0 ( χ1 + κ1 )2 + ξ12 . (3.57) Even if we assume that ε stays constant over small frequency shifts, χ1 , ξ1 , and κ1 in (3.56) will all change with f as a result of fc . We denote their new values as χ 2 , ξ 2 , s is independent of L3 , while and κ 2 at f 2 = f ∓ ∆f . It is obvious from (3.57) that S11 43 s S21 is exponentially decreasing with it. Consequently, L3 should be eliminated from s S21 in (3.57) for a thickness–independent ε determination. Using (3.57), we eliminate s L3 from S21 as ( χ + κ )2 + ξ 2 1 s 1 1 S21 2 2 4κ χ + ξ 1 1 1 2πξ2 2πξ1 λ02 ( χ + κ )2 + ξ 2 λ 2 2 s 0 S21 , = 2 * 4κ χ 2 + ξ 2 2 2 2 (3.58) s s where λ02 = c f 2 and S21 is the measured amplitude of S21 at f 2 . We can easily * s s s and S11 , which is the measured amplitude of S11 at determine ε from (3.58) using S11 * f 2 , by any two–dimensional root–searching algorithm [75]. However, for industrial– based applications it is appropriate to have a well–defined domain for fast numerical computations. In this respect, we decided to use one–dimensional root–finding algorithms, which require a function with only one–variable. We obtained a function which depends on only χ1 as follows. s Using S11 in (3.57), we expressed ξ1 in terms of χ1 and κ1 ( s 2 ξ1 = 2 χ1κ1Λ1 − ( χ12 + κ12 ) , Λ1 = 1 + S11 ) (1 − S s ). 2 11 (3.59) It is obvious from (3.56) that the equation for ξ 2 will be in similar from in (3.59) except s s → S11 . At this stage, we expressed ξ1 and ξ 2 in for χ1 → χ 2 , κ1 → κ 2 , and S11 * functions of χ1 and χ 2 , respectively. As a result, if we write χ 2 in terms of χ1 and ξ1 , we will obtain a function which depends on only χ1 . To this end, using (3.56) and (3.59), we found 44 χ2 = κ 2 Λ 2 ∓ κ 22 Λ 22 + 2 ( χ12 − ξ12 − κ12 ) 2 , Λ2 = s 2 1 + S11 * s 2 . (3.60) 1 − S11 * Accordingly, we can then determine χ1 by substituting ξ1 , ξ 2 , and χ 2 into (3.58) and using any suitable one–dimensional root–finding algorithm [75]. After computing χ1 , we can analytically determine ξ1 , χ 2 , and ξ 2 from (3.59) and (3.60). As a result, ε will be 2 2 ε = ε ' − jε " = χ m − ξm + λ02m λc2 − 2 j χ mξ m , (3.61) where χ m , ξ m , and λ0 m are, respectively, the mean values of χ1 and χ 2 , ξ1 and ξ2 , and λ0 and λ02 . It is noted that the equation in (3.58) can still be employed for ε determination in coaxial–line or free–space measurements where λc → ∞ . For these measurements, χ 2 = χ1 = χ , ξ 2 = ξ1 = ξ , and κ 2 = κ1 = κ . However, since λ02 ≠ λ0 , we can still determine ε from (3.58). s After computing ε from (3.61), L3 can be calculated from S21 in (3.57). Because we use approximations to S11 and S21 , the computed ε and L3 will be fairly accurate but must be refined for precise values. For this purpose, we used the least squares minimization [75], which is given as min ∑ ( Sα α − Pα )2 + ∑ ( Sα * − Pα * ) α 2 for ε , L3 , (3.62) where α = 11 or 21 , Sα and Pα are the measured and predicted amplitudes of S– parameters at f , and the subscript '*' denotes the measured and predicted amplitudes of S–parameters at f 2 . 45 3.3.2 Higher-Order (ZO) Approach for Complex Permittivity Determination It is seen from (3.58) that for the ZO approximation to be applied to our problem one has s s s s to measure different S11 , S11 , S21 , and S21 . If the sample does not yield different * * amplitude–only measurements over small frequency shifts, we have to search for a new frequency that yields different amplitude–only measurements. However, using amplitude–only S–parameter measurements at two widely separated frequencies may not be suitable for the ZO approximation and then the simplified expressions in (3.55) may not be substituted for those in (3.53) and (3.54) since ε can change considerably. In this circumstance, higher–order (HO) approximations for ε should be used. For these approximations, Baker–Jarvis et al. proposed an extended Debye relaxation model [89], which in a compact form is ε( f )= 2 n ∑ An (1 + j 2π Bn f ) , (3.63) n =0 where An and Bn are the unknown complex quantities. The equation in (3.63) can also be written as ε( f )= m ∑ Cn f n = C0 + C1 f + C2 f 2 + C3 f 3 + ..., (3.64) n =0 where Cn ’s are the unknown complex quantities and m is the degree of approximation. The equation in (3.64) will reduce to ZO approximation when m = 0 [88] and linear interpolation when m = 1 [90]. It is clearly seen from (3.64) that HO approximations necessitate an initial guess for ε extraction. Although they are more general than the ZO approximation, the latter gives an insight on electrical behavior of the sample and presents a readily available 46 initial guess for HO approximations for ε . To determine ε using (3.64), we substitute an initial guess for ε into (3.64) determined by the ZO approximation from measurements at f and at an f 2 , which is the nearest frequency to f . Then, we refine this ε by iterating it and L3 using (3.62). 3.4 Measurements In this section, we present measurement results of some liquid materials (methanol, a commercially available antifreeze solution, and a binary mixture solution of ethyl alcohol and distilled water) for validation of the methods given in sections 3.2 and 3.3. We constructed a general purpose waveguide measurement set–up for this purpose as shown in Fig. 3.5. The set–up has a HP8720C VNA connected as a source and measurement equipment, two dielectric plugs and a sample inside a waveguide holder, two coaxial–to– waveguide adapters, and two extra waveguide sections more than 70 mm in length between the calibration plane and adapters. It operates at 8.2–12.4 GHz (X–band). Fig. 3.5. The measurement set–up. 47 We assumed single–mode transmission (TE10) through the sample in this chapter. This condition for empty and sample–filled sections of the waveguide will not be consistent for a sample with ε ' > 4 [92]. In this case, higher–order modes may appear. Using a sample thickness less than one–half guided wavelength of the fundamental mode will suppress these modes [29]. However, this may decrease the measurement accuracy as a result of sample thickness uncertainty. Another option could be using two extra waveguide sections with lengths greater than 2λ0 between the sample and coaxial–to– waveguide adapters [21],[29],[41]. This is because higher–order modes will die out drastically in a short distance away from the sample, and actual measurements are performed near the adapters. We used extra waveguide sections, which are greater than 2λ0 at X–band, to eliminate any higher–order mode in the waveguide set–up in Fig. 3.5. Since our primary concern is to extract ε from amplitude–only S–parameter measurements, any calibration technique which uses complex S–parameter measurements, such as the Thru–Reflect–Line (TRL) technique [93] and the Line– Reflect–Line (LRL) technique [94], is not suitable for validation of the method. Therefore, we calibrated the set–up by the frequency response calibration [95], which only removes the frequency (tracking) error by normalizing the frequency responses of a waveguide short and of a thru standard (extra waveguide sections are connected back–to– back). To verify the measurements, we also calibrated the set–up by the TRL calibration. We used a waveguide short and the shortest spacer in our lab (a 44.38 mm long waveguide section) for reflect and line standards, respectively. The line has a ∓ 70o maximum offset from 90o between 9.7 GHz and 11.7 GHz. Figs. 3.6(a) and 3.6(b), respectively, show a comparison between the TRL calibration and the frequency response 48 calibration with/out aperture smoothing for measurements of waveguide short and thru connections. It is observed that calibration results in Figs. 3.6(a) and 3.6(b) are clearly in agreement with those in the VNA operating manual [95]. We noted from the measurements in Figs. 3.6(a) and 3.6(b) that the frequency response calibration results in more fluctuations than the TRL calibration. This is because it corrects only frequency errors while the TRL calibration performs a full two–port calibration [95]. However, applying aperture smoothing to the measurements obtained from the frequency response calibration allowed us to measure quite similar results for especially the thru connection in Fig. 3.6(b). In addition, for measurements of lossy materials a maximum 0.5 dB difference between the two calibrations for the waveguide short connection in Fig. 3.6(a) will not much affect the ε determination. It is appropriate to discuss measurement uncertainties in the system before conducting measurements. Several factors contribute to the uncertainty in ε determination in waveguide measurements [21], [30] namely: 1) the uncertainty in measured S–parameters; 2) errors in the sample length (and the dielectric plugs); 3) the uncertainty in reference plane positions; 4) guide losses and conductor mismatches; 5) air gaps between the external surface of the sample (and dielectric plugs) and inner walls of waveguides; and 6) higher order modes. All these uncertainties are well–treated in the literature. The reader can refer to [21], [24], [25], [29]–[31], [96]–[99] for details. Care can be taken to limit the uncertainties due to sample position, calibration, and air gaps. Using calibration techniques, uncertainties in reference plane positions can be minimized. Preparing the sample (and dielectric plugs) and using a sample holder with 49 no scratches, nicks, or cracks, and machining the sample (and dielectric plugs) to fit precisely into the waveguide will increase the measurement accuracy and reduce the effect of air gaps [21], [29]. Fig. 3.6. Comparison of the frequency response (R) with/out aperture smoothing and the TRL calibrations for (a) waveguide short and (b) thru connections. 50 We machined two identical polytetrafluoro–ethylene (PTFE) samples (10 mm) for dielectric plugs. We utilized them for holding the liquids in place so that there will be no meniscus formation on surfaces of the liquids. The permittivities of PTFE samples are measured over 9.7–11.7 GHz using explicit formulae derived by Boughriet et al. [31]. Its value is around 2.04 − j 0.0014 over the band. Then, we poured the ethyl alcohol–water solution and antifreeze solutions into the measurement cell (waveguide section) and pressed on the plugs to decrease any meniscus that can form on top of the liquids [25]. Next, we carried out complex S–parameter measurements and collected 801 data points evenly spaced between 9.7 GHz and 11.7 GHz. We also extended the calibration plane to the surface of the PTFE samples using the VNA port extension facility. Next, we measured the ε of the liquids by the plug–loaded two–port transmission–line method (PLTL) [25] using complex S–parameter measurements after the TRL calibration. We also applied time–domain gating to the main transmission properties of measured complex S–parameters to obtain smoother ε measurements. Because frequency response calibration corrects only frequency errors, it produced fluctuations on amplitude–only scattering parameter measurements throughout the frequency band. The time–domain gating facility of the VNA cannot be applied to decrease these fluctuations since it requires phase information. Instead, we applied 20 % aperture smoothing to obtain an even dependency for amplitude–only S–parameter measurements. Since the measured ε of the liquids from (3.50) and (3.51) by our method are not precise, we refined it by using the Newton’s search algorithm [75]. Figs. 3.7 and 3.8 demonstrate the measured ε of the liquids by our method and the PLTL method [25]. It is seen from Figs. 3.7 and 3.8 that measured ε of the liquids from the 51 frequency response and TRL calibrations are completely consistent [33]. Small differences between measured ε by the methods arise because of a maximum 0.5 dB difference in measured S11 by different calibrations. We obtained highly accurate results for both methods since the measured S21 was greater than –30 dB with respect to the reference (thru) signal. As it is known, the main uncertainty in measurements of lossy samples comes from S21 . If it is less than –40 dB (in fact, it is –50 dB for the HP8720C VNA over 8–20 GHz), the level of measurements will be comparable with that of noise present in the system, and thus the uncertainty will significantly increase [21], [25]. Fig. 3.7. Measured ε of an antifreeze solution extracted by the plug–loaded two– port transmission–line method (PLTL) [25] and the proposed method (PM) in section 3.2 without aperture smoothing and with 20 % aperture smoothing. 52 Fig. 3.8. Measured ε of a binary mixture of ethyl alcohol (75%) and water (25%) by the plug–loaded two–port transmission–line method (PLTL) [25] and the proposed method (PM) in section 3.2 without aperture smoothing and with 20 % aperture smoothing. After validation of the proposed method in section 3.2, we also validated our second method in section 3.3. Since the derivations given in section 3.3 are not applicable for a dielectric–sample–dielectric structure, we removed the effect of PTFE samples from measurements using the de–embedding technique [28]. After, we applied our second method to measure the ε of the methanol using amplitude–only S–parameter measurements after applying 10 % and 20 % aperture smoothing to frequency response calibration. In applying our method, firstly ε is computed using (3.58)–(3.62). This ε gives its probable value over 9.7–11.7 GHz. Then, we refined it using (3.64) with m = 1 53 along with (3.62). We continued this refining with larger m values until the refined ε with present and previous m values is approximately the same. Figs. 3.9(a) and 3.9(b) illustrate, correspondingly, the measured real and imaginary parts of the permittivity of the methanol. It is seen from Figs. 3.9(a) and 3.9(b) that the measured ε by the PLTL method and by our method are in good agreement with those in [33]. Small differences between measured ε by the methods arise because of a maximum 0.5 dB difference between the applied calibrations. As discussed before, we obtained highly accurate results for both methods since the measured S21 was greater than –30 dB with respect to the reference (thru) signal. 54 Fig. 3.9. (a) Real part and (b) imaginary part of the measured ε of methanol by the plug–loaded two–port transmission–line method (PLTL) [25] and by the proposed method (PM) in section 3.3 without aperture smoothing and with 10 % and 20 % aperture smoothing. 55 CHAPTER 4 A WAVEGUIDE SAMPLE HOLDER FOR AUTOMATED THICKNESS– INDEPENDENT COMPLEX PERMITTIVITY AND COMPLEX PERMITTIVITY MEASUREMENTS OF GRANULAR AND LIQUID MATERIALS In chapter 3, we introduced two methods for complex permittivity determination of lossy materials using amplitudes of scattering (S–) parameter measurements. Although these methods can easily be implemented by using a relatively inexpensive scalar network analyzer (SNA), they are not sufficient for full electrical characterization (complex permittivity, complex permeability, etc.) of low–to–high–loss materials. In addition, they always require a numerical analysis for extracting the complex permittivity since real and imaginary parts of the complex permittivity cannot be exclusively expressed in terms of amplitudes of S–parameter measurements. A problem encountered in practice in nonresonant methods is the transformation of S– parameter measurements from the calibration plane to end surfaces of the sample [30]. The port extension feature of the vector network analyzers (VNAs) can be used for this transformation; however, they require precise lengths between the calibration plane and the sample, which in some instances are unknown. In addition, thickness– independent complex permittivity, ε , and/or complex permeability, µ , measurements of 56 materials are very attractive since the requirement of the physical thickness measurement of the materials is eliminated. In the literature, the problems pertaining to either calibration plane–invariant or sample length–independent ε and µ measurements of materials have been addressed. For example, Altschuler proposed the determinant of a network (admittance) presentation of materials under test to determine calibration plane– invariant ε in a simple fashion [100]; and for the same purpose, Baek et al. proposed a position insensitive method for ε and µ determination of materials [101]. On the other hand, Sucher and Fox presented several methods for thickness–independent ε determination of materials backed by a short circuit termination [102]. Because the short– circuit termination at different locations away from the sample could result in similar S– parameter measurements as a result of the same reactive termination, another method which eliminates this problem using measurements of broadband and short–circuit terminations was proposed in [103]. Baker–Jarvis et al. were the first researchers who derived equations that are simultaneously invariant to the location of the calibration plane and the sample length [30] for ε determination of materials. Although these equations are useful, the relation between ε and the sample length is not explicit. Therefore, any method using these equations requires a numerical technique for thickness–independent ε determination. In addition, the derived equations are limited to ε extraction of materials. Owing to their relative simplicity, broad frequency coverage, feasibility for analyzing the measurements either in time–domain or frequency–domain, and relatively higher accuracy, nonresonant methods have been widely utilized [29]. When applying these methods for measurements of granular and liquid materials, some form of 57 approximation in the physical nature of the measurement apparatus and/or forward and inverse problem formulations are made [25],[33] or formulations require that the dielectric specimen be low–loss [34]–[36]. To avoid any meniscus formation on the top of liquid samples, generally the measurement set–up is positioned vertically [37]. For the case of granular specimens, planar measurement planes may not be easily achieved [25] and this can degrade the measurement performance. In this chapter, we will propose a waveguide sample holder for calibration–plane invariant thickness–independent complex permittivity, ε , and/or complex permeability, µ , determination of low–, medium–, and high–loss granular and liquid materials. The holder readily lends itself for automated measurements using a VNA. 4.1 Theoretical Analysis The geometry for simultaneous ε and µ measurement of a liquid and/or granular sample with length, Ls , in a waveguide is shown in Fig. 4.1. Fig. 4.1. The complex S–parameter measurements of granular and liquid samples sandwiched between stable and movable dielectric plugs. 58 The sample is sandwiched between stable and movable plugs. It is assumed that the sample and plugs are isotropic and homogenous. Also, we assume that the waveguide operates at the dominant mode (TE10). It is obvious that the structure we consider in Fig. 4.1 is similar to that in Fig. 3.1. Therefore, the derived S–parameters for the five–layer structure in chapter 3 can be utilized in this chapter with some modifications. Assuming that the measurement set–up in Fig. 4.1 is calibrated to the calibration plane (it is assumed that the line standard is an empty waveguide section), by making analogy to the five–layer structure in Fig. 3.1 we can model, respectively, the stable plug, the sample, the movable plug, and air regions as media 1 through 4. As a result, using the expressions in (3.13), (3.14), and (3.21), we derived forward and reverse complex S–parameters for the structure in Fig. 4.1 as Γ s − Γ p ) (1 − Γ p Γ sT p2 ) + ( Γ pT p2 − Γ s ) (1 − Γ p Γ s ) Ts2 ( S11 = , 2 2 2 1 − Γ Γ T 1 − Γ Γ + Γ − Γ Γ T − Γ T ( ) ( p s p ) p s ( s p )( p p s ) s )( ( ) ( )( )( ) (4.1) 1− Γ Γ Γ T 2 − Γ + Γ −T 2 Γ − Γ T 2 p s s p p s p s p s 2 = T , 2 2 2 0 1− Γ Γ T 1− Γ Γ + Γ − Γ T T Γ − Γ p s s s p s s p p p p (4.2) 1 − Γ 2s )(1 − Γ 2p ) TsT pT0 ( S21 = S12 = , (1 − Γ pΓsTp2 ) (1 − Γ pΓs ) + ( Γ s − Γ p ) ( Γ pTp2 − Γs )Ts2 (4.3) S22 ( )( ) ( ) where Γp = ( γ p − γ 0µ p γ p + γ 0µ p , Γs = ) γ p µ − µ pγ s γ p µ + µ pγ s , T p = exp −γ p L p , Ts = exp ( −γ s Ls ) , T0 = exp ( −γ 0 L0 ) , 59 (4.4) (4.5) γp = j 2π λ0 2 2 2 2π 2π λ0 λ λ , γs = j 1− 0 . εµ − 0 , γ 0 = j λ0 λ0 λc λc λc ε pµ p − (4.6) Here, Γ p , T p , Γ s , and Ts are, respectively, the first reflection and transmission coefficients of the movable plug– and the sample–filled cells; T0 is the propagation factor in the air–filled cell; γ 0 , γ p , and γ s represent, respectively, the propagation constants of the air–, movable plug–, and the sample–filled cells; ε p , µ p , ε , and µ are, correspondingly, the relative complex permittivities and permeabilities of the movable plug and the sample; L p and L0 denote the lengths of the movable plug and the air section inside the waveguide; and λ0 and λc correspond to the free–space and cut–off wavelengths. 4.2 The Method In the formulation, it is assumed that we know the electrical properties ( ε p and µ p ) and the length ( L p ) of the movable plug and the length of the waveguide holder Lg = Ls + L p + L0 . Our goal is to determine the constitutive parameters of the sample while its length and the air–section length inside the holder are unknown. Because of the asymmetry of the measurement cell (sample, movable plug, and air) inside the calibration plane, we could not directly apply the Nicolson–Ross–Weir (NRW) technique [22],[23]. However, we obtained a different formulation as follows. It is clear from (4.1)–(4.6) that S11 is independent of L0 while S 22 and S21 are functions of L0 . We firstly eliminated L0 using (4.2) and (4.3) as 60 ( ) 2 2 1 − Γ 2s (1 − Γ p ) Ts2T p2 S21 = . S 22 Λ AΛ B )( ( 2 ) ( (4.7) )( ) Λ A = 1 − Γ p Γ s Γ sT p2 − Γ p + Γ s − T p2 Γ s − Γ p Ts2 , ( )( )( ) ( (4.8) ) Λ B = 1 − Γ p Γ sT p2 1 − Γ p Γ s + Γ s − Γ p Γ pT p2 − Γ s Ts2 . Then, because Γ p and T p are known, we expressed Ts2 in terms of Γ s using the S11 in (4.1) as 1 − Γ p Γ sT p2 ) ( Γ s − Γ p ) − (1 − Γ p Γ s ) S11 ( . Ts = 2 ( Γ pTp − Γs ) ( Γs − Γ p ) S11 − (1 − Γ pΓs ) 2 (4.9) Substituting Ts in (4.9) into (4.7), we find a function depending on Γ s , Γ p , and T p as ( ) F Γs , Γ p ,Tp = ( Ω1 1 − Γ 2s ) ( Γ pΓsTp2 − 1)( Ω2Γ2s + Ω3Γs + Ω2 ) 2 ( Γs − Γ pTp ) ( S11 + Γ p ) Γs − S11Γ p − 1 ( Γ pΓs − 1) 2 2 2 = 0, (4.10) where 2 S21 2 Ω1 = Γ p − 1 , Ω 2 = U − S11 − Γ p S11 − 1 Γ p − S11 T p , U = , S22 ( { 2 ) (( 2 ( ) ) (4.11) } ) Ω3 = ( ( S11 − U ) S11 + 1) Γ p + T p2 − 2 U + 4S11 Γ p + (1 + ( S11 − U ) S11 ) T p2 + U Γ p . (4.12) After some functional analysis, it is observed that among the numerator terms, only Ω 2 Γ 2s + Ω3Γ s + Ω 2 satisfies the equation in (4.10). Therefore, the explicit expression for Γ s will be Γ s(1,2 ) = −Ω 4 ∓ Ω 42 − 4 2 61 , Ω4 = Ω3 . Ω2 (4.13) The correct root of Γ s from (4.13) can be selected by imposing Γ s ≤ 1 . After determining Γ s , Ts can be calculated by substituting that Γ s into (4.9). Then, we can determine L0 using either (4.2) or (4.3). Finally, Ls , ε , and µ will be Ls = Lg − L p − L0 , (4.14) 2 j = ln (Ts ) , 2 Λ 2π Ls 1 µ = µp ε= 4.3 (1 + Γ s ) (1 − Γ s ) (4.15) 1 Λ ε pµ p λ02 − 1 , (4.16) λc2 2 2 1 λ0 λ0 + . µ Λ λc (4.17) Measurements We utilized the general purpose waveguide set–up introduced in chapter 3 for validation of the proposed method in this chapter. Because the proposed method in this chapter requires one of the plugs (left one) in Fig. 3.6 be fixed at the left terminal of the calibration plane, we shifted that plug and changed the configuration of the set–up to be similar to that in Fig. 4.1. We utilized two 10 mm polytetrafluoro−ethylene (PTFE) samples as plugs for holding the liquid samples in place so that there will be no meniscus formation on surfaces of liquid samples. We measured the ε p of the PTFE samples by the plug-loaded two-port transmission line (PLTL) method [25] while only the PTFE sample is present inside the holder in Fig. 4.1, and found ε p ≅ 2.03–j0.0014 over 9.7– 11.7 GHz. We calibrated the set–up by the Thru–Reflect–Line (TRL) technique [93] 62 since it does not necessitate precise or well–defined calibration standards such as the load standard. We used methanol, a commercially available antifreeze solution, and distilled water as test samples to validate the method. Figs. 4.2–4.4 demonstrate the measured ε of these liquids over 9.7–11.7 GHz. Fig. 4.2. (a) Real part and (b) imaginary part of the measured (dashed line) and theoretical (solid line) ε of distilled water. 63 Fig. 4.3. (a) Real part and (b) imaginary part of the measured (dashed line) and theoretical (solid line) ε of methanol. 64 Fig. 4.4. (a) Real part and (b) imaginary part of the measured ε of an antifreeze solution by the plug–loaded two–port transmission line method (PLTL) [25] and by the proposed method. 65 It is seen from Figs. 4.2 and 4.3 that both the real and imaginary parts of the measured ε of the distilled water and methanol are in good agreement with those computed from Debye model [104], [105]. The theoretical curve is obtained using the Debye equation ε ' = ε∞ + ε" = ε s − ε∞ , 1 + ω 2τ 2 (4.18) (ε s − ε ∞ ) ωτ , (4.19) 1 + ω 2τ 2 where ω is the angular frequency; ε ∞ is infinite frequency dielectric constant (or the dielectric constant when ω goes to infinite); ε s is static dielectric constant (or the dielectric constant when ω goes to zero); τ is the relaxation time. In the literature, one frequently finds a ‘relaxation wavelength’ or ‘Sprungwellenlange’ λs , defined as λs = 2π cτ , (4.20) where c is the speed of light. We obtained ε ∞ , ε s , and τ values for the distilled water and methanol from [104] and [105] for room temperature (20–25 Co) and are given in Table 4.1. Table 4.1. The parameters ε ∞ , ε s , and τ for distilled water and methanol. The data are estimated by interpolation from [104],[105] for room temperature (20–25 Co). εs ε∞ τ (ps) Distilled water 78.5 5.2 8.3 Methanol 32.6 5.6 48 Parameters 66 Fig. 4.4 illustrates the measured ε of an antifreeze solution over 9.7–11.7 GHz by our method and by the plug–loaded two–port transmission line method (PLTL) [25]. It is seen from Fig. 4.4 that the real and imaginary parts of the measured ε of the antifreeze solution by both methods are very good agreement with each other. To apply the PLTL method, we used the set–up in Fig. 3.6. We also kept the distances between calibration plane and end surfaces of the plugs minimum so that we could easily measure them. In our proposed method, we eliminated this necessity using (4.7) and (4.9). The proposed method assumes that the sample is homogenous and isotropic. In practice, it may not be homogenous. Naturally–occurring materials (e.g., timber) may have density and grain orientation variations while manmade materials may have density and constituent variations [41]. To evaluate the correctness of extracted constitutive parameters of and/or the homogeneity of samples, as a self–checking feature, the dependency of computed L0 can be drawn over the frequency band. For example, Fig. 4.5 shows this dependency for measurements of commercially available antifreeze solution over 9.7–11.7 GHz. It is observed that standard deviation of the computed L0 is little higher than that of the air section length of another sample holder operating at 3 GHz [41]. The reason for this can be the higher frequencies (approximately 10.7 GHz) used for testing our proposed holder. 67 Fig. 4.5. Dependency of the computed length of air–filled section, L0 , inside the proposed sample holder from commercially available antifreeze solution measurements over frequency. 68 CHAPTER 5 A SIMPLE FREE–WAVE METHOD FOR ELECTRICAL CHARACTERIZATION OF CEMENT–BASED MATERIALS In this chapter, we will present a free–wave method for complex permittivity, ε , measurements of cement–based materials (mortar, concrete, etc.) with different internal contents. These materials are widely used in the construction industry. Mortar contains cement powder, fine aggregate (sand), and water while concrete is composed of mortar and coarse aggregate (gravel); i.e., cement powder, fine aggregate (sand), water, and coarse aggregate (gravel). Measurement and monitoring of performance and properties (compressive and tensile strengths, porosity, permeability, hydration, mixture content, etc.) of these materials are important for evaluation of their quality and integrity. As it was discussed in chapter 2, generally reflection properties of these materials are measured to characterize them. Reflection measurements are very convenient for applications where the sensor (antenna) can only be placed on one side of the material (especially for measurements on pavement, back–filled concrete wall, roadway, etc.). However, any surface roughness of materials can significantly affect reflection measurements more than transmission measurements at high frequencies [20], [65]. In addition, transmission measurements provide longitudinal averaging of variations in material properties, which is particularly important for relatively lossy heterogeneous materials such as cement–based materials and moist coal [86], [106]; they are also more 69 sensitive to the dielectric properties of high–loss samples [86] and offer a wide dynamic range for measurements [86]. Although the results obtained from reflection measurements are very promising [42], [46]–[49], [53], [54], [59], [60], reflection–only measurements do not disclose any new information about cement–based materials, such as periodic assessment of aging of civil structures for structural integrity and public safety [59], after approximately 28th day of curing since reflection properties stay constant. Therefore, transmission measurements can be utilized to gather more information about these materials [20], [64], [66]. Among nonresonant methods, the free–space or free–wave technique appears to be more applicable for non–contact, noninvasive, and in situ applications [65]. In the literature, methods employing free–space measurements use vector network analyzers (VNAs) for highly accurate measurements and their time–gating facilities to reduce the multiple reflections inside the sample and between antennas and the sample [27],[28],[56],[59],[60],[65],[66],[107]–[111]. However, for practical purposes, it is attractive to use the simplest measurement system because using expensive VNAs is not convenient for industrial–based applications [32]. In this dissertation, we present microwave reflection and transmission properties of young/hardened mortar and concrete specimens at X–band (8–12 GHz) by a simple and relatively inexpensive set–up. Firstly, we will illustrate hourly measurements of young mortar specimens with different water–to–cement ratios (w/c) changing between 0.40 and 0.60 with a 0.05 of increment during 20–30 and 44.54 hours. Second, we demonstrate microwave reflection and transmission properties measured from different sides of hardened mortar and concrete specimens with different w/c ratios at X–band. 70 5.1 Theoretical Background Fig. 5.1 demonstrates a general configuration for measuring reflection and transmission properties of cement–materials at far–field. In the theoretical analysis, it is assumed that the sample is isotropic, non–magnetic, and extends to infinity in length in transverse dimensions. Fig. 5.1. The configuration for the measurement of reflection and transmission properties by using a free–space method. Using the S–parameters derived in chapter 3 in (3.17) and (3.18), we can write ( ) ( ) 2 Er Γ 1 − Z S11 = = , Ei 1 − Γ 2 Z 2 2 Et Z 1 − Γ S21 = = , Ei 1 − Γ 2 Z 2 where 71 (5.1) (5.2) ( ) Z = exp − jk0 L ε , Γ= 1− ε , 1+ ε ε = ε ' − jε " , k0 = 2π f . c (5.3) Here, Ei , Er , Et , ε , L , k0 , f , c , Γ , and Z are, respectively, the electric field intensity of incident, reflected, and transmitted waves, the complex relative permittivity and thickness of the sample, the free–space propagation (or phase) constant (free–space wave number), the operating frequency, the velocity of light, the reflection coefficient when the sample is infinite in length, and the propagation factor of the sample. 5.2 The Method Our goal is to determine ε from amplitudes of S–parameters in (5.1) and (5.2). Assuming the sample in Fig. 5.1 shows at least 10 dB attenuation and then applying the single–pass technique, the expressions in (5.1) and (5.2) can be approximated to s S11 = Γ, ( ) s S21 = Z 1 − Γ2 . (5.4) where the letter ‘s’ at the superscript of S11 and S21 in (5.4) denotes simplified expressions. In the simplification, we made use of Z 2 <<1 where i denotes the s s magnitude of the expression 'i' . In (5.4), S11 is independent of L while S21 is a function of it. s s It is obvious that the expressions of S11 and S21 in (5.4) and those in (3.33) and (3.34) are the same for L2 = L4 = 0 and ε 2 = ε 4 = 1 . Therefore, the derived objective function in chapter 3 can be employed for ε determination using L2 = L4 = 0 and ε 2 = ε 4 = 1 . Substituting them into (3.36)–(3.50), we find an objective function for this problem: 72 f 3l (ξ ) = (1 − χ 2 + ξ 2 ) 2 2k0ξ L s − S21 + 4 χ 2ξ 2 exp =0 2 2 (1 + χ ) + ξ (5.5) where k0 = 2π λ0 , λ0 is free–space wavelength, and χ= 2 s S11 −ξ2. (5.6) In the derivation of (5.5) and (5.6) from (3.36)–(3.50), we made use of χ − jξ = Γ, H1 ( χ , ξ ) = 1, H 2 ( k , l ) = 1, k = − χ , l = −ξ . (5.7) Because the ‘one–to–one’ property of the objective function derived for the five– layer structure in Fig. 3.1 is independent on the electrical properties of the dielectric plugs, the objective function in (5.5) will accurately determine a unique ε . This is because we already satisfy the condition that the loss tangent of the sample be greater than that of the medium which is next (left) to the sample (in our problem, it is air). 5.3 The Measurement Set–up The schematic diagram of the free–space measurement set–up used in measurements is shown in Fig. 5.2. The set–up measures amplitudes of the reflection and transmission coefficients (or S11 and S21 ) from electric field strengths of incident, reflected, and transmitted waves. It is composed of rectangular waveguide sections, a microwave source (OSC), a rotary attenuator (ATT), a frequency meter (FM), two directional couplers, two antennas, and three diode detectors. The OSC operates at X–band. Its output power is approximately 10 mW at X–band and is suitable and applicable for most microwave nondestructive applications [112]. The signal from the source is modulated by a 1 kHz 73 signal to measure amplitude–modulated reflected and transmitted signals by a simple reflectometer and an attenuation–meter as shown in Fig. 5.2. Fig. 5.2. The schematic diagram of the measurement set–up. The simple reflectometer is constructed to measure S11 and consists of two square–law detectors and the couplers. While Coupler I measures incident waves, Coupler II measures reflection properties. The attenuation–meter (T–meter) measures S21 and consists of two square–law detectors, Coupler I, a precision attenuator, an amplifier, and an indicator. The ATT is a rotary type attenuator, which is used to change the level of the incident signals and to decrease the level of reflected signals at the source (an isolator is much preferred for preventing the source from any harmful reflections in the measurement system or from the measurement cell; however, in our lab there is no available isolator). It is a FLANN 16110 precision attenuator with 0 to 60 dB attenuation range and 0.1 dB reading accuracy. The FM (or wavemeter) is utilized for highly sensitive frequency measurements. It is a FLANN 16072 tunable cavity resonator with 74 0.10 % accuracy and high Q (5500) when loaded. We used FLANN 16131 multihole directional couplers. They operate at 8.2–12.5 GHz and have a minimum 40 dB directivity and a 20 dB coupling. Standard gain horn antennas are used for matching the impedances of air and waveguide apertures. These antennas have an aperture of 60.5 mm x 45 mm and a gain of nearly 15 dB at X–band. It is noted that the measurement set–up in Fig. 5.2 was constructed from microwave components available in the Electromagnetic Fields and Microwave Techniques laboratory in Electrical and Electronic Engineering Department at Cukurova University, Adana, 01330, Turkey. 5.4 Calibration Procedure The transmit and receive antennas in Fig. 5.2 are separated according to: a) the distance between them fulfills the plane wave condition; b) the center of the samples is exactly matching the center of horn antennas; c) the aperture of the antennas is positioned to be parallel to the transverse dimension of the samples; d) the minimum amount of wave is scattered by the edges of the sample; and e) maximum incident signal is received when there is no sample between antennas to increase the sensitivity of the measurement set– up. The plane wave assumption is satisfied practically by LU >> 2 Dr2 λ where Dr , λ ( = c f ), and LU are the maximum lateral dimension of the radiator (diagonal length for rectangular and square antennas and diameter for cylindrical antennas), the wavelength, and the distance between the radiator (antenna) and each specimen [113]. According to the cross section of used horn antennas, 60.5 mm x 45 mm, the value of Dr 75 is determined approximately 75 mm and maximum LU is calculated as 380 mm at X– band. The transmit and receive antennas are separated from each other approximately one meter. In a recent study [59], it was shown that if the distance between the transmit antenna and the specimen is more than 8 times the maximum wavelength ( LU > 8λmax ) at which measurements are carried out, the plane wave assumption can be applied. According to the minimum frequency at X−band (8.2 GHz) used in our measurement set– up, 8λmax ≅ 292 mm which is much less than the antenna separation in our set–up (samples are located at equal distance from both antennas to reduce multiple reflection between antennas and specimens). Previous investigations show that the edge diffraction might be practically omitted by selecting Ds >> 5λ where Ds is the maximum lateral dimension of the samples (diagonal length for rectangular and square samples and diameter for cylindrical samples) [113]. The cross dimension of prepared samples, which will be discussed in the next chapters, is 150 mm x 150 mm. It is readily noted from this information that the maximum transverse dimension of the prepared samples, Ds =212.1 mm, is greater than 5λmax ≅ 183 mm. The horn antennas with narrow 3−dB beam widths at X−band are used for further decreasing the diffraction [27]. The maximum amount of the received signal is obtained by arranging various components of measurement apparatus; the center and parallel alignment and middle distance of samples according to aperture of the antennas are provided by placing the samples and antennas appropriately. We performed relative measurements. The reference level for reflection measurements is obtained by putting a metal plate (copper) exactly the position at which 76 front face of samples is located, and measuring the reflected signal from this metal plate. Likewise, reference level for transmission measurements is set by measuring the transmitted signal when there is no sample present between horn antennas. Because the output signal of the square-law detector is proportional to S11 2 (or r ) (assuming higher order terms are suppressed by detector [114]), the reflection coefficient of samples is obtained as r = Vsr Vm (5.8) where Vsr and Vm are the dc voltage values of reflected signal from the sample and metal plate (reference), respectively. Likewise, transmission coefficient is measured as t = Vst Vf (5.9) where Vst and V f are the dc voltage values of transmitted signal through the sample and in free-space (reference), respectively. 5.5 Measurements We carried out two different measurements for characterization of cement–based materials. The discussion of the results for each measurement is presented in two subsections. 5.5.1 Measurements of Young Mortar Specimens We present the measurement results of free–space reflection and transmission coefficients of five mortar specimens with varying mixture contents. We introduced a 77 new approach (relative slope approach for reflection properties) to predict the history of the hydration (a chemical process that water molecules bond with cement molecules) process of these specimens. Five mortar specimens are prepared with different w/c ratios (w/c=0.40, 0.45, 0.50, 0.55, and 0.60). The mix proportions are given in Table 5.1. The cement used in the experiment is the ASTM Type I normal Portland Cement, 100 % of the sand mass consists of particles less than 4 mm in diameter. Well–graded, quartzite, natural sand is used. The sand complies with the requirements of ASTM C–33 [115]. The absorption capacity is 0.1 % and its specific gravity (relative density) at saturated surface dry condition is 2.72. Table 5.1. Mass percentages of raw materials of cement samples. Mortar Sand(s) Cement(c) Water (w) s/c w/c I 58.80 29.40 11.80 2 0.40 II 57.96 28.98 13.06 2 0.45 III 57.14 28.57 14.29 2 0.50 IV 56.34 28.17 15.49 2 0.55 V 55.56 27.78 16.66 2 0.60 The thickness of the samples is chosen in such a way that transmission measurements of young mortar samples can be detected. According to the previous study [63], the optimum thickness of samples is selected as 100 mm after initial transmission measurements. The lateral dimensions of the samples are 150 mm x 150 mm. Temperature and relative humidity for curing conditions were kept between 28–32 Co and 50–60 %, respectively, at ordinary laboratory conditions. 78 The hourly measurement of reflection and transmission properties of five young mortar specimens with raw material properties in Table 5.1 has been conducted at X−band by the nondestructive and non–contact free–space measurement set–up in Fig. 5.2. For example, Figs. 5.3(a) and 5.3(b) show the hourly measurements of amplitude of the reflection coefficient, S11 (or r ), at 8.5 GHz of mortar specimens in Table 5.1. It can be seen from Figs. 5.3(a) and 5.3(b) that dependency of reflection properties of each sample is monotonic (decreasing with time), and the higher amplitude of reflection coefficient corresponds to the mortar sample with higher w/c ratio at early ages. This is the result of the effect of both evaporation of free–water from the surface and hydration process [20],[42],[63]. It means that the reflection properties depend on w/c ratio. These measurement results compare well with the previous results of 2–3 day cured mortar and concrete samples at room temperature [20],[42],[63]. Since sensing of the material using microwave energy interrogates the dielectric properties of the material on a molecular level, it can be assumed that the curing process is complete once the dielectric properties of these chemically active materials become constant. Therefore, by periodic monitoring of the reflection properties of the specimens during their curing process, an estimation of the cure–state of the specimens can be ascertained [42]. In this regard, the slope or time rate change of the reflection properties can be taken as a metric of the state and speed (or degree) of hydration process [42]. This is also meaningful if we consider different curing (wet or dry) and environmental conditions (temperature, humidity, etc.) for the measurements of properties of cement– samples because the amplitude of reflection properties changes with them. 79 (a) (b) Fig. 5.3. Hourly reflection properties of young mortar samples with different w/c ratios at 8.5 GHz (a) between 20–30 hours and (b) between 44–54 hours. 80 In this study, we proposed a relative slope approach for reflection properties of young mortar samples to predict the history of the hydration process of these samples with equally increased w/c ratios; i.e., ∆w ∆c = 0.05 , and for this purpose monitored the amplitude of the reflection properties. The proposed approach is reasonable since mortar can be assumed as a homogenous dielectric mixture at relatively low microwave frequencies (even at 10 GHz) [42]. The wavelength corresponding to the frequency used in our measurements (8.5 GHz) is 35.3 mm, which is much greater than the diameter of the maximum sand particles (less than 4 mm). It is demonstrated in Table 5.2 that, considering only the samples whose w/c ratio difference is 0.05, the average absolute temporal amplitude difference in reflection coefficients for mortar samples I and II is much greater than that of other mortar samples. This difference can also be seen from Figs. 5.3(a) and 5.3(b). This means a non–linear relationship between the degree of hydration and w/c ratios for early ages of curing. Table 5.2. The absolute temporal change of reflection coefficient of each young mortar specimens. Sample Number Hours 20–30 44–54 20–54 I – w c =0.40 0.0300 0.0155 0.0543 II – w c =0.45 0.0296 0.0170 0.0570 III – w c =0.50 0.0325 0.0170 0.0578 IV – w c =0.55 0.0271 0.0152 0.0584 V – w c =0.60 0.0271 0.0135 0.0595 81 From the physical point of view, during curing, evaporation of free–water and hydration inside samples occur simultaneously. Because the free–water very near to the cement molecules will yield immediate reaction and hydration, the binding inside each sample will be different depending upon the amount of free–water. All factors being the same, the strength of mortar decreases as the w/c ratio increases. A mortar sample with a lower w/c ratio will gain strength quicker than the one with a higher w/c ratio [19],[42]. From the electrical point of view, it is known that measured reflection properties for a mixture consisting of a relatively low permittivity of dielectric constituent (cement) and a high permittivity of dielectric constituent (water) is expected to increase as a function of increasing water content [116]. In addition, while dielectric properties of free–water are much higher than those of cement powder, dielectric properties of bound– water are similar to those of cement powder [117]. Therefore, a considerably lower amplitude of reflection coefficient of mortar sample I means that a fast chemical reaction or hydration (an indication of chemical binding) occurred inside it. Therefore, we can deduce the history (i.e., quickness) of hydration process (and strength) of mortar samples from our proposed relative slope approach. While reflection measurements show surface characteristics of samples, the transmission measurements show the internal structure (central) and provide longitudinal averaging of variations in sample properties [20]. For example, Figs. 5.4(a) and 5.4(b) show the result of the hourly measurements of amplitude of transmission coefficient, S21 (or t ), at 8.5 GHz of mortar specimens in Table 5.1. It can be seen from Figs. 5.4(a) and 5.4(b) that a higher transmission coefficient is for the mortar sample with a lower w/c ratio (meaning that transmission properties are 82 w/c ratio dependent) and that there is no intersection point observed between these dependencies. In a fresh state of mixture during hardening (setting) time, the heavy ingredients of the mixture tend to move downward (aggregates) and light ingredients of the mixture move upward (water), resulting in bleeding (a special form of segregation) and porosity formation [20],[118]. Pore structure inside cement samples can be thought as an air medium with a relative permittivity of almost one. This means that microwave signals will penetrate much easily through cement samples with higher porosity than the ones with lower porosity. It is well known that hardened cement sample with a bigger w/c ratio will have a higher porosity than the one with a smaller w/c ratio [20],[118]–[121]. The reason is the evaporation of free–water from the surface of samples. On the other hand, since the transmission properties of the mortar sample with lower w/c ratio are higher than those with higher w/c ratio at early ages of curing as shown in Figs. 5.4(a) and 5.4(b), we can conclude that the potential porosity inside the young mortar sample with higher w/c ratio should be less than the one with lower w/c ratio. This is because free water cannot immediately escape from the cement mixture at early stages of curing. As a result, monitoring of transmission properties along with reflection properties can be used for the quality enhancement of cement samples since early age detection of properties is important for quality enhancement of these samples [42]. 83 (a) (b) Fig. 5.4. Hourly transmission properties of young mortar samples with different w/c ratios at 8.5 GHz (a) between 20–30 hours and (b) between 44–54 hours. 84 Because measurements were conducted during only day and evening times, there is no measurement data between 30 and 44 hours (night times). We expect that there are no significant changes of the dependencies of reflection and transmission properties (the higher reflection is for the sample with higher w/c ratio, and the higher transmission is for the sample with lower w/c ratio) since we kept the environmental conditions constant throughout the measurements. However, as it can be seen from Figs. 5.3(a) and 5.3(b) that we predict the average relative distance between the slopes of the reflection properties of mortar samples I and II during 30–44 hours will be greater than the average distance during 20–30 hours and less than the average distance during 44–54 hours. It can be seen from Figs. 5.3 and 5.4 that we observed some variations in reflection and transmission properties. This is because samples are switched for each measurement step and reflection and transmission properties are surface and integral content dependent, respectively. It is noticed that the variations seen in reflection measurements (Fig. 5.3) are greater than those of transmission measurements (Fig. 5.4). It is known that any misalignment of the aperture of sample (angle deviation) yields different reflection and transmission properties, and reflection properties are affected more [20]. It should be noted that the effect of the environment (temperature, humidity, etc.) is minimized throughout measurements by keeping the environmental conditions constant. We also assumed that the characteristics of the measurement setup are stable during measurement. We used two methods for the ε determination of mortar samples. These methods are the graphical and Newton’s methods. The graphical method is used because it shows the general pattern of the dependency of reflection and transmission properties on real 85 and imaginary parts of the ε [20],[26],[74]. For given measured amplitudes of the reflection and transmission coefficients, we can obtain the constant value lines of these coefficients, CR and CT [20],[26],[74]. The necessary and sufficient condition for determining the ε from the measured amplitudes is that there is just one cross point between the lines CR and CT for high–loss samples. Although the graphical method is very clear and useful for primary evaluation of the dielectric properties, the speed of this method is very slow. We utilized the Newton’s method for ε determination of mortar samples using the objective function in (5.5) along with (5.6). We set ξ = 0 as initial guess and searched for the minimum of the objective function. Figs. 5.5 and 5.6 show the ε of young mortar samples in Table 5.1 during early ages of curing (20–30 and 44–54 hours). It is seen that the values of real and imaginary parts of the complex permittivity, ε ' and ε " respectively, of mortar samples are in good agreement with known complex permittivity values of hardened cement samples obtained from reflection measurements by vector network analyzer (VNA) [122] and amplitudes–only reflection and transmission measurements with a simple set–up [20]. It is demonstrated in these figures that a higher w/c ratio corresponds to a higher imaginary (real) part of the permittivity, ε " ( ε ' ). From the physical point of view, since at early ages of curing the free water content will be greater than the bound water content inside mortar samples, a higher content of water will result in bigger losses (or attenuations) and bigger reflections. Additionally, because ε " represents the loss of electric field in the sample and ε ' represents the ability of the sample to store energy 86 [123], one can conclude that transmission (reflection) measurements will be mainly affected by ε " ( ε ' ). From the mathematical point of view, assuming a high–loss case, it is seen from (5.5) that the effect of the imaginary part of the complex permittivity on transmission coefficient is almost directly present in exponential term since the imaginary part of the square root of a complex number is much more dependent on the imaginary part than real part of this complex number. This also validates the strong relation between transmission (reflection) measurements and ε " ( ε ' ) at early ages of curing. From the empirical point of view, the same conclusions can be made. To this end, we apply a newly developed relative slope approach for reflection properties to transmission properties. It was mentioned that the relative difference between reflection properties of mortar samples I and II are much bigger than any other nearly the same relative differences in Fig. 5.3. However, it seen from Fig. 5.4 that all relative differences between transmission properties of mortar samples are almost the same. It is noted that only the samples whose w/c ratio difference is 0.05 are considered for comparisons. In addition, it is observed that whereas reflection properties change from one measurement step to another in Fig. 5.3(a), transmission properties show a monotonic behavior in Fig. 5.4(a). The reason of this change might be the misalignment of the position of surface of the samples with aperture of antennas (angle deviation). Comparing these measurement results with calculated ε ' and ε " values in Figs. 5.5 and 5.6, it is realized that ε ' and ε " follow the change patterns of reflection and transmission properties, respectively. Therefore, we can conclude that ε ' ( ε " ) is mainly affected by reflection (transmission) properties at early ages of curing. 87 (a) (b) Fig. 5.5. (a) Real part and (b) imaginary part of the complex permittivities of young mortar samples with different w/c ratios between 20–30 hours. 88 (a) (b) Fig. 5.6. (a) Real part and (b) imaginary part of the complex permittivities of young mortar samples with different w/c ratios between 44–54 hours. 89 5.5.2 Measurements of Hardened Mortar and Concrete Specimens Measurements of reflection and transmission properties from various sides of hardened mortar and concrete specimens with different w/c ratios are carried out at X–band over 3– 36 months by the free–space set–up in Fig. 5.2. Several cubic mortar specimens with different w/c ratios and dimensions of 150 mm × 150 mm × 150 mm are prepared. The raw materials of the mortar and concrete specimens are shown in Table 5.3. Concrete Mortar Table 5.3. Mixture proportions of mortar and concrete specimens. Coarse aggregate Water/cement Sand/cement I 0.4 1.5 /cement –– II 0.7 1.5 –– I 0.4 1.5 2.0 II 0.7 1.5 2.0 The cement used in the experiment is the ASTM Type I normal Portland Cement, 100 % of the sand mass consists of particles less than 4 mm in diameter. Well–graded, quartzite, natural sand is used. The sand complies with the requirements of ASTM C–33 [115]. The absorption capacity is 0.1% and its specific gravity (relative density) at saturated surface dry condition is 2.72. Coarse aggregates have a maximum size of 16 mm and are natural round shaped obtained from the river. We prepared all specimens with a surface protuberance less than 5 mm by vibrating each specimen inside its metal mould. We removed all molds after 1 day. Then, the specimens were left to cure at ordinary room conditions (a temperature and a relative 90 humidity between 28 and 32 oC and 50 and 60 %, respectively). For comparing the side effects of specimens on reflection and transmission properties, we, respectively, labeled the bottom, right, top, and left sides of specimens with numbers from 1 to 4. At early times of curing, the transmission properties are too low to measure accurately. Therefore, we present transmission properties of mortar and concrete specimens after 3 months. For example, Fig. 5.7 shows the results of monthly measurements of S21 from right sides of mortar and concrete specimens with different w/c ratios at 10 GHz. For easy reference in Figs. 5.7–5.10, whereas ‘M’ and ‘C’ in the figure legend refer to mortar and concrete specimens, the numbers from 1 to 4 denote sides of the specimens, and ‘I’ and ‘II’ mean w/c=0.4 and 0.7, respectively. Fig. 5.7. Results of transmission properties from right sides of mortar and concrete specimens with different w/c ratios. It is seen from Fig. 5.7 that the higher w/c ratio results in bigger transmission properties of mortar (concrete) specimens. The reason is the evaporation of free–water 91 from the surface of samples and formation of pore structure inside samples. It is well known that a hardened cement sample with a bigger w/c ratio will have a higher porosity than the one with a smaller w/c ratio [20],[63],[118]–[121]. A crossover point was observed between transmission properties of the same type specimens with different w/c ratios at 5th days of curing [20]. We did not observe any crossover point between transmission properties of the specimens with different w/c=0.4 and 0.7 ratios during 3–36 months of curing. To monitor the state and degree of aging (curing) inside hardened specimens, we developed a relative difference approach. It uses temporal differences between transmission properties of the specimens and can be applied to: a) the same type specimens with different w/c ratios and b) different type specimens with the same w/c ratio. While a temporal difference demonstrates the state of aging, its tangent relates the degree of aging of specimens. The application of the technique to the same type and different type specimens using the measurements in Fig. 5.7 is shown in Figs. 5.8(a) and 5.8(b), respectively. 92 (a) (b) Fig. 5.8. Application of the relative difference approach to (a) the same type specimens and (b) different type specimens. The temporal dependencies in Fig. 5.8 are investigated as three separate regions. These regions, respectively from first to third, are labeled with plot symbols ‘○’, ‘□’, and ‘∆’. The main conclusions drawn from the measurements in Fig. 5.8 are the following: 93 1) Regions–I and II correspond to periods of aging for specimens whereas Region–III refers to the end of aging since temporal differences change throughout Regions–I and II and are stable and monotonic (constant) during Region–III. It is seen from Fig. 5.8(a) that the aging periods for mortar and concrete specimens are during 3–30 and 3–24 months, respectively. The difference between aging periods comes from the fact that gravel particles are heavier in mass than sand particles and force free–water to escape from the open surfaces of the specimens. As a result, concrete specimens will complete or finalize the aging process sooner than mortar specimens with the same w/c ratio. 2) For Region–I in Fig. 5.8(a), both dependencies significantly increase for mortar specimens from 0.076 to 0.138 (an increment of 0.062) and for concrete specimens from 0.038 to 0.121 (an increment of 0.083) during 3–9 months. This means that tangents of the dependencies are both positive for Region–I. The increments in the dependencies are because the evaporation or desiccation of free–water inside the mortar (concrete) specimen with a higher w/c ratio is faster than that inside the same specimen with a lower w/c ratio. As opposed to this, during Region–I in Fig. 5.8(b), both dependencies decrease throughout 3–6.1 months. This time, the effect is that the evaporation of free–water inside a concrete specimen is faster than that inside a mortar specimen with the same w/c ratio. Accordingly, a concrete specimen with a higher w/c ratio will lose free–water much faster than other specimens with different w/c ratios in Region–I. The reason for the increments and decrements in the dependencies is, as discussed in 1), the difference in mass proportions of gravel and sand particles and the w/c ratio. 3) During Region–II, we observed smaller decreases for mortar specimens from 0.138 to 0.131 (0.007 difference) and for concrete specimens from 0.121 to 0.118 (0.003 94 difference) during 9–30 months in Fig. 5.8(a). In contrast, a small increase in the dependencies is recorded for the ‘M.II.2–C.II.2’ dependency from 0.024 to 0.034 (0.01 difference) and for the ‘M.I.2–C.I.2’ dependency from 0.006 to 0.021 (0.015 difference) during 6.1–30 months. Therefore, a mortar specimen with lower w/c ratio will drop free– water sooner than other specimens with different w/c ratios in Region–II. This can be because of the continuing evaporation of free–water inside mortar specimens as contrary to decreasing evaporation of concrete specimens. 4) We note that the conclusions made for the temporal transmission properties for the right and left sides of mortar and concrete specimens are nearly the same. Only a small difference (<3%) between the dependencies is examined. This demonstrates the integrity and homogeneity of prepared specimens. 5) The temporal transmission properties for the top and bottom sides of mortar and concrete specimens are the same. This clearly proves that the specimens under investigation are inactive materials [14]. 6) The magnitude of transmission properties for the top (or bottom) is less than those for the right (or left) side for each measurement step. This is due to the effect of gravel and sand present inside the specimens after hardening (setting) [63],[118]. Because microwave signals will scatter from the aggregates (scatterer) which moved to the bottom of the specimens during setting, lower transmission properties were measured for specimens with top side is located toward to the microwave signal. 7) While the patterns of the dependencies of transmission properties for the same type specimens (different w/c ratios) stay the same for the top and right sides, the difference between the dependencies of these properties for the top side is more pronounced than 95 those for the right side. This is because of greater scattering effect of gravel particles than that of sand particles. 8) We did not record any significant change between the dependencies for different type specimens (the same w/c ratios) for top and right sides. The reason for this is approximately the same mixture proportions of gravel and sand particles inside mortar (and concrete) specimens with different w/c ratios. As shown in Table 5.3, because w/c ratios of the specimens are much lower than sand–to–cement (s/c) and coarse aggregate– to–cement (ca/c) ratios, the same type specimens will approximately have the same gravel and sand particles. Whereas transmission properties of mortar and concrete specimens change over long service lives and become constant after approximately 30 months, reflection properties of these specimens vary during 1 month after preparation and stay constant since then. Reflection properties of these specimens were analyzed in the literature [20],[42],[47],[63]. Table 5.4 demonstrates measured reflection properties of prepared samples after 3 months of curing. Table 5.4. Reflection properties from various sides of mortar and Concrete Mortar concrete specimens with different w/c ratios after 3 months of curing. Right Left Top Bottom I 0.51 0.50 0.49 0.52 II 0.46 0.45 0.44 0.49 I 0.48 0.47 0.46 0.54 II 0.45 0.44 0.43 0.52 96 The main findings from reflection measurements in Table 5.4 are as follows: 1) The difference between reflection properties from the right (or left) side of mortar specimens with different w/c ratios is greater than that of the concrete specimens. This is because of the fact that cement proportions inside mortar specimens are much greater than those inside concrete specimens because of gravel and the same s/c ratio. While hydration occurs, cement and water molecules chemically combine with each other. This process greatly changes the permittivity of specimens [42]. Because reflection properties are affected by the permittivity changes, we can conclude that the higher cement content inside mortar and concrete specimens the bigger reflections from the surface of these specimens while other ingredients are kept approximately the same. 2) Reflection properties from the bottom of mortar and concrete specimens are greater than those from other sides of the same type specimens for each w/c ratio. This is because of the scattering effect of aggregates. While hardening (or setting), heavy ingredients inside specimens move downward and stays there. Because reflection properties are measured after hardening, not only hydration but also scatterers (aggregates) affect microwave reflection properties. 3) Reflection properties from the top of mortar and concrete specimens are less than those from other sides of the same type specimen for each w/c ratio. This can be explained using the relative permittivity values of pore and aggregates. While hardening (or setting), heavy aggregates force lighter ingredients (free–water molecules) to the top. This force creates pore structures inside mortar and concrete specimens after water evaporates from the sides of the specimens. Because pore structures inside cement 97 samples can be thought as air medium with a relative permittivity of one, microwave signals will penetrate inside samples easier from the top. We utilized the Newton’s method for ε determination of mortar and concrete specimens using the objective function in (5.5) along with (5.6). We set ξ = 0 as initial guess and searched for the minimum of the objective function. Figs. 5.9 and 5.10 demonstrate, respectively, the real and imaginary parts of the ε of mortar and concrete specimens with different w/c ratios and sides using the measurements in Fig. 5.7 and Table 5.4. It is seen from Figs. 5.9 and 5.10 that whereas ε ' stays approximately constant, ε " decreases over 3–36 months for all specimens. Because reflection measurements were constant and transmission properties increased over 3–36 months and ε ' represents the ability of the substance to store energy [123], the results in Figs. 5.9 and 5.10 clearly demonstrate a strong relation between the pattern of reflection properties and ε ' of mortar and concrete specimens [20]. In other words, the higher the reflection properties the bigger ε ' . In addition, since ε " (loss factor) represents the loss of electric field energy in the substance [123], the higher transmission properties correspond to a lower loss factor. Therefore, we also observed a correlation between the change patterns of transmission properties and ε " of specimens [20]. 98 (a) (b) Fig. 5.9. The real part of the ε of mortar and concrete specimens with different w/c ratios and sides. 99 (a) (b) Fig. 5.10. The imaginary part of the ε of mortar and concrete specimens with different w/c ratios and sides. In the light of the conclusions made in this subsection, we can extract some useful information about the reflection and transmission properties of mortar and concrete specimens using the information present in Figs. 5.9 and 5.10. For example, the highest and the lowest ε ' of mortar and concrete specimens correspond to the reflection 100 properties from top and bottom, respectively, for each w/c ratios. As another example, because it is well known that the specimens with higher w/c ratio contain pore structures much more than those with a lower w/c ratio and these structures can be thought as air medium inside specimens, the smallest ε " corresponds to mortar or concrete specimens with the highest w/c ratio for a given side (top, bottom, or etc.). It can be seen from Figs. 5.3, 5.4, 5.7, and 5.8 that we observed some variations in reflection and transmission properties. This is because samples are switched for each measurement step and reflection and transmission properties are surface and integral content dependent, respectively. It is noticed that the variations seen in reflection measurements (Fig. 5.3) are greater than those of transmission measurements (Figs. 5.4, 5.7, and 5.8). A detailed discussion about these variations can be found in subsection 5.5.1. 5.6 Measurement Uncertainties The reliability of reflection and transmission measurements depends on measurement uncertainties. Although we developed a calibration technique to eliminate some of systematic errors, it does not consider MR between the measurement cell and antennas, between antennas, and inside the sample, deviations from the characteristics of directional and square–law detectors, and surface roughness. It is assumed in section 5.2 that the sample has enough attenuation so that MR between the two surfaces of the sample can be neglected. To validate this assumption for measurements of young mortar samples (Figs. 5.3 and 5.4), the dependencies of reflection and transmission (attenuation) properties for lossless sample case presented by 101 (5.1) and (5.2) versus sample thickness, L , are drawn. For example, Fig. 5.11 illustrates such kind of dependencies for the w/c=0.40 mortar sample with ε = 21 − j 0.69 at 54 hours of curing in Fig. 5.6. Fig. 5.11. Dependencies of reflection and transmission properties of w/c=0.40 mortar sample with ε = 21 − j 0.69 on sample thickness, L . The dB values for reflection and attenuation properties in the curves in Fig. 5.11 are obtained R ( dB ) = −20log10 S21 , (5.10) T ( dB ) = −20log10 S21 , (5.11) where R and T , respectively, denote the reflection and attenuation properties in dB. Because L in (5.1) and (5.2) is expressed in complex exponential terms, we expect an oscillatory behavior of reflection and transmission properties with thickness for 102 lossless samples. The oscillations in these dependencies demonstrate MR inside the sample because a small change in thickness results in a large alteration in the s dependencies. It was stated in section 5.2 that for high–loss samples (with no MR) S11 is independent of L . Therefore, a small change in thickness will not result in an oscillatory behavior for reflection properties for high–loss samples. The same situation is true for s S21 since it is dependent upon Γ , which is a function of only ε , and a linear term 2k0ξ L . exp (1 + χ )2 + ξ 2 (5.12) This linear term will yield a constant value for a given L (not an oscillation). Therefore, the oscillatory behavior of the dependencies of S11 and S21 over L for lossless samples will turn to a monotonic behavior for high–loss samples. It is seen from Fig. 5.11 that there are large oscillations in both transmission (attenuation) and reflection properties in the 0–8 cm thickness range. Beyond this thickness, these oscillations decrease with increasing thickness, and their behaviors start to be monotonic. Therefore, whereas 0–8 cm thickness range corresponds to a lossless (or low–loss) behavior, thicknesses greater than 8 cm result in a high–loss behavior. In practice, MR inside samples can be neglected if the sample shows at least 10 dB attenuation [20],[26],[123]. It is seen in Fig. 5.11 that the selected thickness ( L =100 mm) demonstrates approximately 15 dB attenuation. The same monotonic behavior of reflection and transmission dependencies of w/c=0.4 young mortar sample at 54 hours of curing is observed for the same sample at other measurement steps and the other mortar samples at all measurement steps. As a result, we conclude that the selected thickness of mortar samples shows at least 10 dB attenuation and assures the correctness of complex 103 permittivity computations using (5.5). It is noted that determined ε can be refined, if needed, using that ε as an initial guess for the amplitudes of (5.1) and (5.2) in a two– dimensional search algorithm [75]. We can monitor the dependencies of R and T in (5.10) and (5.11) for validation of the determined ε of hardened mortar and concrete specimens in Figs. 5.9 and 5.10. Instead, we apply another technique to demonstrate the validation of this assumption that s s S11 and S21 can, respectively, be approximated to S11 and S21 (or MR inside the specimens can be neglected for our measurements). In the section 5.2, we utilized Z 2 <<1 for simplifying the expressions in (5.1) and (5.2). This condition can be written in an explicit from using (5.3) and (5.7) as 4k0ξ L << 1. exp (1 + χ )2 + ξ 2 (5.13) The dependency of (5.13) versus L is used for correctness of the determined ε in Fig. 5.9 and 5.10. It is clearly seen from (5.13) that for a known f both χ and ξ are in effect. It is noted from (5.7) that ξ is highly dominated by ε " . Because determined ε ' stays approximately constant in Fig. 5.9 and ε " changes with time in Fig. 5.10, monitoring the effect of ξ on (5.13) will help us validate the foregoing assumption. It was proven in chapter 3 that χ < 0 and ξ < 0 . Since it is easy to satisfy (5.13) when ξ is large for a constant χ , we have to take into consideration the worst case (the smallest ξ ) in order to prove the foregoing assumption for all ξ values. From Figs. 5.9(a) and (b), the lowest and the highest ξ values are, respectively, obtained from the mortar specimen with w/c=0.7 and from the concrete specimen with 104 w/c=0.4 after 36 months of curing because ξ is proportional to ε " . For example, Fig. 5.12 shows the dependency of Z 2 over L for ε = 6.95 − j 0.22 and ε = 11.18 − j 0.60 . Fig. 5.12. Dependency of Z 2 versus specimen thickness, L , for ε = 6.95 − j 0.22 and ε = 11.18 − j 0.60 . It is clearly seen from Fig. 5.12 that the dependencies of Z 2 decrease with L . At L =150 mm, Z 2 =0.075 for ε = 6.95 − j 0.22 and Z 2 =0.0034 for ε = 11.18 − j 0.60 . Because Fig. 5.12 shows the dependencies for the lowest and the highest ξ , the dependencies of Z 2 for other hardened mortar and concrete specimens will lie between the curves in Fig. 5.12. As a result, we can conclude that MR inside the specimens can be neglected for all prepared hardened specimens. Since MR between antennas depend upon MR between sample surfaces (assuming this effect is already omitted), its effect can also be neglected. Because relative 105 measurements are performed, the effects of MR between antenna and sample and the apparatus uncertainties are already small and/or eliminated. Another effect might be the finite directivity of the directional coupler used in the measurement set–up in Fig. 5.2. Finite directivity reflects a possibility of a linkage between the forward and reverse ways of a directional coupler [14]. The oscillator signal coupled into the detector output for reflection measurements (because of the finite directivity of coupler more than 40 dB) shows that its value is much small compared with the reflected signal value and can be neglected [14]. On the other hand, the accuracy of S11 and/or S21 measurements might be reduced by the deviations from the square–law characteristics of the crystal diode detectors used in our set–up. It is well known that the square–law characteristic may be altered if the level of measured voltage value is too high. Considering this, we selected the thickness of the young mortar samples and hardened mortar and concrete samples in such a way that both the level of transmission measurements is in the range of crystal diode characteristics and the levels are high enough to be distinguishable for different w/c ratio samples. Finally, S11 and S21 in (5.1) and (5.2) account only for specular reflection, which occurs for smooth surfaces. When the surface is rough, the impinging energy will be scattered in angels other than the specular angle of reflection, thereby reducing energy in the specularly reflected and transmitted components. As a result, the surface protuberance might change reflection and transmission measurements, and its effect must be analyzed. The modified S11 , which takes into account the surface roughness, can be written as [67] r S11 = ρ s S11 , 106 (5.14) where ρ s is the correlation factor and the letter ‘r’ at the superscript of S11 in (5.14) denotes the expression for surface roughness. The expression of ρ s is given as [67] πσ h cosθi λ0 ρs = exp −8 2 , (5.15) where σ h is the standard deviation of the surface height in the first Fresnel zone of the illuminating antenna and θi is the angle of incidence. The Rayleigh criterion is commonly used as a test for surface roughness (critical height) of materials [67] as hc = λ0 8cosθi . (5.16) The height, h , of a given rough surface is defined as the minimum to maximum surface protuberance. The surface is considered smooth if h < hc and rough if h > hc . It can be seen from equation (5.16) that hc reaches minimum value for a given wavelength λ0 at θi = 0 . The minimum value of hc in (5.16) is between 4–5mm at X–band. In practice, cement–based samples have surface roughness with heights less than 5 mm, and thus can be assumed that they are smooth [20]. In addition, most cement–based structures have surfaces relatively having larger curvatures so that at X–band they are locally quite smooth. As a result, for elaborately prepared samples the surface protuberance effect can be omitted. 107 CHAPTER 6 A SIMPLE AND RELATIVELY INEXPENSIVE MICROCONTROLLER–BASED MICROWAVE FREE–SPACE MEASUREMENT SYSTEM FOR INDUSTRIAL– BASED APPLICATIONS Microwave measurements lend themselves to automation because many readings and considerable computation are often needed to achieve reasonable accuracies. Therefore, microwave measurements, especially broadband microwave measurements, can be made considerably faster and more accurately when they are automated [124]. Automated systems have been demonstrated to provide some principal advantages over manual ones in microwave measurements: a) Increased speed b) Computational enhancement c) Repeatability d) More complex measurements Any measurement system that would benefit from any of the above advantages is a likely candidate for automation [124]. The introduction of inexpensive microprocessors, the availability of programmable sources and measurement instruments have made it possible to build custom automated network analyzers automated network analyzers at a substantially lower cost than commercial system purchase price [125]. 108 In this chapter, we will present a microcontroller based microwave free–space (free–wave) measurement system for electrical characterization (complex permittivity, ε , determination) of lossy materials. This system is very suitable for industrial–based applications since the total cost to construct a microwave measurement system for electrical characterization of lossy materials is kept at a minimum. It is composed of a microwave section and an electronic section. While the microwave section measures the amplitudes of reflection and transmission measurements of samples, the electronic section calculates the ε using those measurement data. In this chapter, we also present a calibration procedure, which not only takes into account of diffraction effects from the sample edges but also undesired reflections from the surrounding environment, multiple– path reflections from the ground, and the losses in the measurement system. We validate the whole measurement system by amplitude only measurements of a fresh sample, which is poured into a wooden container, at 10 GHz. 6.1 The Measurement System The proposed microcontroller–based microwave free–space measurement system is shown in Fig. 6.1. It measures the amplitudes of reflection and transmission coefficients ( r ( S11 ) and t ( S21 )) at discrete frequencies and computes the complex permittivity, ε , of lossy sample (or a measurement cell which contains or holds the sample under test) using a general purpose microcontroller (µC). It can be separated into two main sections: a) microwave section and b) electronic section. Since the microwave section is already discussed in chapter 5, we will only describe the electronic section. 109 Fig. 6.1. The schematic diagram of the microcontroller–based microwave free–space measurement system. The electronic section gathers and processes measurement data to determine ε and outputs the result. It consists of a µC, an analog–to–digital converter (ADC), an electrically erasable programmable read–only memory (EEPROM), a liquid crystal display (LCD), and a printer (Fig. 6.1). The circuit board of the electronic section is shown in Fig. 6.2. The µC commands every operation in this section. We employed a multi–purpose 8 bit AT89S8252–24PI µC from ATMEL for our project. It has a 8K Bytes flash memory, a 2K Bytes EEPROM, and a 24 MHz internal crystal frequency. It is a powerful 110 µC, which provides a highly–flexible and cost–effective solution to many control applications. An 18.432 MHz external crystal frequency is used to operate this µC. Fig. 6.2. The circuit board of the electronic section. Since the microwave detector output is analog, it can readily be used as an input signal to any ADC. In our preliminary studies, we employed an 8 bit ADC (ADC 0808) for converting the analog signal from the microwave section into a digital signal for processing in the µC [79]. Then, we added amplifiers between the microwave section and the µC as a buffer to increase the sensitivity [78]. However, since the resolution or the accuracy of an 8 bit ADC is not suitable for precise ε determination, in this dissertation, we employ a 12 bit LTC1298 ADC to process the data. Its operating range is adjusted between 0 and 1000 mV. This sets a resolution or a quantile level less than 250 µV. 111 We utilized an external 24LC512 EEPROM (512 Kbit) to save and transport the measurement data for post–processing. Chip select and clock digital signals are used to access this EEPROM by the µC, and a serial transfer is utilized to acquire/write the data (Fig. 6.1). The LCD from Hantronix has a 16x2 character display (two lines where each line can show at most 16 characters). An enable and a read/write digital signals are used to control it by the µC (Fig. 6.1). A general purpose HP 6 MP Laserjet printer is connected for printing r , t , and ε . A common interface standard (centronics) is utilized to communicate with the printer and to print the results. We employed this standard because the majority of printers use this basic handshaking procedure. The computed ε can be saved into the internal 2KB EEPROM of the µC, stored into the external EEPROM, demonstrated on the LCD display, or printed using the printer. 6.2 Calibration Procedure The microwave source of the measurement set–up in Fig. 6.1 was operated at least two hours before measurements to stabilize source power. Care was taken to minimize the interference to measurements. For the set–up in Fig. 6.1, we can write Y (ω ) = U (ω ) H (ω ) . (6.1) Here, U (ω ) , H (ω ) , and Y (ω ) correspond to the expected or theoretical signal in the frequency domain, the system transfer function, and the measured signal in the frequency domain. We applied a calibration technique similar to the Thru–Reflect–Line (TRL) technique [93]. Its schematic diagram is shown in Fig. 6.3. 112 Fig. 6.3. The schematic diagram of the calibration technique. Error terms denoted by Ed (ω ) , E pf (ω ) , E pr (ω ) , EDf (ω ) , EDr (ω ) , and E f (ω ) correspond to errors of directivity, forward and reverse port match, forward and reverse diffraction, and frequency response (tracking), respectively. The r and t are the actual amplitudes of reflection and transmission properties of specimens. The error terms EDf (ω ) and EDr (ω ) include not only diffraction effects from sample edges but also undesired reflections from the surrounding environment, multiple–path reflections from the ground, and the loss in the measurement system. In the analysis, firstly it is assumed that E f (ω ) = 1 , E pf (ω ) = 0 , and E pr (ω ) = 0 throughout the frequency domain. Before finding the error terms, the transmit and receive antennas are separated as follows: a) the distance between them fulfills the plane wave condition, b) the center of the samples is exactly matching the center of horn antennas, and c) maximum incident signal is received when there is no sample between antennas to increase the sensitivity of the measurement set–up. All these criteria were discussed in detail in Section 5. 113 Next, a highly reflective metal plate (copper) having the same cross section and surface properties as of prepared specimens is placed at the middle distance of the antennas, and amplitudes of incident– I m (ω ) , reflected– R m (ω ) , and transmitted– T m (ω ) signals of the metal plate are measured. Using the diagram in Fig. 6.3 and (6.1), we can write Γ m (ω ) H r (ω ) = R m (ω ) − EDf (ω ) EDr (ω ) − Ed (ω ) , (6.2) Z m (ω ) H t (ω ) = T m (ω ) − EDf (ω ) , (6.3) where Γ m (ω ) , Z m (ω ) , H r (ω ) , and H t (ω ) are the theoretical amplitudes of reflection and transmission coefficients for the metal plate, and transfer functions for reflected and transmitted signals of the system, respectively. For each frequency of interest, we set Γ m (ω ) = 1 and Z m (ω ) = 0 . Accordingly, we obtain H r (ω ) = R m (ω ) − EDf (ω ) EDr (ω ) − Ed (ω ) , (6.4) T m (ω ) = EDf (ω ) . (6.5) Then, we removed the metal plate between antennas and measured amplitude of the incident signal in free–space, I f (ω ) . Since the level of the incident signal changed because of the mismatch produced by removing the plate, its level is firstly returned to I m (ω ) using a rotary attenuator. Afterwards, we measured reflected– R f (ω ) and transmitted– T f (ω ) signals in free–space. Using the diagram in Fig. 6.3 and (6.1), we can find Γ f (ω ) H r (ω ) = R f (ω ) − EDr (ω ) T f (ω ) − Ed (ω ) , (6.6) Z f (ω ) H t (ω ) = T f (ω ) − EDf (ω ) , (6.7) 114 where Γ f (ω ) and Z f (ω ) are the theoretical amplitudes of reflection and transmission coefficients for free–space, respectively. For each frequency of interest, we set Γ f (ω ) = 0 and Z f (ω ) = 1 . Consequently, we find EDr (ω ) = R f (ω ) − Ed (ω ) , (6.8) H t (ω ) = T f (ω ) − EDf (ω ) . (6.9) T f (ω ) To obtain the calibrated r and t , we have to determine the transfer functions of the system, H r (ω ) and H t (ω ) . We already know H t (ω ) from (6.5) and (6.9). However, the determination of H r (ω ) requires another step. To this end, first, we determined Ed (ω ) by measuring the amplitude of reflected signals when a matched waveguide load is connected to the terminal of the waveguide section where the transmit antenna is connected in Fig. 6.3. Next, we calculated EDr (ω ) from (6.8). Then, we determine H r (ω ) from (6.4) using EDf (ω ) , EDr (ω ) , and Ed (ω ) . Finally, we placed measurement cells and measured amplitudes of the incident– I s (ω ) , reflected– R s (ω ) , and transmitted– T s (ω ) signals of them. We also set I s (ω ) equal to I m (ω ) by using the attenuator. Since measured signals from a square–law detector are proportional to square of the reflection (transmission) coefficient, α r 2 ( α 2 t 2 ) where α ( α 2 ) is a constant of proportionally, the calibrated r and t are obtained by substituting H r (ω ) and H t (ω ) as 115 12 R s (ω ) − EDf (ω ) EDr (ω ) + Ed (ω ) r (ω ) = m R (ω ) − EDf (ω ) EDr (ω ) + Ed (ω ) , (6.10) 12 T s (ω ) − EDf (ω ) t (ω ) = f T (ω ) − EDf (ω ) . (6.11) Because we divide amplitudes of measured signals in (6.10) and (6.11), the constant proportionalities will cancel each other. This means that we do not need to determine them. Since we leveled the source to produce constant incident signals for reflection and transmission measurements, we can assume that E pf (ω ) = 0 and E pr (ω ) = 0 in the signal block diagram in Fig. 6.3. In addition, we also utilized a matched waveguide load connected to the waveguide section to which the receive antenna is connected to realize E pr (ω ) = 0 . Finally, we can presume that E f (ω ) = 1 by using constant incident power for each frequency of interest. 6.3 Algorithm The complex scattering (S–) parameters derived for the five–layer structure in chapter 3 will be utilized in this chapter for ε determination of lossy materials by the µC based free–space measurement system. To apply the derived objective function, f (ξ ) , in (3.50) to the our problem in this chapter, we let λc → ∞ . Our goal is to code a program into the µC which determines ε by using r and t at one frequency. While writing the program into the µC, we paid attention on three important facts. These are: 1) the implementation of the square–root and exponential functions in (3.34) and (3.38) for a wide range of values, 2) increasing the speed of 116 iterations (code optimization), and 3) organizing the variables to occupy less space (memory optimization). It is proven in chapter 3 that for a lossy medium 3, −1 < χ ≤ 0 and −1 < ξ ≤ 0 . However, in some cases, while performing the computations for ε , the square–root in (3.38) may be negative which results in an imaginary χ . This situation adversely affects computations in µCs since they do computations using real numbers. As a result, to resolve this problem, we propose two solutions depending on the applied numerical method. As a first solution, for numerical methods which require two initial guesses for ξ such as bisection and secant methods [75], ξ1 = 0 and ξ 2 = ξ min can be used where ξ min is obtained by iterating ξ 2 = −1 until the first real χ is achieved. As a second solution, for numerical methods which only necessitate one initial guess such as Newton’s method [75], it is observed that ξ3 = 0 initial guess accurately determines the correct ξ . We coded three programs using different one–dimensional root–finding numerical methods (bisection, secant, and Newton’s methods) based on the objective function in (3.50). The simplified algorithm for each program is shown in Fig. 6.4. The programs firstly acquire measurement data from the microwave section, and then assign a suitable domain for the solution using ξ1 = 0 and ξ 2 = ξ min (for bisection and secant methods) or using ξ3 = 0 (for Newton’s method). Next, the tolerance for computations, TOL , and the total number of iterations, N p , for evaluation of ε are set. Each program halts whenever a) N p is reached or b) f (ξ ) ≤ TOL . Finally, ε is computed using (3.51) and (3.52) and the result is shown. 117 Fig. 6.4. The simplified algorithm for each program. To implement the exponentials and square roots in (3.34) and (3.38), Taylor series expansions to the degree N for the exponential and to the second degree for the square root are used. They are N Xn eX = ∑ , n = 0 n! M 1 s2 + X X = ∑ sn = n−1 2 sn−1 n = 0 , (6.12) where X , n , s−1 , N , and M are the value of which the exponential or square root is evaluated, index number, initial guess for the square–root, and total iteration numbers, respectively. Because N , M , N p , and TOL set the resolution or accuracy of computations, we firstly computed the exponential and square–root using (6.12) for 118 different values of N and M . It is observed that N =50 and M =50 granted highly accurate results. Next, for precise computations we set TOL = 10−4 . Then, we assigned N p =50 after some computations using different frequency, f , and sample thickness, L , ( N =50 and M =50). Each program is coded using a private basic compiler (BASCOM) from MCS Electronics. This compiler can perform calculations using a wide range of variable set. The set can change from an 8 bit unsigned variable to a 32 bit signed variable. We obtained HEX files after compiling the programs by this compiler. Then, these files are transferred into the µC using a serial programming mode. It is clear from (3.34)–(3.38) that the determination of ε in the µC is a difficult task. AT89S8252–24PI µC has only 8K Bytes Flash Memory, and we have to allocate this limited memory for variables and calculations. To evaluate the written program in BASCOM, we show the compiler results (the memory footprint of the variables, total ROM memory space allocated, and other specific information such as crystal frequency, ROM start, and RAM start) of the written program which uses Newton’s method for the ε determination in Fig. 6.5. It is assumed that the thicknesses and complex permittivities of left and right dielectric plugs are equal, and λc → ∞ . 119 Fig. 6.5. The memory footprint, total ROM space allocated, and some information related to the program, which uses Newton’s method for ε , written in BASCOM. It is clearly seen from Fig. 6.5 that the total allocated memory space for computations and variables is approximately 8K Bytes, which is the maximum available space in the µC Flash Memory. This clearly shows the difficulty of coding a program into the µC. We observed that the total allocated space for the programs, which use bisection and secant methods, is much less than that for the program which utilizes the Newton’s method. This is because the Newton’s method requires the derivative of the objective function, f (ξ ) , for computations of ε while bisection and secant methods do not [75]. 120 6.4 Results 6.4.1 Validation of Programs and the Microwave Section Before carrying out measurements, we performed some computations for validation of programs coded into the µC. The permittivity results obtained from the programs for a three–layer structure (air–sample–air) are compared with those of the papers [66],[79] using the measured R and T at f =57.5 GHz of a concrete sample with L =30 mm for different times of its aging in [66]. We included the determined ε from [66] as a reference data for comparison. Here, R and T are the dB values of r and t and are expressed as T = −20log t and R = −20log r . Since we use a three–layer structure for validation of the programs, we employ the objective function in (5.5) and (5.6). The results of the comparison are given in Table 6.1. Table 6.1. The comparison of the permittivity results. Measurements Complex permittivity, ε T R [66] [79] µC 73.6 6.44 7.73–j1.28 7.76–j1.27 7.76–j1.27 68.4 6.83 6.98–j1.11 6.97–j1.12 6.97–j1.12 46.5 7.15 6.49–j0.71 6.51–j0.72 6.51–j0.72 28.1 7.19 6.49–j0.42 6.49–j0.42 6.49–j0.42 It is seen from Table 6.1 that results of the µC programs are in good agreement with those in [66],[79]. Because all µC programs compute approximately the same ε 3 , we demonstrated only one result in Table 6.1 as a representative for all programs. To validate computed ε by all µC programs, we firstly substituted them into (5.1) and (5.2) 121 and then computed R and T . Next, we compared the measured R and T in Table 6.1 and computed Rc and Tc by all µC programs. The result is shown in Table 6.2. Table 6.2. The comparison of the measured R and T from [66] and computed Rc and Tc by all µC programs. Measurements Computed Results Comparison (%) T R Tc Rc ∆T ∆R 73.6 6.44 73.547 6.449 0.072 0.149 68.4 6.83 68.382 6.844 0.027 0.207 46.5 7.15 46.086 7.152 0.892 0.030 28.1 7.19 27.710 7.183 1.387 0.101 It is seen from Table 6.2 that the measured R and T are computed Rc and Tc are very good agreement with each other. There are two important results we can conclude from the results in Table 6.2. First, the simplified expressions of S11 (or r ) and S21 (or t ) in (5.4) can be approximated to those in (5.1) and (5.2) provided that the sample under test indicates at least 10 dB attenuation ( T >10 dB). It is seen from Table 6.1 that the sample shows at least 10 dB attenuation during all its aging period (from 1 week to 14 months). Second, the programs written in µC are very accurate (less than 2 % difference between measured and computed values in Table 6.2) and thus can readily be used in industrial−based applications whenever the accuracy and low–cost are key concerns. We discussed the accuracy and drawbacks of the microwave section of the set−up in Fig. 6.1 in chapter 5. We also validated that section in chapter 5 by amplitude–only measurements of reflection and transmission properties of young mortar and hardened mortar and concrete specimens with different mixture contents. 122 6.4.2 Validation of Whole Measurement System We tested the whole measurement system using a fresh cement paste sample with a 0.375 water–to–cement ratio (w/c). Cement paste is comprised of the mixture of the cement powder and water. High–density particle boards are used as container for holding the fresh cement paste sample. The boards have a cross section of 150 mm x 150 mm and a thickness of L2 = L4 ≅ 11 mm. The complex permittivity of the boards is assumed 2.80 − j 0.20 at 10 GHz [126]. The distance between two inner container walls is L3 ≅ 30 mm. The cement paste mixture with different w/c=0.375 is prepared as follows. Firstly, in order to eliminate any possible effect between cement mixtures and wooden containers, a nylon thin film is placed into the container. Then, the prepared cement mixture is vibrated to achieve homogenous internal structure. Next, it is poured into the container, and the container is vibrated so that there will be no gap between the container walls and the mixture. This process is continued until the mixture covers the top of the container. We calibrated the whole set−up in Fig. 6.1 using the calibration procedure in section 6.4 before measurements. Then, amplitude–only measurements of the container with the cement paste sample are conducted at X–band. For example, Fig. 6.6 shows hourly measurement of these amplitudes at 10 GHz during 2–26 hours of curing. 123 Fig. 6.6. Hourly measurement of r and t at 10 GHz for the cement paste sample with w/c=0.375. It is seen from Fig. 6.6 that r ( t ) decreases (increases) with time. In addition, t is much lower than r . It is an expected result since over time free–water molecules inside the sample evaporates and this circumstance allows more microwave signals to pass through the sample. We removed the measurement cell between antennas after each measurement time and than located it back at the middle distance of antennas. It is obvious from Fig. 6.6 that reflection properties have more oscillatory behavior over 2–26 hours of curing than transmission properties. This is because the measurement cell for each measurement may not be located at the same place (theoretically, at the middle distance of antennas). This result is in agreement with the predictions discussed in chapter 5. We observed that the measurements are in good agreement with those in the literature [42],[47],[122]. 124 We determined the ε of the cement paste sample from the µC program, which uses Newton’s method, using the measured r and t in Fig. 6.6. The result of the ε determination is shown in Fig. 6.7. Fig. 6.7. Determined ε of the cement paste sample using the measurements in Fig. 6.6. It is seen from Fig. 6.7 that both real and imaginary parts ( ε ' and ε " ) of the complex permittivity of the cement sample decrease with time. This is completely in agreement with the conclusions made in chapter 5 that ε ' ( ε " ) is dominated by reflection (transmission) measurements. Therefore, it is no surprise that both ε ' and ε " decrease with time since, in Fig. 6.6, reflection properties decrease while transmission properties increase with time. The numerical method used for ε determination of the cement sample was Newton’s method. To compare its speed, we also computed ε by bisection and secant methods. It is noted that Newton’s method computes ε much faster than do secant and 125 bisection methods, since it performs fewer iterations and its convergence to the solution is quadratically [75]. As a measure, programs using Newton’s and secant methods needed approximately 10 and 32 iterations, respectively, while the program using bisection method did not result in a solution before maximum iteration number, N p = 50 . In addition, while simulation times of computations were firstly around 4 seconds for Newton’s method [78], in this study, it is noted that these times are around 1 second. This is because we defined a highly restrictive and well–behaved domain for computations. 126 CHAPTER 7 CONCLUSION AND FUTURE WORK 7.1 Conclusion In this dissertation, we investigated the application of microwave techniques for electrical characterization of materials. In the first part of this dissertation, we derived forward and reverse complex scattering (S−) parameters for a five−layer structure using conventional multiple−reflections models. The derived S−parameter expressions can also represent a four− or three−layer structure if appropriate substitutions for the electrical and thickness of discarded layers are provided. This is important for materials characterization regime since each structure (three−, four− and five−layer) finds applications in different circumstances. Then, we proposed two methods for complex permittivity determination of lossy materials. While the first method utilizes amplitude only reflection and transmission S−parameter measurements at one fixed frequency for complex permittivity determination of lossy materials, the second one uses the same measurements at different frequencies for thickness−independent complex permittivity extraction of lossy materials. For the first method, we derived an objective function, which depends on only one variable, for permittivity determination of lossy materials sandwiched between two low−loss dielectric plugs. We defined the domain for computations for this function and showed that a unique solution exits for given physical, dimensional, and measurement parameters in the domain. Therefore, this function readily permits fast and dynamic 127 computations as well as unique complex permittivity extraction of lossy materials. For the second method, we simulated the change in the complex permittivity of lossy materials for a change in frequency using two approximations (zero−order and higher−order). We derived an objective function, which depends on only one unknown quantity, for complex permittivity computations using the zero−order approximation. This function also provides an initial guess for the higher−order approximations. We expanded the power series of the extended Debye relaxation model to simulate the change in permittivity over frequency for higher−order approximations. For the validation of both methods, we constructed a general purpose rectangular waveguide test cell, and calibrated the cell by two different calibration techniques. While the first technique corrects only the frequency errors (frequency response calibration), the latter performs a full two−port calibration (the Thru−Reflect−Line calibration). It is shown that the extracted complex permittivity of three test liquid samples (methanol, binary mixture of ethyl alcohol and distilled water, and a commercially antifreeze solution) by the proposed method, which uses frequency response calibration measurements, is in good agreement with that determined from the full two−port calibrated measurements. As a result, the proposed methods are good candidates for industrial−based applications since both of them use one−variable objective functions for fast and dynamic computations, and a very simple calibration which can easily be implemented by inexpensive microwave instruments such as a scalar network analyzer. In the second part of the dissertation, a waveguide sample holder for automated complex permittivity and complex permeability determination of granular and liquid materials, which are positioned into two low−loss dielectric plugs, is presented. There are 128 three key features of the holder: a) it takes into account the effect of low−loss dielectric plugs on measurements; b) it eliminates the dependency of the calibration plane on measurements so that there is no need to fully fill the calibration plane with the unknown sample; and c) it automatically measures the thickness of liquid samples so that a prior physical measurement or knowledge of the sample thickness is not necessary. For validation of the sample holder, we measured the complex permittivity of three test liquids (methanol, distilled water, and a commercially available antifreeze solution) and observed that the proposed sample holder is superior to the available sample holders reported in the literature. In the third section of the dissertation, reflection and transmission properties of cement−based materials measured at X−band (8.2−12.4 GHz) by a free−space microwave set−up are illustrated. Firstly, a simple and relatively inexpensive free−space set−up is described, and a simple calibration procedure before measurements is presented. Then, we demonstrated the reflection and transmission properties of young mortar specimens with different water−to−cement (w/c) ratios over 20−54 hours of curing after sample preparation. We introduced a new approach (relative slope approach for reflection properties) to predict the history of the hydration (a chemical reaction between water molecules and cement powder) process of these specimens. The main results concluded from the measurements are as follows: a) the mortar with a lower w/c ratio will gain hydration (and strength) quicker than the one with a higher w/c ratio (a non–linear relationship exits between the degree of hydration and w/c ratios at early ages of curing), and b) early age transmission measurements can be used to monitor the porosity level inside cement samples, and this monitoring can be very useful for the 129 quality enhancement of these samples. Next, we showed the reflection and transmission properties of hardened mortar and concrete specimens with different w/c ratios over 3−36 months of curing after sample preparation. We introduced a new approach (relative difference approach for transmission measurements) to monitor the aging inside these specimens. The main results of the measurements are: a) concrete specimens will complete the aging process sooner than mortar specimens with the same w/c ratio as a result of heavy aggregates; b) the concrete specimen with a higher w/c ratio will exhibit faster aging during approximately 3−6.1 months; and c) mortar specimen with a lower w/c ratio will display faster ageing during approximately 6.1−30 months. Measurement uncertainties of the simple measurement set−up are also discussed. The final portion of the dissertation presents a simple and relatively inexpensive microwave sensor for complex permittivity determination of lossy materials in industrial−based applications. We firstly presented the electronic and microwave sections of the sensor. Then, we introduced a newly developed calibration procedure, which is similar to the well−known Thru−Reflect−Line calibration technique, to improve the accuracy of measurements. This calibration corrects the diffraction effect from sample edges, which is very important for free−space measurements, as well as undesired reflections from the surrounding environment, multi−path reflections from the ground, and the loss in the measurement system. Next, we coded three programs into a general purpose microcontroller for complex permittivity determination and validated them by the measurement data taken from the literature. Finally, we measured the reflection and transmission properties of a fresh cement paste sample, which is poured into a wooden container, at X−band for validation of the sensor. Because the sensor is noncontacting, 130 nondestructive, noninvasive, and uses amplitudes of reflection and transmission properties, it will find many applications for electrical and physical characterization of materials in industry. 7.2 Future Work The waveguide sample holder discussed in chapter 4 is based on reflection and transmission complex S−parameter measurements. In some applications such as the measurements of low−loss granular and liquid materials, reflection measurements cannot accurately measure the electrical properties of specimens. In this circumstance, a sample holder based on solely transmission complex S−parameter measurements can be developed. In addition, another remedy for the same problem is to develop a waveguide sample holder for measurements of low-loss specimens backed by a metal conductor. Both of these sample holders are currently under investigation. The prediction and estimation of compressive strength of cement−based materials are the key research fields in construction industry. It was shown that amplitudes of the microwave reflection properties over time can be correlated to the compressive strength of these materials. Although this correlation is verified by measurements, amplitudes of reflection properties of two cement−based materials with close w/c ratios will result in nearly the sample reflection properties. This is because the resolution of reflection measurements from highly reflective materials such as fresh cement−based materials decreases. As a result, the performance of estimating the accurate correlation decreases. Therefore, there is a need for accurate compressive strength estimation of these materials in the construction industry. In chapter 5, we generally focused on reflection and 131 transmission properties of young mortar specimens over their early curing periods and hardened mortar and concrete specimens during their late curing periods. It was observed that the temporal dependencies of reflection and transmission properties of young specimens intersect each other at approximately the 5th day of curing. This intersection can be used as a metric for compressive strength development in cement−based materials. 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