# Microwave cavity perturbation measurements of complex dielectric permittivity and complex magnetic permeability

код для вставкиСкачатьMicrowave Cavity Perturbation Measurements of Complex Dielectric Permittivity and Complex Magnetic Permeability A thesis submitted by Mi Lin In partial fulfillment of the requirements for the degree of Master of Science In Electrical Engineering TUFTS UNIVERSITY May 2006 © Mi Lin ADVISER: Professor Mohammed Nurul Afsar UMI Number: 1433639 UMI Microform 1433639 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 Acknowledgements The author would like to thank Professor Mohammed N. Afsar and Dr. Young Wang for their guidance during the graduate study at Tufts University. The author would also like to thank Mr. Paul Dee for his courteous helps in the design and construction of various mechanical parts for the cavity resonators. Finally, the author would like to thank his parents for their supports in study, in life and in everything else. ii Abstract Measurement of complex permittivity and permeability of dielectric and magnetic material plays an important role in microwave technology. Knowing properties of material is the first step in all kind of design when the material is used. This thesis has examined the rectangular cavity perturbation method as it used for material complex permittivity and complex permeability measurement. A wide range of material’s permittivity and permeability have been successfully determined based on the rectangular cavity perturbation technique. Also, a modified rectangular cavity perturbation technique has been introduced and a new set of formulas have been derived in accommodating with a special measurement case namely when the material sample under measurement has insufficient sample length comparing to the height of cavity resonator. These newly derived formulas are more general than the standard forms and have been proved to be accurate in yielding consistent results by measuring different samples with different lengths. iii Contents Chapter 1 Introduction 1 Chapter 2 Cavity Perturbation Technique 5 Chapter 3 Experimental Results 13 Chapter 4 The New Cavity Perturbation Technique 31 Chapter 5 Conclusion 40 Measured Resonant Frequencies for Each Designed Rectangular Cavity Resonator 42 B The General TRL Calibration Technique and Debye’s Equation 45 C Calculation of Ne Due to Image Effect 53 Appendices A Publications 55 Bibliographies 56 iv List of Tables Table 2.1 The dimension of each designed rectangular cavity resonator. Table 2.2 The operating modes for each designed rectangular cavity resonator. Table 3.1 Theoretical resonant frequencies, measured resonant frequencies of empty cavity and measured resonant frequencies of the cavity loaded with empty capillary tube. Table 3.2 Measured real and imaginary parts of complex permittivity of six chemical liquids. Table 4.1 Comparison of complex permittivity calculated by using (4.5a-4.5b) and (4.10a-4.10b) with Ne at frequency 4.5463 GHz. Table B.1 A discrete measured S-Parameters data retrieved from VNA. Table B.2 Debye’s parameters for chlorobenzene from literature. v List of Figures Figure 1 The vector diagram of complex permittivity. Figure 2.1 Geometry of a rectangular waveguide. Figure 2.2a Side view of rectangular waveguide for cavity resonator. Figure 2.2b Front view of rectangular waveguide for cavity resonator. Figure 2.2c The coupling plate for rectangular cavity resonator. Figure 2.3 A rectangular cavity resonator for a particular frequency band. Figure 2.4 Changed in resonant frequency and Q-factor of a cavity resonator due to a sample insertion. Figure 3.1 Liquid samples measurement setup. Figure 3.2 Measured resonant frequencies of empty rectangular cavity. Figure 3.3 Complex permittivity of cyclohexane measured using cavity perturbation technique and the waveguide transmission line technique at the X-band. Figure 3.4 Complex permittivity of chlorobenzene measured using cavity perturbation technique and calculated data by the Debye equation at the Xband. Figure 3.5 Complex permittivity of ethyl alcohol measured using cavity perturbation technique at the X-band. Figure 3.6 Complex permittivity of 1-4dioxane measured using cavity perturbation technique at the X-band. Figure 3.7 Complex permittivity of benzene measured using cavity perturbation technique at the X-band. Figure 3.8 Complex permittivity of acetone measured using cavity perturbation technique at the X-band. Figure 3.9 Dielectric samples measurement setup. Figure 3.10 Measured complex permittivity of boron nitride. Figure 3.11 Measured complex permittivity of magnesium fluoride. vi Figure 3.12 Ferrite samples measurement setup. Figure 3.13 Measured real parts of permittivity of 99.9% pure YIG, gadoliniumsubstituted YIG and aluminum-substituted YIG. Figure 3.14 Measured complex permeability of 99.9% pure YIG. Figure 3.15 Measured complex permittivity of nickel ferrite. Figure 3.16 Measured complex permeability of nickel ferrite. Figure 3.17 Measured complex permeability of strontium ferrite composite sample #1 Figure 3.18 Measured complex permeability of strontium ferrite composite sample #2 Figure 4.1 Polarization of the material sample with the external applied electric field As a result, a depolarizing field of magnitude. Figure 4.2 Variations of Ne for the normalized sample length to the height of the cavity. Figure 4.3 The new cavity measurement set up. Figure A.1 Measured resonant frequencies for H band rectangular waveguide cavity resonator. Figure A.2 Measured resonant frequencies for C band rectangular waveguide cavity resonator. Figure A.3 Measured resonant frequencies for X band rectangular waveguide cavity resonator. Figure A.4 Measured resonant frequencies for Ku band rectangular waveguide cavity resonator. Figure A.5 Measured resonant frequencies for K band rectangular waveguide cavity resonator. Figure B.1 Block diagram of a network analyzer measurement of a two port device. Figure B.2 Block diagram for Thru connection. Figure B.3 Signal flow graph for Thru connection. Figure B.4 Block diagram for Reflect connection. vii Figure B. 5 Signal flow graph for Reflect connection. Figure B.6 Block diagram for Line connection. Figure B. 7 Signal flow graph for Line connection. Figure B.8 Resonant frequency generated from discrete data lists in table B.1. Figure B. 9 Calculation of resonant frequency and Q factor for the discrete data lists in table B.1. Figure C.1 Image effect due to metallic ground plates. viii Chapter 1 Introduction Dielectric and Magnetic Properties of Matter Propagation of electromagnetic waves in a media or in a material is totally described mathematically by Maxwell’s equations: ∇× E = − ∂B ∂t (1.1a) ∇× H = − ∂D +J ∂t (1.1b) ∇•D = ρ (1.1c) ∇•B = 0 (1.1d) where J is the current density E is the electric field intensity D is the electric flux density H is the magnetic intensity field B is the magnetic flux density p is the charge density The unique solution to a particular circumstance is decided by the boundary conditions and the electromagnetic properties of the material. The permittivity (ε), permeability (µ) and conductivity (σ) are commonly referred as the electromagnetic properties of a material. They relate the six parameters of Maxwell equation in following 1 way: D =ε ⋅E (1.2a) B = µ⋅H (1.2b) J =σ ⋅E (1.2c) At high frequencies, especially in microwave frequency and frequency beyond the effect of conductivity is ignorable. This is true for most non-conductive material. Therefore, the mainly interested electromagnetic properties of a material to be studied in this thesis in high frequency are the permittivity and permeability. Permittivity which is a complex quantity describes the interaction of a material with an electric field. Complex permittivity is also called complex dielectric constant. The notation for the complex permittivity is given as follow: ε = ε ∗ = ε 0ε r (1.3a) ε r = ε r' − jε r'' (1.3b) where ε0 = 1 × 10 −9 , ε 0 is the free space permittivity and has a unit of F/m. 36π The real part of complex permittivity (εr') is a measure of how much energy from an external electric field is stored in a material. The imaginary part of permittivity (εr '') is called the loss factor and is a measure of how dissipative or lossy a material is to an external electric field. The ratio of the imaginary part of complex permittivity to the real part of complex permittivity is called loss tangent ( tan δ). A simple vector diagram of complex permittivity is illustrated in figure 1. 2 Figure1. The vector diagram of complex permittivity Permeability which is also a complex quantity describes the interaction of a material with a magnetic field. Under an applied external static field the complex magnetic permeability is well known as non-diagonalized tensor: (1.4) where µ ∗ , µ z∗ and k ∗ are all complex numbers. µ ∗ is the principal direction transverse component of the magnetic permeability. µ z∗ is the parallel component. k ∗ is the offdiagonal transverse component. With the absent of external applied static fields, the complex magnetic permittivity reduces to: µ = µ = µ ∗ = µ0 µr (1.5a) µ r = µ r' − jµ r'' (1.5b) where µ 0 = 4π × 10 −7 , µ 0 is the free space permeability and has a unit of H/m. The real part of complex permeability (μr') is a measure of how much energy from 3 an external magnetic field is stored in a material. The imaginary part of permittivity (μr '') is a measure of how loss of material when subject to an external magnetic field. Review of Measurement Techniques in Microwave Frequency Ranges Measurement of complex permittivity and permeability of dielectric and magnetic material plays an important role in microwave technology. Knowing properties of material is the first step in all kind of design when the material is used. Quite a few techniques such as cavity perturbation [1], open-ended coaxial probes [2], free space [3], waveguide transmission line [4] and dispersive fourier transform spectrometer [5] have been developed for material permittivity and permeability measurement. Nevertheless, the cavity perturbation method has a reputation as one of most simple and accurate techniques for measuring dielectric and magnetic properties of material over years. The earliest treatment of cavity perturbation theory was given by Bethe and Schwinger [6]. Following them, many researchers have been studied and improved cavity perturbation technique. Up to nowadays, many theories related to it have been proposed and experimentally verified. The cavity perturbation technique used to carry out the measurement tasks as described in chapter 3 is mainly follow K. T. Matthew’s method [1]. Then, a new cavity perturbation technique derived from K. T. Matthew’s method [1] will be presented in chapter 4. 4 Chapter 2 Cavity Perturbation Technique Rectangular Waveguide The geometry of a rectangular waveguide is shown in figure 2.1. The rectangular waveguide can propagate TE and TM modes of electromagnetic waves, but not TEM waves. Rectangular waveguide have cutoff frequencies below that electromagnetic wave the propagation is not possible. The cutoff frequency for a particular mode in rectangular waveguide is determined by the following equation: ( f c ) mn = 1 2π uε 2 ⎛ mπ ⎞ ⎛ n π ⎞ ⎟ ⎟ +⎜ ⎜ ⎝ a ⎠ ⎝ b ⎠ 2 (2.1) So, the corresponding wavelength is then: (λc ) mn = 2 2 ⎛m⎞ ⎛n⎞ ⎜ ⎟ +⎜ ⎟ ⎝ a ⎠ ⎝b⎠ 2 (2.2) Figure 2.1 Geometry of a rectangular waveguide. 5 The TE10 mode is the dominant mode (lowest cutoff frequency) of a rectangular waveguide with a >b. Design of Rectangular Cavity Resonators The cavity resonator is formed by using a section of rectangular copper waveguide with both ends covered by a pair of flat copper coupling plates. Figure 2.2a2.2c illustrates the structure of a designed rectangular cavity resonator. To cover the measurement frequency from 4.5 to 26.5 GHz which composes H, C, X, Ku and K bands, five different dimension pieces of waveguide are used, namely WR 187, WR 137, WR 90, WR 62 and WR 42. In order to get the sample in and out of the cavity without disassembling it, a narrow non-radiated slot was opened at the top surface of the cavity wall. Table 2.1 lists the dimension of each designed rectangular cavity resonator. Table 2.2 lists the operating modes and corresponding frequencies for each designed band of rectangular cavity resonators. The narrow opened slot has negligible effect on changing the geometrical configuration of electromagnetic fields inside of the cavity. Figure 2.3 shows a complete constructed rectangular waveguide resonator. Figure 2.2a Side view of rectangular waveguide for cavity resonator. 6 Figure 2.2b Front view of rectangular waveguide for cavity resonator. Figure 2.2c The coupling plate for rectangular cavity resonator. Figure 2.3 A rectangular cavity resonator for a particular frequency band. 7 The resonant frequency of rectangular cavity resonator for the TEmnL or TMmnL mode is determined f mnL = C 2π u r ε r 2 2 ⎛ mπ ⎞ ⎛ nπ ⎞ ⎛ Lπ ⎞ ⎜ ⎟ +⎜ ⎟ +⎜ ⎟ ⎝ a ⎠ ⎝ b ⎠ ⎝ d ⎠ 2 (2.3) For the resonator shown in figure 2.3 with b < a < d, the lowest resonant frequency will be the TE101 mode, corresponding to the TE10 dominant waveguide mode. Table 2.1 lists the dimension of each designed rectangular cavity resonator. And, table 2.2 lists the operating modes and the corresponding resonant frequencies for each designed rectangular cavity resonator. The measured resonant frequencies for each designed rectangular cavity resonator can be found in appendix A. Table 2.1 The dimension of each designed rectangular cavity resonator. Waveguide b (mm) a (mm) d (mm) WR 187 WR 137 WR 90 WR 62 WR 42 22.16 15.84 10.12 8.04 4.38 47.56 34.91 22.80 15.78 10.69 91.43 91.32 89.45 90.01 79.91 Coupling hole diameter (mm) 7.56 5.47 4.50 3.95 2.80 Opened slot width (mm) 3.34 3.25 3.10 2.15 1.75 Volume (mm^3) Recommend frequency range (GHz) Cutoff frequency(fc) (GHz) 96360.78 50497.62 20639.34 11419.68 3742.28 3.95-5.85 5.85-8.20 8.20-12.40 12.4-18.0 18.0-26.5 3.15 4.30 6.557 9.486 14.05 8 Table 2.2 The operating modes and the corresponding resonant frequencies for each designed rectangular cavity resonator. m n 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 L WR 187 2 3 WR 137 3 4 WR 90 3 4 5 6 WR 62 5 6 7 8 9 WR 42 6 7 8 9 10 11 12 fmnL (GHz) λg (mm) 4.551 5.826 91.43 60.95 6.538 7.851 60.88 45.66 8.282 9.395 10.658 12.021 59.63 44.73 35.78 29.82 12.641 13.796 15.048 16.374 17.757 36.00 30.00 25.72 22.50 20.00 17.990 19.221 20.550 21.960 23.434 24.964 26.537 26.64 22.83 19.97 17.76 15.98 14.53 13.315 Measurement Theory When a small material sample is inserted in a resonant cavity, it will cause the complex frequency shift. The amount of complex frequency shift is given by Waldron [7] as − δω = ω (εr −1)εo ∫ E ⋅ Eo*dν + (µr −1)µo ∫ H ⋅ Ho*dν Vs Vs ∫ Do ⋅ E + Bo ⋅ H dv * o * o (2.4) Vc where δω is the complex frequency shift; Do, Eo, Bo and Ho are the unperturbed cavity fields, and E and H are the fields in the interior of the sample. εr = εr′ - j εr′′ and µr = µr′ - 9 jµr′′. Vc and Vs are the volumes of the cavity and the sample, respectively. The fundamental idea of cavity perturbation is that the change in the overall geometrical configuration of the electromagnetic fields upon the introduction of a material sample should be small. This indicates that percentage change in the real part of resonant frequency must be small [1]. Therefore, the constraint on equation (2.4) is that δω << ω. For a sample placed at the electric field maximum position, equation (2.4) can be simplified to δω − ω (ε r = − 1) ∫ E ⋅E * o max E ⋅E dv dν (2.5) Vs 2 ∫ o o Vc For a sample placed at the magnetic field maximum position, equation (2.4) can be simplified to δω − = ω (µ r − 1) ∫ H ⋅ H * o max dν Vs 2 ∫ H o ⋅H o (2.6) dv Vc For a rectangular cavity operated at TE10L mode, the total fields can be written as Lπz d (2.7) = − jE o max πx Lπz sin cos Z TE a d (2.8) = − j π E o max L πz πx cos sin Kηa a d (2.9) E y = E H x H z o max sin πx a sin where a is the broader dimension of waveguide, d is the length of cavity, ZTE = (Kη/β), is wave impedance of transverse electromagnetic fields, η is the intrinsic impedance of the material filling the waveguide. For air filling, η equals 377 Ω. K and β are the 10 wavenumber and propagation constant of waveguide. And, the complex frequency shift can separate into real and imaginary parts as [1] δω ω where ≈ δ f f and ⎡ δfo fs ⎡ 1 ⎤ + jδ ⎢ ⎥ ⎣ 2Q o ⎦ f = o s 1 δ ⎢ ⎣ 2Q 0 ⎤ ⎥ = ⎦ s − f f (2.10) o s 1 ⎡ 1 1 ⎤ − ⎢ ⎥ 2 ⎣ Q s Q 0 ⎦ fo and Qo are the resonance frequency and the Q-factor of the cavity without sample inserted, respectively; fs and Qs are the quantities with the sample inserted. If the perturbation condition is satisfied, we can assume E=Eo, H=Ho. After performing integration and rearranging, equation (2.4) becomes [1] ε r' − 1 = (fo ⎛ 1 − fs )Vc 2 fs Vs 1 ⎞ V ⎟⎟ c − ε r'' = ⎜⎜ ⎝ Q s Q o ⎠ 4V s (2.11a) (2.11b) Equation (2.6) becomes [1] ⎛ λ2g + 4a 2 ⎞ ( f o − f s ) Vc ⎟ u −1 = ⎜ ⎜ 8a 2 ⎟ f s Vs ⎝ ⎠ (2.12a) ⎛ λ 2g + 4a 2 u r'' = ⎜ ⎜ 16a 2 ⎝ (2.12b) ' r ⎞⎛ 1 1 ⎟⎜ − ⎟⎜⎝ Qs Qo ⎠ ⎞ Vc ⎟⎟ ⎠ Vs where λg = 2d/L, is the guided wavelength and L=1,2,3,… . When volumes and resonant frequencies and quality factors are measured, complex permittivity and complex permeability can be calculated analytically. Figure 2.4 demonstrates the resonant frequency and Q-factor changed due to insertion of a material sample into an empty 11 cavity resonator. Figure 2.4 Changed in resonant frequency and Q-factor of a cavity resonator due to a sample insertion. 12 Chapter 3 Experimental Results To calibrate the set up, a custom built thru-reflect-line (TRL) calibration technique is applied to it. The TRL calibration kit consists of a flat ground plate used as short circuit standard and a piece of waveguide with a known waveguide length used as the line standard. The length of line is calculated using the following equation: λ gn = 2 ( λ gl × λ gh ) ( λ gl + λ gh ) (3.1) where λ gl and λ gh are wavelengths of the low frequency end and the high frequency end at the corresponding frequency band. The general TRL calibration technique given by David M. Pozar [8] can be found in appendix B. Since at each designed band waveguide cavity resonator has multiple resonant frequencies, TRL calibration was re-used in each resonant frequency range, and the measurement was separately performed in each resonant frequency range. The source of microwave signal and the scattering parameters measurement equipment is an Agilent Vector Network Analyzer model 8510C. The VNA allows measurement at discrete frequency point only, and offer several sets of measurement points. 201 points set is chosen to carry out for all measurements. Since for cavity perturbation measurements of complex permittivity and complex permeability are based on the accurate calculation of resonant frequency and Q factor, a curve fitting technique is applied to the discrete measured data. The curve fitting technique will improve 13 resonant frequency and Q factor calculation for a discrete measured data. An example of a discrete measured data and calculation of resonant frequency and Q factor using curve fitting are illustrated in appendix B. This chapter presents complex permittivity and complex permeability measured results of a wide range of materials. These materials are categorized as chemical liquids, dielectric solids, ferrites and custom made ferrite powerepoxy composite ceramics. 14 Liquids The study of complex permittivity of chemical liquids at microwave frequencies becomes more important than ever as the tendency of microwave technologies applied to biomedical and biochemical engineering. For example, microwave dielectric heating is rapidly becoming an established procedure in synthetic chemistry due to its several advantages over conventional heating for chemical conversions [9]. Breast cancer detection utilizing microwave imaging technique is one of rapid growing research areas in biomedical application. One of motivations of developing microwave imaging technique for breast cancer detection is the significant contrast in the dielectric properties at microwave frequencies of normal and malignant breast tissue [10]. This section presents measurement results of complex permittivity of a number of different chemical liquids at the X-band. Some comparisons of measured results with published data measured using the waveguide transmission line technique and with calculated data by the Debye equation [11] were made. Figure 3.1 shows the measurement setup. The length of rectangular cavity is 89.45 mm. The waveguide cavity operates at the TE103-6 modes, which correspond to the theoretical resonant frequencies of f103-6 at 8.2863 GHz, 9.3880 GHz, 10.6823 GHz, and 12.0547 GHz. Figure 3.1 Liquid samples measurement setup. 15 Figure 3.2 shows the measured resonant frequencies of empty waveguide cavity. The measured resonant frequencies are at 8.2860 GHz, 9.3860 GHz, 10.6620 GHz, and 12.0260 GHz. 0 -10 S21 (dB) -20 -30 -40 -50 -60 -70 -80 -90 8.0 9.0 10.0 11.0 12.0 f (GHz) Figure 3.2 Measured resonant frequencies of empty rectangular cavity. A capillary tube with negligible loss tangent was chosen as liquid sample holder. However, ft and Qt should replace the fo and Qo in equation (2.11a) and (2.11b) for the calculation of εr′ and εr′′ in this measurement, where ft and Qt are the resonance frequency and Q-factor of cavity loaded with an empty capillary tube, respectively. Table 3.1 lists and compares the theoretical resonant frequencies, measured resonant frequencies of empty cavity and measured resonant frequencies of the cavity loaded with empty capillary tube. It is seen that the capillary tube has little effect on the resonant frequencies. 16 Table 3.1 Theoretical resonant frequencies, measured resonant frequencies of empty cavity and measured resonant frequencies of the cavity loaded with empty capillary tube. Theoretical resonant frequencies (GHz) Measured empty cavity resonant frequencies (GHz) Measured empty cavity resonant frequencies with capillary tube (GHz) 8.2863 9.3880 10.6823 12.0547 8.2860 9.3860 10.6620 12.0260 8.2466 9.3450 10.6000 11.9510 Measured resonant frequency different in percentage (with and without capillary tube) 0.476% 0.437% 0.582% 0.624% The measurements were performed at room temperature of 22 0 C. The capillary tube containing the liquid sample was placed at electric field maximum position for each measurement. Figure 3.3 illustrates measured complex permittivity of cyclohexane using cavity perturbation technique compared with published data measured by the waveguide transmission line technique at the X-band. Figure 3.4 illustrates measured complex permittivity of chlorobenzene using cavity perturbation technique, and comparing with calculated data by the Debye equation. An illustration of using Debye’s equation for calculation of chemical liquid can be found in appendix B. They all show quite well match, especially for the imaginary part. Cavity perturbation technique measured the real part of complex permittivity of cyclohexane and chlorobenzene are slightly large than published data measured by the-waveguide transmission line technique and calculated data using Debye’s equation. These consistent differentiations are possible due to the effect of capillary tube in the cavity since the capillary tube also shows some degree of dielectric constant in the X- band. Therefore, capillary tube, the sample holder’s effect should be studied and taken into account for more accurate cavity perturbation technique measurements. Figures 3.5-3.8 show the measured complex permittivity for ethyl alcohol, 1-4 dioxane, benzene and acetone, respectively. Table 3.2 lists the measured real and imaginary parts of complex permittivity of all the six chemical liquids. 17 Table 3.2 Measured real and imaginary parts of complex permittivity of six chemical liquids. Frequency (GHz) 8.2362 9.3345 10.5877 11.9380 Frequency (GHz) 8.2340 9.3320 10.5860 11.9350 Frequency (GHz) 8.2342 9.3317 10.5852 11.9352 Cyclohexane ε' 2.21 2.18 2.24 2.22 Ethyl Alcohol ε' 4.87 4.78 4.65 4.73 Benzene ε' 2.47 2.49 2.48 2.46 ε'' 0.00026 0.0031 0.011 0.00072 ε'' 2.30 2.35 2.00 1.97 ε'' 0.0067 0.013 0.0063 0.0012 Frequency (GHz) 8.2165 9.3145 10.5660 11.8945 Frequency (GHz) 8.2352 9.3321 10.5860 11.9355 Frequency (GHz) 8.1725 9.2665 10.5415 11.8785 Chlorobenzene ε' 4.92 4.83 4.79 4.57 1-4 Dioxane ε' 2.34 2.45 2.41 2.44 Acetone ε' 18.15 16.91 15.72 16.43 ε'' 1.38 1.43 1.58 1.56 ε'' 0.014 0.022 0.029 0.020 ε'' 1.83 1.92 2.42 2.99 3 2.5 2 real (cavity technique) 1.5 imaginary (cavity technique) 1 real (in-waveguide technique) imaginary (in-waveguide technique) 0.5 0 8 9 10 f (GHz) 11 12 Figure 3.3 Complex permittivity of cyclohexane measured using cavity perturbation technique and the waveguide transmission line technique at the X-band. 18 6 5 4 Real (cavity technique) Imaginary (cavity technique) 3 real (Debye) 2 imaginary (Debye) 1 0 8 9 10 11 12 f (GHz) Figure 3.4 Complex permittivity of chlorobenzene measured using cavity perturbation technique and calculated data by the Debye equation at the X-band. 8 7 6 5 4 3 2 1 0 real imaginary 8 9 10 f (GHz) 11 12 Figure 3.5 Complex permittivity of ethyl alcohol measured using cavity perturbation technique at the X-band. 19 3 2.5 2 1.5 Real 1 Imaginary 0.5 0 8 9 10 f (GHz) 11 12 Figure 3.6 Complex permittivity of 1-4dioxane measured using cavity perturbation technique at the X-band. 3 2.5 2 Real Imaginary 1.5 1 0.5 0 8 9 10 f (GHz) 11 12 Figure 3.7 Complex permittivity of benzene measured using cavity perturbation technique at the X-band. 20 20 17.5 15 12.5 10 7.5 5 2.5 0 Real Imaginary 8 9 10 f (GHz) 11 12 Figure 3.8 Complex permittivity of acetone measured using cavity perturbation technique at the X-band. 21 Dielectric Solids This section presents the cavity perturbation measurement results of complex permittivity of boron nitride and magnesium fluoride. The measurement was performed at C, X, Ku, and K four frequency bands ranges from 4.5 to 26.5 GHz. Boron nitride has very high thermal conductivity, good thermal shock resistance and high electrical resistance. It is widely used in electronics such as: heat sinks, substrates coil forms or microcircuit packaging. Magnesium fluoride is commonly used for UV windows, lenses and polarizers due to its optical properties. However, not much information of complex permittivity about them can be found at microwave and millimeter wave range. Figure 3.9 shows the measurement setup. Figure 3.9 Dielectric samples measurement setup. Figure 3.10-3.11 demonstrates measured complex permittivity of boron nitride (Grade XP with parallel orientation) and magnesium fluoride. The measurements show that both boron nitride and magnesium fluoride are very low loss dielectric materials. Boron nitride has real part of permittivity about 4, and magnesium fluoride has real part of permittivity about 5 in the frequency tested. 22 Permittivity 5 4 3 2 1 0 -1 real imaginary 5 10 15 20 Frequency (GHz) 25 30 Permittivity Figure 3.10 Measured complex permittivity of boron nitride. 7 6 5 4 3 2 1 0 -1 real imaginary 5 10 15 20 Frequency (GHz) 25 30 Figure 3.11 Measured complex permittivity of magnesium fluoride. 23 Ferrites Ferrite materials have found a diverse range of applications in modern microwave and telecommunication systems. Ferrite materials such as nickel ferrite and yttrium iron garnet ceramics with relative high permittivity, low loss and anisotropic properties are often used in microwave filter, resonator, circulator or phase shifter designs. In order to achieve the best design, the accurate measurement of their dielectric and magnetic properties becomes important. This section presents measurement results of complex permittivity and permeability of three types of YIG specimen and one nickel ferrite sample All of the samples have been measured in demagnetized state. The YIGs under measurement are 99.9% pure YIG (4πMs = 1800 ±5% Gauss), Aluminum-Substituted YIG (4πMs = 1000 ±5% Gauss) and Gadolinium-Substituted YIG (4πMs = 1200 ±5% Gauss). The nickel ferrite sample is a commercially available isotropic soft nickel ferrite material provided by TransTech, Adamstown, MD. (With the manufacturer data: 4πMs = 5000 ±5% Gauss, ∆H ≤200 Oe at 9.4 GHz). Figure 3.12 shows the measurement setup. Figure 3.12 Ferrite samples measurement setup. Figure 3.13 demonstrates measured real parts of permittivity of three types of YIG. 24 Measurements show real parts of permittivity of 99.9% pure YIG, gadolinium-substituted YIG and aluminum-substituted YIG graduate increase with frequency. The imaginary parts of permittivity values for YIG specimen are extremely low, and therefore can’t be measured due to the limited sensitivity of the cavity resonator. Real part of permittivity 18 17 16 15 14 YIG Gd-YIG Al-YIG 13 12 2 7 12 17 Frequency (GHz ) 22 27 Figure 3.13 Measured real parts of permittivity of 99.9% pure YIG, gadoliniumsubstituted YIG and aluminum-substituted YIG. Figure 3.14 illustrates that the 99.9% pure YIG has a magnetic resonance near 5 GHz. No magnetic resonances were found for Aluminum-Substituted YIG and Gadolinium-Substituted YIG in the frequency range of testing. Real parts of permeability of them are all around 1. Since pure and substituted versions of YIG are all soft ferrites, the natural magnetic resonances should occur well below 30 GHz. The measurement results demonstrated in this paper show well agreement with that. 25 Complex Permeability 1.2 1 0.8 0.6 real imaginary 0.4 0.2 0 3 5 7 9 11 Frequency (GHz) 13 15 Figure 3.14 Measured complex permeability of 99.9% pure YIG. Figure 3.15 shows the complex permittivity of nickel ferrite. Measurements show that nickel ferrite has real part of permittivity around 12.5 and slowly increases with frequency. Figure 3.16 shows the complex permeability of nickel ferrite. The figure clearly shows magnetic resonance of nickel ferrite occurs around 13 GHz. Complex Permittivity 20 15 10 real imaginary 5 0 2 7 12 17 Frequency (GHz ) 22 27 Figure 3.15 Measured complex permittivity of nickel ferrite. 26 Complex Permeability 1.4 0.9 real 0.4 imaginary -0.1 6 8 10 12 14 16 18 20 22 24 26 Frequency (GHz ) Figure 3.16 Measured complex permeability of nickel ferrite. 27 Strontium Ferrite Powder Composite Ceramics Magnetic composite materials are also inspired the interests of researchers over years. The composite materials have several advantages, such as composite materials are easy to prepare in the desired shape and to machine by molding. Also by varying the compositions in a composite material, the real and imaginary parts of complex dielectric permittivity and complex magnetic permeability properties, as well as the position of resonance frequency vary accordingly. This section presents complex permittivity and permeability measurement results of two custom made strontium ferrite powder-epoxy composite ceramic samples. The two strontium ferrite composite ceramic samples with slightly different chemical composition and specific gravity were prepared at Tufts University. The strontium ferrite samples were prepared by mixing strontium ferrite powder with epoxy resin and a very small amount of peroxide hardener. The mixtures were prepared homogeneously and baked in an oven. These specimens were then flattened and polished and cut into specific shapes to accommodate different measurement techniques. The first strontium ferrite composite ceramic sample (Sample #1) has powder and epoxy resin ratio of 1: 2.3, and 1.9 g/cm for specific gravity. The second composite ceramic sample (Sample #2) has a slightly higher concentration of strontium ferrite powder as it compared to sample #1. It has powder and epoxy resin ratio of 1: 2.0, and 1.8 g/cm for specific gravity. 28 4.5 Complex Permittivity 4 3.5 3 real 2.5 imaginary 2 1.5 1 0.5 0 0 5 10 15 20 25 30 Frequency (GHz) Figure 3.17 Measured complex permeability of strontium ferrite composite sample #1 4.5 Complexer Permittivity 4 3.5 3 2.5 real 2 imaginary 1.5 1 0.5 0 -0.5 0 5 10 15 20 25 30 Frequency (GHz) Figure 3.18 Measured complex permeability of strontium ferrite composite sample #2 Figures 3.17-3.18 shows complex permittivity measurement results for strontium ferrite sample #1 and sample # 2. Measurements show that strontium ferrite sample #1 and sample #2 has real part of permittivity about 3.6 and 4.2 respectively in these frequency region. These real parts of permittivity values are higher than the permittivity values of original strontium ferrite powder in the frequency range tested. The strontium ferrite powder has the average real permittivity 2.65 from 8 to 26.5 GHz [18]. In 29 previous measurements, Afsar et al show that when the concentration of strontium ferrite powder in the mixture is increased, the absorption value at the resonance frequency increases. Measurements also show that the imaginary part of permittivity values for both samples gradually increase as frequency increases. For the strontium ferrite, it is expected to have the magnetic resonance to appear around 56GHz. It is therefore the real part of permeability values for both samples are expected to be about unity in these frequency regions. The present measurements also reveal similar values to original strontium ferrite powder values and that are all around unity. 30 Chapter 4 The New Cavity Perturbation Technique The fundamental cavity perturbation condition is that the change in the overall geometric configuration of the electromagnetic fields inside the resonant cavity upon the introduction of a material sample should be small [1]. In other words, it requires that the amount of the complex frequency shift be small. In rectangular cavity perturbation measurements, the measurement is also limited by the sample shape requirement. It is required to have the length of the sample under measurement equal to the height of the cavity resonator for the formulas to work out. However, very often fabricating a long and thin sample becomes quite challenging task especially for some materials are hard and (or) fragile in nature. Therefore, it is necessary to extend the cavity perturbation measurement technique to include the case when the sample length is less than the height of the cavity resonator. In this chapter, cavity perturbation formulas have been expanded to include the situation mentioned above when the sample length is insufficiently long comparing to the height of the cavity. In order to check the validity of these new equations, complex permittivity values of different samples with different sample lengths have been measured and compared. Brief Review of the Cavity Perturbation Theory The frequency shift due to the insertion of a small material sample into a resonant cavity is given by Waldron [7], [12] as 31 δω =− ω (ε ∗ r ) ( ) − 1 ε o ∫ E ⋅ Eo*dν + µ r∗ − 1 µo ∫ H ⋅ H o*dν Vs Vs ∫ Do ⋅ E + Bo ⋅ H dv * o * o (4.1) Vc Where Do, Eo, Bo and Ho are the unperturbed cavity field quantities, and E and H are the fields in the interior of the sample. ε r∗ = εr′ - j εr′′ and µ r∗ = µr′ - jµr′′. Vc and Vs are the volumes of the cavity and the sample, respectively. δω/ω is the amount of the complex frequency shift that can be separated into real and imaginary parts as [1] δω ω δ f where f ⎡ and δfo ≈ fs ⎡ 1 ⎤ + jδ ⎢ ⎥ ⎣ 2Q o ⎦ f = o s s 1 δ ⎢ ⎣ 2Q 0 ⎤ ⎥ = ⎦ − f f (4.2) , o s 1 ⎡ 1 1 ⎤ − ⎥ ⎢ 2 ⎣ Q s Q 0 ⎦ fo and Qo are the resonance frequency and the Q-factor of the cavity without sample inserted, respectively; fs and Qs are the quantities with the sample inserted. For a sample placed at the electric field maximum, equation (4.1) can be simplified to δω ω (ε = − ∗ r − 1) ∫ E ⋅ E * o max dν Vs 2 ∫ E o ⋅E o dv (4.3) Vc A rectangular cavity operates at TE10L mode, the total electric fields can be written as E o = E o max sin πx a sin Lπz d (4.4) where a is the broader dimension of waveguide, d is the length of cavity and L=1,2,3,… . L is an integer. 32 If the perturbation condition is satisfied, and the length of the inserted material sample equals the height of the cavity resonator, then it’s valid to assume that E = Eo, the electric field in the interior of the sample equals the unperturbed field. Along with this assumption, substituting (4.2) and (4.4) into (4.3), evaluating the equation, and then separating and equal the real and imaginary parts of it, the equation (4.3) becomes [1] (fo − fs 2 fs ε ' r −1 = ε '' r ⎛ 1 1 = ⎜⎜ − Q o ⎝ Q s )V c Vs ⎞ Vc ⎟⎟ ⎠ 4V s (4.5a) (4.5b) Equations (4.5a)-(4.5b) are precise solutions for calculating the complex permittivity when the perturbation condition is satisfied [13]. However, the accuracy of the equations (4.5a)-(4.5b) are also affected by the length of the inserted material sample. In other words, these equations require that the inserted sample length equals the height of the cavity. If the amount of the complex frequency shift is still small but either the length of the inserted material sample is shorter than the height of the cavity or the sample is only partially filling the cavity height, then the validity of the previous assumption E = Eo is not longer true. Subsequently, evaluating equations (4.5a)-(4.5b) will yield incorrect complex permittivity results. Assessment of Complex Permittivity of a Material with the Sample Length Less Than the Height of the Cavity When the inserted sample length is shorter than the height of the cavity resonator, a new estimation of E, the electric field in the interior of the sample, must be carried out and new equations for calculating complex permittivity must be derived. To evaluate E, 33 one additional assumption need to be made besides the fundamental perturbation condition. Namely, the electromagnetic fields inside the resonant cavity are assumed to be static fields. This assumption is most likely to be accurate here. So, when the sample is inserted and placed at the electric field maximum position of the cavity, we have an electrostatic situation. The sample will be partially polarized by the applied electric field, E0. Therefore, the sample will behave like a dipole with a length equal to the length of the sample as illustrated in Figure 4.1. Figure 4.1 Polarization of the material sample with the external applied electric field As a result, a depolarizing field of magnitude. E d = − NP ε (4.6) o will point in the opposite direction of the applied electric field. Where N is the depolarizing factor and P is the polarization magnitude. In a linear medium, the electric polarization is linearly related to the applied electric field as ( ) P = ε r∗ − 1 ε o ⋅ E o (4.7) Therefore, the electric field in the interior of the material sample is the total vector sum of the applied field and the depolarizing field. That is to say, now 34 = E E + E o (4.8) d By substituting (4.8) along with (4.4), (4.6) and (4.7) into (4.3) and then evaluate the right hand-side equation, equation (4.3) becomes δω = − ω [1 − (ε ∗ r ) ] − 1 N ( ε r∗ − 1) ∫ E o ⋅ E o* max d ν Vs 2 ∫ E o ⋅ E o dv (4.9) Vc Again substituting (4.2) in (4.9), evaluating the equation, and then separating and equal the real and imaginary parts of it, equation (4.9) becomes [ ] ( f s − f o ) = {N (ε ' )2 − 2ε ' − ε '' + 1 + (− ε ' fs r r r r ) ]⎛⎜⎜ 2 VV 1⎛ 1 1 ⎞ ⎟ = ε r'' − 2 N ε r' − 1 ε r'' ⎜⎜ − 2 ⎝ Q s Qo ⎟⎠ [ )} ⎛ V ⎞ + 1 ⎜⎜ 2 s ⎟⎟ ⎝ Vc ⎠ ( ⎝ s c ⎞ ⎟⎟ ⎠ (4.10a) (4.10b) The value of the depolarizing factor N depends on the geometrical dimension of the sample, and has been calculated by a number of researchers [14-16]. According to Bozorth [14], for a prolate ellipsoid with semi-axis h, b and c, (h > b =c) that is being polarized along the longer axis, N is given by 1 ⎡ ⎤ m + m2 −1 2 ⎥ 1 ⎢ m −1 ln N= 2 1 ⎥ m −1 ⎢ 2 12 2 m − m −1 2 ⎥⎦ ⎢⎣2 m −1 ( ) ( ) ( ( ) ) (4.11) Where m = h/b. For cavity perturbation measurements of permittivity, the sample usually is prepared in a rod or a cylinder rather than in prolate ellipsoid shape. With a small volume of the sample size, a prolate ellipsoid can be substituted by a cylinder with the same volume for the calculation of the N value. The error of this substitution will be very insignificant [17]. Thus far, the value of N used in (4.10a) and (4.10b) to calculate complex 35 permittivity has been computed for the sample in the absence of the surrounding cavity walls. The value of N will be slightly affected by the present of the cavity conducting surfaces due to the image effect. By including the effect of the images of the dipole sample to the cavity, following Parkash and Abhai [17], the new N value will now be Ne = N πh 2H cot πh (4.12) 2H Ne is the effective depolarizing factor. H is the height of the cavity and h is the length of the inserted material sample. An illustration of how to calculate the image effect given by Parkash and Abhai can be found on appendix C. Figure 4.2 illustrates the calculated values of Ne versus the normalized sample length to the height of the cavity for a cylinder substituted prolate ellipsoid with different diameters. The diameters of the cylinders illustrated here are 1.1 mm, 1.25mm, 1.5mm. As noticed, for all three cases Ne approach zero as h approaches H, and Ne equals zero when h equals H. Notice that when Ne equals zero, equations (4.10a)-(4.10b) will be reduced to the standard forms, (4.5a)-(4.5b). 1.4 1.2 Ne 1 Ne (b=1.1 mm) Ne (b=1.25 mm) Ne (b =1.5 mm) 0.8 0.6 0.4 0.2 0 0 0.2 0.4 h/H 0.6 0.8 1 1.2 Figure 4.2 Variations of Ne for the normalized sample length to the height of the cavity. 36 Experimental Results and Discussion The cavity resonator is formed by using a section of WR 187 rectangular copper waveguide with both ends covered by a flat copper coupling plate. The length of the WR 187 waveguide is 91.52 mm and the diameter of the coupling hole is 5.52 mm. In order to get the sample in and out of the cavity without disassembling it, a non-radiated slot was opened at the top surface of the cavity wall. The width of the non-radiated open slot is 3.48 mm. The WR 187 waveguide cavity operates at the TE102 mode, which correspond to the theoretical resonant frequency of f102 at 4.5488 GHz. The actual measured resonant frequency of the empty cavity is 4.5463 GHz. The agreement between the theoretical and measured values for the cavity resonant frequency is excellent. The difference is only 0.055%. A small piece of rod-shaped styrofoam with both ends attached by double-sided scotch tape is positioned firmly at the electric field maximum point. The cylinder sample is inserted and placed at the center of the styrofoam by using a tweeter with narrow-ended tips. The introduction of the styrofoam is found has very little effect on the empty cavity’s resonant frequency and the Q-factor. But, when calculating complex permittivity, the resonant frequency and the Q-factor measured with the styrofoam are used as “empty” cavity quantities. Figure 4.3 shows the new cavity measurement set up. Figure 4.3 The new cavity measurement set up. 37 To calibrate the set up, a custom built thru-reflect-line (TRL) calibration technique is applied to it. The TRL calibration kit consists of a short circuit and a piece of waveguide with a known waveguide length. The measurement results obtained by applying the custom designed TRL calibration have been proved to be very accurate by testing a number of material complex permittivity and permeability for both cavity perturbation and waveguide transmission line techniques [13] [18]. The resonance frequency and the Q-factor used in calculation of complex permittivity are determined from measured S-parameters from a Vector Network Analyzer (VNA). The Vector Network Analyzer used to carry out this experiment task allows measurement at discrete frequency points only (Agilent model 8510 C with the 801 maximum measurement points over a defined frequency range). One of key factors for obtaining accurate measurement results using cavity perturbation technique is the ability to precisely resolve the resonant frequency and the Q-factor with and without a material sample inserted. Therefore, a curve fitting technique is applied to the discrete measured S-parameters data each time. The curve fitting technique will improve the accuracy in determination of the resonant frequency and the Q-factor. The samples under test are Teflon, Magnesium Fluoride and Plexiglas. Each sample was prepared with several different sample lengths. The diameters of the same type sample with different sample lengths are made to be identical. The diameter of Teflon and Magnesium Fluoride samples are 1.25 mm. The diameter of Plexiglas samples are 1.26 mm. 38 Table 4.1 Comparison of complex permittivity calculated by using (4.5a-4.5b) and (4.10a-4.10b) with Ne at frequency 4.5463 GHz. (Cavity height = 22.1 mm). Teflon 22.1 15 10 5 εr′ 2.08 2.06 2.02 1.86 εr′′ ----- From Eqs (10a-b) with Ne ε r′ εr′′ 2.08 -2.07 -2.07 -2.04 -- Magnesium Fluoride 22.1 15 7 5.08 4.87 3.89 ---- 5.08 5.09 5.06 ---- Plexiglas 4.1 1.81 0.10 2.51 0.22 Material sample Length of sample (mm) From Eqs (5a-b) The measured permittivity results are tabulated in Table 4.1. The table lists and compares the permittivity results calculated using the standard form and the newly derived equations. The imaginary parts of permittivity values for Teflon and Magnesium Fluoride are extremely low, and therefore can’t be measured due to the limited sensitivity of the cavity resonator. However, the real parts of permittivity values of Teflon and Magnesium Fluoride with different together with Ne show much consistent and accurate result than using the standard equations. Therefore, the newly derived equations should be used for the calculation of complex permittivity of materials when the sample has insufficient length comparing to the height of cavity resonator. 39 Chapter 5 Conclusion This thesis has examined the rectangular cavity perturbation technique as it used for material complex permittivity and complex permeability measurement. A wide variety range of material’s permittivity and permeability have been successfully determined. There are two main error sources associated with this permittivity and permeability measurement task. First of all, the sample may only be placed at the maximum electric or magnetic field positions with the accuracy of the experiment’s eyesight. As derived from the cavity perturbation theory, the standard cavity perturbation measurement formulas (equations 2.11(a-b) and 2.12 (a-b)) assume that a sample places at the maximum field location. A sample not situated at the maximum field location under measurement causes some degrees of error in obtaining the permittivity and permeability results. Another measurement error source is related with calculating the resonant frequency and the Qfactor from the measured discrete S-Parameter points. Due the finite measurement frequency points of a Vector Network Analyzer, a curve fitting technique is applied to the measured S-Parameter data each time. Applied a curve fitting to the discrete measured SParameter data reduces the inaccuracy in resolving the resonant frequency and the Qfactor, but not totally eliminate the error associated with that. Finally, a modified rectangular cavity perturbation technique has been introduced and a new set of formulas have been derived in accommodating with a special measurement case namely when the material sample under measurement has insufficient sample length comparing to the 40 height of cavity resonator. These newly derived formulas are more general than the standard forms and have been proved to be accurate in yielding consistent results by measuring different samples with different lengths. 41 Appendix A Measured Resonant Frequencies for Each Designed Rectangular Cavity Resonator -45.0 -55.0 -65.0 l S21 l dB -75.0 -85.0 -95.0 -105.0 -115.0 -125.0 -135.0 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 f (GH) Figure A.1 Measured resonant frequencies for H band rectangular waveguide cavity resonator. -40.0 -50.0 l S21 l dB -60.0 -70.0 -80.0 -90.0 -100.0 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 f (GHZ) Figure A.2 Measured resonant frequencies for C band rectangular waveguide cavity resonator. 42 0 -10 -20 S21 (dB) -30 -40 -50 -60 -70 -80 -90 8.0 9.0 10.0 11.0 12.0 f (GHz) Figure A.3 Measured resonant frequencies for X band rectangular waveguide cavity resonator. -30.0 -40.0 l S21 l dB -50.0 -60.0 -70.0 -80.0 -90.0 12.4 13.4 14.4 15.4 16.4 17.4 f (GHz) Figure A.4 Measured resonant frequencies for Ku band rectangular waveguide cavity resonator. 43 -30.0 -40.0 l S21 l dB -50.0 -60.0 -70.0 -80.0 -90.0 -100.0 17.5 18.5 19.5 20.5 21.5 22.5 23.5 24.5 25.5 26.5 f (GHz) Figure A.5 Measured resonant frequencies for K band rectangular waveguide cavity resonator. 44 Appendix B The General TRL Calibration Technique The S parameters of DUT can be measured using the setup shown in figure B.1. For a cascade network measurement, it’s convenient to convert S parameters measurement to ABCD parameter measurement first and then convert the results back to S parameters. Figure B.1 Block diagram of a network analyzer measurement of a two port device. Figure B.2-B.7 show general error characterization for TRL calibration technique. Figure B.2 Block diagram for Thru connection. 45 Figure B.3 Signal flow graph for Thru connection. Figure B.4 Block diagram for Reflect connection. Figure B.5 Signal flow graph for Reflect connection. Figure B.6 Block diagram for Line connection. 46 Figure B.7 Signal flow graph for Line connection. The ABCD parameters for the DUT can be found as follow: (B.1) Table B.1 A discrete measured S-Parameters data retrieved from VNA. Real part Imaginary part l S21 l in dB 4.11E-05 1.44E-05 1.58E-05 1.14E-05 4.75E-05 3.53E-05 5.74E-06 2.35E-05 1.66E-05 3.12E-05 2.81E-05 2.71E-05 3.84E-05 2.50E-05 1.66E-05 3.79E-05 2.35E-05 3.09E-05 2.29E-05 3.59E-05 2.99E-05 2.63E-05 2.44E-05 8.24E-06 3.14E-05 3.37E-05 4.01E-05 4.85E-05 2.95E-05 3.52E-04 3.61E-04 3.72E-04 3.74E-04 3.76E-04 3.53E-04 3.81E-04 3.67E-04 3.64E-04 4.02E-04 3.92E-04 3.93E-04 3.83E-04 4.03E-04 3.80E-04 3.93E-04 4.06E-04 4.04E-04 4.25E-04 4.22E-04 4.09E-04 4.27E-04 4.21E-04 4.36E-04 4.29E-04 4.23E-04 4.31E-04 4.47E-04 4.39E-04 -69.021 -68.853 -68.588 -68.535 -68.419 -69.003 -68.377 -68.689 -68.768 -67.881 -68.119 -68.088 -68.293 -67.885 -68.402 -68.070 -67.808 -67.841 -67.428 -67.470 -67.736 -67.375 -67.492 -67.199 -67.323 -67.453 -67.271 -66.949 -67.140 Frequency point (MHz) 8185.000 8185.375 8185.750 8186.125 8186.500 8186.875 8187.250 8187.625 8188.000 8188.375 8188.750 8189.125 8189.500 8189.875 8190.250 8190.625 8191.000 8191.375 8191.750 8192.125 8192.500 8192.875 8193.250 8193.625 8194.000 8194.375 8194.750 8195.125 8195.500 47 3.90E-05 1.92E-05 4.04E-05 3.38E-05 3.21E-05 4.90E-05 1.26E-05 4.17E-05 1.98E-05 3.90E-05 2.60E-05 4.28E-05 5.03E-05 3.82E-05 4.14E-05 4.20E-05 3.33E-05 4.08E-05 3.46E-05 3.07E-05 3.63E-05 6.23E-05 5.16E-05 5.11E-05 4.28E-05 3.23E-05 6.08E-05 5.12E-05 3.72E-05 3.15E-05 4.93E-05 6.66E-05 5.51E-05 7.27E-05 6.87E-05 6.47E-05 5.50E-05 4.42E-05 3.56E-05 5.92E-05 7.62E-05 6.00E-05 7.28E-05 6.53E-05 6.66E-05 6.98E-05 4.83E-05 5.42E-05 5.66E-05 7.68E-05 5.59E-05 6.74E-05 8.13E-05 4.42E-04 4.49E-04 4.60E-04 4.69E-04 4.74E-04 4.65E-04 4.68E-04 4.44E-04 4.79E-04 4.79E-04 4.84E-04 4.72E-04 4.97E-04 4.90E-04 5.00E-04 5.14E-04 5.26E-04 5.06E-04 5.26E-04 5.09E-04 5.34E-04 5.51E-04 5.17E-04 5.56E-04 5.46E-04 5.67E-04 5.40E-04 5.49E-04 5.62E-04 5.66E-04 5.87E-04 5.95E-04 5.51E-04 5.82E-04 5.80E-04 6.11E-04 6.46E-04 6.42E-04 6.38E-04 6.33E-04 6.17E-04 6.28E-04 6.47E-04 6.73E-04 6.55E-04 6.83E-04 6.87E-04 6.69E-04 6.87E-04 7.03E-04 7.30E-04 7.20E-04 7.38E-04 -67.065 -66.940 -66.716 -66.559 -66.457 -66.602 -66.588 -67.005 -66.388 -66.361 -66.282 -66.484 -66.037 -66.174 -65.998 -65.750 -65.561 -65.895 -65.565 -65.853 -65.431 -65.120 -65.695 -65.063 -65.231 -64.907 -65.305 -65.164 -64.989 -64.932 -64.590 -64.457 -65.132 -64.633 -64.666 -64.228 -63.769 -63.834 -63.886 -63.936 -64.130 -64.000 -63.724 -63.403 -63.631 -63.261 -63.241 -63.461 -63.233 -63.005 -62.709 -62.814 -62.582 8195.875 8196.250 8196.625 8197.000 8197.375 8197.750 8198.125 8198.500 8198.875 8199.250 8199.625 8200.000 8200.375 8200.750 8201.125 8201.500 8201.875 8202.250 8202.625 8203.000 8203.375 8203.750 8204.125 8204.500 8204.875 8205.250 8205.625 8206.000 8206.375 8206.750 8207.125 8207.500 8207.875 8208.250 8208.625 8209.000 8209.375 8209.750 8210.125 8210.500 8210.875 8211.250 8211.625 8212.000 8212.375 8212.750 8213.125 8213.500 8213.875 8214.250 8214.625 8215.000 8215.375 48 5.84E-05 5.63E-05 6.41E-05 7.04E-05 8.45E-05 8.21E-05 5.87E-05 9.81E-05 7.57E-05 7.69E-05 8.74E-05 8.14E-05 8.02E-05 1.00E-04 1.13E-04 8.34E-05 1.29E-04 1.02E-04 9.41E-05 1.25E-04 9.97E-05 1.03E-04 1.26E-04 1.04E-04 1.00E-04 1.27E-04 1.21E-04 1.39E-04 1.38E-04 1.49E-04 1.42E-04 1.37E-04 1.41E-04 1.52E-04 1.86E-04 1.54E-04 1.73E-04 1.77E-04 1.76E-04 1.88E-04 2.06E-04 2.08E-04 1.72E-04 2.54E-04 2.51E-04 2.58E-04 2.58E-04 2.89E-04 2.96E-04 2.93E-04 3.23E-04 3.40E-04 3.66E-04 7.19E-04 7.53E-04 7.43E-04 7.86E-04 7.58E-04 7.89E-04 7.88E-04 7.94E-04 8.25E-04 8.27E-04 8.46E-04 8.53E-04 8.74E-04 8.75E-04 8.81E-04 9.17E-04 9.05E-04 9.45E-04 9.69E-04 9.51E-04 1.00E-03 1.01E-03 1.00E-03 1.03E-03 1.05E-03 1.05E-03 1.05E-03 1.10E-03 1.13E-03 1.16E-03 1.16E-03 1.19E-03 1.20E-03 1.22E-03 1.25E-03 1.27E-03 1.31E-03 1.32E-03 1.36E-03 1.39E-03 1.43E-03 1.45E-03 1.49E-03 1.54E-03 1.56E-03 1.61E-03 1.66E-03 1.72E-03 1.76E-03 1.80E-03 1.84E-03 1.91E-03 1.98E-03 -62.839 -62.439 -62.550 -62.062 -62.354 -62.007 -62.041 -61.938 -61.632 -61.610 -61.404 -61.340 -61.131 -61.104 -61.026 -60.713 -60.780 -60.444 -60.230 -60.364 -59.959 -59.876 -59.921 -59.687 -59.561 -59.475 -59.524 -59.091 -58.868 -58.613 -58.612 -58.426 -58.389 -58.204 -57.957 -57.858 -57.567 -57.522 -57.247 -57.032 -56.809 -56.682 -56.463 -56.154 -56.044 -55.769 -55.498 -55.183 -54.977 -54.800 -54.581 -54.229 -53.916 8215.750 8216.125 8216.500 8216.875 8217.250 8217.625 8218.000 8218.375 8218.750 8219.125 8219.500 8219.875 8220.250 8220.625 8221.000 8221.375 8221.750 8222.125 8222.500 8222.875 8223.250 8223.625 8224.000 8224.375 8224.750 8225.125 8225.500 8225.875 8226.250 8226.625 8227.000 8227.375 8227.750 8228.125 8228.500 8228.875 8229.250 8229.625 8230.000 8230.375 8230.750 8231.125 8231.500 8231.875 8232.250 8232.625 8233.000 8233.375 8233.750 8234.125 8234.500 8234.875 8235.250 49 3.80E-04 4.08E-04 4.52E-04 4.82E-04 5.07E-04 5.37E-04 5.79E-04 6.28E-04 6.87E-04 7.64E-04 8.00E-04 8.87E-04 1.01E-03 1.11E-03 1.20E-03 1.38E-03 1.56E-03 1.77E-03 2.04E-03 2.41E-03 2.84E-03 3.40E-03 4.09E-03 5.02E-03 6.21E-03 7.76E-03 9.61E-03 1.17E-02 1.35E-02 1.43E-02 1.36E-02 1.18E-02 9.67E-03 7.68E-03 6.05E-03 4.75E-03 3.82E-03 3.09E-03 2.54E-03 2.09E-03 1.77E-03 1.48E-03 1.25E-03 1.12E-03 9.74E-04 8.44E-04 7.27E-04 6.45E-04 5.70E-04 5.39E-04 4.70E-04 4.30E-04 4.00E-04 2.03E-03 2.11E-03 2.17E-03 2.24E-03 2.36E-03 2.43E-03 2.55E-03 2.65E-03 2.76E-03 2.88E-03 3.03E-03 3.17E-03 3.34E-03 3.52E-03 3.74E-03 3.94E-03 4.18E-03 4.45E-03 4.72E-03 5.05E-03 5.40E-03 5.79E-03 6.15E-03 6.52E-03 6.76E-03 6.82E-03 6.37E-03 5.19E-03 2.92E-03 -1.32E-04 -3.34E-03 -5.66E-03 -6.96E-03 -7.41E-03 -7.37E-03 -7.04E-03 -6.65E-03 -6.19E-03 -5.77E-03 -5.38E-03 -5.02E-03 -4.70E-03 -4.41E-03 -4.14E-03 -3.90E-03 -3.69E-03 -3.47E-03 -3.32E-03 -3.18E-03 -3.04E-03 -2.89E-03 -2.76E-03 -2.64E-03 -53.683 -53.359 -53.105 -52.790 -52.338 -52.067 -51.661 -51.294 -50.923 -50.525 -50.084 -49.660 -49.138 -48.649 -48.126 -47.584 -47.000 -46.397 -45.776 -45.049 -44.295 -43.457 -42.628 -41.695 -40.747 -39.715 -38.764 -37.872 -37.217 -36.920 -37.084 -37.663 -38.480 -39.435 -40.415 -41.414 -42.307 -43.195 -44.007 -44.778 -45.476 -46.138 -46.779 -47.354 -47.913 -48.448 -48.995 -49.404 -49.803 -50.213 -50.660 -51.069 -51.465 8235.625 8236.000 8236.375 8236.750 8237.125 8237.500 8237.875 8238.250 8238.625 8239.000 8239.375 8239.750 8240.125 8240.500 8240.875 8241.250 8241.625 8242.000 8242.375 8242.750 8243.125 8243.500 8243.875 8244.250 8244.625 8245.000 8245.375 8245.750 8246.125 8246.500 8246.875 8247.250 8247.625 8248.000 8248.375 8248.750 8249.125 8249.500 8249.875 8250.250 8250.625 8251.000 8251.375 8251.750 8252.125 8252.500 8252.875 8253.250 8253.625 8254.000 8254.375 8254.750 8255.125 50 3.35E-04 3.25E-04 2.71E-04 2.58E-04 2.49E-04 2.18E-04 2.13E-04 1.94E-04 1.94E-04 1.69E-04 1.57E-04 1.46E-04 1.42E-04 -2.55E-03 -2.44E-03 -2.37E-03 -2.30E-03 -2.20E-03 -2.14E-03 -2.06E-03 -2.02E-03 -1.94E-03 -1.89E-03 -1.83E-03 -1.78E-03 -1.73E-03 -51.807 -52.165 -52.463 -52.726 -53.098 -53.363 -53.689 -53.862 -54.215 -54.421 -54.711 -54.968 -55.188 8255.500 8255.875 8256.250 8256.625 8257.000 8257.375 8257.750 8258.125 8258.500 8258.875 8259.250 8259.625 8260.000 fo of empty cavity with capillary tube - 30. 0 8185 - 35. 0 8195 8205 8215 8225 8235 8245 8255 - 40. 0 l S21 l dB - 45. 0 - 50. 0 - 55. 0 - 60. 0 - 65. 0 - 70. 0 - 75. 0 f ( GHz) Figure B.8 Resonant frequency generated from discrete data lists in table B.1. 51 Figure B.9 Calculation of resonant frequency and Q factor for the discrete data lists in table B.1. Debye’s Equation Debye’s equation is used to carry out the computation of complex permittivity of materials as follows: ε = ε∞ + εs − ε∞ 1 + jωτ (B.2) where ε is the complex permittivity εs is the static permittivity ε∞ is the permittivity at the infinite frequency τ is the relaxation time Table B.2. Debye’s parameters for chlorobenzene from literature Materials ε∞ εs τ (pico-sec) Chlorobenzene 2.3 5.54[19] 10.3[19] 52 Appendix C Calculation of Ne Due to Image Effect Figure C.1 Image effect due to the metallic ground plates. Let A= the area of cross section of sample, 2h = length of sample and 2H = height of cavity resonator. The total depolarizing electric field due to the polarization of the sample and its images can be written as (C.1) where 53 (C.2) is the magnitude of the polarization of the first image A1 B1. Use the standard result (C.3) lead to (C.4) Therefore, Ne can be defined as: (C.5) 54 Publications [1] Mi Lin, Yong Wang, and Mohammed N. Afsar, “Precision measurement of complex permittivity and permeability by microwave cavity perturbation technique”, presented at The Joint 30th International Conference on Infrared and Millimeter Waves and 13th International Conference on Terahertz Electronics, Williamsburg, Virginia, September, 2005. [2] Mi Lin and Mohammed N. Afsar, “Measurement of dielectric and magnetic characteristics of nickel ferrite and strontium ferrite composite from 4.5 to 26.5 GHz frequency range”, accepted by IEEE IMTC 2006 – IEEE Instrumentation and Measurement Technology Conference, Sorrento, Italy, April, 2006. [3] Mi Lin and Mohammed N. Afsar, “Cavity perturbation measurement of dielectric and magnetic properties of ferrite materials in microwave frequency range”, accepted by IEEE INTERMAG 2006 – IEEE International Magnetics Conference, San Diego, California, May, 2006. [4] Mi Lin and Mohammed N. Afsar, “A new cavity perturbation technique for accurate measurement of dielectric parameters”, accepted by IEEE MTT-S International Microwave Symposium, San Francisco, California, June, 2006. 55 Bibliographies [1] K. T. Matthew and U. Raveendranath, “Cavity Perturbation Techniques for Measuring Dielectric Parameters of Water and Other Allied Liquids,” Sensors Update, vol. 7, iss.1, pp185-210, Weinheim, Fed. Rep. of Germany, WILLEY, 2000. [2] Kjetil Folgero and Tore Tjomsland, Permittivity measurement of thin layers using open-ended coaxial probes, Meas.Sci.Technol. 7, 1996, pp. 1164-1173. [3] Rene Grignon, Mohammed N. Afsar, Yong Wang, and Saquib Butt, “Microwave broadband free-space complex dielectric permittivity measurements on low loss solids,” IEEE Instrumentation and Measurement Technology Conference, Colorado, USA, pp. 865-870, May 2003. [4] Yong Wang and Mohammed N. Afsar, Measurement of complex permittivity of liquids using waveguide techniques, Progress In Electromagnetics Research, PIER 42, 131-142, 2003. [5] Mohammed N. Afsar and Kenneth J. Button “Millimeter-wave Dielectric Measurement of Materials”, Proceeding of the IEEE, vol.73, no.1, January 1985. [6] H.A. Bethe and J. Schwinger, NRDC report D1-117, Cornell University, Ithaca, NY, 1943. [7] R. A. Waldron, “Perturbation theory of resonant cavities,” Proc. Inst. Elec. Eng., vol. 107C, p. 272, 1960. [8] David M. Pozar, Microwave Engineering, Second Edition. [9] Camelia Gabriel, Sami Gabriel, Edward H. Grant, Ben S. J. Halstead, and D. Michael P. Mingos, “Dielectric parameters relevant to microwave dielectric heating,” Chemical Society Reviews, vol.27, pp213-223, 1998. [10] Elise C. Fear, Xu Li, Susan C. Hagness, and Maria A. Stuchly, “Confocal Microwave Imaging for Breast Cancer Detection: Localization of Tumors in Three Dimensions,” IEEE Trans. Biomed. Eng., vol. 49, no.8, pp812-822, Aug. 2002. [11] Fabienne Duhamel, Isabelle Huynen, and Andre Vander Vorst, “Measurements of complex permittivity of biological and organic liquids up to 110 GHz,” IEEE MTT-S digest, 1997, pp.107-110. [12] R. A. Waldron, “Theory of guided electromagnetic waves,” London, England: Van 56 Nostrand, 1970, ch. VI, pp. 292-318. [13] Mi Lin, Yong Wang, and Mohammed N. Afsar, “Precision measurement of complex permittivity and permeability by microwave cavity perturbation technique,” presented at 30th Int.Conf. Infrared and Millimeter Waves, Virginia USA, 2005 [14] R. M. Bozorth and D. M. Chapin, “Demagnetizing factors of rods,” J. Appl. Phy., vol. 13, pp. 320-326, May.1942. [15] J.A. Osborn, “Demagnetizing factors of the general ellipsoid,” phy. Rev., vol.67, no.11, pp. 351-357, Jun. 1945. [16] A. Sihvola, P.Y.Oijala, S.Jarvenpaa, and J.Avelin, “Polarizabilities of platonic solids,” IEEE Trans. Antennas Propagat., vol. 52, no.9, Sept. 2004. [17] A. Parkash, J.K.Vaid, and A. Mansingh, “Measurement of dielectric parameters at microwave frequencies by cavity-perturbation technique,” IEEE Trans. Microwave Theory Tech., vol.MTT-27, no.9, Sept. 1979. [18] Adil Bahadoor, Yong Wang, and Mohammed N. Afsar, “Complex permittivity and permeability of barium and strontium ferrite powders in X, Ku, and K-band frequency ranges,” J. Appl. Phys. Vol.97,10F105, May. 2005. [19] V.P. Pawar and S.C. Mehrota, “Dielectric relaxation study of liquids having chloro- group with associated liquids,” Journal of solution chemistry, Vol. 31, July 2002. 57

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