• 沒有找到結果。

時域反射量測技術改良及於水土混和物之應用

N/A
N/A
Protected

Academic year: 2021

Share "時域反射量測技術改良及於水土混和物之應用"

Copied!
242
0
0

加載中.... (立即查看全文)

全文

(1)

國 立 交 通 大 學

土木工程學系

博 士 論 文

時域反射量測技術改良及於水土混和物之應用

Improved Time Domain Reflectometry Measurements and Its

Application to Characterization of Soil-Water Mixtures

研 究 生:鐘志忠

指導教授:林志平 教授

(2)

Improved Time Domain Reflectometry Measurements and Its

Application to Characterization of Soil-Water Mixtures

研 究 生:鐘志忠 Student:Chih-Chung Chung

指導教授:林志平 Advisor:Chih-Ping Lin

國 立 交 通 大 學

土 木 工 程 學 系

博 士 論 文

A Thesis

Submitted to Department of Civil Engineering College of Engineering

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Civil Engineering

July 2008

HsinChu, Taiwan, Republic of China

中華民國九十七年七月

(3)

時域反射量測技術改良及於水土混和物之應用

學生:鐘志忠

指導教授:林志平 博士

國立交通大學土木工程學系

中文摘要

時域反射法(Time domain reflectometry, TDR)為可量測物體的視介電度(Apparent dielectric constant)、導電度(Electrical conductivity)以及介電頻譜(Dielectric spectroscopy) 特性之新穎技術,且具現地量測及多工(Multi-function)優勢,因此近來被廣泛應用至土 壤或混擬土等材料性質量測。然而目前TDR 視介電度及導電度量測方法有待釐清之處, 且 TDR 的介電頻譜量測方法過於複雜,高頻量測結果可靠性不足,所以本研究的目的 在於發展改良TDR 量測方法,並提出 TDR 感測器製作原則,提供正確穩定的量測資料, 以應用於土水混和物之電學性質探討。 TDR 視介電度可由不同走時分析方法求得,但此一量測值缺乏實際物理意義,因此 本研究採用數值模擬方法,探討視介電度及其等效對應頻率受材料導電度、介電頻散特 性及纜線阻抗等因子影響,瞭解視介電度量測實務限制,進而提倡介電頻譜量測之重要 性。本文並以實際量測與數值模擬,分析介電頻譜量測靈敏度與可靠度,探討介電頻譜 量測誤差來源與可能改善方法。鑑於TDR 介電頻譜高頻量測限制,本研究發展 TDR 頻 率域相位速度分析方法,模擬分析結果顯示此一方法能有效提供材料高頻的相位速度, 且於土壤含水量量測應用,不受導電度、土壤種類與纜線電阻影響,具有極大發展潛力。

(4)

本研究另外針對TDR 導電度量測,建立考慮纜線電阻之 TDR 導電度量測理論與系統誤 差率定方法,配合模擬分析與實驗結果,並考量穩定反應所需之資料擷取時間長度,證 實所提理論與率定方法可克服系統電阻以及系統誤差影響,提供材料正確導電度量測。 本研究另採用3D 電磁波模擬分析工具,以及量測靈敏度理論推導,提供 TDR 感測 器製作原則。為達到深層土壤(岩石)性質探討目的,研發之 TDR 圓錐貫入器 (TDR penetrometer)可同時提供材料之視介電度及導電度量測。配合此一 TDR 圓錐貫入器,於 石門水庫進行水庫底層淤泥性質探討,基於底泥導電度量測結果,可推估底層淤泥含量 與工程物理性質。由於近來台灣水庫因洪颱事件而產生大量入庫泥沙之危機,本研究進 一步利用視介電度與感測器製作研究成果,發展新式走時分析方法以及 TDR 相位速度 分析方法,研發高精度懸浮泥沙濃度量測技術,期以TDR 多工多點的優勢,建置 TDR 自動化懸浮泥沙濃度量測。唯未來研究將建議持續現場量測驗證,以提供穩定量測資料。

(5)

Improved Time Domain Reflectometry Measurements and Its

Application to Characterization of Soil-Water Mixtures

Student: Chih-Chung Chung Advisor: Dr. Chih-Ping Lin

Department of Civil Engineering National Chiao Tung University

ABSTRACT

Time domain reflectometry (TDR) can be used to measure apparent dielectric constant, electrical conductivity, and dielectric permittivity as a function of frequency. This relatively new technique is gaining popularity in characterization of engineering materials, such as suspended suspension, soil, concrete, etc, due to its versatility and applicability in field measurements. However, some disputes about the measurement methods for apparent dielectric constant and electrical conductivity (EC) have not been resolved. And dielectric spectroscopy remains relatively difficult in practice. The objectives of this study were to investigate and improve the TDR measurement techniques, provide guidelines for TDR probe design, and, as an application example, apply TDR to characterization of soil-water mixture.

Since the apparent dielectric constant derived from various travel time analyses dose not have clear physical meanings, this study first investigated the influence factors, such as electrical conductivity, dielectric dispersion, and cable resistance, and associated effective frequencies. The applicability and limitations of travel time analyses are revealed with emphasis on the importance of dielectric spectroscopy. The dielectric spectrum, although

(6)

sensitivity and reliability of dielectric spectroscopy to identify the source of uncertainty and provide preferred guidelines. A novel approach to obtain dielectric permittivity at high frequencies was proposed based on the frequency domain phase velocity.

The TDR EC measurement is more straightforward, but methods accounting for the cable resistance remain controversial; and the effect of TDR recording time has been underrated when long cables are used. A comprehensive full waveform model and the DC analysis were used to show the correct method for taking account of cable resistance and guideline for selecting proper recording time. In addition, a system error in typical TDR EC measurements was identified and a countermeasure was proposed, leading to a complete and accurate procedure for TDR EC measurements.

Following the studies on TDR dielectric permittivity and EC measurements, this study further investigated the factors associated with probe designs for both types of measurements. The sensitivity of TDR measurements as affected by the probe parameters was discussed to provide guidelines for probe design. In addition, a penetrometer type of TDR probes was developed to allow simultaneous measurements of dielectric permittivity and electrical conductivity during cone penetration for measurements at depths.

Although the aforementioned TDR measurement methodology was originally developed with soil applications in mind, the sediment problems in Shihmen Reservoir manifested by the Typhoon Aere in 2004 provides imperative opportunities for TDR applications. The TDR measurement techniques were adapted for characterization of soil-water mixtures. TDR penetrometer was integrated with the Marchetti dilatometer (DMT) and the TDR/DMT probe was pushed into the bottom mud to determine simultaneously, the solid concentration, stiffness and stress state of the bottom mud. A novel TDR probe and measurement procedure were further developed for accurate monitoring of suspended sediment concentration in fluvial and reservoir environment.

(7)

誌謝

本論文得以順利完成,首先感謝我的指導老師林志平博士亦師亦友的指導,讓我在 研究與為人處事上獲益匪淺,在此致上萬分感謝。另外在求學期間,承蒙方永壽老師、 單信瑜老師、黃安斌老師、廖志中老師以及潘以文老師指導,受益良多,在此由衷感謝 各位老師。 論文口試期間,承蒙中興大學蘇苗彬老師、香港科技大學王幼行老師、經濟部水利 署水文技術組組長洪銘堅博士、交通大學電信系張志楊老師、以及廖志中老師、潘以文 老師以及葉克家老師撥冗蒞臨指導,針對學生論文給予寶貴的建議,在此亦致上敬意。 感謝各位學長姐以及學弟妹在課業與研究上的協助幫忙,尤其平時相處時候大家相 互扶持,一起歡樂也一起努力,這些回憶感激有你們的參與。特別是本門的伙伴們,士 弘、宗盛、瑛鈞、吉勇、致翔、培熙、志龍、逸龍、俊宏、証傑、和翰、仁弘、育嘉、 文欽、奕全、浚昇、瑋晉、岳勳、哲毅、永政、智棟以及新進學弟妹,有了你們的參與, 才能如此豐富這一段日子,特別感謝。 爸爸、媽媽、弟弟與妹妹,這些年來因求學在外,無法時時刻刻在家裡陪伴,感謝 你們在背後的支持與鼓勵,也感謝女友伶蓉這一路上的陪伴與支持,最後將本論文獻予 各位,謝謝。 民國九十七年七月 于 新竹

(8)

TABLE OF CONTENTS

中文摘要...i ABSTRACT ...iii 誌謝...v TABLE OF CONTENTS...vi LIST OF TABLES ...x

LIST OF FIGURES ...xi

LIST OF SYMBOLS...xx 1 Introduction ...1 1.1 Motivation ...1 1.2 Objectives ...2 2 Literature Review ...5 2.1 Introduction ...5

2.2 Electromagnetic Properties of Materials ...5

2.2.1 Basic Electromagnetic Properties...5

2.2.2 Dielectric Behavior of Water and Soil Solid ...13

2.2.3 Interfacial Polarization of Soil-water Mixture...15

2.2.4 Dielectric Mixing Model ...18

2.3 TDR Principle and Analysis ...25

2.3.1 Basics of TDR ...25

2.3.2 TDR Travel Time Analysis...28

2.3.3 TDR Electrical Conductivity Analysis ...34

2.3.4 TDR Waveform Modeling...37

(9)

2.3.6 TDR Probe Development and Performance ...45

3 TDR Dielectric Permittivity Analysis and Influence Factors...50

3.1 Implication of Travel Time Analysis ...50

3.1.1 Travel Time Calibration and Effective Frequency ...50

3.1.2 Evaluation of Influence Factors...51

3.2 Dielectric Spectrum Analysis ...65

3.2.1 Sensitivity Analysis and Reliability of Dielectric Spectroscopy...66

3.2.2 Frequency Domain Phase Velocity Method ...74

3.2.2.1 Principle of Frequency Domain Analysis of Phase Velocity...75

3.2.2.2 Proof of Concept...79

4 TDR EC Analysis ...89

4.1 Comprehensive Method of EC Analysis ...89

4.1.1 DC Lumped Circuit Model...89

4.1.2 Theoretical Assessment of EC Measurement ...92

4.1.3 Effect of Recording Time ...101

4.1.4 Experimental Verifications ... 110

4.2 Calibration of EC Measurement ... 112

4.2.1 Clarification of Reflection Coefficient in EC Measurement ... 112

4.2.2 TDR System Error... 115

4.2.3 Calibration Method and Verification ...120

5 TDR Probe Design ...129

5.1 Probe Rods Configuration and Boundary Effect...129

5.2 Sensitivity of Travel Time and EC Measurement...134

5.3 Development of TDR Penetrometer ...138

(10)

5.3.2 Calibration Method for Ka and EC ...141

5.3.3 Evaluation of TDR Penetrometer Performance...142

5.3.4 Simulated Penetration Tests...152

6 Applications: Characterization of Soil-water Mixture ...159

6.1 Introduction ...159

6.2 Characterization of Basal Sediment ...160

6.2.1 Ka/EC—Sediment Concentration Relationship by TDR Penetrometer.... ...161

6.2.2 Field Testing ...163

6.3 Characterization of Suspension ...170

6.3.1 Dielectric Spectrum Analysis of Suspended Sediment ...171

6.3.2 Theoretical Development of TDR SSC Measurement ...173

6.3.2.1 Dielectric Mixing Model for Suspended Sediment ...173

6.3.2.2 Sensitivity-Resolution Analysis...175

6.3.2.3 TDR Probe Design for SSC Measurement ...176

6.3.2.4 Temperature Effect and Correction Method...178

6.3.3 Evaluation of Measurement Performance ...180

6.3.3.1 Effect of Water Salinity ...183

6.3.3.2 Travel time - SSC Rating Curve...185

6.3.3.3 Effect of Soil Type and Particle Size...187

6.3.3.4 Effect of Lead Cable Length ...189

6.3.4 SSC Measurements using Frequency Domain Phase Velocity Method ... ...190

7 Conclusions ...196

(11)
(12)

LIST OF TABLES

Table 2-1 Relative permeability data for selected materials (modified from Cheng, 1989)

...7

Table 2-2 Electrical conductivities of materials (modified from Cheng, 1989)...8

Table 2-3 Relative permittivity of some often used materials [modified from Cheng, 1989]...12

Table 3-1 TDR system parameters ...53

Table 3-2 Cole-Cole parameters for material used in numerical simulations (modified from Friel and Or, 1999)...53

Table 3-3 The calibrated probe length (m) obtained from the air-water calibration for different cable lengths and methods of travel time analysis...58

Table 3-4 Volumetric mixing parameters ...80

Table 4-1 Simulation parameters...94

Table 4-2 Cole-Cole parameters for material used in numerical simulations ...94

Table 4-3 Fitted Kp (m-1)from laboratory measurements using a Campbell Scientific TDR100 ...124

Table 4-4 Percentage errors between the TDR EC measurements and conductivity meter EC measurements ...125

Table 5-1 Probe types with different conductor configurations. ...140

Table 6-1 Comparisons of derivation range due to salinity effect for each probe types ...184

(13)

LIST OF FIGURES

Fig. 2-1 Polarization mechanisms in single component materials (modified from Santamarina et al., 2001) ...13 Fig. 2-2 Frequency response of permittivity and loss factor for a hypothetical dielectric showing various contributing phenomena (Ramo et al., 1994)...13 Fig. 2-3 Surface-related polarization (modified from Santamarina et al., 2001) ...17 Fig. 2-4 Qualitative representation of dielectric properties of wet soils as a function of frequency (modified from Hilhorst and Dirkson, 1994)...17 Fig. 2-5 Dielectric spheres are guests in the dielectric background host. (after Sihvola, 1999)...20 Fig. 2-6 The geometry of an ellipsoid. The semi-axis ax, ax,and ax fix the Cartesian

co-ordinate system. (after Sihvola, 1999)...22 Fig. 2-7 Anisotropic ellipsoid (with permittivity εi ) in anisotropic environment

erε0). Ei refers to the internal electric field. (after Sihvola, 1999) ...22 Fig. 2-8 TDR system configuration...27 Fig. 2-9 A typical TDR response...27 Fig. 2-10 Illustration of various methods of travel time analysis: (a) locating the end

reflection by the dual tangent (A point) and single tangent (B point) methods; (b) the derivative methods locates the end reflection by the apex of the derivative (C point) (modified after Robinson et al., 2005)...33 Fig. 2-11 The optimal frequency range in which the dielectric permittivity is dominated by water content and least affected by electrical conductivity and dielectric dispersion due to soil-water interaction. (modified after Lin, 2003b) ...34 Fig. 2-12 Equivalent circuit of a cascade of uniform section for TDR system. (after Lin,

(14)

2003a) ...41 Fig. 2-13 The flow chart of the spectral algorithm (modified from Lin, 1999) ...42 Fig. 2-14 The comparison of TDR modeling with and without consideration of cable

resistance (after Lin and Tang, 2007) ...44 Fig. 2-15 Relative electric field intensity and energy storage density cross-sections for a variety of TDR probe designs. Configurations include (a) two rods, (b) three rods, (c) three rods with the center rod twice the diameter of the outer rods, (d) five rods, (e) parallel plates, and (f) parallel plates with the right-hand plate twice the length of the left-hand plate. (after Kirkby, 1996)...48 Fig. 3-1 The apparent dielectric constants as affected by electrical conductivity in (a) the non-dispersive case and (b) the dispersive case. ...55 Fig. 3-2 The equivalent frequency and frequency bandwidth corresponding silt loam case

...56 Fig. 3-3 Measurements in water with various cable lengths. ...59 Fig. 3-4 The apparent dielectric constants as affected by cable length in (a) the

non-dispersive case and (b) the dispersive case. ...60 Fig. 3-5 The equivalent frequency and frequency bandwidth corresponding to silt loam case ...61 Fig. 3-6 The apparent dielectric constants and frequency bandwidth by changing the

dielectric relaxation frequency while keeping other Cole-Cole parameters constant in (a) water and (b) silt loam. ...63 Fig. 3-7 The relationship between Ka from the dual tangent method and frequency

bandwidth due to different influences ...65 Fig. 3-8 Estimated frequency-dependent dielectric properties of a silt loam (after Lin,

(15)

Fig. 3-9 Equivalent capacitance and extra length for fringing effect...69 Fig. 3-10 Estimated dielectric spectrum of tap water from open-end and shorted-end

coaxial probe (modified after Tang, 2007)...70 Fig. 3-11 TDR abs(S11) response with different cable length as measuring the (left) tap water and (right) silt loam ...70 Fig. 3-12 Normalized sensitivity of abs(S11) due to εdc as measuring the (a) tap water and (b) silt loam...71 Fig. 3-13 Normalized sensitivity of abs(S11) due to ε∞ as measuring the (a) tap water and (b) silt loam...72 Fig. 3-14 Normalized sensitivity of abs(S11) due to length of probe (L) as measuring the (a) tap water and (b) silt loam...73 Fig. 3-15 Normalized sensitivity of abs(S11) due to impedance of probe (Zp) as measuring the (a) tap water and (b) silt loam...74 Fig. 3-16 Field configuration of SASW method and illustration of recording data

(modified after Foti, 2000) ...77 Fig. 3-17 Unwrapping process of the angle of cross-power spectrum for the SASW

method ...77 Fig. 3-18 (a) typical TDR waveform, and (b) its derivative...78 Fig. 3-19 The TDR frequency domain phase velocity method analogous to the SASW method ...78 Fig. 3-20 (a) The phase angle of the cross-spectral density (Δφ) of two characteristic

signals before unwrapping, (b) the results after unwrapping compared with theoretical values, and (c) the measured frequency domain phase velocity (Vph) compared with the theoretical values ...82

(16)

(Δφ) and (b) the error percentage of phase velocity compared with the theoretical one as a variety of EC. ...83 Fig. 3-22 The synthetic dielectric dispersion due to soil-water interaction and soil water content using four-component dielectric mixing model...85 Fig. 3-23 The estimated frequency domain phase velocity at 1GHz in term of dielectric constant (Ka) as affected by a variety of soil water content and soil type...86 Fig. 3-24 The estimated apparent dielectric constant (Ka) at 1GHz from frequency

domain phase velocity analysis for As = 200 and the apparent dielectric constants estimated by the single tangent method as affected by EC of free water (σfw) and soil water content ...87 Fig. 3-25 The estimated apparent dielectric constant (Ka) at 1GHz from frequency

domain phase velocity analysis for As = 200 and the apparent dielectric constants estimated by the single tangent method as affected by cable length and soil water content ...88 Fig. 4-1 (a) The mutli-section transmission line model of the TDR measurement system, (b)

the associated DC circuit model, and (c) a typical TDR waveform showing definition of reflection coefficient ρ...91 Fig. 4-2 Effect of cable resistance on TDR waveforms: (a) measured TDR waveforms compared with that predicted by the full waveform model in this study; (b) measured TDR waveforms in Fig. 5b of Castiglione and Shouse (2003). ...97 Fig. 4-3 Illustration of the nonlinear relationship between the steady state reflection

coefficient with 200-m RG-58 cable and that without cable resistance. ...99 Fig. 4-4 The estimated EC using the actual probe constant in three different methods

compared with the numerically-controlled true EC. ...100 Fig. 4-5 The estimated EC using the fitted probe constant in three different methods

(17)

compared with the numerically-controlled true EC. ...101 Fig. 4-6 Examples showing how (a) EC, (b) Zp and L of waveguide, and (c) dielectric permittivity affect the time required to reach the steady state, with time expressed as the time that includes multiples of roundtrip travel time in the probe section (t0) .106 Fig. 4-7 Recording time required to reach the steady state for probes (a) short-circuited, (b) in water of two electrical conductivities, and (c) open in air...107 Fig. 4-8 The effect of recording time, expressed as the time that includes multiples of roundtrip travel time in the probe section, on the estimated EC using series resisters model with (a) Rcable measured and β fitted, (b) Rcable and β fitted, (c) Castiglione-Shouse method with actual β determined, and (d) Castiglione-Shouse method with β fitted...108 Fig. 4-9 The effect of recording time, expressed as multiples of roundtrip travel time in the lead cable, on the estimated EC using series resisters model with (a) Rcable measured and β fitted, (b) Rcable and β fitted, (c) Castiglione-Shouse method with actual β determined, and (d) Castiglione-Shouse method with β fitted...109 Fig. 4-10 The effect of recording time, expressed as multiples of roundtrip travel time in the lead cable, on the estimated probe constant β using (a) series resisters model, and (b) Castiglione-Shouse method ... 110 Fig. 4-11 Electrical conductivity measured by TDR compared with that measured by YSI conductivity meter. ... 112 Fig. 4-12 Theoretical values of EC-associated reflection coefficient ρ′∞ for three distinct electrical conductivities in the case of (a) zero cable resistance and (b) non-zero cable resistance. ... 115 Fig. 4-13 (a) The relationship between instrument reflection coefficient ρ and

(18)

source voltage) due to imperfect amplitude calibration at the 50 Ω level; (b) the effect of instrument error on TDR EC in high EC range and (c) in low EC range ... 118 Fig. 4-14 (a) Effect of 20m RG-58 cable resistance on steady-state reflection coefficient

ρ′∞ and the scaled steady-state reflection coefficient by the Castiglione-Shouse method and series resistors model; (b) deviation of the estimated EC from true EC for the Castiglione-Shouse method and series resistors model using actual probe constant

... 119 Fig. 4-15 (a) Original TDR waveforms from a Tektronix 1502C and (b) the associated corrected waveforms using calibration equation Eq. [4-10]...126 Fig. 4-16 (a) Original TDR waveforms from a Campbell Scientific TDR100 and (b) the associated corrected waveforms using calibration equation Eq. [4-10]. ...127 Fig. 4-17 TDR EC measurements made by a Campbell Scientific TDR100 (a) without reflection coefficient calibration and (b) with reflection coefficient calibration....128 Fig. 5-1 Setup of the electromagnetic simulation in Ansoft HFSS® for a three-rod TDR probe. ...131 Fig. 5-2 Details of the metal wires connecting the probes and the coaxial cable. ...132 Fig. 5-3 E-field plots of a three-rod shielded TDR probe on the yz plane with probe ends open (left) and shorted (right)...132 Fig. 5-4 E-field plots of a three-rod unshielded TDR probe on the yz plane with probe ends open (left) and shorted (right). ...133 Fig. 5-5 E-field plot of the L-shaped three-rod shielded TDR probe on the xz plane..133 Fig. 5-6 E-field plots of a two-rod shielded TDR probe on the yz plane with probe ends open (left) and shorted (right)...134 Fig. 5-7 E-field plots of a two-rod unshielded TDR probe on the yz plane with probe ends open (left) and shorted (right)...134

(19)

Fig. 5-8 The sensitivity of the travel time due to Ka and probe length ...136

Fig. 5-9 The sensitivity of the EC due to probe length and impedance ...137

Fig. 5-10 Interpretation of the TDR waveform to determine apparent dielectric constant. ...140

Fig. 5-11 The TDR waveforms of probes with different number of conductors...145

Fig. 5-12 The TDR waveforms of probes with different conductor spacing (width)...146

Fig. 5-13 Spatial weighting function for dielectric constant Ka. ...148

Fig. 5-14 Spatial weighting function for electrical conductivity σ. ...149

Fig. 5-15 Prototype of the TDR penetrometer ...150

Fig. 5-16 Ka calibration of TDR penetrometer with known values materials...151

Fig. 5-17 σ calibration of TDR penetrometer with known EC liquids...152

Fig. 5-18 Correlation between √Ka and volumetric water content...155

Fig. 5-19 Correlation between √σ and volumetric water content...155

Fig. 5-20 The Ka obtained from TDR penetrometer vs. that from MRP...157

Fig. 5-21 The electrical conductivity obtained from TDR penetrometer vs. that from MRP...158

Fig. 6-1 Soil classification and density estimation based on DMT (after Marchetti and Crapps, 1981) ...161

Fig. 6-2 Relationship between dielectric constant and sediment concentration...162

Fig. 6-3 Relationship between electrical conductivity and sediment concentration ....163

Fig. 6-4 (a) The photo of TDR/DMT probe and (b) the schematic illustration of TDR/DMT probe ...165

Fig. 6-5 Operation of TDR/DMT from a barge...166

Fig. 6-6 The test locations ...166

(20)

Fig. 6-8 The interpreted TDR test results...169 Fig. 6-9 The DMT test results ...169 Fig. 6-10 The inversion result (model free) of dielectric spectrum of a variety of

suspended sediment concentration (modified after Tang, 2007)...173 Fig. 6-11 Theoretical measurement resolution of soil volume affected by sampling

interval (dt) and probe length (L) ...177 Fig. 6-12 Six types of TDR probe for SSC measurement ...182 Fig. 6-13 Particle size distribution and relative density (Gs) of Shihmen clay, ChiChi silt, and silica silt ...182 Fig. 6-14 The salinity effect for (a) travel time Δτ and (b) estimated error of the 70cm open probe using the derivative method...185 Fig. 6-15 Rating curve of travel time Δτ and Shihmen clay volumetric volume (SS)

within background water with two salinity contents, and error bar represents experimental data with 2 standard deviation ...186 Fig. 6-16 The measurement error from rating curve of travel time Δτ and Shihmen clay volumetric volume within background water with two salinity contents...187 Fig. 6-17 Rating curve of travel time Δτ with Shihmen clay, silica silt and ChiChi silt, and error bar represents experimental data with 2 standard deviation ...188 Fig. 6-18 The measurement error from rating curve of travel time Δτ with Shihmen clay, silica silt and ChiChi silt...189 Fig. 6-19 Rating curve comparison for silica silt with QR320 lead cable length = 2m,

15m, and 25m ...190 Fig. 6-20 Rating curve of Ka and Shihmen clay volumetric volume (SS) within

background water with two salinity contents, and error bar represents experimental data with 2 standard deviation...191

(21)

Fig. 6-21 The measurement error from rating curve of Ka and Shihmen clay volumetric volume within background water with two salinity contents...192 Fig. 6-22 Rating curve of Ka with Shihmen clay, silica silt and ChiChi silt, and error bar represents experimental data with 2 standard deviation...193 Fig. 6-23 The measurement error from rating curve of Ka with Shihmen clay, silica silt and ChiChi silt...194 Fig. 6-24 Rating curve comparison for silica silt with QR320 lead cable length = 2m,

(22)

LIST OF SYMBOLS

A: Resistance correction factor

As: Specific surface area of soil particle (m2 g-1) B: Magnetic flux density (Wb m-2)

c: Light velocity in vacuum (2.998 * 108 m sec-1) E: Electrical field (N C−1or volt m-1)

Fm: Magnetic force (N)

Fr: Volumetric percentage of inclusion H: Magnetic field (A m-1)

H(f): TDR system function

f: Frequency (Hz)

fbw: Frequency bandwidth (Hz)

frel: Relaxation frequency (Hz) J: Current density (A m-2)

j: −1

Ka: Apparent dielectric constant

Kp: Geometric factor of TDR probe (m-1)

Kss: Apparent dielectric constant of soil solid

Kw: Apparent dielectric constant of water

L: Probe length (m)

M: Magnetization (A m-1)

Nx: Depolarization factor in the x-direction of Cartesian coordinate system P: Polarization vector (C m-2)

q: Electric charge (-1.602 * 10−19 C)

Rcable: Cable resistance (Ω)

RL: Load resistance (Ω)

Rsample: Sample resistance (Ω) S11: System scatter function

SS: Volumetric percentage of suspended sediment m

S): Normalized sensitivity by physical parameter m

t0: Time difference between selected point and start reflection point (sec)

(23)

ts: True travel time in the measuring probe (sec) u: velocity vector (m sec-1)

vr: Reflection voltage (volt)

vs: Source pulse level of TDR pulser (volt)

v0: Input step pulse level (volt)

v∞: Steady state of TDR waveform (volt) Va: Apparent velocity (m sec-1)

Vph: Phase velocity (m sec-1)

Zc: Characteristic impedance (Ω)

Zin: Input impedance (Ω)

ZL: Terminal impedance (Ω)

Zp: Characteristic impedance in air (Ω)

ZS: Source impedance (Ω)

α: Constant shape factor for dielectric mixing model α(f): Attenuation constant (m-1)

αR: Resistance loss factor (sec-1/2)

β: Probe constant (S m-1)

β(f): Imaginary part of propagation constant (m-1)

γ: Propagation constant (m-1)

γd: Dry density of soil (g cm-3)

γs: Gravity density of soil solid (g cm-3)

γw: Gravity density of water (g cm-3)

Δt: Apparent TDR travel time in the measuring probe (sec) Δτ: Measured TDR travel time = t0 + ts (sec)

Δφ: Phase angle of cross-spectral density

ε: Absolute permittivity (F m-1)

ε0: Permittivity of free space (8.854*10-12 F m-1)

εr: Relative permittivity or dielectric constant

εr’: Real part of relative permittivity

εr’’: Imaginary part of relative permittivity

ε’dc: Real part of relative permittivity lower than resonant frequency

ε’∞: Real part of relative permittivity higher than resonant frequency

(24)

εr*: Equivalent complex permittivity

εrii: Imaginary part of equivalent complex permittivity

ξ: Factor accounting possible spread in relaxation frequency

η0: Intrinsic impedance (≒ 120π Ω)

μ: Magnetic permeability (H m-1)

μ0: Magnetic permeability in free space (4π*10-7 H m-1)

μr: Relative magnetic permeability

μe: Electron mobility

ρ: Reflection coefficient

ρ’: Reflection coefficient without system error

ρe: Charge density of drift electrons (c m-1)

ρopen: Reflection coefficient in open-end case

ρscale: Scaled reflection coefficient

ρw: Density of water (g cm-3)

ρ∞: Steady state of reflection coefficient

ρ∞, air: Steady state of reflection coefficient in air

ρ∞, SC: Steady state of reflection coefficient in shorted-end case

ρ∞, scale: Scaled reflection coefficient by series resistors model σ: Electrical conductivity (S m-1)

σdc: DC conductivity (S m-1) σeff: Effective conductivity (S m-1)

σGT: Electrical conductivity estimated by Giese and Tiemann method (S m-1) σw: Electrical conductivity of water(S m-1)

χe: Electric susceptibility

χm: Magnetic susceptibility ω: Soil water content

(25)

1 Introduction

1.1 Motivation

Electrical properties are highly correlated with the physical and mechanical properties of a composite material and are more conveniently measured or monitored. Hence, electrical methods hold great potential to characterize composite materials, such as soil, concrete, and suspensions. For instance, direct current (DC) resistivity method is commonly applied to measure resistivity (reciprocal of electrical conductivity, EC) of soils and concrete. However, resistivity alone is not uniquely related to complex composition of a composite material. Electrical properties include dielectric constant and electrical conductivity, and dielectric constants of composite materials are often functions of frequency due to interactions between phases.

The technique of Time Domain Reflectometry (TDR) was firstly applied for fault detection in transmission lines in the 50’s. TDR is composed of a pulser and a transmission line. The pulser sends a step pulse into the transmission line and the reflection signal (or the waveform) caused by impedance mismatch along the transmission line is recorded by the sampler in the time domain. By analyzing the reflected signal, the fault in the transmission line can be easily located. Topp et al. (1980) adopted the TDR technique for the estimation of the soil moisture content, in which apparent dielectric constant (Ka) was defined by the apparent travel time in the TDR waveform and experimentally related to soil water content. TDR was found also capable of determining the EC through the steady state of signal (Giese

and Tiemann, 1975; Topp et al., 1988; Zegelin et al., 1989). Furthermore, combining

transmission line theory and spectral analysis of reflected signals leads to estimation of dielectric permittivity at various frequency (i.e. dielectric spectroscopy) (Heimovaara, 1992;

(26)

geotechnical engineering, agriculture engineering, and environmental engineering.

TDR is gaining popularity in characterizing materials based on electrical properties due to its versatility (simultaneous measurements of apparent dielectric constant, electrical conductivity, and dielectric spectrum) and applicability in field measurements. However, improvement of TDR measurements remains an active research topic. First of all, various travel time analyses have been proposed to determine the apparent dielectric constant. But the physical meaning of the apparent dielectric constant is not clear. How actual electrical properties and system parameters affect the apparent dielectric constant have not been extensively investigated. The dielectric spectrum, although more informative, is difficult to be reliably obtained. Works remain to be done to increase the stability and frequency range of dielectric spectroscopy. The TDR EC measurement is straighter forward, but methods accounting for the cable resistance remain controversial, and the effect of TDR recoding time has been underrated when long cables are used. The probe design also plays an important role for each type of TDR measurement. The sensitivity of TDR measurement as affected by the probe parameters and design of probe type for geotechnical and hydrological applications require further study.

Since methodologies and probe design of TDR technology remain disputes so far, measurements of dielectric properties using TDR may lead improprieties in engineering implementation; therefore, this study will present improved TDR measurements and its applications to the characterization of materials.

1.2 Objectives

The objectives of this study were to investigate and improve the TDR measurement techniques, provide guidelines for TDR probe design, and, as an application example, apply TDR to characterization of soil-water mixture.

(27)

Chapter 2 of this thesis firstly reviews electromagnetic properties of materials and TDR methods, including Ka, EC, and dielectric spectroscopy. Chapter 3 deals with measurement of dielectric permittivity. It first investigates the effects of EC, dielectric dispersion, and cable resistance on the apparent dielectric constant and associated effective frequency. The applicability and limitations of travel time analysis are revealed with emphasis on the importance of dielectric spectroscopy. The second part of the chapter 3 examines the sensitivity and reliability of dielectric spectroscopy to identify the source of uncertainty and provides guidelines. A novel approach to obtain the dielectric permittivity at high frequency, where dielectric spectroscopy is most uncertain, based on the frequency domain phase velocity.

In chapter 4, a comprehensive full waveform model and the DC analysis were used to show the correct method for taking account of cable resistance and guideline for selecting proper recording time. In addition, a system error in typical TDR EC measurements was identified and a countermeasure was proposed, leading to a complete and accurate procedure for TDR EC measurements. Meanwhile, chapter 5 investigated the factors associated with probe designs for both types of measurements. The sensitivity of TDR measurements as affected by the probe parameters was discussed to provide guidelines for probe design. In addition, a penetrometer type of TDR probes was developed to allow simultaneous measurements of dielectric permittivity and electrical conductivity during cone penetration for measurements at depths.

Although the aforementioned TDR measurement methodology was originally developed with soil applications in mind, the sediment problems in Shihmen Reservoir manifested by the Typhoon Aere in 2004 provides imperative opportunities for TDR applications. The TDR measurement techniques were adopted for characterization of soil-water mixtures. Therefore, in the first part of chapter 6, TDR penetrometer was integrated with the Marchetti dilatometer

(28)

(DMT) and the TDR/DMT probe was used to determine the solid concentration, stiffness and stress state of the bottom mud. A novel TDR probe and measurement procedure were further developed for accurate monitoring of suspended sediment concentration (SSC) in fluvial and reservoir environment.

(29)

2 Literature Review

2.1 Introduction

TDR is a sensing technology based on electromagnetic wave. A TDR device sent out an electromagnetic pulse into a transmission line connected to a sensing waveguide (or probe), and records the reflected signal from the sensing waveguide. The reflected signal contains information related to the electromagnetic properties of the medium surrounding the sensing waveguide. The electromagnetic properties and principle of TDR technology are reviewed in the following.

2.2 Electromagnetic Properties of Materials

2.2.1 Basic Electromagnetic Properties

Electromagnetic properties of a material include: magnetic permeability, electrical conductivity, and dielectric permittivity. These properties will be briefly introduced in this section.

Magnetic permeability

When a charge q (which is negative for electrons) is in motion in a magnetic field H (ampere m-1). The charge q would experience a force called the magnetic force, Fm. The characteristics of Fm can be described by defining a vector field quantity, the magnetic flux density B, thus the magnetic force can be expressed as (Cheng, 1989):

B q

(30)

On the other hand, as predicted by the Biot-Savart law, moving charges q generates magnetic field, H. The magnetization M (ampere per meter, A m-1) of a material depends on the field H and the magnetic properties of the medium. Therefore, magnetic field or magnetic field intensity H can define as (Cheng, 1989):

M B H = −

0

μ [2-2]

where the μ0 (= 4π*10-7H m-1) is the permeability of free space. The use of the vector H can write a curl equation relating the magnetic field and the distribution of free current in any medium.

As the magnetic properties of the medium are linear and isotropic, the magnetization is directly proportional to the magnetic field intensity (Cheng, 1989):

H

Mm [2-3]

where χm is a dimensionless quantity called magnetic susceptibility. Therefore, by substituting Eq. [2-3] into Eq. [2-2] yields (Cheng, 1989):

H H H

B0(1+χm) =μ0μr =μ [2-4]

where μr is another dimensionless quantity called relative permeability of the medium. The parameter μ is the absolute permeability (or sometimes just permeability) of the medium.

For simple media (linear, isotropic, and homogeneous), χm and μr are constants.

(31)

as iron, nickel, and cobalt, μr could be very large. The relative permeability of some selected materials is listed in Table 2-1.

Table 2-1 Relative permeability data for selected materials (modified from Cheng, 1989)

Material Relative Permeability (μr)

Nickel 250 Cobalt 600 Iron (pure) 4,000 Aluminum 1.000021 Magnesium 1.000012 Palladium 1.00082 Titanium 1.00018 Bismuth 0.99983 Gold 0.99998 Copper 0.99999 Water 1 Electrical conductivity

Consider N number of charges q across a surface with velocity u, it is convenient to

define a vector point function, volume current density, or simply current density, J, in amperes

per square meter (Cheng, 1989)

Nqu

J = [2-5]

Since the conduction currents are the results of the drift motion of charges carried under the influence of an applied electric field E (newtons per coulomb, N C−1 or, equivalently, volts per meter, V m−1), it can be justified that for most conducting materials the average drift velocity is directly proportional to the electric field intensity. For metallic conductors

(32)

E

u =−μe [2-6]

where μe is the electron mobility measured in (m2V-1s-1). Therefore, substituting Eq. [2-6] into [2-5], the current density, J, can be written as:

E

J =−ρeμe [2-7]

where ρe = Nq is the charge density of the drifting electrons and is a negative quantity. Eq.

[2-7] then can be rewritten as:

E

J =σ [2-8]

where the proportionality constant, σ =−ρeμe, is a macroscopic constitutive parameter of the medium called electrical conductivity (siemens per meter, S m-1) (Cheng, 1989). Table 2-2

shows the electrical conductivities of some frequently used materials.

Table 2-2 Electrical conductivities of materials (modified from Cheng, 1989)

Material Conductivity, S m-1 Silver 6.17 x 107 Copper 5.80 x 107 Aluminum 3.54 x 107 Iron 107 Seawater 4 Fresh water 10-3 Distilled water 2 x 10-4 Dry soil 10-5 Glass 10-12 Fused quartz 10-17

(33)

Dielectric permittivity

Before the discussion of the dielectric permittivity, phenomenon of polarization should be introduced. Polarization arises when a force displaces a charge from some equilibrium position. However, polarization cannot occur instantaneously given that charges possess inertia; therefore, polarization is a dynamic phenomenon with a characteristic time-scale

(Santamarina et al., 2001).

In the static case (or zero frequency), when the dielectric properties of the medium are linear and isotropic, the polarization is directly proportional to the electric field intensity, E,

and the proportionality constant is independent of the direction of field, thus the polarization vector P can be written as:

E

P0χe [2-9]

where ε0 ( ≒ 1/36π * 10-9 farad per meter, F m-1) is the permittivity of free space, χe is a dimensionless quantity called electric susceptibility. A dielectric medium is linear if χe is independent of E and homogenous if χe is independent of space coordinates.

Therefore, a new fundamental field quantity, called the electric flux density or electric displacement, D, can be defined with polarization vector P as:

E E E P E D r e ε ε ε χ ε ε = = + = + = 0 0 0 (1 ) [2-10]

where εr is a dimensionless quantity known as relative permittivity or dielectric constant of the medium. The coefficient ε = ε0εr is the absolute permittivity of the medium and is

(34)

used materials (Cheng, 1989).

For single component and homogenous materials (like fluid) may experience three types of polarization mechanisms: electronic, ionic, and dipolar. Electronic polarization occurs when the externally applied electric field causes a shift in the atom’s positive and negative charges, as shown in Fig. 2-1a. Equilibrium is attained when the internal Coulomb attractive force produced by the charge separation balances the applied force. When a charge separation occurs, it essentially has a microscopic electric dipole. Ionic polarization occurs in molecules composed of positively and negatively charged ions (cations and anions). An externally applied electric field again results in a microscopic separation of charge centers thus resembling a dipole of charge, as shown in Fig. 2-1b. Dipolar polarization, on the other hand, occurs in materials that possess permanent, microscopic separations of charge center. In the absence of an applied electric field, these permanent dipoles are randomly oriented. In the presence of an applied electric field, these permanent dipoles tend to rotate to align with the applied field as shown in Fig. 2-1c(Lin, 1999).

In the dynamic case, polarization mechanisms display one of two characteristic spectra: resonance or relaxation. Polarization mechanisms that trigger restoring forces, in general, display a resonance spectrum (This is the case of electronic and ionic polarization). The complex permittivity for resonance is described as (Santamarina et al., 2001)

res res dc r r r j j ω β ω ω ε ε ε ε ε ε + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + = − = ∞ ∞ 1 2 1 ' ' ' '' ' [2-11]

where ε'dc and ε'∞ are the real permittivity at frequencies much lower and higher than the resonant frequency ωres, ω is radian frequency, which equals 2πf (f is frequency), j = √-1, and

(35)

freedom system.

If the polarization does not restore forces or if damping prevails over inertial forces, the material exhibits a relaxation spectrum. This is the case for molecular, spatial, and double layer polarizations. A typical relaxation equation as well-known Debye's equation is

rel dc r r r j j ω ω ε ε ε ε ε ε + − + = − = ∞ ∞ 1 ' ' ' '' ' [2-12a]

where ωrel is the characteristic relaxation frequency, and

(

)

2 / 1 ' rel dc r ω ω ε ε ε ε + − + = ∞ ∞ [2-12b]

(

)

(

)

2 / 1 / " rel rel dc r ω ω ω ω ε ε ε + − = ∞ [2-12b]

As discussed above, Fig. 2-2 shows frequency response of permittivity and loss factor for a hypothetical dielectric by various contributing phenomena. Polarizations form different mechanisms accumulate towards lower frequency. As a result, the real permittivity ε’r increases with decreasing frequency, and resonant mechanisms shows a peak near resonance.

After introducing the polarization mechanism and the dielectric permittivity, the effective imaginary permittivity combining polarization losses and conduction losses can be defined as:

0 '' '' ωε σ ε ε dc r eff = + [2-13]

(36)

expressed in terms of the “effective alternating current (AC) conductivity”: 0 ''ωε ε σ σeff = dc + r [2-14]

Therefore, it is convenient in electric field analysis to combine the dielectric loss and conductive loss terms. The resulting equivalent complex permittivity becomes

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − = − = 0 " ' ' * ωε σ ε ε ε ε ε dc r r ii r r r j j [2-15]

in which εrii = εr"+σdc/(ωε0)is the equivalent imaginary part of the permittivity.

Table 2-3 Relative permittivity of some often used materials [modified from Cheng, 1989]

Material Relative Permittivity

Air 1.0 Glass 4 - 10 Mica 6.0 Oil 2.3 Polyethylene 2.3 Rubber 2.3 – 4.0 Distilled water 80 Dry soil 3 - 4 Teflon 2.1 Sea water 72

(37)

+

Direction of applied field

-+ (a) Electronic polarization (resonance) + -(b) Ionic polarization (resonance) (c) Orientation polarization of dipolar molecules (relaxation) Before After

+

+

Direction of applied field

--+ (a) Electronic polarization (resonance) + -(b) Ionic polarization (resonance) (c) Orientation polarization of dipolar molecules (relaxation) Before After

Fig. 2-1 Polarization mechanisms in single component materials (modified from Santamarina et al., 2001)

Fig. 2-2 Frequency response of permittivity and loss factor for a hypothetical dielectric showing various contributing phenomena (Ramo et al., 1994)

2.2.2 Dielectric Behavior of Water and Soil Solid

Dielectric behavior of water

The frequency dependence of the dielectric permittivity of pure water, εw, is given by

(38)

constant, high-frequency limit, and relaxation frequency of εw, respectively. In addition to the dependence on frequency, the dielectric permittivity of water is also temperature dependent because the dielectric loss of the orientational polarization results from thermal effects.

Stogryn (1971) determined the high-frequency dielectric constant for pure water to be εw

= 4.9. At high frequency, the contribution of the dielectric constant is from the electronic and ionic polarizations, which are a mechanical effect rather than a thermal effect. The dependence of εw∞ on temperature is so weak that for computational purpose εw∞ may be

considered a constant, thus

9 . 4 = ∞ w ε [2-16]

Stogryn (1971) obtained an expression for ωwrel by fitting a polynomial to the data

reported by Grant et al. (1957) as:

( )

1.1109 1010 3.824 1012 6.938 1014 2 5.096 1016 3

2πτwr T = × − − × − T+ × − T − × − T [2-17]

where τwr= 1/ωwrel , T is in °C. The relaxation frequency of pure water, fwr =1/(2πτwr), lies in the microwave region where fwr(0°C) ≈ 9 GHz and fwr(20°C) ≈ 17 GHz. Klein and Swift (1977) generated a regression fit for εwdc(T) from dielectric measurements conducted at 1.43 GHz and 2.65 GHz this resulted in

( )

T 88.045 0.4147T 6.295 10 4T2 1.075 10 5T3 wdc − − + × × + − = ε [2-18]

(39)

(

)

(

)

(

)

(

1 4.58 10 3 25 1.19 10 5 25 2 2.8 10 8 25 3

)

54 . 78 ) (T = ⋅ − ⋅ − T − + ⋅ − T − − ⋅ − Twdc ε [2-19]

in which the εwdc of water would be equals 78.54 when temperature is 25 °C.

Dielectric behavior of soil solid

Soil is a three-phase system consisting of air, solid particles, and water. The dielectric permittivity of air is approximately equal to 1.0 (i.e. no polarization in a free space). The conductivity of air is equal to 0. Solid particles in a soil are non-polar materials. Their dielectric polarization is only due to electronic and ionic polarization mechanisms, which have relaxation frequencies above 1 THz (1012 Hz). Therefore, they have a low value of dielectric permittivity (εr ≈ 5), and are nearly lossless, independent of frequency and temperature at frequencies less than 1 THz (Weast, 1986).

2.2.3 Interfacial Polarization of Soil-water Mixture

The dielectric property of each soil phase can be described by the dielectric mechanisms mentioned above. The heterogeneity of soil-water mixture, however, adds to the complexity of its dielectric properties. There are three major effects due to this heterogeneity: bound water polarization, double layer polarization, and the Maxwell-Wagner effect.

The bound water polarization results from the fact that water can be bounded to the soil matrix as shown Fig. 2-3a. The degree of binding varies from unbound or free water at a great distance from the matrix surface, to heavily bound or absorbed water. If water becomes bounded to the soil matrix, it is not capable of doing as much work and hence looses energy.

(40)

Considering the composition of each phase of soil-water mixture, double layer polarization of soil-water mixture could be different in three conditions: low particle concentration in deionized water, low particle concentration in an electrolyte, and high particle concentration (Santamarina et al., 2001). For the condition of low particle concentration in deionized water, double layer polarization is due to the relative displacement of the double layer counterion cloud with respect to the charged particle in response to an electric field, as shown in Fig. 2-3b. If the polarization occurs, the movement of charges leaves one end of the particle with an excess of surface charges and the other end with and excess of counterions.

For the case of low particle concentration in an electrolyte, the double layer polarization is hindered in an electrolyte because displaced ions are replaced by the diffusion of ions and out of the bulk solution, as shown in Fig. 2-3c. Double layer polarization is also hindered in high particle concentration since ions in the double layer can move from one particle to a neighboring particle in response to the applied electric field, as shown in Fig. 2-3d. This situation develops even if the bulk fluid is deionized water.

The Maxwell-Wagner effect is the most important phenomenon that affects the low-frequency end of the dielectric spectrum of soils or soil-water mixtures. The Maxwell-Wagner effect is a macroscopic phenomenon that depends on the differences in dielectric properties of the soil constituents, as shown in Fig. 2-3e. It is a result of the distribution of conducting and non-conducting areas in the soil matrix. This interfacial effect is dominant at frequencies less than 150 MHz, below the frequencies where bound water relaxation plays a dominant role (Hilhorst, 1998).

A qualitative representation of the dielectric properties of wet soils is presented in Fig. 2-4

(Hilhorst and Dirkson, 1994). The dielectric spectrum can be roughly divided into two parts

(41)

bound water relaxation and the lower frequencies are dominated by the Maxwell-Wagner effect. The TDR frequencies lie from the higher end of the Maxwell-Wagner effect to the lower end of free water relaxation.

+

+

Direction of applied field

--+ + -εr1 σ1 εr2 σ2 + + + + --εr1 σ1 εr2 σ2 (a) Bound water

polarization

(b) Double layer polarization In deionized water

(c) Double layer polarization hindered by electrolyte

(d) Hindered by particle-particle

Interaction (surface conduction) (e) Maxewll-Wanger (relaxation)

(42)

2.2.4 Dielectric Mixing Model

The dielectric behavior of composite materials depends on the compositions. Great attempts on describing bulk dielectric properties of mixtures in terms of their compositions have been proposed through dielectric mixing modeling. In this section, several mixing models will be discussed from different aspects.

Classical mixing model

Figure 2-5 shows spherical inclusions with permittivity εi occupy random positions in the environment of permittivity εe. Let the fraction Fr of the total volume be occupied by the inclusion phase, and the volume fraction 1-Fr left for the host. Thus, a classical mixture rule with spherical inclusions called Maxwell Garnett mixing formula (Maxwell Garnett, 1904)

with the polarizability expression gives:

(

i e

)

r e i e i e r e eff F F ε ε ε ε ε ε ε ε ε − − + − + = 2 3 [2-20]

where εeff is the effective permittivity of a mixture, and this formula is in wide use in the very diverse fields of application. However, this analysis treated the media as a plain and pure dielectric, and no charge flow took place when fields were incident on the materials. A bold and straightforward application of the Maxwell Garnett mixing formula gives for the complex effective permittivity of lossy materials:

(

)

(

)

(

)

(

[

(

)

)

(

)

]

e r i r e r i r e i e i e e r e e eff eff eff F F j F F j j F j j ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ′′ + + ′′ − − ′ + + ′ − ′′ − ′′ − ′ − ′ ′′ − ′ + ′′ − ′ = ′′ − ′ = 2 1 2 1 3 [2-21]

(43)

in which the inclusions are assumed to be spherical, and complex permittivity, which includes the electrical conductivity, of the environment isεeejσ/ω =εe′ − jεe′′, and the same with the complex permittivity of the inclusion as εii′− jεi′′. If the volume fraction of the inclusion phase is small, the effective conductivityσeff =ωεeff′′ , calculated from Eq. [2-21], is:

(

)

2 2 2 2 / 2 9 ω σ ε ε σ ε σ i e i i r e eff F + + ′ = [2-22]

However, the derivation of the Maxwell Garnett mixing formula was based on the algebraic dependence of the internal filed, and it is known that for time-dependent field, losses entail exponential attenuation of the field amplitudes which can be considerable if the extent of the lossy medium is large compared with the penetration depth. Hence the requirements of allowed use of the Maxwell Garnett mixing formula for time-dependent fields is that the inclusion size must not be larger than the skin depth of the wave in the lossy medium 2/

(

ωμiσi

)

with μi being the magnetic permeability of the inclusion material.

(44)

Fig. 2-5 Dielectric spheres are guests in the dielectric background host. (after Sihvola, 1999)

Mixing model for anisotropic mixtures

The original Maxwell Garnett mixing formula basically has an assumption that the inclusion and the background material are isotropic. However, mixtures such as soil and suspensions may have anisotropic inclusions and/or anisotropic background material.

Now we consider inclusions as ellipsoids, as shown in Fig. 2-6, the depolarization factor

Nx (the factor in the ax-direction of Cartesian co-ordinate system) is

(

) (

)(

)(

)

∞ + + + + = 0 2 2 2 2 2 z y x x z y x x a s a s a s a s ds a a a N [2-23]

where s is the integration variable, whose value is between zero and infinity. For other depolarization factor Ny (Nz), interchange ay and ax (az and ax) in the above integral. Collecting those in a single dyadic (the elementary dyadic analysis could refer to Appendix A), the depolarization dyadic for an ordinary ellipsoid reads:

(45)

(

)

(

)

∞ − = + + = = 0 2 1 2 , , 2 det det I s A I s A ds A v v N L z y x i i i i [2-24]

where vi is unit vectors along the three orthogonal eigendirections, and the symmetric and positive-definite dyadic

= = z y x i i i i v v a A , , [2-25] with detA =axayaz

If the ellipsoid, which is anisotropic with permittivity εi, is located in an anisotropic material εerε0, and exposed to a uniform external electric field Ee, as shown in Fig. 2-7. The internal electric field Ei can be shown to be

(

)

[

e i e

]

e e

i L E

E = ε + ′⋅ ε −ε −1⋅ε ⋅ [2-26]

where the transformed depolarization dyadic L′, which is that of real geometry of the ellipsoid after it has been transformed affinely by the anisotropy of the environments, can be calculated from:

(

)

(

)

∞ − + + ⋅ = ′ 0 2 1 2 det 2 det r r r s A s A ds A L ε ε ε [2-27]

(46)

anisotropic, it is natural to accept that a mixture composed of reciprocal materials must display reciprocal electromagnetic behavior. Therefore, the Maxwell Garnett equation can be adapted in the following from:

(

)

(

)

[

1 1

]

1 1 − − − ⋅ − + − + = e Fr i e Fr e L eff ε ε ε ε ε [2-28]

Fig. 2-6 The geometry of an ellipsoid. The semi-axis ax, ax,and ax fix the Cartesian co-ordinate system. (after Sihvola, 1999)

Fig. 2-7 Anisotropic ellipsoid (with permittivity εi) in anisotropic environment (εerε0).

Ei refers to the internal electric field. (after Sihvola, 1999)

Generalized mixing model

(47)

electromagnetic literatures, and it is also known by other names: Polder-van Santen formula and de Loor formula. The essence of the Bruggeman mixing rule is the absolute equality between phases in the mixture (Sihvola, 1999), and it for the case when the inclusions are randomly oriented ellipsoids is

(

)

(

)

= + − − + = z y x j eff j i eff eff e i e eff N f , , 3 ε ε ε ε ε ε ε ε [2-29]

where Nj are the depolarization factors as shown in Eq. [2-24]. Dobson et al. (1985) used the de Loor formula for determining the dielectric constant εm of the four-component mixing for a soil mixture, which is composed of dry soil solids, bound water in the Stern layer, bulk water in the Gouy layer, and air. In additional, Dobson et al. (1985) assumed that the ellipsoid depolarization factors lead to Nj = (0,0,1), εeff has a potential range of εsεeff≤ εm , thus the εm can be written as:

(

)

(

)

(

)

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − + − + − + = 1 1 1 3 2 2 2 3 a s a bw s bw fw s fw s a a s bw bw s fw fw s m V V V V V V ε ε ε ε ε ε ε ε ε ε ε ε ε ε [2-30]

in which Vi refers to the volume fractions of the inclusions, and subscripts s, a, fw and bw refer to dry soil solids, air, free Gouy layer water, and Stern layer water, respectively. Hallikainen

et al. (1985) found a relationship between εs and soil gravity density γs from an empirical

(48)

Dobson et al. (1985) also adopted the Birchak formula which is also called Power-law

model [Birchak et al. 1974] or refractive index model, to provide a four-phase semi-empirical dielectric mixing model for soil mixture,

α α α α α α ε ε ε ε ε ε s a a fw fw bw bw i s i i m =

V =V +V +V +V [2-32]

in which V is the volumetric fraction of the soil component i, the subscripts s, a, fi w and bw refer to the solid soils, air, free water and bound water, respectively. The exponent α is a constant shape factor, and Mironov et al. (2004) presented a generalized refractive mixing dielectric model based on α = 0.5.

Heimovaara et al. (1994b) and Lin (2003b) formulated the four-component dielectric

mixing equation based on the Dobson’s semi-empirical formula, in terms of physical parameters of soil as:

(

)

α

(

)

α α α α θ ε γ γ ε δγ ε δγ θ ε γ γ ε a s d bw s d fw s d s s d m A A ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + + − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 1 [2-33]

where γd is soil bulk dry density, γs is the gravity density of solid, the δγdAs product represents the volumetric bound water content, δ is the average thickness of the bound water, and As is the specific surface of the soil.

(49)

2.3 TDR Principle and Analysis

2.3.1 Basics of TDR

The basic principle of TDR is the same as radar. But instead of transmitting a 3-D wave front, the electromagnetic wave in a TDR system is confined in a waveguide. Fig. 2-8 shows a typical TDR measurement setup composed of a TDR device and a transmission line system. A TDR device generally consists of a cable tester (or pulse generator), a sampler, and an oscilloscope. The transmission line system consists of a leading coaxial cable and a measurement waveguide. The pulse generator sends an electromagnetic pulse along a transmission line and the oscilloscope is used to observe the returning reflections from the measurement waveguide due to impedance mismatches.

The propagation behavior of electromagnetic wave is determined by the Maxwell’s Equation (Cheng, 1989). The propagation behavior of electromagnetic wave can be controlled by two major parameters: Propagation Constant γ, and Characteristic Impedance

Zc. For a coaxial transmission line, the equation of these two parameters can be written as

(Ramo et al., 1994) β α ε π γ j c f j r = + = 2 ∗ [2-34]

(

)

∗ ∗ = = r p r c Z a b Z ε ε ε μ π 1 2 / ln 0 0 [2-35]

where c is the velocity of the light ( = 2.998*108 m/s), a and b are the radii of the outer and inner conductors, α and β are the attenuation coefficient and phase constant, respectively. Zp is the impedance of an ideal air-filled coaxial transmission line or probe, and it is a function of

(50)

dielectric permittivity of measured material as shown in Eq. [2-15]. The propagation constant is a function of equivalent complex dielectric permittivity of measured material. The real part of the propagation constant represents the attenuation of the wave. The imaginary part of the propagation constant is the spatial frequency, which gives the velocity of wave propagation when divided by temporal frequency (2π f). The characteristic impedance is an intrinsic property of the transmission line, and it is controlled by cross-sectional geometry of the transmission line and the equivalent complex dielectric permittivity of measured material. For a line with sections having different impedances, reflection and transmission of waves can occur at the section interfaces.

Fig. 2-9shows a traditional TDR measured response, which contains several reflections due to impedance mismatches, and a reflection coefficient ρ can be defined at a mismatch interface as: i c i c i c i c r Z Z Z Z v v , 1 , , 1 , 0 + − = = + + ρ [2-36]

where the vr represents the reflection voltage, v0 is input voltage form step generator, and Zc,i and Zc,i+1 are characteristic impedance for ith section and (i+1)th section, respectively.

(51)

Pulse generator t V Oscilloscope

TDR device

vs Sampler Coaxial cable Measurement waveguide

Transmission

line system

v0 Pulse generator t V t V Oscilloscope

TDR device

vs Sampler Sampler Coaxial cable Measurement waveguide

Transmission

line system

v0

Fig. 2-8 TDR system configuration

Fig. 2-9 A typical TDR response

數據

Fig. 2-2    Frequency response of permittivity and loss factor for a hypothetical dielectric  showing various contributing phenomena (Ramo et al., 1994)
Fig. 2-6    The geometry of an ellipsoid. The semi-axis a x , a x , and a x  fix the Cartesian  co-ordinate system
Fig. 2-10    Illustration of various methods of travel time analysis: (a) locating the end reflection  by the dual tangent (A point) and single tangent (B point) methods; (b) the derivative methods  locates the end reflection by the apex of the derivative
Fig. 2-14    The comparison of TDR modeling with and without consideration of cable  resistance (after Lin and Tang, 2007)
+7

參考文獻

相關文件

Thoughts: The discovery of this epitaph can be used by the author to write a reference to the testimony of the book Tuyuan Cefu, to fill the lack of descriptions

(In Section 7.5 we will be able to use Newton's Law of Cooling to find an equation for T as a function of time.) By measuring the slope of the tangent, estimate the rate of change

◦ 金屬介電層 (inter-metal dielectric, IMD) 是介於兩 個金屬層中間,就像兩個導電的金屬或是兩條鄰 近的金屬線之間的絕緣薄膜,並以階梯覆蓋 (step

which can be used (i) to test specific assumptions about the distribution of speed and accuracy in a population of test takers and (ii) to iteratively build a structural

Using this formalism we derive an exact differential equation for the partition function of two-dimensional gravity as a function of the string coupling constant that governs the

In this study, we compute the band structures for three types of photonic structures. The first one is a modified simple cubic lattice consisting of dielectric spheres on the

Programming languages can be used to create programs that control the behavior of a. machine and/or to express algorithms precisely.” -

A marble texture can be made by using a Perlin function as an offset using a Perlin function as an offset to a cosine function. texture = cosine( x +