Electric characterization of sands with heterogeneous saturation distribution

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Electric

characterization of

sands

with heterogenous

saturation

distribution

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Electric characterization of sands

with heterogenous saturation distribution

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Propositions belonging to the thesis:

Electric characterization of sands with heterogenous saturation distribution

1. More than any other method, the Propagation Matrices method provides physical insight and a simple mathematical treatment of multisection transmission lines. (Chapters 3 and 5 of this thesis)

2. The frequency dependence of the permittivity of a mixture is not only determined by the permittivity and volume fractions of its components, but also by their geometrical distribution. (Chapter 6 of this thesis)

3. The arithmetic mean and the CRIM mixing law characterize the same layered media at two limits, the low and high frequency limits respectively.

(Chapter 6 of this thesis)

4. True scientific progress is achieved only when physical experiments and theoretical developments drive each other.

5. Our fear of being wrong inhibits our creativity.

6. Many problems would not exist if people understood that nationalisms, patriotisms and faiths are circumstantial and not causal.

7. Contrary to what the “Real Academia de la Lengua Espa˜nola” dictionary says, the present day meaning of the word “discusi´on” is an escalating argument rather than a discussion. This reflects part of the nature of the Spanish.

8. Bureaucracy is a vile steeplechase and most participants feel as if they joined the race with two broken legs!

9. Women liberation was not meant to get us out from behind vacuum cleaners to the cover of Hustler.

10. The meaning of life is to come to peace with its finiteness.

These propositions are considered defendable and as such have been approved by the supervisors, Prof.dr.ir. J.T. Fokkema and Dr.ir. E.C. Slob

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Stellingen behorende bij het proefschijft:

Elektrische karakterisering van zanden met heterogene saturatieverdelingen

1. Meer dan welke methode ook geeft de propagatie matrices methode fysisch inzicht en een eenvoudige wiskundige behandeling van aaneensgeschakelde transmissielijnen. (Hoofdstukken 3 en 5 van dit proefschrift)

2. De frequentie-afhankelijkheid van de permittiviteit van een mengsel wordt niet alleen bepaald door de permittiviteitwaardes en de afzonderlijke volume fracties van de componenten, maar ook door hun geometrische verdeling.

(Hoofdstuk 6 van dit proefschrift)

3. Het arithmetrische gemiddelde en de CRIM mengregel karakteriseren het-zelfde materiaal voor twee limietgevallen, respectievelijk de laag- en hoog frequentie limiet. (Hoofdstuk 6 van dit proefschrift)

4. Werkelijke wetenschappelijke vooruitgang wordt slechts geboekt wanneer fysische experimenten en theoretische ontwikkelingen elkaar voortstuwen.

5. Onze angst het verkeerd te hebben smoort onze creativiteit in de kiem.

6. Veel problemen zouden niet bestaan wanneer mensen zouden begrijpen dat nationalismen, pattriottismen and geloofsovertuigingen een gevolg zijn van omstandigheden en geen oorzaak.

7. Tegensteld aan wat het woordenboek van de ”Real Academia de la Lengua Espa˜nola” zegt, betekent het woord “discusi´on” in dagelijks gebruik eerder ”escalerende redetwist” dan ”discussie”. Dit is een afspiegeling van de Spaanse natuur.

8. Bureaucratie is een laaghartige hordenloop en voor de meeste deelnemers voelt het alsof ze met twee gebroken benen aan de race meedoen!

9. Vrouwenbevrijding was niet bedoeld om ons van achter de stofzuigers weg te krijgen naar de omslag van Hustler.

10. De betekenis van het leven is vrede te vinden in haar eindigheid.

Deze stellingen worden verdedigbaar geacht en zijn als zonadig goedgekeurd door de promotors, Prof.dr.ir. J.T. Fokkema en Dr.ir. E.C. Slob

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Electric characterization of sands

with heterogenous saturation distribution

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universtiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College van Promoties,

in het openbaar te verdedigen op 2 november 2004 te 15:30 uur door

Ainhoa GONZ´ALEZ GORRITI

Licenciada en Ciencias F´ısicas,

especializaci´on F´ısica de la Tierra y el Cosmos Universidad Complutense de Madrid,

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Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. J.T. Fokkema

Toegevoegd promotor: Dr.ir. E.C. Slob

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof.dr.ir. J.T. Fokkema Technische Universiteit Delft, promotor

Dr.ir. E.C. Slob Technische Universiteit Delft, toegevoegd promotor Prof.dr. K. Holliger Swiss Federal Institute of Technology

Prof.dr. A. Sihvola Helsinki University of Technology Prof.dr.ir. L. Ligthart Technische Universiteit Delft Prof.dr. S.M. Luthi Technische Universiteit Delft Dr. J. Bruining Technische Universiteit Delft

Hans Bruining heeft in belangrijke mate bijgedragen aan de totstandkoming van dit proefschrift.

Copyright c_{2004 by A.G. Gorriti}

Cover design: Jes´us G. Gorriti

ISBN 90-9018709-X

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A mi familia, en especial a mi abuelo. Y a Jara y a Ron, in memoriam.

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Financial support

The research reported in this thesis is financially supported by the Netherlands Organisation of Scientific Research (NWO) under contract number 809.62.013, which support is gratefully acknowledged.

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Uno no es lo que es por lo que escribe, sino por lo que ha le´ıdo∗_.

J.L. Borges (1899-1986)

∗_{Traduction of the author: One is not what he is because of what he has written,} but because of what he has read.

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CONTENTS ix

Introduction

A proper understanding of the electrodynamic response of soils will lead to an improvement in the forward and inverse modelling of electromagnetic (EM) geo-physical techniques.

1.1 Statement of the problem

Electromagnetic waves have been beautifully described by J.C. Maxwell (1873). He related the material response to EM fields through constitutive parameters that are representative for the macroscopic electromagnetic properties of the media. These parameters are the electrical permittivity and conductivity, and the magnetic permeability.

Constitutive parameters can be treated from two different points of view. On one hand the macroscopic and empirical one, in which case they are derived from appro-priate experiments. On the other the microscopic and theoretical one, where they are derived from a microscopic model in which hypotheses about the interaction between elementary building blocks of matter are made and the constitutive pa-rameters follow from an appropriate spatial averaging, (Hippel, 1954). Comparison between models and experiments should lead to a better understanding of these relations.

As geophysicists we are interested in the imaging capabilities of electromagnetic waves. The goal is to disturb the subsurface of the Earth with EM fields and infer an image from the response to these fields. It is then essential to know how the materials that are commonly found in the subsurface of the Earth react to applied EM fields. In the literature, these materials have been studied from both the macroscopic and microscopic point of views, but in any case, models are validated through experimental results.

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2 Chapter 1. Introduction

We think that the best methodology to study rocks and soils is, not to have a pre-conceived idea about their microscopic properties, but to test them with appropriate experiments. That way we are probing the material at a reasonable macroscopic scale and no assumptions, apart from the constitutive relations defined by Maxwell, are needed. Taking into consideration the complexity of rocks and soils this is an advantage.

In order to study the EM properties of rocks and soils, convenient experiments have to be performed. These have to be accurate and reliable. Furthermore, the materials are very complex as they are composed of several constituents and there is coexistence of phases (solid, liquid and gas). Due to this complexity, many different parameters can have an impact on the properties under study and they must be under control. Much of this thesis is, therefore, devoted to the design and calibration of a tool and methodology to study the response of these materials to an applied EM field, and to accurately reconstruct their permittivity values. We have restricted our study to the complex permittivity because most materials of geophysical interest are non-magnetic, while the conductivity is incorporated in the complex permittivity.

From reliable and accurate experimental results theoretical models can be derived and existing ones can be validated. In fact, in the design of any experiment an a-priori model of the response of the material is needed (in our case, it would be the constitutive relations postulated by Maxwell) and it has to be verified. This has to be done with an accurate tool or otherwise errors could camouflage its validity. Currently, the computer modelling algorithms used in EM exploration discretise the Earth’s interior in blocks. The permittivity assigned to each block is generally the one that happens to lie beneath each grid point, or an average of the properties of the different materials coexisting in a specific block. But how this average should be taken is not clear in all situations; what happens in case of time-varying fields? How does this affect the averaging?

On the other hand, most Mixing Laws compute permittivity from the volume frac-tion and permittivities of the constituents. Theoretically, they are valid for quasi-static fields and differ for different types of media. Nevertheless, they are generally applied over a much wider spectrum and the limitation of their application beyond the frequency for which they have been derived is not clear. It is also unclear how the different compositions of the same components and volume fractions lead to different results.

With this thesis we want to validate the effective medium theories in the particular case of the electrodynamic response of soils. Existing experimental evidence does not favor any model in particular, which supports our questioning.

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1.2. Scientific framework 3

We have chosen to investigate the permittivity of partially saturated sands when the same saturation is distributed differently. We also study its frequency dependency. In that way, very simple experiments can be used to test the relevance of the heterogeneities and the validity of considering the response of sands with a single parameter.

1.2 Scientific framework

This section contains relevant literature that frames our work into the appropriate scientific context. This thesis deals with theoretical and experimental issues con-cerning permittivity, therefore, the literature is very varied and extensive. Distinct scientific disciplines study permittivity at different frequency ranges, depending on the scale of their interest. Even different nomenclatures are used; in the low frequency range it is referred to as dielectric constant, in optics as the complex refraction index and in telecommunication as the complex propagation factor. For simplicity, we only include the most relevant publications for our work and we have divided them into categories; theoretical models, experiments and validation of models through experiments.

Theoretical models

The beginning of the systematic investigation of the dielectric properties can be established with the works of Mossotti (1850) and Clausius (1879), as they at-tempted to correlate the dielectric constant with the microscopic structure of the materials. They considered the dielectric to be composed of conducting spheres in a non-conducting medium, and succeeded in deriving a relation between the dielectric constant and the volume fraction occupied by the conducting particles . Debye (1929), in the beginning of the 20th century, realized that some molecules had permanent electric dipole moments and that it gave rise to the macroscopic dielectric properties of the materials. He succeeded in extending the Clausius-Mossotti theory to take into account the permanent moments of the molecules. His theory, later extended by Onsager (1936) and Kirkwood (1939), linked the disper-sion of permittivity to the characteristic time needed for the permanent molecular dipoles to reorient following an alternating EM field. He deduced that the time lag between the average orientation of the moments and the field becomes noticeable when the frequency of the applied field is of the order of the reciprocal relaxation time.

Debye’s model is still being used for polar liquids, where the dipoles are relatively far away from each other. However, the dielectric behavior of solids deviates sig-nificantly from Debye’s theory of relaxation. Cole & Cole (1941) pioneered the

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first approach to interpret the non-Debye relaxation of materials by means of a superposition of different relaxation times and Jonscher (1983) postulated that the relaxation behavior at molecular level is intrinsically non-Debye due to the cooper-ative molecular motions, the ”many-body interactions” approach.

All these researchers modelled the macroscopic behavior of dielectric materials com-posed of a single phase and a single type of micro-particles, that is, they did not describe the electromagnetic properties of mixtures. However, most materials are indeed mixtures and researchers have struggled, and still do, to characterize their physical properties with effective parameters.

The literature on the effective properties of materials and mixtures is very vast. Some comprehensive reviews are Wang & Schmugge (1980), Shutko & Reutov (1982), Dobson et al. (1985), Chelidze & Guguen (1999a), Sihvola (1999) and Choy (1999). Most of the models for the permittivity of mixtures only consider the volume fraction of the constituents and their permittivities. They are valid for macroscopic homogeneous and isotropic media and quasi-static fields. With the advent of numerical computing many theoretical studies are being published but they lack experimental corroboration.

Experiments

Each part of the spectrum has a specific physical measuring principle, and each principle has several techniques. In the microwaves the standing waves methods are used, and among these there are transmission line methods, wave guides, open resonators, closed cavities, etc. And every field of interest requires different set-ups for the specific characteristics of the samples; human tissues, seeds, polymers, wood, ceramics, composites, rocks, etc. The technique chosen depends on the frequency of interest and the sample requirements. Afsar et al. (1986) reviews the existing techniques for the microwave region.

The permittivity of porous media is usually measured from very low frequencies up to the giga-Hertz region, where it becomes constant for most natural materials. This is also the range of interest for many field applications; geo-radar and Time Domain Reflectometry (TDR) in the shallow subsurface, and dielectric logging tools for petroleum reservoir characterization.

In the low frequency regime, up to hundreds of mega-Hertz, the so-called direct methods are being used. Generally, the material is placed between two parallel plates and the impedance or admittance of this capacitor is measured so that the permittivity is calculated directly from these measured quantities, see for e.g. Shen et al. (1987) and Bona et al. (1998).

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1.2. Scientific framework 5

In the high-frequency regime, the relation between the permittivity and the mea-sured quantities is no longer linear for reasonably sized sample holders and more complex set-ups have to be implemented. For broad band measurements, reflec-tion and/or transmission methods are used. Resonant-cavity methods are limited to a few frequency points and for medium- to low-loss samples. The reflection method gives good results for low- and medium-loss samples but not for high-loss samples, contrary to the transmission method that works best for relatively high-loss samples. The S-parameter method, combines the reflection and transmission methods and overcomes the disadvantages of both. It has become very popular in the study of dielectric properties. It is common to place the material in a coaxial transmission line (Shen (1985) and Nguyen (1998)), or in a coaxial-circular wave guide (Taherian et al. (1991)) and measure the S-parameters of the set-up with a Network Analyzer.

The reconstruction of permittivity from measured S-parameters can be done ana-lytically or via an optimization procedure. Analytical methods compute the elec-trical permittivity from analytical expressions involving the S-parameters. Until now there were no straight forward methods, the S-parameters had to be com-pensated, in one way or an other (Rau & Wharton (1982), Kruppa & Sodomsky (1971), Freeman et al. (1979), Shen (1985) and Chew et al. (1991)). Then dif-ferent reconstruction formulas were used: Nicolson & Ross (1970), Weir (1974) and Stuchly & Matuszewsky (1978). In this thesis we develop a novel method, the Propagation Matrices Method, that reconstructs the permittivity from the mea-sured S-parameters of the tool. It provides a representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler than existing methods. It also enables us to show how all reconstruction formulas correspond to the same method, and, therefore, suffer from the same instabilities. The EM properties of the sample can also be computed via an optimization procedure. Belhadj-Tahar et al. (1990) and Taherian et al. (1991) applied this method to a coaxial-circular waveguide, and we have used it for our measurements. In this thesis we compare all existing reconstruction methods.

Validation of models through experiments

Different researchers have compared measured permittivities of different soils, but their results are mainly qualitative and certainly not conclusive. In the high fre-quency band, the following researchers have published studies over a broad variety of soils, different water contents and temperature: Shutko & Reutov (1982), Hal-likainen et al. (1985) Dobson et al. (1985) and Chelidze & Gueguen (1999b). To our knowledge, up to this thesis, there has not been any study of heterogeneously saturated samples.

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Permittivity states of soils

Introduction

Basic electromagnetic equations

Experimental design

Tool for accurate permittivity measurements

Reconstruction Methods

Permittivity states of mixe-phase;

2 and 3 component mixtures

' '

Representations for coaxial transmission lines The Propagation Matrices representation

' EM field equations

' '

Technical design and measurement characteristics Forward model and calibration

' ' Analytical Optimization ' ' ' Saturation Techniques 2-layer samples Interpretation of results

1

3

2

6

5

4 Conclusions and Recommendations

7

% % % Resonant frequencies Propagation Matrices Compensation Parameters Preliminaries TOOL= Evidenceof permittivitystates forwardandinversemodels calibratedcoaxTL +

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1.3. Outline of the thesis 7

1.3 Outline of the thesis

A schematic structure of this thesis is shown in Figure 1.1. This introductory chapter is followed by the basic electromagnetic equations, presented in Chapter 2. It contains Maxwell’s theory of propagation of EM waves and the constitutive relations.

The next three chapters deal with the design and calibration of the tool to probe the electrodynamic response of sands and the reconstruction of permittivity. Chapter 3 reviews various experimental designs for measuring the permittivity of soil samples and justifies our choice for a coaxial transmission line. We also introduce a new notation (Propagation Matrices) to describe these set-ups. This notation provides a representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler than existing methods. It also enables us to prove the independence of the transmission and reflection measurements.

The characteristics and calibration of our specific tool are presented in Chapter 4. It is a customized coaxial transmission line that allows for fluid flow through the sample, and whose S-parameters can be determined in the frequency range from 300MHz to 3GHz. We show the calibration needed for the forward model to be in very good agreement with the measured data, and its high sensitivity (it can detect relative changes in permittivity in the order of 1%).

Chapter 5 contains the comparison of existing reconstruction techniques and the novel one based in the Propagation Matrices. All the reconstruction methods lead to a permittivity value per frequency without a pre-defined model of frequency dependency. We show the advantages and disadvantages of these reconstructions (analytical and optimized) and how we can use the tool as a resonator for low-loss materials. The Propagation Matrices Representation shows again valuable since its analytical inversion identifies all existing inversions equal to either one of the two fundamental and independent solutions. This explains why indistinctively of the expression used, the reconstruction of permittivity from analytical expressions always suffers from the same problems at resonant frequencies and with low-loss materials. We successfully reconstruct the relative permittivity of air within ±1% error.

After this thorough study of the tool, in Chapter 6 we investigate the effect of the distribution of constituents with measured and modelled experiments. We perform experiments with sands composed of loose quartz grains; dry, partially and fully saturated. The saturation technique has an effect in the reconstructed permittivity and extra care is needed for proper measurements. We perform gravity drainage experiments so that the same sample can be measured at different saturation levels.

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Samples saturated in that way result in distinctive 2-layer samples, whose recon-structed permittivity exhibit anomalies due to the interface. These samples can be represented by a 2-layer model very accurately, if the width of the layers is found from the phase of the transmission coefficients. Hence, we can use the tool to monitor the movement of the fluid front. With the aid of numerical experiments, we show that the distribution of layers has a strong impact on the permittivity, and that heterogeneities as small as λ/100 are detectable.

We also compare our experiments with existing Mixing Laws and find the best mixing formula for partially and fully homogeneously saturated sands to be a Power Law equation. However, depending on the saturation level and whether we are interested in the real or imaginary parts of the permittivity, different exponents are needed. We think that our results on multilayered samples can explain the variety of exponents encountered in this thesis and in the existing literature. Depending on the desired precision of the model, 2-layer samples can be modelled via a Power Law with three constituents (permittivity and volume fraction of solid grains, air, water) or two (permittivity and volume fraction of the two layers). The first is less accurate and requires different exponents and the second one is very accurate and uses a single exponent, plus it has a clear physical sense. In Chapter 6 we also present numerical and experimental examples that contradict the theory derived by del Rio & Whitaker (2000a).

Finally, we conclude this thesis with Chapter 7 where we also make recommenda-tions for future research. We also include five appendices that contain relevant equations and derivations. Appendix A shows a general solution of the EM wave equation in cylindrical structures, and Appendix B presents the solution of these waves in coaxial and circular waveguides. It also contains information on how to modify the tool to measure solid cores, in a combination of coaxial-circular waveg-uide. Appendix C shows the equations of wave propagation through a transmission line; the reformulation of Maxwell’s wave equation in terms of circuit parameters and voltage and current waves. And Appendix D gives a short introduction to the representation of 2-port networks with scattering matrices. The last appendix, (Appendix E) is a short summary of the theoretical models that try to explain the EM behavior of materials and mixtures.

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From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics.

Richard P. Feynman (1918-1988)

Chapter 2

Basic electromagnetic equations

In this chapter the theoretical background of electromagnetic waves propagating through confined regions is presented. We introduce Maxwell’s equations, the constitutive relations and the generalized wave equations.

2.1 The electromagnetic field equations

We use the vector notation and to specify position, we employ the coordinates {x, y, z} with respect to a Cartesian reference frame with origin O and three mutu-ally perpendicular base vectors {ˆx, ˆy, ˆz_{} of unit length each. In the indicated order,} the base vectors form a right-handed system. The time coordinate is denoted by t and the position is specified by the vector x = xˆx+ yˆy+ zˆz.

The electromagnetic phenomena under study occur in a determined domain D in IR3_{. The filling material of D is characterized by electric parameters, in the case of} free space by the electric permittivity of free space ε0 and magnetic permeability

µ0. Together they determine the velocity of light in free space as

c0= 1 √_ε 0µ0 def = 299792458[m/s].

The value of c0 has been defined, while the value of the free space magnetic

permeability is fixed and given by

µ0 = 4π × 10−7[H/m].

Hence the exact value for the free space permittivity must be determined from these two. It is given by

ε0=

1

µ0c20 ≈ 8.854 × 10

−12_[F/m].

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10 Chapter 2. Basic electromagnetic equations

When the medium is different from free space but linear, isotropic, locally reacting and time-invariant, the electric permittivity, ε(x, t), the conductivity σ(x, t) and the magnetic permeability µ(x, t) may depend on position due to inhomogeneities of the medium and on the history of the applied fields in the medium (relaxation processes in time)

The electromagnetic field is characterized by two field quantities:

E is the electric field strength [V/m], H is the magnetic field strength [A/m],

and the action of sources is represented by

Je is the source volume density of electric current [A/m2_],

Ke is the source volume density of magnetic current [V/m2].

2.1.1 Basic equations for electromagnetic fields

We start with the basic equations for electromagnetic fields in vacuum, where they assume their simplest form. Sources of electromagnetic field are essentially composed of matter and are therefore introduced in the volume densities of electric and magnetic currents that describe the electromagnetic action of matter. The induced parts of these volume densities then describe the reaction of a piece of matter to an electromagnetic field.

− ∇ × H + ε0∂tE = −Jmat, (2.1)

∇_{× E + µ}₀_∂_tH _{= −K}mat, (2.2) In a vacuum domain, the material volume densities {Jmat, Kmat_{} are zero-valued.} In the presence of matter, we distinguish between the active part and the passive part. The active part describes the external source behavior, that generates the field. The volume densities of electric and magnetic currents are denoted Je and Ke, respectively. The induced, or passive, part describe the reaction of matter to the presence of an electromagnetic field and are generally field dependent. They are denoted Ji and Ki, respectively,

Jmat= Ji+ Je Kmat= Ki+ Ke, (2.3) Ji= J + ∂tP Ki = µ0∂tM, (2.4)

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2.1. The electromagnetic field equations 11

where

J is the volume density of electric current [A/m2], P is the electric polarization [C/m2_],

M is the magnetization [A/m].

Further it is customary to introduce the quantities

D= ε0E+ P , (2.5)

B= µ0(H + M ). (2.6)

where

D is the electric flux density [C/m2], B is the magnetic flux density [T].

Upon substituting equations (2.5) and (2.6) into equations (2.1) and (2.2), we arrive at the two Maxwell equations in matter,

− ∇ × H + J + ∂tD = −Je, (2.7)

∇_{× E + ∂}_t_B _{= −K}e_, _(2.8) From Maxwell’s equations two other equations can be derived that are not indepen-dent of Maxwell’s equations, the compatibility equations. This means that field quantities that satisfy Maxwell’s equations by definition also satisfy the compati-bility equations. Compaticompati-bility equations are obtained by applying the divergence operator (∇·) to equations (2.7) and (2.8), respectively. This results in,

∇_{· (J + ∂}_t_D_{) = −∇ · J}e_, _(2.9) ∂t∇· B = −∇ · Ke. (2.10)

Historically, the volume density of electric charge is introduced as,

ρ = ∇ · D. (2.11)

Obviously, from the above equations it is clear that there is a relation between charge and electric current. The relation between charge and current densities is clear if we combine equations (2.9) and (2.11) and assume that there are no sources of electric current present

∇_{· J + ∂}_t_{ρ = 0,} _(2.12) known as the charge conservation law.

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2.2 The constitutive relations

The constitutive relations provide information about the environment in which electromagnetic fields occur. It is customary to relate the quantities {J, D, B} to {E, H} through the constitutive parameters, which are representative for the macroscopic electromagnetic properties of the media. For general media, which are linear, locally reacting, time invariant, instantaneously reacting and isotropic, the constitutive parameters are scalars. If the medium is inhomogeneous the coef-ficients change with position, therefore their dependence on x, but if the medium is homogeneous they become constants.

J(x, t) = σ(x)E(x, t), (2.13) P(x, t) = χe(x)E(x, t), (2.14)

M(x, t) = χm(x)H(x, t), (2.15)

where

σ is the electrical conductivity [S/m], χe is the electric susceptibility [-],

χm is the magnetic susceptibility [-].

Substituting equations (2.14) and (2.15) into equations (2.5) and (2.6) give the relations between {D, B} and {E, H}

D(x, t) = ε(x) E(x, t) = ε0εr(x)E(x, t) (2.16)

B(x, t) = µ(x)H(x, t) = µ0µr(x)H(x, t), (2.17)

where

ε is the (absolute) permittivity [F/m], εr is the relative permittivity [-],

µ is the (absolute) permeability [H/m], µr is the relative permeability [-],

and can be expressed as

ε = ε0(1 + κe) or εr = 1 + κe, (2.18)

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2.2. The constitutive relations 13

where

κe is the relative electric susceptibility [-],

κm is the relative magnetic susceptibility [-].

We restrict ourselves to variations in the electric and magnetic parameters, such that they are piecewise constant functions of position. Then substitution of equa-tions (2.13)-(2.17) to obtain Maxwell’s equaequa-tions in the electric and magnetic field strengths only, gives

− ∇ × H + σE + ε∂tE = −Je, (2.20)

∇_{× E + µ∂}_t_H _{= −K}e_, _(2.21) And the compatibility relations read

σ∇ · E + ε∂t∇· E = −∇ · Je, (2.22)

µ∂t∇· H = −∇ · Ke. (2.23)

Only causal solutions of the differential equations (2.20) and (2.21) are acceptable from a physical point of view. Assuming that the sources start to act at the instant t = 0, causality of the time behavior of the electromagnetic field is then ensured by putting the field values of E and H equal to zero prior to t = 0.

Constitutive parameters can be treated from two different points of view. On one hand the macroscopic and empirical one; they are then derived from appropriate ex-periments. And on the other the microscopic and theoretical one; in this case they are derived from a microscopic model in which hypotheses about the interaction between elementary building blocks of matter are made and the constitutive pa-rameters follow from an appropriate spatial averaging. Comparison between models and experiments should lead to a better understanding of these relations.

Material with relaxation

The constitutive parameters of a material with relaxation have to show the effect of this type of causal behavior. Assuming a linear, time invariant, locally reacting and isotropic media the relations between {J, P , M} and {E, H} are replaced by a time convolution

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14 Chapter 2. Basic electromagnetic equations J(x, t) = Z t t0₌₀ σ(x, t0_{)E(x, t − t}0)dt0, (2.24) P(x, t) = Z t t0₌₀ χe(x, t0)E(x, t − t0)dt0, (2.25) M(x, t) = Z t t0₌₀ χm(x, t0)H(x, t − t0)dt0, (2.26) where {t0_{∈ R; t}0 _{> 0} and}

σ conduction relaxation function [S/ms], χe relaxation function [s−1],

χm magnetic relaxation function [s−1].

Expressing the constitutive relations by a time convolution, it is mathematically taken into account the fact that the values of the fields {E, H} between the instant t −t0 (cause) and the instant t (effect) contribute to the values of {J, P , M}. The present reaction of the material is influenced by the history of the electromagnetic fields. Boltzmann (1876) was the first one to use time convolutions to mathemat-ically describe mechanical relaxation processes in the study of solid deformations. However, the fact that the relaxation process is time invariant requires the medium to return to its original state after some time so there is no permanent deformation of state, and thus, this is only valid for small disturbances, as was already realized by Boltzmann.

Since the electric field (E) and volume density of electric current (J ) are related by measurable quantities,

J = σE and E = %J , (2.27) the conductivity (σ) by measuring J for an applied E and the resistivity (%) by mea-suring E for an applied J , both σ and % (σ = 1/%) should be causal time-functions. They cannot be minimum-phase functions, because it would not represent a strictly passive medium, which is required for natural media in thermodynamic equilibrium. Their real and imaginary parts should however obey the Kramer-Kroning causality relations, discussed in Subsection 2.4.1. The same argument holds for χe and χm.

The relations between {D, B} and {E, H} read

D(x, t) = ε0E(x, t) + ε0 Z t t0₌₀ κe(x, t0)E(x, t − t0)dt0, (2.28) B(x, t) = µ0H(x, t) + µ0 Z t t0₌₀ κm(x, t0)H(x, t − t0)dt0. (2.29)

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2.3. Boundary conditions 15

2.3 Boundary conditions

Upon crossing the interface of two adjacent media that differ in their constitutive parameters, the electric and magnetic field strengths will in general vary discon-tinuously. Since all physical quantities have bounded magnitudes, the relevant discontinuities are restricted to finite jump discontinuities. Because of their dis-continuous behavior, the electric and magnetic field strengths are no longer con-tinuously differentiable in a domain containing (part of) an interface, and therefore equations (2.7) and (2.8) cease to hold. Since we assume time invariance for the properties of the media, the non-differentiability is restricted to the dependence on the spatial derivatives. The electromagnetic field equations must therefore be supplemented by conditions that interrelate the field values at either side of the interface, the so-called boundary conditions.

Let S denote the interface and assume that S has a unique tangent plane ev-erywhere. Let ν denote the unit vector along the normal to S such that upon traversing S in the direction of ν we pass from the domain D(1) to the domain D(2)_{, D}(1) _{and D}(2) _{being located on either side of S (see Fig. 2.1).}

Figure 2.1: Interface between two media with different electromagnetic properties.

The partial derivatives perpendicular to S meet functions that show a discontinuity across S, which would lead to interface Dirac distributions located on S. These would physically represent impulsive interface sources. In the absence of such sources, the absence of such interface impulses in the partial derivatives across S must be enforced. Looking at Maxwell’s equations we see that the tangential components are differentiated in the direction normal to the interface, so impose the following boundary conditions:

ν_{× H} _{is continuous across S,} (2.30) ν_{× E} _{is continuous across S,} (2.31) and looking at the compatibility equations we see that the normal components

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are differentiated in the direction normal to the interface and hence we have the conditions:

ν_{· (J + ∂}tD) is continuous across S, (2.32)

ν _{· B} _{is continuous across S.} (2.33) These conditions allow the continuous exchange of electromagnetic energy between the two domains across the interface S. Notice that equations (2.32) and (2.33) are not independent conditions, they are conditions corresponding to the compatibility equations (2.9)-(2.10), which are themselves direct consequences of the Maxwell equations (2.7)-(2.8), for electromagnetic fields.

At the surface of an electrically impenetrable object (it cannot sustain in its interior a non-identically vanishing electric field while the boundary condition of the continuity of the tangential part of the electric field strength on its boundary surface ∂D is maintained) a boundary condition of the explicit type can be given

lim

h↓0ν× E(x + hν, t) = 0 for any x ∈ ∂D, (2.34)

Electrically impenetrable materials arise as limiting cases of materials whose con-ductivity and/or permittivity go to infinity.

2.4 The electromagnetic field equations

in the complex frequency domain

To obtain the Maxwell equations in the frequency domain a Laplace transformation with respect to time is carried out.

Time Laplace transformation

Switching on the sources at the instant t = 0 the time domain T in which the source affects the field is defined as

T = {t ∈ R; t > 0}. (2.35) The complement T0 of the domain T and the boundary ∂T between the two domains are defined according to

T0= {t ∈ R; t < 0}, (2.36) ∂T = {t ∈ R; t = 0}. (2.37)

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2.4. EM Field Equations in the Complex Frequency Domain 17

The characteristic function χT(t) of the set T is introduced as

χT(t) = {1, 1/2, 0} when t ∈ {T , ∂T , T0} , (2.38)

and the Laplace transform of a function f (x, t) in space-time, defined for t ∈ T , is ˆ

f (x, s) = Z

t∈Rexp(−st)χ

Tf (x, t)dt, (2.39)

where s is the Laplace transformation parameter, and must satisfy the condition Re(s) > 0.

The inverse Laplace transformation can be carried out explicitly by evaluating the Bromwich integral in the complex s plane:

1 2πj

Z s0+j∞

s=s0−j∞

exp(st) ˆf (x, s)ds = χT(t)f (x, t). (2.40)

The path of integration is parallel to the imaginary s-axis in the right half of the complex s-plane where ˆf is analytic.

Symbolically equation (2.39) is written as

ˆ

f (x, s) = Lf(x, t), (2.41) and the Laplace transform of a partial differentiation of a function, ∂tf (x, t), equals

L∂tf (x, s) = sf (x, t),ˆ (2.42)

assuming zero initial conditions.

2.4.1 Constitutive parameters: Kramers-Kronig causality relations

Considering a medium with relaxation we can carry out the Laplace transformation of equations (2.24)-(2.29) and obtain

ˆ J(x, s) = ˆσ(x, s) ˆEk(x, s), (2.43) ˆ D(x, s) = ˆε(x, s) ˆEk(x, s), (2.44) ˆ B(x, s) = ˆµ(x, s) ˆHj(x, s), (2.45)

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18 Chapter 2. Basic electromagnetic equations where ˆ σ = Z ∞ t0₌₀exp(−st)σ c(x, t0)dt0, (2.46) ˆ ε = ε0(1 + ˆκe) κˆe(x, s) = Z ∞ t0₌₀exp(−st)χ e(x, t0)dt0, (2.47) ˆ µ = µ0(1 + ˆκm) ˆκm(x, s) = Z ∞ t0₌₀exp(−st)χ m(x, t0)dt0. (2.48)

The complex frequency-domain conduction, electric and magnetic relaxation func-tions are the Laplace transforms of causal funcfunc-tions of time. Therefore their real and imaginary parts for imaginary values of s = jw, with w ∈ R, satisfy the Kramers-Kronig causality relations (see de Hoop, 1995)

For a general relaxation function

ˆ

κ(x, s) = Z ∞

t0₌₀exp(−st)κ(x, t

0_)dt0_, _(2.49)

for s = jw it can be separated into its real ˆκ0 and imaginary ˆκ00 parts

ˆ

κ(x, jw) = ˆκ0_{(x, w) − jˆκ}00_{(x, w) for w ∈ R .} (2.50) The Kramers-Kronig causality relations are given by

ˆ κ00_{(x, w) = −}1 π Z ∞ w=−∞ κ0_{(x, w}0₎ w0_{− w} dw 0 for w ∈ R , (2.51) ˆ κ0(x, w) = 1 π Z ∞ w=−∞ κ00_{(x, w}0₎ w0_{− w} dw 0 for w ∈ R , (2.52) and imply that κ0 _{and κ}00 _{form pairs of Hilbert transforms. Thus, whether the}

non-dissipative part κ0 is known experimentally or it is determined by some theory, it is possible to construct the dissipative part κ00 _{by taking the Hilbert transform.}

Similarly, if the response of κ00_{is known as a function of frequency, then the Hilbert}

transform can infer a constraint on κ0. Although this property is not used in this thesis, it allows to control complex effective medium models, since they must obey these relations. It also shows the strong coupling of the real and imaginary parts of the permittivity.

From the decomposition of the Laplace transformed relaxation function into its real and imaginary parts it follows that κ0 _{is and even function of w and κ}00 _{is an odd}

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κ0_{(x, −w) = κ}0(x, w) _{for all w ∈ R ,} (2.53) κ00_{(x, +w) = −κ}00(x, w) _{for all w ∈ R .} (2.54) The inverse of κ, 1/κ, must also be causal and obey the Kramers-Kronig causality relations. They cannot be minimum phase functions when they represent strictly passive or active media for all frequencies.

2.4.2 Maxwell equations in the frequency domain

The Maxwell equations in the frequency domain for linear, locally reacting, time invariant and isotropic media are obtained Laplace transforming equations (2.20) and (2.21)

− ∇ ×Hˆ + ˆη ˆE _{= −}Jˆe, (2.55) ∇_×Eˆ+ ˆζ ˆH _{= − ˆ}Ke, (2.56) and the compatibility equations (2.9) and (2.10)

∇_{· (ˆη}Eˆ_{) = −∇ ·}Jˆe, (2.57) ∇_{· (ˆ}ζ ˆH_{) = −∇ · ˆ}Ke, (2.58) where

ˆ

η = ˆσ + sˆε is the transverse admittance per length of the medium, or total conductivity

ˆ

ζ = sˆµ is the longitudinal impedance per length of the medium

When the medium is instantaneously reacting the constitutive coefficients ˆσ, ˆε and ˆ

µ are independent of s.

Assuming a homogeneous medium, applying the curl (∇×) to equation (2.55) and combining it with equation (2.57) we end up with a second order differential equation for the electric field, and similarly for the magnetic field

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20 Chapter 2. Basic electromagnetic equations ∇2Eˆ_{− ˆγ}2Eˆ_{= ∇ × ˆ}Ke+ ˆζ ˆJe₋1 ˆ η∇(∇ · ˆ Je), (2.59) ∇2Hˆ _{− ˆγ}2Hˆ _{= −∇ ×}Jˆe+ ˆη ˆKe₋1 ˆ ζ ∇_{(∇ · ˆ}_Ke_), _(2.60) where ˆ γ2 = ˆη ˆζ. (2.61)

Both equations have the same structure and are known as wave equations. Maxwell’s equations determine then, the propagation and the form of the elec-tromagnetic waves. Of course, this is very general, and to obtain the exact form of the waves we have to impose the source characteristics as well as the boundary conditions.

In the rest of the thesis, we restrict ourselves to the special case of propagation along transmission lines. For that purpose it is easier to work in the angular-frequency do-main, where the Laplace parameter s = jw and the transverse admittance and the longitudinal impedance of a medium with conductive, and dielectric and magnetic losses are now

ˆ η(jw) = jwˆε∗, (2.62) ˆ ζ(jw) = jw ˆµ∗, (2.63) where ˆ ε∗_{= ˆ}_ε0_{− j(ˆε}00₊ σdc

w ) is the complex permittivity,

ˆ ε0 _{= ε}

0εˆ

0

r is the real part of the permittivity,

or dielectric constant,

ˆ

ε00= ε0εˆ

00

r is the imaginary part of the permittivity,

or dielectric loss, and the term ˆε00₊σdc

w

accounts for dielectric and conductive losses,

ˆ

µ∗ _{= ˆ}_µ0_{− j ˆ}_µ00 _{is the complex permeability,}

ˆ

ε∗ and ˆµ∗ cannot be minimum phase functions if they have to represent strictly passive or active media for all frequencies, therefore

ˆ

ε0 > 0 and εˆ00_{≥ 0,} (2.64) ˆ

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and we choose the positive branch of the square roots, so that

<(ˆγ ≥ 0). (2.66)

For non-magnetic materials (ˆµ∗ _{= µ}

0) the propagation constant becomes

ˆ γ2_{= −w}2εˆ∗µˆ∗ _{= −}w 2 c2 0 ˆ ε0_r_{− j} ˆ ε00_r+ σdc wε0 , (2.67)

and the Maxwell’s equations (2.55) and (2.56), the compatibility equations (2.57) and (2.58), and the wave equations (2.59) and (2.60) remain the same, taking into account the new expressions for the medium parameters.

The propagation constant in free space is then

ˆ γ₀2_{= −w}2ε0µ0 = −w 2 c2 0 , (2.68)

It is possible to define a source-free domain when the source field at the boundary of the domain is known. Then the problem reduces to solving Maxwell’s equations in a source-free domain

∇2_Eˆ_{− ˆγ}2_Eˆ _{= 0,} _(2.69) ∇2Hˆ _{− ˆγ}2Hˆ = 0. (2.70) Along the thesis, the hat that indicates that we are in the frequency domain has been removed for simplicity in the notation.

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Problems worthy of attack prove their worth by fighting back. Paul Erdos (1913-1996)

Chapter 3

Experimental design

This chapter is introduced with a short discussion on the very broad field of per-mittivity measurements, followed by a more specific one devoted to porous media and the reasoning for choosing a coaxial transmission line as the tool for the mea-surements present in this thesis. It also includes an introduction to the notation we have developed to treat this type of configurations and several characteristics of the measurements.

3.1 Measuring the permittivity of dielectric materials

Permittivity is the property that determines the behavior of dielectric materials to an applied EM field, as seen in Chapter 2. Any non-metal, and even metals as a limiting case, can be considered dielectrics. The permittivity is, then, the key to understand the behavior and composition of matter from an EM point of view, and therefore, to many applications. The microscopic arousal of this property is not the topic of this thesis and it can be found in any elementary electromagnetic text book. I would specially recommend von Hippel (1954), as he presents most of the existing microscopic phenomena and their theories in a very comprehensive way. Distinct scientific disciplines study this property at different frequency ranges, de-pending on the scale of their interest. Even different nomenclatures are used, in the low frequency range it is referred as the dielectric constant, in optics as the com-plex refraction index and in telecommunication as the comcom-plex propagation factor. We will stick to the complex permittivity, since that is how it appears in Maxwell equations, and no matter what part of the spectrum you deal with, they are the governing equations of any electromagnetic process.

Measuring at different frequency ranges involves the use of different experimental techniques. From direct current to, approximately 1 MHz, bridges and resonant

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24 Chapter 3. Experimental design

cuits are used. From hundreds of MHz up to 1011Hz we are in the microwave region. The microwaves, from decimeter to millimeter wavelengths, are the waves of the spectrum with more scientific and engineering applications. These include, among others, medical (tissue characterization for imaging), agriculture (seeds health, root intake), food and pharmaceutical industries, electrical engineering (telecommuni-cation, circuit components, etc), and of course, geophysics (logs, soil saturation, geo-radar). At these frequencies standing waves methods of measurements are used. Afsar et al. (1986) review the existing techniques for the microwave region. From the infrared to the ultraviolet ([1012_{− 10}16] Hz) we are in the geometrical optics field and permittivity is determined by reflection and transmission measure-ments. Molecular chemists master the upper end of the microwaves and the infrared since their wavelengths are comparable to the size of the molecules. Above 1016Hz the size of the atoms and the molecules and their separation become comparable to the incident wavelength, we are in the X-ray region and interference techniques are used by solid physicists to study the fundamental material properties, which often are tensors. CT-scans also operate in this range. Further up, we enter the Gamma ray region (above 1019 Hz), where quantum effects are of relevance and particle physicists try to reveal the fundamentals of the atoms in particle accelerators. Each part of the spectrum has a specific physical measuring principle, and each principle has several techniques. For example, in the microwaves the standing waves methods are used and among these there are transmission line methods, waveguides, open resonators, closed cavities, etc, and every field of interest requires different set-ups for the specific characteristics of the samples; human tissues, seeds, polymers, wood, ceramics, composites, rocks, etc. The technique chosen will then depend on the frequency of interest and the sample requirements. Multi-frequency or single frequency measurements will also affect this selection. In the next section we justify our choice.

3.1.1 Measuring the permittivity of porous media

The permittivity of porous media is usually measured from very low frequencies up to the giga-Hertz region, where it becomes constant for most natural materials. This is also the range of interest for many field applications; geo-radar and Time Domain Reflectometry (TDR) in the shallow subsurface, and dielectric logging tools for petroleum reservoir characterization.

On the low frequency regime, up to hundreds of mega-Hertz, the so-called direct methods are being used. Generally, the material is placed between two parallel plates and the impedance or admittance of this capacitor is measured so that the permittivity is calculated directly from these measured quantities, see for e.g. Shen et al. (1987) and Bona et al. (1998).

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3.1 Measuring the permittivity of dielectric materials 25

Most theories modelling the EM behavior of porous media are developed for static fields (see Appendix E). EM fields can be considered quasi-static below 1 MHz. However, most field applications operate on the decimeter spectrum of the mi-crowaves (0.3-3 GHz), and the static fields theories are being used in many appli-cations, since on the high frequency regime, there are yet no theoretical models to explain and predict the behavior of the media. We think that this extrapolation must have its limitations as the particles present in the medium cannot behave in the same way when a static field or a rapidly alternating one is applied. But, this interaction is very complex and it is almost impossible to theoretically model it. We will instead, study the limitations of such an extrapolation from an experimental point of view in Chapter 6. First we need to obtain reliable permittivity measure-ments, which is already a difficult task and to which this first part of this thesis is devoted.

On the high-frequency regime, the relation between the permittivity and the mea-sured quantities is no longer linear for reasonably sized sample holders and more complex set-ups have to be implemented. For broad band measurements, reflec-tion and/or transmission methods are used. Resonant-cavity methods are limited to a few frequency points determination and for medium to low-loss samples. The reflection method gives good results for low and medium-loss samples but not for high-loss samples, contrary to the transmission method that works best for relatively high-loss samples. The S-parameter method, combines the reflection and transmis-sion methods and overcomes the disadvantages of both. It has become very popular in the study of dielectric properties. It is common to place the material in a coaxial transmission line, Shen (1985) and Nguyen et al. (1998), or coaxial-circular wave guide, Taherian et al. (1991), and measure the S-parameters of the set-up with a Network Analyzer.

Since we want to study the behavior of the permittivity of sands at different degrees of saturation with different fluids in the microwave part of the spectrum for applica-tion purposes, we need broad band measurements and a standing wave device. We chose a coaxial transmission line, where the propagation of the TEM mode only is ensured, to simplify the modelling and for better accuracy. The tool is connected to the two ports of an ANA (Automated Network Analyzer) that makes sweeps in frequency, and measures the response of the tool from 300 KHz to 3 GHz on both ports, determining its full S-parameters matrix.

The obtention of the permittivity from S-parameters measurements is discussed extensively in Chapter 5, but here, I would like to introduce the three main theo-retical representations for this type of measurements; full wave, transmission line and 2-port networks, plus a new approach, a combination of the transmission line representation with the traditional method of propagation matrices. This new rep-resentation has a clear advantage above the other three as we will show in the next section.

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26 C h ap te r 3. E xp er ime n ta l d es ig n

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3.2. Different representations for a coaxial transmission line 27

3.2 Different representations for

a coaxial transmission line

In the literature different authors have approached the problem of propagation of EM waves along coaxial transmission lines differently. Here is a summary of the three main traditional representations plus the new one we developed.

Full wave representation

This approach is very convenient for waveguides in which the sole propagation of TEM waves cannot occur, either by its shape or by the frequency of operation. It is then necessary to model all the modes propagating along the guide solving Maxwell equations for the particular set-up, and to impose boundary conditions on the interfaces of the different sections. In Appendix B we present their solution for a coaxial waveguide together with the cut-off frequencies for the different modes. The fields are represented by series expansions.

Belhadj-Tahar et al. (1990) and Taherian et al. (1991) use this representation as they model the response of a coaxial-circular waveguide. Each section of the line is characterized by its EM properties, the electrical permittivity ε∗ _{and the magnetic}

permeability µ∗_{, see Figure 3.1a.}

In contrast, when the propagation of only the TEM mode can be ensured, it is simpler to treat the coaxial waveguide using characteristics of lumped element equivalent circuits.

Transmission Line representation

In a coaxial waveguide operated below cut-off only the TEM mode propagates and it is then possible to express the waves in terms of voltage and current. Full treatment and equations can be found in Appendix C. Each section of the line is characterized by its impedance Zn and propagation constant γn, and they are

determined by the transmission line parameters, see Figure 3.1b. As explained in Appendix C, if such a representation is chosen, recursive expressions for the total reflection and transmission of the line can be computed via equations (C.30) and (C.31).

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2-port network: Scattering matrix representation

Electrical engineers use the scattering matrix ([S]) representation of 2-port net-works to relate the reflected voltage (V−) to the incident voltage (V+) on circuit components with two distinct ports or connections, V− = [S]V+. It is a way to describe the reflection/transmission response of electrical components and their in-teraction, see Appendix D. An N-multisectional transmission line can be treated as N 2-port networks connected in series (Figure 3.1c) and each section is represented by its own scattering matrix [Sn]. The total response of the line is obtained by the

Redheffer’s star product (Redheffer, 1961) of the individual matrices. Shen (1985) used this approach to compute the permittivity of soil samples. We discuss it in more detail in Subsection 5.2.4.

These three representations provide the same solution with a different set of equa-tions. In a multisection transmission line, the sample is placed in a single section and it is the properties of the sample that we would like to obtain from those equations. The simpler the equations the smaller the propagation of error through numerical computations. Moreover, the representation in which the interactions of the sections is clear is most desirable for a thorough understanding. However, these three representations provide rather cumbersome expressions, series expansions for the full wave, recursive equations for the transmission lines and tedious Redheffer’s star products for the 2-port network. It is then, relatively easy to loose track of the section interactions and the relevant features. With this in mind, we have devel-oped a new representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler.

Propagation Matrices: a combined representation

In physics it is common to represent the down-going and up-coming wave fields in a layered medium in matrix notation, for a seismic example see Claerbout (1968). This technique is known as the Propagation Matrices method. It can be applied to any medium composed of layers of different properties where waves are propagating, such as a succession of sections of a coaxial transmission line filled with different dielectrics. This formulation, for the specific case of a transmission line, is presented in the next section. In essence, it consists of rewriting the Transmission Line formulation (Appendix C) in matrix form, and relating the S-parameters to the total reflection and transmission of the line seen as a 2-port network (Appendix D).

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3.2 Different representations for a coaxial transmission line 29

3.2.1 Propagation Matrices: a combined representation

Operating the line below the cut-off frequency, only the TEM mode will propagate and a Transmission Line formulation can be adopted. The solution to the transmis-sion line equations (C.2) and (C.3) in between two interfaces n − 1 and n, placed at zn−1 and at znrespectively, can be written as

Vn(z) = Vn+e−γn(z−zn−1)+ Vn−e−γn(zn−z), (3.1) In(z) = 1 Zn [V_n+e−γn(z−zn−1 _{− V}− n e−γn(zn−z)], (3.2)

where γn is the propagation constant of section n given by equation (C.4) and Zn

is the impedance given by equation (C.8). At any n-th interface, the fields must be continuous, and this can be expressed in a propagation matrices notation as

e−γndn ₁ 1 Zne −γndn ₋ 1 Zn V_n+ V− n = 1 e−γn+1dn+1 1 Zn+1 − 1 Zn+1e −γn+1dn+1 V_n+1+ V_n+1+ , (3.3)

or in its compact form as

ML_nVn= MRn+1Vn+1, (3.4) where Vn= V+ n V_n− , (3.5) and ML_n = e−γndn ₁ 1 Zne −γndn ₋ 1 Zn and MR_n = 1 e−γndn 1 Zn − 1 Zne −γndn . (3.6)

Then, for a multisectional line, as shown in Fig. 3.1, it is easy to relate the fields at the first interface, n = 1 to those at the last one, n = N , via the expression:

V1 = (ML1)−1 "_{N −1} Y n=2 MR_n(ML_n)−1 # MR_NVN. (3.7)

Now, we can group its elements into three main sections with four interfaces, see Fig. 3.2.

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Figure 3.2: Simplified 3 sections model of a multisectional coaxial transmission line

The sample holder is in its central part P , anything at its left is included in L and at its right in R, as follows

P = MR_p(ML_p)−1, where p stands for the sample holder (3.8)

L = (ML₁)−1 "_n=p−1 Y n=2 MR_n(ML_n)−1 # , (3.9) R =   n=N −1 Y n=p+1 (ML_n)−1MR_n  MR_N, (3.10)

and equation (3.7) can now be rewritten as

V1= LP RVN. (3.11)

But, V1 and VN are related to the S-parameters of the line, and that is what

effectively the Network Analyser is measuring. Then, when an electromagnetic wave impinges from Port 1, equation (3.11) transforms into

1 S11 = LP R S12 0 , (3.12)

and when it does from Port 2

1 S22 = ´L ´P ´R S21 0 . (3.13)

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3.2 Different representations for a coaxial transmission line 31

Note, that we can keep the same formulation for this case, if we use a mirror image of the transmission line. So L0 corresponds to the sections between Port 2 and the sample holder, represented now by P0. And the sections from the sample holder to Port 1 are included in R0. Notice that P = P0, when it is filled with a homogeneous material, because on both cases it is composed of only the sample holder. To avoid writing mirror equations and since the properties derived for the reflection and transmission coefficients of one specific pair are shared by the other pair, from now on, we will use Γ to represent both reflections, S11 or S22, and Υ

for the transmissions, S12 or S21. The transmission coefficients are always equal

to each other: [S12 = S21]always, since the path the waves travel is the same, but

the reflection coefficients are equal only when the multisectional line is perfectly symmetric and filled with homogenous materials: [S11= S22]symmetric line.

Having expressed the full reflection/transmission of TEM waves along a transmis-sion line in a propagation matrices formulation enables us to compute the electric permittivity and magnetic permeability of any sample filling the sample holder. We show this method in Chapter 5, together with other techniques, but first, we illustrate this representation for an ideal case, in the next section.

The Propagation matrices representation consists in rewriting the Transmission Line formulation (Appendix C) in matrix form, and relating the S-parameters to the total reflection and transmission of the line seen as a 2-port network (Appendix D). It is, therefore, very difficult to dissociate these terms. However, as a rule, when talking about S-parameters, we will be referring to the reflection/transmission measure-ments at the reference planes of a transmission line.

3.2.2 Combined representation of the ideal case

Let us consider the simplest configuration possible, where the material of study is placed in region 2 of a coaxial transmission line, such as the one in Figure 3.3, and has a length dp. Regions 1 and 3 are filled with air so that their impedance is Z0.

The walls of the line are made of perfect conducting metals and the measurement planes are at the interfaces of the material. In practice, the commercial transmission lines available (perfectly continuous cylinders filled up with air and with appropriate connectors), are the closest set-ups to the ideal case. Their special configuration, however, is responsible for uncertainties of sample length and position, Baker-Javis et al. (1990). In any case, in many situations, the sample specimens are not suitable for these type of fixtures, e.g. loose sands, large sample sizes. Moreover, the commercial fixtures don’t allow for any special measurement feature, e.g. fluid flow. In those cases, the transmission line has to be customized and, in fact, many transition sections between the reference measurement planes and the sample are needed. The problem with these measurements lies in the difficulty of removing the effect of these extra sections, and we treat this extensively in the following chapters,

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but for the moment, we focus on this simple case, as it proves to be very useful in the theoretical understanding of this type of problems. It is this configuration that most of the published methods consider for the determination of permittivity from transmission and reflection measurements, Weir (1974), Stuchly & Matuszewsky (1978) and Ligthart (1983).

REGION 1 REGION 2 REGION 3

Ref. plane 1 Ref. plane 2

a b p d

e

₀

e

₀

e*

Figure 3.3: Simplified coaxial transmission line

For such a configuration, we can write the extended form of equation (3.12) as:

1 Γ = 1 4 1 Z0 1 −Z0 e+γpdp_{+ e}−γpdp _Z p(e+γpdp− e−γpdp) 1 Zp(e +γpdp_{− e}−γpdp₎ _e+γpdp_{+ e}−γpdp 1 1 1 Z0 − 1 Z0 Υ 0 , (3.14)

where Z0 is the impedance of sections 1 and 3, and Zp, γp and dp are, respectively,

the impedance, propagation constant and length of section 2. It is clear from equation (3.14) that the input on reference plane 1 is equal to the product of 3 propagation matrices, first to go from port 1 into the sample holder, then the propagation in the sample holder itself, and finally from the sample holder into port 2, and the output at reference plane 2. As the reference planes coincide with the sample holder interfaces, L and R have no exponential terms but only an impedance contrast, as the source and receiver cannot be placed on the actual sample.

Electric characterization of sands with heterogeneous saturation distribution

Electric

characterization of

sands

with heterogenous

saturation

distribution

Electric characterization of sands

with heterogenous saturation distribution

Electric characterization of sands

with heterogenous saturation distribution

Contents

Chapter 1

Introduction

1.1

Statement of the problem

1.2

Scientific framework

Permittivity states of soils

Introduction

Basic electromagnetic equations

Experimental design

Tool for accurate permittivity measurements

Reconstruction Methods

Permittivity states of mixe-phase;

2 and 3 component mixtures

1

3

2

6

5

4

Conclusions and Recommendations

7

1.3

Outline of the thesis

Chapter 2

Basic electromagnetic equations

2.1

The electromagnetic field equations

2.2

The constitutive relations

2.3

Boundary conditions

2.4

The electromagnetic field equations

in the complex frequency domain

Chapter 3

Experimental design

3.1

Measuring the permittivity of dielectric materials

3.2

Different representations for

a coaxial transmission line

e

e

e*