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Dielectric Constant Measurements Using an Open Ended Coaxial Probe and a Vector Network Analyzer Edición Única

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(1)INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY División de Electrónica, Computación, Información, y Comunicaciones. Dielectric Constant Measurements Using an Open-Ended Coaxial Probe and a Vector Network Analyzer. Presentada como requisito parcial para obtener el grado de. Maestrı́a en Ciencias en Ingenierı́a Electrónica con Especialidad en Sistemas Electrónicos. Por. Ing. José Luis Montes Marrero Monterrey, N.L., Agosto de 2007. i.

(2) Dielectric Constant Measurements Using an Open-Ended Coaxial Probe and a Vector Network Analyzer. por. Ing. José Luis Montes Marrero. Tesis Presentada al Programa de Graduados de la Escuela de Tecnologı́as de Información y Electrónica como requisito parcial para obtener el grado académico de. Maestro en Ciencias especialidad en. Sistemas Electrónicos.. Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey Agosto de 2007.

(3) Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey Escuela de Tecnologı́as de Información y Electrónica Programa de Graduados. Los miembros del comité de tesis recomendamos que la presente tesis de José Luis Montes Marrero sea aceptada como requisito parcial para obtener el grado académico de Maestro en Ciencias, especialidad en: Sistemas Electrónicos.. Comité de tesis:. Dr. Frantz Bouchereau Lara Asesor de la tesis. Dr. Sergio Omar Martı́nez Chapa. Dr. Graciano Dieck Assad. Sinodal. Sinodal. Dr. Graciano Dieck Assad Director del Programa de Graduados. Agosto de 2007.

(4) Dedico esta tesis a mis padres, José Luis y Susana, quienes incondicionalmente me han apoyado a lo largo de mi vida. Gracias por brindarme su amor y su esfuerzo. A mis hermanas, Susana y Ana Lucı́a, con quienes he compartido gratos momentos de mi vida. A Behtzua, por ser mi compañera en esa búsqueda incejable de la verdad y la belleza..

(5) Reconocimientos. A mi asesor y maestro, Dr. Frantz Bouchereau Lara, por su orientación y apoyo para la realización de este trabajo. Persona muy preparada e investigador incansable. A mi sinodal y maestro, Dr. Graciano Dieck Assad, por su guı́a y apoyo académicos. A mi sinodal, maestro y director de departamento, Dr. Sergio Omar Martı́nez Chapa, por proporcionarme la oportunidad de pertenecer a la cátedra de investigación de BioMEMS, donde me fue posible contribuir al desarrollo de ciencia y tecnologı́a en nuestro paı́s, que tanto lo necesita. A mis compañeros de departamento, Marco Guevara, Carlos Dı́az, Luis Saracho, Luis Peraza, por representar una compañı́a reconfortante durante las largas jornadas de trabajo. A mis compañeros de casa, Igmar y Enrique, por ser buenos amigos y darme su apoyo moral. A la Unión Ciclista Borregos Tec, donde he hecho muy buenas amistades y porque me permitió mantener contacto con la naturaleza y el deporte.. José Luis Montes Marrero Instituto Tecnológico y de Estudios Superiores de Monterrey Agosto 2007. vi.

(6) Dielectric Constant Measurements Using an Open-Ended Coaxial Probe and a Vector Network Analyzer. José Luis Montes Marrero, M.C. Instituto Tecnológico y de Estudios Superiores de Monterrey, 2007. Thesis advisor: Dr. Frantz Bouchereau Lara. Nowadays, acute leukemia patients who have finished a chemotherapy treatment have to remain under observation, being subjects of periodic bone-marrow extractions in order to know if re-incidence of the disease exists. This procedure is harmful, risky and expensive for patients, and it provokes big expenses in money and resources for health institutes. This work tries to simplify the leukemia-reincidence diagnostic reducing the time it requires and introducing the possibility of performing it without the need of an invasive biopsy, based on the fact that in presence of cancer, human tissues show an increase on the dielectric constant parameter. For this, a method that allows for measuring such electric parameter and that has already been studied for breast cancer detection was implemented, which makes use of an open-ended coaxial probe and a Vector Network Analyzer (VNA). In this scheme, the probe is immersed in a bone marrow sample for obtaining reflection measurements of electromagnetic fields emitted by the analyzer. In order to estimate electrical properties of the tissue under observation, reflection measurements are processed by means of mathematical algorithms and electromagnetic simulations which were implemented within this work. In-vivo leukemia-reincidence detection will be possible by the use of micro-machined coaxial probes manufactured with emerging nanotechnologies. Non-invasive methods will be possible by modeling bone, muscle, fat and skin tissues as a whole, and by uncoupling their electromagnetic response from obtained measurements of the coaxial-probe in contact with the skin of a patient’s extremity..

(7) Contents. Reconocimientos. vi. Abstract. vii. List of Tables. x. List of Figures. xi. Chapter 1 Introduction. 1.1 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Theoretical Background. 2.1 Electrical Properties of Biological Materials . . . . . . . . . . 2.1.1 Debye Parameters. . . . . . . . . . . . . . . . . . . . . 2.2 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Plane Waves. . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Impedance Types. . . . . . . . . . . . . . . . . . . . . 2.2.3 Wave Impedance for the TEM-Mode. . . . . . . . . . 2.2.4 A note about Wave Velocities and Dispersion. . . . . . 2.2.5 Wave Propagation Inside Coaxial Lines. . . . . . . . . 2.3 S-Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite-Differences Time-Domain Method for Electromagnetics 2.4.1 General Description. . . . . . . . . . . . . . . . . . . . 2.4.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . 2.4.3 2D BOR FDTD. . . . . . . . . . . . . . . . . . . . . . 2.4.4 Numerical Stability. . . . . . . . . . . . . . . . . . . . 2.4.5 Waveguide Source Conditions. . . . . . . . . . . . . . 2.4.6 Excitation: The gated Gaussian pulse. . . . . . . . . . 2.4.7 Analytical Absorbing Boundary Conditions. . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. Chapter 3 Procedure for Measuring Electric Properties of Materials Using Precision Open-Ended Coaxial Probes and a Vector Network Analyzer. 3.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Coaxial Probe’s Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Time Gating and Probe’s Length. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 De-embedding Model for the Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii. 1 1 2 2 3 3 4 5 5 6 7 7 8 10 14 14 15 20 22 23 25 27. 30 30 30 31 32 33.

(8) . . . . . . . . . . . . . . .. 33 34 36 36 37 39 40 40 42 45 46 47 49 49 50. Chapter 4 Simulation Results. 4.1 FDTD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 S -parameters and RFM coefficients obtention . . . . . . . . . . . . . . . . . . . . . . .. 52 52 57. Chapter 5 Conclusions. 5.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 65 66. Bibliography. 67. 3.3. 3.4. 3.5 3.6 3.7. 3.2.1 General Description of the Model. . . . . . . . . . . 3.2.2 Obtention of the Model S -Parameters. . . . . . . . . RFM of the Probe . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 RFM Definition. . . . . . . . . . . . . . . . . . . . . 3.3.2 Obtention of the RFM Coefficients. . . . . . . . . . . 3.3.3 RFM Inversion . . . . . . . . . . . . . . . . . . . . . FDTD Simulations . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 General Description of the Simulation Scheme. . . . 3.4.2 Specifications. . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Obtention of the Reflection Coefficient Using FDTD. 3.4.4 Obtention of the Aperture Admittance. . . . . . . . 3.4.5 Implementation Issues. . . . . . . . . . . . . . . . . . VNA Calibration . . . . . . . . . . . . . . . . . . . . . . . . Summarized Pre-Measurement Steps . . . . . . . . . . . . . Summarized Measurement Protocol . . . . . . . . . . . . . .. ix. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ..

(9) List of Tables. 2.1. Real part of the complex-dielectric constant for some healthy bone marrow samples at different frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 3.1 3.2. Best fit αnp coefficients for Y M odel in (3.4) . . . . . . . . . . . . . . . . . . . . . . . . Best fit βmq coefficients for Y M odel in (3.4) . . . . . . . . . . . . . . . . . . . . . . . .. 37 37. 4.1 4.2. Debye parameters used in simulations, obtained from [5]. . . . . . . . . . . . . . . . . Convergence test for merit function (3.3) used along with Matlab’s fmincon tool. . .. 54 59. x.

(10) List of Figures. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8. 3.1. 3.2 3.3. 3.4. 3.5 3.6. 3.7. 3.8. 3.9. Phase fonts of (a) TEM and (b) plane waves. . . . . . . . . . . . . . . . . . . . . . . . Coaxial geometry showing Cartesian and polar coordinates. . . . . . . . . . . . . . . . Field lines for the (a) TEM and (b) TE11 modes of a coaxial line. . . . . . . . . . . . . Arbitrary N -port microwave network. . . . . . . . . . . . . . . . . . . . . . . . . . . . N -port network - shifting of the reference plane. . . . . . . . . . . . . . . . . . . . . . Yee cell geometry for Cartesian coordinates in three dimensions. . . . . . . . . . . . . (a) Yee cell geometry for use in FDTD-method with 3D-cylindrical coordinates. (b) 3D-Cell projection into the plane r-z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian pulse described by (2.28) for a simulation with ∆t = 0.067 ps, ∆x = 28.75µm, and ρ = 5. Amplitude at truncation level is equal to 1.375×10−11 . . . . . . . . . . . . Overview of the dielectric characterization method using a precision open-ended coaxial probe. Reflection coefficient data referenced to the connector/calibration plane is acquired by the VNA for later de-embeding it to the aperture plane. . . . . . . . . . . Schematic multisection construction of the probe . . . . . . . . . . . . . . . . . . . . . Sketch of Okoniewski’s [7] VNA screen showing the gated time-domain trace of the reflection coefficient measured with the precision probe immersed in water. Arrows mark the small reflections arising from the discontinuities at the airline section, while the flags on the trace mark the center and the span of the time gate used to remove the connector reflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-D BOR FDTD model of the idealized probe used in the derivations of the deembedding model and RFM. The radii of the inner and outer conductors are denoted as a and b2 , respectively. The outer diameter of the probe is denoted as b1 . . . . . . . . . . . . . . Illustration of the simulated plane within the probe, using a Body-of-Revolution approach. Incident wave traveling to the probe’s aperture. (a) Ex component of incident wave. (b) Negligible Ez component of incident wave inside the probe. (c) Hy component of incident wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflected and transmitted waves after the incident wave reached the probe’s aperture. (a) Ex component of reflected wave. (b) Ez component of transmitted wave. (c) Hy component of reflected wave. Transmitted wave also possesses Ex and Hy components, but they are not visible with the used scale. . . . . . . . . . . . . . . . . . . . . . . . . Obtention of the TEM mode radial electric fields for a wave traveling inside the coaxial probe, with the objective of recording these fields and using them as a compact hard source. The simulation is long enough to allow multi-mode fields to decay. . . . . . . . Geometry of the problem described in [24] . . . . . . . . . . . . . . . . . . . . . . . . .. xi. 6 8 10 11 13 18 21 27. 31 32. 33. 39 41. 42. 43. 44 47.

(11) 3.10 Plot for the Hφ component in a simulation of the coaxial probe immersed in water, for a frequency of interest of 2 GHz. The necessary simulation steps (to obtain Γa and Y ) are accomplished before the wave reflected from the radial boundary (upper boundary) reaches the probe again, avoiding data contamination. . . . . . . . . . . . . . . . . . . 3.11 Diagram of the steps involved in a dielectric constant measurement. . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13. 4.14. Complex permittivity for water at 20◦ C for frequencies from 0 to 8 GHz, obtained from its Debye parameters [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex permittivity for methanol at 20◦ C for frequencies from 0 to 8 GHz, obtained from its Debye parameters [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex permittivity for ethanol at 20◦ C for frequencies from 0 to 8 GHz, obtained from its Debye parameters [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into water at 20◦ C, compared to the one computed through FDTD. . . . . . . . . . . Imaginary part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into water at 20◦ C, compared to the one computed through FDTD. . . . . . Relative error of the YModel admittance predicted by the RFM (3.4) vs. YFDTD admittance computed with BOR-FDTD simulations in water (obtained with (4.1)). . . Real part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into methanol at 20◦ C, compared to the one computed through FDTD. . . . . . . . . Imaginary part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into methanol at 20◦ C, compared to the one computed through FDTD. . . . Relative error of the YModel admittance predicted by the RFM (3.4) vs. YFDTD admittance computed with BOR-FDTD simulations in methanol (obtained with (4.1)). Real part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into ethanol at 20◦ C, compared to the one computed through FDTD. . . . . . . . . . Imaginary part of the aperture admittance predicted by the RFM (3.4) for the probe immersed into ethanol at 20◦ C, compared to the one computed through FDTD. . . . . Relative error of the YModel admittance predicted by the RFM (3.4) vs. YFDTD admittance computed with BOR-FDTD simulations in ethanol (Obtained with (4.1)). Coaxial line filled with water and borosilicate glass, used for comparison of FDTDcomputed reflection coefficient and the analytically obtained reflection coefficient for two contiguous infinite transmission lines. . . . . . . . . . . . . . . . . . . . . . . . . . Simulation for further test of the FDTD scheme developed. Gaussian pulse traveling inside the transmission line (only radial electric component is shown). a) Incident Gaussian pulse traveling toward the discontinuity, b) transmitted and reflected waves traveling away from the discontinuity. . . . . . . . . . . . . . . . . . . . . . . . . . . .. xii. 48 51. 53 54 55 56 57 58 60 61 61 62 62 63. 63. 64.

(12) Chapter 1. Introduction.. This work started as part of a project that is looking for alternative methods of leukemia re-incidence detection, looking to alleviate some of the difficulties that nowadays biopsies present. Many medical diagnostic procedures require invasive interventions for extracting tissue samples, causing risks and uncomfortabilities, as well as elevated costs for patients and health institutes. Nowadays, acute-leukemia patients who are subject of chemotherapy treatments need to remain under monitoring for the detection of a potential re-incidence, being forced to abide for periodic bone-marrow extractions, a process that is repeated monthly during the first year, every two months during the second year, and twice during third, fourth and fifth year. The samples obtained this way are analyzed in a microscope in order to detect an increase of immature white cells, which is an indicator of the presence of the illness. Prior work has been performed looking to detect substances whose quantities in blood stream or in bone-marrow could be related with the presence of leukemia. Also, it has been suggested by published data in the literature that a substantial contrast exists between the dielectric properties of normal and cancerous tissues [4][5][23], so it could be possible to use dielectric constant measurements for early detection and diagnostic of cancer [5].. 1.1. Problem Description.. In a common bone marrow biopsy, cells are collected through a hollow needle inserted into the outer edge of a hipbone, or a small cylinder of bone containing marrow is removed with a special needle. The marrow sample is then examined with a microscope for the presence of leukemic cells. This is a painful and uncomfortable procedure for patients, as well as an slow process for doctors. The approach treated in this work takes advantage of the fact that in presence of leukemia, bone marrow’s cells exhibit an increase in capacitance, causing the permit-. 1.

(13) tivity of the medium to increase and its conductivity to decrease significantly [4]. Then it could be possible to make a permittivity test in a patient’s bone marrow sample to determine with a good degree of accuracy if the disease is present or not. For this, a method of dielectric spectroscopy based on the use of an open-ended coaxial probe and a Vector Network Analyzer (VNA) is proposed. This method allows the measurement of an admittance at the probe’s aperture while it is immersed or in contact with a tissue characterized by an unknown complex-permittivity. This permittivity can be subsequently estimated with the use of an appropriate model.. 1.2. Contribution.. The method for measuring dielectric properties in different human tissues using an open-ended coaxial probe and a VNA was fully implemented, aimed for being used along with a VNA that handles a maximum frequency of 8 GHz. Since the existing precision probes are designed for a frequency limit of 20 GHz ([7], [18], [5], [14]), this implementation includes the definition of a longer coaxial probe that is adequate for the smaller frequency limit.. 1.3. Organization.. Chapter 2 consists in a brief theoretic basis that is needed in order to understand the dielectric constant measurement method treated in this work. It basically consists of transmission line theory, since the coaxial probe is a section of one, as well as a description of simulation tools needed to emulate the behavior of waves traveling within this line while immersed in a medium-under-test that causes a discontinuity: these tools are of extreme importance for the calibration process of the system. Chapter 3 describes the complete measurement method and the way it was implemented, along with all the implementation issues that had to be overcome. It also summarizes the steps required for performing a permittivity measurement. Chapter 4 presents some tests and simulations that were performed in order to prove the correct implementation of the method. Chapter 5 includes future work and recommendations, as well as the conclusions of this research.. 2.

(14) Chapter 2. Theoretical Background.. 2.1. Electrical Properties of Biological Materials. In general, the electromagnetic (EM) properties of matter may be described in terms of its permittivity , permeability µ, and conductivity σ. These quantities express the macroscopic properties of a substance without giving any thought to the microscopic (atomic and molecular) structure. For biological materials, the electrical properties are summarized by the following two parameters: 1. Conductivity σ (also called electric loss), which relates the in-phase movement of charges (conduction currents) to the electric field. 2. Permittivity  (or dielectric constant), which relates the out-of-phase movement of charges (displacement currents) to the electric field. The electrical properties of biological materials and the operating frequency determine the EM interaction mechanisms. Biological materials are regarded as lossy dielectrics, which are frequently macroscopically or microscopically heterogeneous [12]. Lossy dielectrics have a complex relative permittivity whose imaginary component is given by:. 00 =. σ , ω0. (2.1). where ω = 2πf is the radian frequency of the applied field. For time harmonic (steadystate sinusoidal time dependence) fields, the loss is taken into account by considering the permittivity  to be complex. It is represented by. ∗ = 0 (0 − j00 ) .. 3. (2.2).

(15) F requency(M Hz) 50 1×102 1×103 3×103 8.5×103 1×104. r (real part) 6.8-7.7 (horse) 6.8-7.7 (horse) 4.3-7.8 (horse) 4.2-5.8 (horse) 4.4-5.4 (horse) 5.8 (human). Table 2.1: Real part of the complex-dielectric constant for some healthy bone marrow samples at different frequencies.. Physically, 0 is a measure of the relative amount of polarization that occurs for a given applied electric field, and 00 is a measure of both the friction associated with changing polarization and drift of conduction charges. The values of both dielectric constant and conductivity vary substantially with frequency. The permittivity of biological tissues depends on the type of tissue (e.g. skin, fat, or muscle), water content, temperature, and frequency. The permittivity and frequency may also determine how far the EM wave penetrates into the body. Table 2.1 shows some representative values for the real part of the dielectric constant measured in bonemarrow samples [9].. 2.1.1. Debye Parameters.. There are simple relations that allow the calculation of the real and imaginary parts of the complex relative premittivity for many materials as a function of frequency provided that the real part of the complex permittivity is known at zero (static) frequency (denoted by s ) and at very large (ideally infinity) frequency (denoted by ∞ ). These relations are obtained from the well-known Debye equation for the complex dielectric constant, which states that. ˆ(ω) = 0 − j00 = ∞ +. s − ∞ , 1 + jωτ0. (2.3). where τ0 is the relaxation time in seconds [17]. This equation can be used to estimate the real and imaginary parts of the complex relative permittivity (complex dielectric constant) for many gases, liquids, and solids. In this work, it is used to obtain the constitutive parameters of the liquids in which the coaxial probe is immersed for calibration purposes. 4.

(16) 2.2. Wave Propagation. For a given electromagnetic boundary-value problem, many field configurations that satisfy the wave equations, Maxwell’s equations, and the boundary conditions usually exist [3]. All these different field configurations (or solutions) are referred to as modes. In transmission lines and waveguides, different types of wave propagation and modes can exist. Transmission lines that consist of two or more conductors may support transverse electromagnetic (TEM) waves, characterized by the lack of longitudinal field components (Ez = Hz = 0, if z is taken as the direction of propagation). Waveguides support transverse electric (TE) and/or transverse magnetic (TM) waves, characterized by the presence of longitudinal magnetic or electric field components, respectively. The propagation mode of interest for this work is the TEM, since it is the only mode present in the coaxial probe if the cutoff frequency for a TE11 mode, the dominant waveguide mode, is never reached. In addition, this type of waves possesses a uniquely defined voltage, current, and characteristic impedance. This is because at microwave frequencies the measurement of voltage or current is difficult (or impossible), unless a clearly defined terminal pair is available. Such a terminal pair may be present in the case of TEM-type lines (such as coaxial cable, microstrip, or stripline), but does not exist for non-TEM lines (such as rectangular, circular, or surface waveguides). [22] 2.2.1. Plane Waves.. In the TEM mode, both E (Electric) and H (magnetic) field intensities are at every point in space contained in a local plane, referred to as equiphase plane, that is independent of time. In general, the orientations of the local planes associated with the TEM wave are different at different points in space. In other words, at point (x1 , y1 , z1 ) all the field components are contained in a plane. At another point (x2 , y2 , z2 ) all field components are again contained in a plane; however, the two planes need not be parallel (Fig. 2.1a). If the space orientation of the planes for a TEM mode is the same (equiphase planes are parallel), as shown in Fig. 2.1b), then the fields form plane waves. In other words, the equiphase surfaces are parallel planar surfaces. If in addition to having planar equiphases the field has equiamplitude planar surfaces (the amplitude is the same over each plane), then it is called a uniform plane wave; that is, the field is not a function of the coordinates that form the equiphase and equiamplitude planes [3].. 5.

(17) Figure 2.1: Phase fonts of (a) TEM and (b) plane waves.. 2.2.2. Impedance Types.. The various types of impedance definitions involved in this work, along with their notations, are: q. • η = µ/: intrinsic impedance of the medium. This impedance is dependent only on the material parameters of the medium, but is equal to the wave impedance for plane waves (µ - permeability of the medium,  - permittivity of the medium). • Zw = Et /Ht = 1/Yw : wave impedance. This impedance is a characteristic of the particular type of wave. TEM, TM, and TE waves each have different wave impedances, which may depend on the type of line or guide, the material, and the operating frequency (E - Electric field, H - Magnetic field ). q. • Z0 = 1/Y0 = L/C: characteristic impedance. Characteristic impedance is the ratio of voltage to current for a traveling wave. Since voltage and current are 6.

(18) uniquely defined for TEM waves, the characteristic impedance of a TEM wave is unique. TE and TM waves, however, do not have a uniquely defined voltage and current, so the characteristic impedance for such waves may be defined in various ways [22].. 2.2.3. Wave Impedance for the TEM-Mode.. The wave impedance of a TEM mode can be found as the ratio of the transverse electric and magnetic fields: Ex Ztem = = Hy. r. µ = η, . the other pair of transverse field components give −Ey Ztem = = Hx. r. µ = η. . This wave impedance is the same as that for q a plane wave in a lossless medium. It is also identical to the intrinsic impedance η = µ/ of the medium. In general, this is true, not only for uniform plane waves but also for plane and TEM waves; however, it is not true for TE or TM modes [3]. 2.2.4. A note about Wave Velocities and Dispersion.. The speed of a TEM plane wave in a medium (the velocity at which a plane wave √ would propagate in that medium, 1/ µ) and the phase velocity (the speed at which a constant phase point travels vp = ω/β) are identical, while for other types of guided wave propagation the phase velocity may be greater or less than the speed of light. If the phase velocity and attenuation of a line or guide are constants that do not change with frequency, then the phase of a signal that contains more than one frequency component will not be distorted. If the phase velocity is different for different frequencies, then the individual frequency components will not maintain their original phase relationships as they propagate down the transmission line or waveguide, and signal distortion will occur. Such an effect is called dispersion, since different phase velocities allow the ”faster” waves to lead in phase relative to the ”slower” waves, and the original phase relationships will gradually be dispersed as the signal propagates down the line. In such a case, there is no single phase velocity that can be attributed to the signal as a 7.

(19) whole. However, if the bandwidth of the signal is relatively small, or if the dispersion is not too severe, a group velocity can be defined in a meaningful way. This velocity then can be used to describe the speed at which the signal propagates [22]. 2.2.5. Wave Propagation Inside Coaxial Lines.. The coaxial line (Fig. 2.2) can support TE and TM waveguide modes in addition to a TEM mode. In practice, these modes are usually cutoff (evanescent), and they only have a reactive effect near discontinuities or sources, where they are excited. It is important then to be aware of the cutoff frequency of the lowest order waveguide-type modes to avoid their propagation. Deleterious effects may otherwise occur, due to the superposition of two or more propagating modes with different propagation constants. Additionally, in order to use an aperture admittance model for our coaxial probes, we need a section of them to show single-mode propagation [24]. Avoiding the propagation of higher order modes sets an upper limit on the size of a coaxial cable; this ultimately limits the power handling capacity of a coaxial line. The TE11 mode is the dominant waveguide mode of the coaxial line, so it is of primary importance.. Figure 2.2: Coaxial geometry showing Cartesian and polar coordinates.. For TE modes, Ez = 0, and Hz satisfies the wave equation:. ∇2 Hz + k 2 Hz = 0.. 8. (2.4).

(20) If Hz (ρ, φ, z) = hz (ρ, φ) e−jβz , (2.4) can be expressed in cylindrical coordinates as δ2 1 δ2 1 δ + 2 2 + kc2 hz (ρ, φ) = 0, + 2 δρ ρ δρ ρ δφ !. (2.5). where kc2 = k 2 − β 2 . The general solution to this equation is given by the product hz (ρ, φ) = (A sin nφ + B cos nφ) (CJn (kc ρ) + DYn (kc ρ)) ,. where Jn (x) and Yn (x) are the Bessel functions of first and second kinds, respectively [2]. The boundary conditions are that. Eφ (ρ, φ) = 0,. ρ = a, b.. (2.6). Finding Eφ from Hz , using the relations between transverse fields and longitudinal components for a circular waveguide [22], we obtain. Eφ =. jωµ (A sin nφ + B cos nφ) (CJn0 (kc ρ) + DYn0 (kc ρ)) e−jβz . kc. (2.7). Applying (2.6) to (2.7) yields two equations: CJn0 (kc a) + DYn0 (kc a) = 0,. CJn0 (kc b) + DYn0 (kc b) = 0.. Since this is a homogeneous set of equations, the only nontrivial C 6= 0, D 6= 0 solution occurs when the determinant is zero. Thus we must have. Jn0 (kc a) Yn0 (kc b) = Jn0 (kc b) Yn0 (kc a) . 9. (2.8).

(21) This is a characteristic (or eigenvalue) equation for kc . The values of kc that satisfy (2.8) then define the TEnm modes of the coaxial line. Equation (2.8) is a trascendental equation, which must be solved numerically for kc . An approximate solution that is often used in practice [22] is. kc =. 2 . a+b. (2.9). Once kc is known, the propagation constant or cutoff frequency can be determined with the equation. fc =. ckc √ . 2π r. (2.10). The coaxial probe used in this work has a cutoff frequency of approximately 28 GHz, as will be shown later. Field lines for the TEM and TE11 modes of the coaxial line are shown in Figure 2.3.. Figure 2.3: Field lines for the (a) TEM and (b) TE11 modes of a coaxial line.. 2.3. S-Parameters.. A representation in accordance with direct measurements that considers incident, reflected and transmitted waves, is given by the scattering matrix. The scattering matrix 10.

(22) Figure 2.4: Arbitrary N -port microwave network.. for an N -port network provides a complete description of the network as seen at its N ports, relating the voltage waves incident on the ports to those reflected from the ports. Consider the N -port network shown in Figure 2.4, where Vn+ is the amplitude of the voltage wave incident on port n, and Vn− is the amplitude of the voltage wave reflected from port n [22]. The scattering matrix, or S matrix, is defined in relation to these incident and reflected voltage waves as       . V1− V2− .. . VN−. . . S11.      S21  =  .   ..  . . . .    ,  . S12 . . . S1N V1+   ..   + V2 .   .     ... SN 1 . . .. SN N. V− = SV+ .. A specific element of the S matrix can be determined as. Sij =. Vi− Vj+. . Vk+ =0. 11. for. k6=j. VN+.

(23) This means that Sij is found by driving port j with an incident wave of voltage Vj+ , and measuring the reflected wave amplitude, Vi− , coming out of port i. The incident waves on all ports except the j th port are set to zero, which means that all ports should be terminated with matched loads to avoid reflections. Thus, Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads (also denoted as Γ in the rest of this work), and Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads. Because the S parameters relate amplitudes (magnitude and phase) of traveling waves incident on and reflected from a microwave network, phase reference planes must be specified for each port of the network. It can be shown how the S parameters are transformed when the reference planes are moved from their original locations. Consider the N-port microwave network shown in Figure 2.5, where the original terminal planes are assumed to be located at zn = 0 for the nth port, and where zn is an arbitrary coordinate measured along the transmission line feeding the nth port. The scattering matrix for the network with this set of terminal planes is denoted by S. Now consider a new set of reference planes defined at zn = ln , for the nth port, and let the new scattering matrix be denoted as S 0 . Then in terms of the incident and reflected port voltages we have that V− =. SV + , (2.11). V. 0−. 0. = SV. 0+. ,. where the unprimed quantities are referenced to the original terminal planes at zn = 0, and the primed quantities are referenced to the new terminal planes at zn = ln . Now from the theory of traveling waves on lossless transmission lines we can relate the new wave amplitudes to the original ones as Vn0+ =. Vn+ ejθn ,. Vn0−. Vn− e−jθn ,. (2.12) =. where θn = βn ln is the electrical length of the outward shift of the reference plane of. 12.

(24) Figure 2.5: N -port network - shifting of the reference plane.. port n. Writing (2.12) in matrix form and substituting into (2.11a) gives . ejθ1. 0 ejθ1.      . ... ejθN. 0. . . e−jθ1. . 0 e−jθ1.   h  i  0−  V = [S]      . ... e−jθN. 0.  h i   V 0+ .  . Multiplying by the inverse of the first matrix on the left gives  h. V 0−. i. e−jθ1. . 0 e−jθ1.    =  . ... . −jθN. 0. . e−jθ1. . 0 e−jθ1.        [S]     . ... e. . e−jθN. 0.  h i   V 0+ .  . Comparing with (2.11b) shows that . e−jθ1. 0 e−jθ1.   [S ] =     0. ... 0. e−jθN. . . e−jθ1 e−jθ1.        [S]     . ... 0. 13. 0. e−jθN.     ,  .

(25) 0 = e−2jθn Snn , meaning that the phase of Snn is which is the desired result. Note that Snn shifted by twice the electrical length of the shift in terminal plane n, because the wave travels twice over this length upon incidence and reflection. Actually, changing the points along the transmission lines at which the S -parameters are measured introduces only phase changes in the parameters [21].. 2.4. 2.4.1. Finite-Differences Time-Domain Method for Electromagnetics General Description.. The Finite-Differences Time-Domain Method for Electromagnetics (FDTD) consists of a numerical solution to Maxwell’s equations, which govern the relations and variations of the electric and magnetic fields, charges and currents associated with electromagnetic waves. It was created by K. Yee in 1966 [27], and presents the following advantages over frequency-domain and finite-element solutions: • FDTD uses no linear algebra. It avoids the difficulties with linear algebra that limit the size of frequency-domain integral-equation and finite-element electromagnetics models. • FDTD is accurate and robust. The sources of error can be bounded according to the electromagnetic wave interaction problem. • FDTD treats impulsive behavior accurately. A single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steady-state response at any frequency within the excitation spectrum. In our application, this is important because we can uniquely define an excitation source that contains the full bandwidth of interest. • FDTD treats nonlinear behavior naturally. • FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than the potentially complex reformulation of an integral equation. This is very useful in this application because it is possible to approximate the exact geometry of the probes with minimal effort.. 14.

(26) • FDTD takes advantage of computing technology. FDTD matrices operations can be easily parallelized and implemented in special hardware, like a video card (as in the case of EMphotonics FASTfdtd). Additionally, memory capacities are increasing rapidly, allowing for more electromagnetic fields to be calculated in a single simulation. The FDTD code developed for this work takes advantage of Matlab’s matrix handling optimality, instead of using iterative solutions like the ones implemented in Fortran and in other high-level programming languages in the past [17].. 2.4.2. Fundamentals.. Maxwell’s equations. The differential form of Maxwell’s equations is the most widely used representation to solve boundary-value electromagnetic problems. It is used to describe and relate the field vectors, current densities, and charge densities at any point in space at any time. For these expressions to be valid, it is assumed that the field vectors are singlevalued, bounded, continuous functions of time and exhibit continuous derivatives [3]. A complete description of the field vectors at any point (including discontinuities) at any time requires not only Maxwell’s equations in differential form but also the associated boundary conditions. Consider a region in space that has no electric or magnetic current sources, but may have materials that absorb electric or magnetic field energy. Also, consider the medium in this region to be isotropic, linear and nondispersive (this is, a material having fieldindependent, direction-independent and frequency-independent electric and magnetic properties [25]), then the time-dependent Maxwell’s equations in differential form are given by:. ∂B + ∇ × E = −M, ∂t. (2.13). ∂D − ∇ × H = −J, ∂t. (2.14). B = µH,. (2.15). D = E,. (2.16). 15.

(27) where all of these field quantities, and also µ and  are assumed to be given functions of space and time. The definitions and units of the quantities are: • E = Electric field intensity (volts/meter), • H = Magnetic field intensity (amperes/meter), • D = Electric flux density (colombs/square meter), • B = Magnetic flux density (webers/square meter), • J = Electric current density (amperes/square meter), • M = Equivalent magnetic current density (volts/square meter), •  = Permittivity of the medium (farads/meter), and • µ = Permeability of the medium (henries/meter), where J and M can act as independent sources of E−field and H−field energy, Jsource and Msource . Allowing for materials with isotropic, nondispersive electric and magnetic losses that attenuate E− and H−fields via conversion to heat energy, we have [25]:. J = Jsource + σE; M = Msource + σ ∗ H,. (2.17). where • σ: electric conductivity (siemens/meter), • σ ∗ : equivalent magnetic loss (ohms/meter). Finally, substituting (2.15), (2.16), and (2.17) into (2.13) and (2.14), and writing out the vector components of the curl operators in Cartesian coordinates, we obtain the following system of six coupled scalar equations:. ". #. 1 ∂Ey ∂Ez ∂Hx = − − (Msourcex + σ ∗ Hx ) , ∂t µ ∂z ∂y. 16. (2.18).

(28) ". #.  1 ∂Ez ∂Ex  ∂Hy = − − Msourcey + σ ∗ Hy , ∂t µ ∂x ∂z. ∂Hz ∂t. ". #. 1 ∂Ex ∂Ey = − − (Msourcez + σ ∗ Hz ) , µ ∂y ∂x. ". (2.20). #. ∂Ex 1 ∂Hz ∂Hy = − − (Jsourcex + σEx ) , ∂t  ∂y ∂z. ". (2.21). #.  ∂Ey 1 ∂Hx ∂Hz  = − − Jsourcey + σEy , ∂t  ∂z ∂x. ∂Ez ∂t. (2.19). ". (2.22). #. 1 ∂Hy ∂Hx = − − (Jsourcez + σEz ) .  ∂x ∂y. (2.23). This system of partial differential equations represents the basis of the FDTD numerical algorithm for electromagnetic wave interactions with general three-dimensional objects. The next section explains a way to solve it. Yee algorithm. The Yee algorithm solves for electric and magnetic fields in time and space using the coupled Maxwell’s curl equations (system (2.18) to (2.23) for the 3D - Cartesian coordinates case). To accomplish this, the algorithm uses a cubic cell, in which every E component is surrounded by four circulating H components, and every H component is surrounded by four circulating E components. The cell is shown in Figure 2.6, where the notation for specifying a space point is: (i, j, k) = (i∆x, j∆y, k∆z).. Here, ∆x, ∆y, and ∆z are, respectively, the lattice space increments in the x, y, and z coordinate directions, and i, j, and k are integers. A function u of space and time. 17.

(29) Figure 2.6: Yee cell geometry for Cartesian coordinates in three dimensions.. evaluated at a discrete point in the grid and at a discrete point in time is denoted as u(i∆x, j∆y, k∆z, n∆t) = uni,j,k ,. where ∆t is the time increment, assumed uniform over the observation interval, and n is an integer. In order to solve the space and time partial derivatives in (2.18) to (2.23), Yee used centered finite-difference expressions. For example, the following is the central-difference expression for the first partial space derivative of u in the x-direction, evaluated at the fixed time tn = n∆t: h i uni+1/2,j,k − uni−1/2,j,k ∂u (i∆x, j∆y, k∆z, n∆t) = + O (∆x)2 , ∂x ∆x. h. i. where O (∆x)2 represents second order accuracy [25]. The first time partial derivative. 18.

(30) of u, evaluated at the fixed space point (i, j, k) yields: n+1/2. n−1/2. u − ui,j,k ∂u (i∆x, j∆y, k∆z, n∆t) = i,j,k ∂t ∆t. h. i. + O (∆t)2 .. For both expressions, Yee chose the i + 1/2, i − 1/2, n + 1/2, n − 1/2 notation in order to interleave the E and H components in the space lattice at intervals of ∆x/2, ∆y/2, and ∆z/2, and at intervals of ∆t/2 in time, for purposes of implementing a leapfrog algorithm [17]. Now the system (2.18) to (2.23) can be solved applying the ideas above. Consider for example expression (2.21), after applying a substitution of central differences for the time and space derivatives, at Ex (i, j + 1/2, k + 1/2, n):. n+1/2. n−1/2. Ex |i,j+1/2,k+1/2 − Ex |i,j+1/2,k+1/2 ∆t 1 i,j+1/2,k+1/2. =. Hz |ni,j+1,k+1/2 − Hz |ni,j,k+1/2 Hy |ni,j+1/2,k+1 − Hy |ni,j+1/2,k · − ∆y ∆z !. −. Jsourcex |ni,j+1/2,k+1/2. −. σi,j+1/2,k+1/2 Ex |ni,j+1/2,k+1/2. .. (2.24). Using simple algebra, we can obtain the updating equation for the Ex component in 3D and for Cartesian coordinates, yielding:. . 1− . n+1/2. Ex |i,j+1/2,k+1/2 =  . + . ∆t. 1. 1+. σi,j+1/2,k+1/2 ∆t 2i,j+1/2,k+1/2  n−1/2 σi,j+1/2,k+1/2 ∆t  Ex |i,j+1/2,k+1/2 2i,j+1/2,k+1/2. . . i,j+1/2,k+1/2  σi,j+1/2,k+1/2 ∆t  + 2i,j+1/2,k+1/2. Hz |ni,j+1,k+1/2 − Hz |ni,j,k+1/2 · ∆y. Hy |ni,j+1/2,k+1 − Hy |ni,j+1/2,k − Jsourcex |ni,j+1/2,k+1/2 , − ∆z !. 19. (2.25).

(31) using what is called a semi-implicit approximation: n+1/2. Ex |ni,j+1/2,k+1/2. =. n−1/2. Ex |i,j+1/2,k+1/2 + Ex |i,j+1/2,k+1/2 2. .. The rest of the updating equations can be obtained in a manner similar to (2.25), maintaining the same second-order accuracy of the method. The complete set of equations can be found in [17]. 2.4.3. 2D BOR FDTD.. It is possible to take advantage of the rotational symmetry shown by some structures in order to obtain an FDTD algorithm that is more efficient, memory-saving, and that allows the definition of a circular pattern without the need of a staircase type profile. This is the case for some types of antennas and transmission lines, as well as for the coaxial probes considered in this work. For developing a 2D-Body-Of-Revolution FDTD code, an adequate Yee cell has to be defined. Consider Figure 2.7, where cylindric coordinates are used instead of the Cartesian ones. A method described in [25] and in [26] uses Fourier series of sines and cosines in order to introduce the analytical azimuthal field variation:. E =. ∞ X. (eu cos mφ + ev sin mφ) ,. m=0. H =. ∞ X. (hu cos mφ + hv sin mφ) ,. m=0. where m is the mode number, u denotes the Fourier coefficients for the cosinusoidal dependence, and v denotes the coefficients for the sinusoidal dependence. In this expansion, E and H are dependent on r, φ, z, and t, while eu , ev , hu , and hv are dependent on r, z, and t. The dependence of the Fourier coefficients on m is not explicitly expressed, since the algorithm is designed to provide a solution one mode at a time. The updating equations for fields involved in TEM-mode propagation without azimuthal variation. 20.

(32) are:. Ern+1 (i, j) −. 1. =. 1. 1+. σr ∆t 20 r σr ∆t 20 r. ∆t 0 r  σr ∆t + 2 0 r. Ezn+1 (i, j) = +. 1−. 1− 1+. =. ·. n+1/2. Hφ. σz ∆t 20 z σz ∆t 20 z. m∆t 0 z  σz ∆t + 2 0 z. n+1/2 (i, j) Hφ. . Ern (i, j) n+1/2. (i, j) − Hφ ∆z. (i, j − 1).  − . 1. m∆t 0 r  σr ∆t + 2 0 r. Hzn+1/2 (i, j) , ri+1/2. Ezn (i, j) . ∆t. n+1/2. ri+1/2 Hφ Hrn+1/2 (i, j) +  0 σz ∆t  ·  ri 1 + 2z0 z. n−1/2 (i, j) Hφ. n+1/2. (i, j) − ri−1/2 Hφ ri ∆r. (i − 1, j). ∆t Ern (i, j) − Ern (i, j) Ezn (i + 1, j) − Ezn (i, j) − , · − µ0 µ φ ∆z ∆r #. ". where ri = (i − 1/2)∆r and r1/2 = r0 = 0 and the fields associated with coordinate (i,j) are enclosed within the box shown in Figure 2.7b [26].. Figure 2.7: (a) Yee cell geometry for use in FDTD-method with 3D-cylindrical coordinates. (b) 3D-Cell projection into the plane r-z.. 21.  ,.

(33) 2.4.4. Numerical Stability.. Cell size. A fundamental constraint in the FDTD method is that the cell size must be much smaller than the smallest wavelength, in order to obtain accurate results. Since FDTD is a volumetric computational method, if some portion of the computational space is filled with penetrable material, the wavelength in the material must be used to determine the maximum cell size. To understand the reason why the cell size must be much smaller than a wavelength, consider that at any particular time step the FDTD grid is a discrete spatial sample of the field distribution. From the Nyquist sampling theorem, there must be at least two samples per spatial period (wavelength) in order for the spatial information to be adequately sampled. However, since the sampling is not exact, and because the smallest wavelength is not precisely determined, more than two samples per wavelength are required. Another related consideration is grid dispersion error. Due to the approximations inherent in FDTD, waves of different frequencies will propagate at slightly different speeds through the grid. This difference in propagation speed also depends on the direction of propagation relative to the grid. For accurate and stable results, the grid dispersion error must be reduced to an acceptable level, which can be readily accomplished by reducing the cell size [17]. Finally, for the important characteristics of the problem geometry to be accurately modeled, an even smaller cell size may be required. This is the case for accurately modeling the fringing fields rising at the aperture of the coaxial probes considered in this work. Time step size. The size of the time step has to be chosen according to the Courant condition [17]. To understand the basis for this, consider a plane wave propagating through an FDTD grid. In one time step any point on this wave must not pass through more than one cell, because during one time step FDTD can propagate the wave only from one cell to its nearest neighbors. To determine this time step constraint, a plane wave direction is picked so that it propagates most rapidly between field point locations. This direction will be perpendicular to the lattice planes of the FDTD grid. For a grid of dimension d (where d=1,2, or 3), with all cell sides equal to ∆u, it can be shown that with v the. 22.

(34) maximum velocity of propagation in any medium in the problem, ∆u v∆t ≤ √ d. (2.26). for stability. However, exceptions to this occur, for example for conducting materials (σ > 0), which require smaller time steps than the Courant limit in order to achieve stable calculations. This is usually not a problem, because in most calculations the time step size is set by the speed of light in free space. However, the short wavelength inside highly conducting materials may require much smaller FDTD cells than in surrounding free space/dielectric-filled regions [17]. For a 2D-BOR-FDTD simulation, the numerical stability bound for the time-step used can be empirically represented as. ∆t ≤. ∆x , sv. (2.27). √ where s ≈ m + 1 for m > 0, and s = 2 for m = 0. It is also important to consider that the stability of the algorithm is very sensitive to the way the field components near the axis are computed [25]. 2.4.5. Waveguide Source Conditions.. There are some special considerations about sourcing an incident numerical wave in an FDTD model of a dielectric waveguide or transmission line. • The waveguide or transmission line usually supports a number of distinct propagating modes that have substantially different spatial distributions of E and H. It may be difficult to simulate the numerical excitation of one particular mode without inadvertently exciting some or all of the rest. The coaxial line can also support TE and TM waveguide modes in addition to a TEM mode, so it is important to be aware of the cutoff frequency of the lowest order waveguide-type modes, in order to avoid their propagation. [22]. • Waveguides/transmission lines exhibiting cutoff phenomena can be subject to reactive fields loitering in the vicinity of a source and of a discontinuity. The distance between a numerical source and the interaction structure of interest within the. 23.

(35) waveguide must be selected to allow substantially complete decay of these reactive fields. • The use of a wide-band pulsed source in a waveguide/transmission line exhibiting cutoff phenomena may require substantially prolonging the time-stepping of the FDTD simulation to permit the very slowly propagating spectral energy just above the waveguide cutoff frequency to reach the interaction structure of interest, if these spectral components are of interest in the simulation. A simple way to excite an incident wave in an FDTD model of a waveguide or transmission line is to specify a pulsed E hard-source distribution in a transverse cross section. That is, some or all of the E components located in and tangential to the transverse source plane are provided with a space-time variation that coincides with that of the desired propagating mode, and are independent of the presence of any other numerical waves in the grid. In certain cases, the complete transverse field distribution of the desired propagating mode is known analytically, allowing the pulsed numerical wave that is launched to immediately represent the desired mode without unduly generating undesired propagating modes and below-cutoff reactive fields. In other cases, the transverse distribution of the desired mode is initially unknown or can only be approximated. There are two options in this situation: 1. Implement an approximation of the true mode. The impact of specifying a subset of the complete transverse distribution is to possibly generate undesired numerical waves that represent multimoding and reactive fields. A substantial buffer length of waveguiding structure may then be required in the model to permit the reactive fields to decay. 2. Use ”bootstrapping.” This means running a preliminary FDTD model of the waveguide that is sufficiently electrically long to decay all undesired reactive and multimode fields. The transverse field distribution at the far end of this preliminary model is stored in a data file and is subsequently used in all production runs as a compact hard source. Effectively, the preliminary FDTD run is used to solve for the correct transverse field distribution. The principal concern with hard-source conditions is that its effective source impedance is zero. As a result, numerical waves properly reflected from a simulated discontinuity further down the waveguide will nonphysically retroreflect from the hard source if it is still operating. This nonphysical retroreflection is total or partial, depending on 24.

(36) whether the hard source completely or partially specifies the entire transverse source plane. A good way to eliminate this problem is to turn off the hard source before any reflecting numerical waves reach its position. After a pulsed source waveform is smoothly decayed to zero, the source plane is removed and replaced by standard Yee cells by the time the reflecting numerical waves reach its original position. The reflecting numerical waves can then freely propagate through what was the source plane to reach an ABC (Absorbing Boundary Condition) representing the extension of the waveguiding system to infinity. This procedure requires elongation of the simulated waveguiding system between the source plane and the first reflective discontinuity to permit the complete evolution and return to zero of the pulsed source waveform before arrival of the initial reflection at the source plane. A second problem occurs when the bandwidth of the pulsed source is so wide that there is significant spectral energy below cutoff of the waveguiding system. The nonpropagating reactive fields that are generated by the source never leave its vicinity. Upon turning off the hard source and replacing it with normal Yee updating, a spurious transient is generated in the fields loitering about the source. This transient can have sufficient spectral content above cutoff to propagate down the wave guide an indefinite distance, thereby contaminating field data along the entire guide. To avoid this problem, the bandwidth of a hard source that is turned off should be carefully controlled to minimize spectral content below cutoff [25]. 2.4.6. Excitation: The gated Gaussian pulse.. In this work we have considered a gated Gaussian pulse because it provides a smooth roll off in frequency content, it is simple to implement, and also because it is necessary that the exciting field vanishes before reflections from the end of the line return to the excitation plane [18]. This excitation is expressed by the discrete Gaussian function, which is controlled by two numerical constants, α and b, where b denotes the half pulse width (Fig. 2.8). It has the form:. 2. g(n) = e−α(n∆t−n0 ∆t) , 2 (n−n )2 0. g(n) = e−α∆t. 25. .. (2.28) (2.29).

(37) 2. b is thus defined as b = |n − n0 |. Set truncation level at e−ρ , so that. 2 b2. e−α∆t. 2. = e−ρ , 2. ⇒ α∆t = A =. (2.30)  2. ρ b. , 2. ⇒ g(n) = e−A(n−n0 ) .. Here, the truncation level should be comparable with the precision of numbers. For single precision (6 significant digits), recommended value is ρ = 4, what corresponds to truncation at e−16 . For double precision (12 significant digits), ρ = 5, which is the value we used. The pulse width in terms of b is determined according to the desired width of the excitation frequency spectrum. The Fourier transform of the Gaussian pulse is a Gaussian function of frequency: 2 2 G(f ) ∼ = e−π f /α .. We require that the spectral value at the highest frequency of interest is at 0.3 of the maximum. This is:. 2. e−πfmax /α = 0.3,. −. 2 π 2 fmax α. = ln(0.3),. ⇒ A=. 2 π 2 fmax ∆t2 ρ = ln(10/3) b.  2. q. ⇒ b=. ρ ln(10/3) πfmax ∆t. 26. .. ,.

(38) Figure 2.8: Gaussian pulse described by (2.28) for a simulation with ∆t = 0.067 ps, ∆x = 28.75µm, and ρ = 5. Amplitude at truncation level is equal to 1.375×10−11 .. 2.4.7. Analytical Absorbing Boundary Conditions.. A basic consideration with FDTD modeling of electromagnetic wave interaction problems is that many are defined in ”open” regions where the spatial domain of the computed field is unbounded in one or more coordinate directions. Clearly, no computer can store an unlimited amount of data, and therefore, the field computation domain must be limited in size. The computation domain must be large enough to enclose the structure of interest, and a suitable boundary condition on the outer perimeter of the domain must be used to simulate its extension to infinity. Absorbing Boundary Conditions (ABCs) cannot be directly obtained from the numerical algorithms for Maxwell’s curl equations. Principally, this is because these systems employ a central spatial-difference scheme that requires knowledge of the field which is one-half space cell to each side of an observation point. Central differences cannot be implemented at the outermost planes of the space lattice, since by definition there is no information concerning the fields at points one-half space cell outside of these planes. Although backward finite differences could conceivably be used here, these are generally of lower accuracy for a given space discretization and have not been used in major FDTD software [25]. We require an ABC to be present at the plane of excitation of the probe, once the. 27.

(39) incident Gaussian pulse has reached negligible values, so that it is possible to observe the complete development of the reflected wave from the discontinuity without getting it distorted by non-physical reflections at boundaries of the problem space. Also, it is desirable to have ABCs at other boundaries of the FDTD mesh for keeping the simulation space small for achieving shorter computation times. MUR ABC. There are many different schemes for accomplishing an absorbing condition. However, rather than provide the theoretical basis of these ABCs, a popular and easily applied ABC is presented. This is commonly called the Mur absorbing boundary condition [20], or more particularly first or second order Mur ABC, depending on the order of the approximation used to estimate the field on the boundary. A first order condition looks back one step in time and into the space one cell location¡ a second order condition looks back two steps in time and inward two cell locations, etc. Mur introduced a simple and successful finite-difference scheme for implementing ABCs. As an example, this scheme is shown for the BOR grid at the z = 0 boundary. Let W |i,0 represent a cylindrical component of E or H located in the Yee grid at z = 0 and tangential to this boundary. Mur implemented the partial derivatives of the Engquist-Majda oneway wave equations [10] as numerical central differences expanded about an auxiliary gridpoint (i, 1/2). For a square grid, ∆r = ∆z = ∆, and the second order Mur ABC at z=0 can be written as. n−1 W |n+1 i,0 = − W |i,1 +. +.  c∆t − ∆  n+1 W |i,1 + W |n−1 i,0 c∆t + ∆.   (c∆t)2 2∆  n W |i,0 + W |ni,1 + W |ni+1,0 − 2 W |ni,0 + W |ni−1,0 c∆t + ∆ 2∆ (c∆t + ∆) . + W |ni+1,1 − 2 W |ni,1 + W |ni−1,1 .. (2.31). For this application, first order Mur ABC was implemented, expressed as. n−1 W |n+1 i,0 = − W |i,1 +.   c∆t − ∆  n+1 2∆  n n−1 W |i,1 + W |i,0 + W |i,0 + W |ni,1 . (2.32) c∆t + ∆ c∆t + ∆. A limitation of this type of ABCs is that they are designed for absorbing only waves that are traveling in a direction that is transverse to the plane of absorption. Other 28.

(40) consideration is that of determining the distance between the object and the outer boundary. The farther from the object the outer boundary is located the better the absorption of the outward traveling waves. This is due to these waves becoming more like plane waves as they travel farther from the structure that radiates them. MUR ABCs can be used in planes z = 0 and z = zmax of the coaxial probe’s mesh with no problem, leaving only the radial boundary at the top of the FDTD mesh without an absorbing condition, causing the waves reaching that boundary to reflect. This reflections should not be a problem as long as the problem space is long enough in the radial direction to avoid contamination of the area of interest. Behavior of the Mur ABCs is shown later.. 29.

(41) Chapter 3. Procedure for Measuring Electric Properties of Materials Using Precision Open-Ended Coaxial Probes and a Vector Network Analyzer.. This chapter consists of an explanation on how we have implemented the method of open-ended coaxial probes-based method for the measurement of complex electric permittivity of materials in the range of 500 MHz - 8 GHz, with considerations we had to make due to the 8 GHz maximum frequency achieved by our available VNA. We begin with a full description of the system including an explanation of the construction of the probes; then we include a description of calibration steps that have to be accomplished before a measurement can be done, as well as the necessary steps for performing a final dielectric constant measurement.. 3.1. System Description. 3.1.1. General Description.. The VNA-probe system is shown in Fig. 3.1. Here, the probe is attached to the VNA and placed in contact with a sample. Two different planes of the probe are appreciated: the connector-plane is the one connected to the VNA through the standard 50 Ω testing cable, while the aperture-plane is the one in contact with the material-undertest (a tissue for example) characterized by an unknown relative complex permittivity. When a measurement is performed, the VNA measures a reflection coefficient Γc that is referenced to the connector plane, so there is a need for post-processing steps which allow for a compensation of the propagation characteristics of the probe, translating (or de-embedding) the results to a reflection coefficient Γa that is referenced to the aperture plane, which is the one of interest. Once this reflection coefficient is obtained, its value can be converted to an aperture admittance and finally to the permittivity of the immersion/contact medium. Thus, two pre-measurement steps are necessary. The first one, the de-embedding step, 30.

(42) Figure 3.1: Overview of the dielectric characterization method using a precision open-ended coaxial probe. Reflection coefficient data referenced to the connector/calibration plane is acquired by the VNA for later de-embeding it to the aperture plane.. consists of determining the parameters that accurately translate Γc to Γa , a process that has to be performed for each probe due to variations in the manufacturing process. The second step consists of finding the parameters that correctly relate the measured reflection coefficient to the aperture admittance of an idealized probe and this in turn to the permittivity of the sample that it is in contact with. This is achieved by means of a Rational Function Model (RFM). Since an idealized probe is assumed, it is necessary to perform this step only once for a given probe construction. 3.1.2. Coaxial Probe’s Characteristics.. The probe’s dimensions, constitutive materials and geometry specifications are a result of several previous studies [5], [18]. Okoniewski et al. [7] designed a stainless-steel borosilicate-glass-filled coaxial probe that is hermetically sealed at the aperture, making it robust for hospital environments. The materials are chemically inert and thermally matched, and it is flange-free to ensure good contact with tissue samples across the whole aperture. The outer conductor walls near the probe aperture are thickened for increased accuracy, and the aperture diameter is set to 3.0 mm to provide optimum sensitivity in both the high- and low-permittivity tissue measurements. It is constructed with several sections of high-quality 50 Ω coaxial lines (Figure 3.2), which greatly reduces the cost. The first section following the connector is a 24.5-cm-long coaxial line filled with low-loss porous silica (r = 1.72). The last section (at the probe’s. 31.

(43) aperture) is a 2.3-mm-long coaxial line filled with a borosilicate glass (r = 3.75), and an air-filled 2.9-mm-long line is used as the transition between the two glass lines, working as an impedance transformer. At the aperture plane, the radii of the inner and outer conductors are 0.3 and 1.5 mm, respectively, while the probes outer radius is 2.1 mm. The borosilicate glass is hermetically sealed to the stainless-steel conductors. The thermal expansion coefficients of the borosilicate and stainless steel are low and matched (3.0 to 3.4 ×10−6 and 5.0 ×10−6 , respectively). The inner conductor of the probe is protruding slightly to achieve more accurate low-permittivity measurements.. 25.05 Figure 3.2: Schematic multisection construction of the probe. 3.1.3. Time Gating and Probe’s Length.. The minimum length of the probe is dictated by the time resolution of the VNA, because of the time/space filter (time gate) that is applied when performing the measurements: time gating allows to remove unwanted reflections at the connector. During this procedure, the measured reflection coefficients are transformed into the time domain, the connector reflection is electronically gated out, and the data is then transformed back into the frequency domain. An sketch of Okoniewski’s VNA screen showing the gated time-domain trace of the reflection coefficient measured with the probe immersed in water is shown in Fig. 3.3. Since our VNA’s maximum frequency is only 8 GHz (the one used by Okoniewski et al. reaches 20 GHz), and because time/distance resolution for a VNA is on the order of 150mm/Frequency Span (GHz), we would sacrifice accuracy by using the same probe’s length (∼ = 10 cm) used in [7], loosing valuable reflection coefficient data during the filtering. Thus we decided to fabricate a probe that is 20/8. 32.

(44) times longer, taking care not to affect the 50 Ω impedance matching between sections. Since the impedance of a transmission line is not dependent on its length, the low-loss porous silica section was the only section elongated, leaving the air section and the borosilicate glass section unmodified.. Figure 3.3: Sketch of Okoniewski’s [7] VNA screen showing the gated time-domain trace of the reflection coefficient measured with the precision probe immersed in water. Arrows mark the small reflections arising from the discontinuities at the airline section, while the flags on the trace mark the center and the span of the time gate used to remove the connector reflections.. 3.2 3.2.1. De-embedding Model for the Probe General Description of the Model.. As explained earlier, the probe’s de-embedding process needs to be applied once for each manufactured probe. Though multisection construction of the probes greatly reduces the cost of manufacturing, it complicates the calibration procedures, as it gives raise to parasitic capacitances and inductances in the transition regions. The design of the air transition is such that the two parasitic effects should cancel out; however, due to variations in the manufacturing process (particularly welding), this cancellation is not perfect [7]. A reflection-coefficient de-embedding model is needed to compensate for the propagation characteristics of the multisection probe. This model permits the extraction of the desired reflection coefficient at the aperture plane (Γa ) (see Fig. 3.1) from the one measured at the calibration plane (Γc ). This approach for 33.

(45) the de-embedding model treats the probe as a two-port microwave network, wherein S -parameters are used to relate the reflection coefficient at the two planes as follows:. Γc = S11 +. S12 S21 Γa . 1 − S22 Γa. (3.1). Once the S -parameters are determined, in the way that is explained in the next section, (3.1) is inverted for using the de-embedding model as a data processing tool:. Γa =. 1 S21 , S22 + ΓSc12−S 11. (3.2). where Γa , the S -parameters, and Γc are functions of frequency (with upper and lower limits defined by the VNA’s band of operation). 3.2.2. Obtention of the Model S -Parameters.. To determine the unknown frequency-dependent S -parameters in (3.1), we need to obtain data measurements Γc for the probe attached to the VNA and theoretical values of Γa obtained by simulation using FDTD techniques. The connector-plane reflection coefficient Γc is measured for each probe in air and in three reference liquids, whose permittivities should span those of the tissues to be measured in the application. For breast-cancer detection, for example, where specimens’ permittivity ranges from very low values (adipose tissue), to very high values (fibroglandular tissue and tumors), liquids like deionized water (r ∼ = 5) = 80), methanol (5 ≤ r ≤ 80), and ethanol (r ∼ can be effectively used. Instead of positioning the probe exactly on the surface of the liquid samples, it has been shown that it is possible to make a fully immersion of the tip of the probe into the liquids without significant effects on the relevant sensing characteristics of the probe [7]. Okoniewski et al. recommend to make this reflection coefficient measurements on three different days and record them together with the temperature of the immersion medium. This way, three sets of data containing four measurements each (air and three liquids) are generated for each probe, and are used to solve a data fitting problem. The theoretical counterpart of the data observations (i.e. Γa ) is obtained with FDTD simulations of an idealized probe immersed in each of the liquids and air, at the corresponding temperature of measurement. These simulations are explained later in this. 34.

(46) chapter. A Matlab optimization function, fmincon, with an appropriate least-squares (LS) merit function was programmed to determine the S -parameters that optimally fit one of the three measurement data sets (Γc ) to the corresponding simulation data set (Γa ), for the frequency range of interest and for each probe. For each frequency point within the range of interest, the merit function proportioned to fmincon was. 2. χ =. 4 X i=1. BΓai Γci − A + 1 − CΓai . . 2. ,. (3.3). where A = S11 , B = S12 S21 and C = S22 , all of them complex-valued. Since the transmission S -parameters (S12 and S21 ) appear as a product in (3.1), they can be treated as a single quantity. The three standing variables were decomposed into their real and imaginary parts because fmincon can’t manipulate complex variables directly, carrying six independent parameters for each frequency, ar , ai , br , bi , cr and ci . The gradient of the cost function in 3.3 was also determined and given to fmincon as an input, in order to achieve a better convergence of the LS solution. The gradient expressions are calculated as:.    4 X ∂χ2 BΓa (BΓa )∗ = − Γ∗c − ar − jai + − Γc − ar + jai + ∂ar 1 − CΓa 1 − (CΓa )∗ i=1 ". !#!.    4 X (BΓa )∗ ∂χ2 BΓa = j Γc − ar + jai + − j Γ∗c − ar − jai + ∂ai 1 − CΓa 1 − (CΓa )∗ i=1 ". 4 X ∂χ2 = ∂br i=1. Γ∗a − 1 − (CΓa )∗. ". Γa + − 1 − CΓa . 4 X ∂χ2 = ∂bi i=1. ". ". jΓ∗a 1 − (CΓa )∗. #". Γ∗c. #". (br + jbi )Γa Γc − A + 1 − CΓa. (br − jbi )Γ∗a − A + 1 − (CΓa )∗. !#. !#!. ∗. (br + jbi )Γa Γc − A + 1 − CΓa. 35. ,. !#. ,. !#!. ,.

Figure

Figure 2.1: Phase fonts of (a) TEM and (b) plane waves.
Figure 2.2: Coaxial geometry showing Cartesian and polar coordinates.
Figure 2.3: Field lines for the (a) TEM and (b) TE 11 modes of a coaxial line.
Figure 2.4: Arbitrary N -port microwave network.
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