ceg_gorriti_20041102.

ceg_gorriti_20041102.
Electric characterization of
sands with heterogenous
saturation distribution
Ainhoa G. Gorriti
Electric characterization of sands
with heterogenous saturation distribution
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 Española” dictionary says,
the present day meaning of the word “discusión” 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
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 hetzelfde 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
Española” zegt, betekent het woord “discusión” 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
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 GONZÁLEZ GORRITI
Licenciada en Ciencias Fı́sicas,
especialización Fı́sica de la Tierra y el Cosmos
Universidad Complutense de Madrid,
geboren te Madrid, Spanje.
Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.ir. J.T. Fokkema
Toegevoegd promotor: Dr.ir. E.C. Slob
Samenstelling promotiecommissie:
Rector Magnificus
Prof.dr.ir. J.T. Fokkema
Dr.ir. E.C. Slob
Prof.dr. K. Holliger
Prof.dr. A. Sihvola
Prof.dr.ir. L. Ligthart
Prof.dr. S.M. Luthi
Dr. J. Bruining
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, toegevoegd promotor
Swiss Federal Institute of Technology
Helsinki University of Technology
Technische Universiteit Delft
Technische Universiteit Delft
Technische Universiteit Delft
Hans Bruining heeft in belangrijke mate bijgedragen aan de totstandkoming van
dit proefschrift.
c
Copyright 2004
by A.G. Gorriti
Cover design: Jesús G. Gorriti
ISBN 90-9018709-X
Printed in the Netherlands.
A mi familia, en especial a mi abuelo.
Y a Jara y a Ron, in memoriam.
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.
Uno no es lo que es por lo que escribe,
sino por lo que ha leı́do∗ .
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.
CONTENTS
ix
Contents
1 Introduction
1
1.1
Statement of the problem . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Scientific framework . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Basic electromagnetic equations
9
2.1
The electromagnetic field equations
. . . . . . . . . . . . . . . . .
2.2
The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 12
2.3
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
EM Field Equations in the Complex Frequency Domain . . . . . . . 16
3 Experimental design
9
23
3.1
Measuring the permittivity of dielectric materials . . . . . . . . . . 23
3.2
Different representations for a coaxial transmission line . . . . . . . 27
3.3
Γ and Υ: independent measurements . . . . . . . . . . . . . . . . . 34
3.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Tool for accurate permittivity measurements
37
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2
Technical design and measurement characteristics . . . . . . . . . . 39
4.3
Forward Model and Calibration . . . . . . . . . . . . . . . . . . . . 43
4.4
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
x
Contents
5 Reconstruction methods for permittivity
from measured S-parameters
55
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2
Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3
Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4
Analytical vs Optimization . . . . . . . . . . . . . . . . . . . . . . 87
5.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Permittivity states of mixed-phase;
two and three component sands
91
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2
Saturation technique and permittivities . . . . . . . . . . . . . . . . 93
6.3
Averaging ε∗12 and ε∗21 over averaging the S-parameters . . . . . . . 111
6.4
2-layer samples
6.5
Anomalies of layered samples . . . . . . . . . . . . . . . . . . . . . 117
6.6
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Conclusions and recommendations
143
A General solution for the wave equation in polar coordinates
147
A.1 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.2 General solution for the wave equation . . . . . . . . . . . . . . . . 148
B Waveguides
151
B.1 Maxwell’s equations in polar coordinates . . . . . . . . . . . . . . . 151
B.2 Coaxial Waveguide
. . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.3 Coaxial-Circular Waveguide . . . . . . . . . . . . . . . . . . . . . . 160
Contents
C Transmission Lines
xi
163
C.1 Transmission Line Equations . . . . . . . . . . . . . . . . . . . . . 163
C.2 Local Reflection and Transmission coefficients . . . . . . . . . . . . 166
C.3 Line parameters for an ideal section . . . . . . . . . . . . . . . . . 168
C.4 Global Reflection and Transmission coefficients . . . . . . . . . . . 169
D Scattering Matrix
171
D.1 S-parameters and the total reflection and transmission
coefficients of a transmission line . . . . . . . . . . . . . . . . . . . 172
D.2 Unitary property of the Scattering Matrix . . . . . . . . . . . . . . 173
E Permittivity models
175
E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
E.2 Classical approaches to pure materials . . . . . . . . . . . . . . . . 176
E.3 Bounds for the effective permittivity of mixtures . . . . . . . . . . . 178
E.4 Mixing Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
E.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Bibliography
185
Summary
191
Samevatting
195
Acknowledgements
199
About the author
201
xii
Contents
Twenty years from now you will be more disappointed by the
things that you didn’t do than by the ones you did do. So
throw off the bowlines. Sail away from the safe harbor. Catch
the trade winds in your sails. Explore. Dream. Discover.
Mark Twain (1835-1910)
Chapter 1
Introduction
A proper understanding of the electrodynamic response of soils will lead to an
improvement in the forward and inverse modelling of electromagnetic (EM) geophysical 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 appropriate 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 parameters 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.
1
2
Chapter 1. Introduction
We think that the best methodology to study rocks and soils is, not to have a preconceived 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 fraction and permittivities of the constituents. Theoretically, they are valid for quasistatic 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.
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 concerning 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 attempted 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 ClausiusMossotti theory to take into account the permanent moments of the molecules. His
theory, later extended by Onsager (1936) and Kirkwood (1939), linked the dispersion 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 significantly from Debye’s theory of relaxation. Cole & Cole (1941) pioneered the
4
Chapter 1. Introduction
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 cooperative molecular motions, the ”many-body interactions” approach.
All these researchers modelled the macroscopic behavior of dielectric materials composed 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 setups 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).
1.2. Scientific framework
5
In the high-frequency regime, the relation between the permittivity and the measured quantities is no longer linear for reasonably sized sample holders and more
complex set-ups have to be implemented. For broad band measurements, reflection 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 highloss 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 analytically or via an optimization procedure. Analytical methods compute the electrical permittivity from analytical expressions involving the S-parameters. Until
now there were no straight forward methods, the S-parameters had to be compensated, in one way or an other (Rau & Wharton (1982), Kruppa & Sodomsky
(1971), Freeman et al. (1979), Shen (1985) and Chew et al. (1991)). Then different 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 measured 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 frequency band, the following researchers have published studies over a broad variety
of soils, different water contents and temperature: Shutko & Reutov (1982), Hallikainen 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.
6
Chapter 1. Introduction
Permittivity states of soils
Introduction
2
Basic electromagnetic equations
'
3
Experimental design
Technical design and measurement characteristics
Forward model and calibration
Reconstruction Methods
%
'
Analytical
'
Optimization
%
%
Permittivity states of mixe-phase;
2 and 3 component mixtures
'
'
'
7
Saturation Techniques
2-layer samples
Interpretation of results
Conclusions and Recommendations
Figure 1.1: Schematic outline of the thesis.
Evidence of
permittivity states
6
Resonant frequencies
Propagation Matrices
Compensation Parameters
calibrated coax TL
+
forward and inverse models
Tool for accurate permittivity measurements
'
'
5
Representations for coaxial transmission lines
The Propagation Matrices representation
TOOL =
'
'
4
EM field equations
Preliminaries
1
1.3. Outline of the thesis
1.3
7
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.
8
Chapter 1. Introduction
Samples saturated in that way result in distinctive 2-layer samples, whose reconstructed 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 recommendations 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 waveguide. 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.
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 mutually perpendicular base vectors {x̂, ŷ, ẑ} 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 = xx̂ + yŷ + zẑ.
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
def
= 299792458[m/s].
ε0 µ0
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
≈ 8.854 × 10−12 [F/m].
µ0 c20
9
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
H
is the electric field strength [V/m],
is the magnetic field strength [A/m],
and the action of sources is represented by
Je
Ke
2.1.1
is the source volume density of electric current [A/m2 ],
is the source volume density of magnetic current [V/m2 ].
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 ∂t E = −J mat ,
∇ × E + µ0 ∂t H = −K
mat
(2.1)
,
(2.2)
In a vacuum domain, the material volume densities {J mat , K mat } 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 J e and
K e , 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 J i and K i , respectively,
J mat = J i + J e
J i = J + ∂t P
K mat = K i + K e ,
(2.3)
K i = µ0 ∂t M ,
(2.4)
2.1. The electromagnetic field equations
11
where
J
P
M
is the volume density of electric current [A/m2 ],
is the electric polarization [C/m2 ],
is the magnetization [A/m].
Further it is customary to introduce the quantities
D = ε0 E + P ,
(2.5)
B = µ0 (H + M ).
(2.6)
where
D
B
is the electric flux density [C/m2 ],
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 + ∂t D = −J e ,
e
∇ × E + ∂t B = −K ,
(2.7)
(2.8)
From Maxwell’s equations two other equations can be derived that are not independent of Maxwell’s equations, the compatibility equations. This means that field
quantities that satisfy Maxwell’s equations by definition also satisfy the compatibility equations. Compatibility equations are obtained by applying the divergence
operator (∇·) to equations (2.7) and (2.8), respectively. This results in,
∇ · (J + ∂t D) = −∇ · J e ,
e
∂t ∇ · B = −∇ · K .
(2.9)
(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,
known as the charge conservation law.
(2.12)
12
2.2
Chapter 2. Basic electromagnetic equations
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 coefficients 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
χe
χm
is the electrical conductivity [S/m],
is the electric susceptibility [-],
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) = ε0 εr (x)E(x, t)
(2.16)
B(x, t) = µ(x)H(x, t) = µ0 µr (x)H(x, t),
(2.17)
where
ε
εr
µ
µr
is
is
is
is
the
the
the
the
(absolute) permittivity [F/m],
relative permittivity [-],
(absolute) permeability [H/m],
relative permeability [-],
and can be expressed as
ε = ε0 (1 + κe )
or
εr = 1 + κe ,
(2.18)
µ = µ0 (1 + κm )
or
µr = 1 + κm ,
(2.19)
2.2. The constitutive relations
13
where
κe
κm
is the relative electric susceptibility [-],
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 equations (2.13)-(2.17) to obtain Maxwell’s equations in the electric and magnetic field
strengths only, gives
− ∇ × H + σE + ε∂t E = −J e ,
e
∇ × E + µ∂t H = −K ,
(2.20)
(2.21)
And the compatibility relations read
σ∇ · E + ε∂t ∇ · E = −∇ · J e ,
e
µ∂t ∇ · H = −∇ · K .
(2.22)
(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 experiments. 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 parameters 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
14
Chapter 2. Basic electromagnetic equations
J (x, t) =
P (x, t) =
Z
t
t0 =0
Z t
t0 =0
M (x, t) =
Z
t
t0 =0
σ(x, t0 )E(x, t − t0 )dt0 ,
(2.24)
χe (x, t0 )E(x, t − t0 )dt0 ,
(2.25)
χm (x, t0 )H(x, t − t0 )dt0 ,
(2.26)
where {t0 ∈ R; t0 > 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 mathematically 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 measuring 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) = ε0 E(x, t) + ε0
Z
t
t0 =0
B(x, t) = µ0 H(x, t) + µ0
Z
κe (x, t0 )E(x, t − t0 )dt0 ,
t
t0 =0
κm (x, t0 )H(x, t − t0 )dt0 .
(2.28)
(2.29)
2.3. Boundary conditions
2.3
15
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 discontinuously. Since all physical quantities have bounded magnitudes, the relevant
discontinuities are restricted to finite jump discontinuities. Because of their discontinuous behavior, the electric and magnetic field strengths are no longer continuously 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 everywhere. 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
ν ×E
is continuous across S,
is continuous across S,
(2.30)
(2.31)
and looking at the compatibility equations we see that the normal components
16
Chapter 2. Basic electromagnetic equations
are differentiated in the direction normal to the interface and hence we have the
conditions:
ν · (J + ∂t D)
ν ·B
is continuous across S,
is continuous across S.
(2.32)
(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 ν × E(x + hν, t) = 0
h↓0
for any x ∈ ∂D,
(2.34)
Electrically impenetrable materials arise as limiting cases of materials whose conductivity 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 T 0 of the domain T and the boundary ∂T between the two
domains are defined according to
T 0 = {t ∈ R; t < 0},
∂T = {t ∈ R; t = 0}.
(2.36)
(2.37)
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 , T 0 } ,
(2.38)
and the Laplace transform of a function f (x, t) in space-time, defined for t ∈ T , is
fˆ(x, s) =
Z
exp(−st)χT f (x, t)dt,
(2.39)
t∈R
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∞
exp(st)fˆ(x, s)ds = χT (t)f (x, t).
(2.40)
s=s0 −j∞
The path of integration is parallel to the imaginary s-axis in the right half of the
complex s-plane where fˆ is analytic.
Symbolically equation (2.39) is written as
fˆ(x, s) = Lf (x, t),
(2.41)
and the Laplace transform of a partial differentiation of a function, ∂t f (x, t), equals
L∂t fˆ(x, s) = sf (x, t),
(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
Jˆ(x, s) = σ̂(x, s)Ê k (x, s),
(2.43)
D̂(x, s) = ε̂(x, s)Ê k (x, s),
(2.44)
B̂(x, s) = µ̂(x, s)Ĥ j (x, s),
(2.45)
18
Chapter 2. Basic electromagnetic equations
where
σ̂ =
Z
∞
exp(−st)σc (x, t0 )dt0 ,
(2.46)
t0 =0
ε̂ = ε0 (1 + κ̂e )
Z
κ̂e (x, s)
µ̂ = µ0 (1 + κ̂m )
=
Z
κ̂m (x, s) =
∞
exp(−st)χe (x, t0 )dt0 ,
(2.47)
exp(−st)χm (x, t0 )dt0 .
(2.48)
t0 =0
∞
t0 =0
The complex frequency-domain conduction, electric and magnetic relaxation functions are the Laplace transforms of causal functions 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
κ̂(x, s) =
Z
∞
exp(−st)κ(x, t0 )dt0 ,
(2.49)
t0 =0
for s = jw it can be separated into its real κ̂0 and imaginary κ̂00 parts
κ̂(x, jw) = κ̂0 (x, w) − jκ̂00 (x, w) for w ∈ R .
(2.50)
The Kramers-Kronig causality relations are given by
κ̂0 (x, w) =
∞
κ0 (x, w0 ) 0
dw for w ∈ R ,
0
w=−∞ w − w
1 ∞
κ00 (x, w0 ) 0
dw for w ∈ R ,
π w=−∞ w0 − w
κ̂00 (x, w) = −
1
π
Z
Z
(2.51)
(2.52)
and imply that κ0 and κ00 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 κ00 is an odd
function of w
2.4. EM Field Equations in the Complex Frequency Domain
κ0 (x, −w) = κ0 (x, w)
00
00
κ (x, +w) = −κ (x, w)
for all w ∈ R ,
for all w ∈ R .
19
(2.53)
(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)
e
− ∇ × Ĥ + η̂ Ê = −Jˆ ,
e
∇ × Ê + ζ̂ Ĥ = −K̂ ,
(2.55)
(2.56)
and the compatibility equations (2.9) and (2.10)
e
∇ · (η̂ Ê) = −∇ · Jˆ ,
e
∇ · (ζ̂ Ĥ) = −∇ · K̂ ,
(2.57)
(2.58)
where
η̂ = σ̂ + sε̂
is the transverse admittance per length of the medium,
or total conductivity
ζ̂ = sµ̂
is the longitudinal impedance per length of the medium
When the medium is instantaneously reacting the constitutive coefficients σ̂, ε̂ and
µ̂ 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
20
Chapter 2. Basic electromagnetic equations
1
e
e
e
∇2 Ê − γ̂ 2 Ê = ∇ × K̂ + ζ̂ Jˆ − ∇(∇ · Jˆ ),
η̂
(2.59)
1
e
e
e
∇2 Ĥ − γ̂ 2 Ĥ = −∇ × Jˆ + η̂ K̂ − ∇(∇ · K̂ ),
ζ̂
(2.60)
γ̂ 2 = η̂ ζ̂.
(2.61)
where
Both equations have the same structure and are known as wave equations.
Maxwell’s equations determine then, the propagation and the form of the electromagnetic 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 domain, 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
η̂(jw) = jwε̂∗ ,
(2.62)
∗
(2.63)
ζ̂(jw) = jwµ̂ ,
where
ε̂∗ = ε̂0 − j(ε̂00 +
σdc
w )
is the complex permittivity,
0
ε̂0 = ε0 ε̂r
is the real part of the permittivity,
or dielectric constant,
00
ε̂00 = ε0 ε̂r
is the imaginary part of the permittivity,
or dielectric loss, and the term ε̂00 + σwdc
accounts for dielectric and conductive losses,
µ̂∗ = µ̂0 − j µ̂00
is the complex permeability,
ε̂∗ and µ̂∗ cannot be minimum phase functions if they have to represent strictly
passive or active media for all frequencies, therefore
ε̂0 > 0
0
µ̂ > 0
and
and
ε̂00 ≥ 0,
00
µ̂ ≥ 0,
(2.64)
(2.65)
2.4. EM Field Equations in the Complex Frequency Domain
21
and we choose the positive branch of the square roots, so that
<(γ̂ ≥ 0).
(2.66)
For non-magnetic materials (µ̂∗ = µ0 ) the propagation constant becomes
σdc
w2 0
00
γ̂ = −w ε̂ µ̂ = − 2 ε̂r − j ε̂r +
,
wε0
c0
2
2 ∗ ∗
(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
γ̂02 = −w2 ε0 µ0 = −
w2
,
c20
(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 Ê − γ̂ 2 Ê = 0,
2
2
∇ Ĥ − γ̂ Ĥ = 0.
(2.69)
(2.70)
Along the thesis, the hat that indicates that we are in the frequency domain has
been removed for simplicity in the notation.
22
Chapter 2. Basic electromagnetic equations
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 permittivity 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 measurements 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, depending 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 complex refraction index and in telecommunication as the complex 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 cir23
24
Chapter 3. Experimental design
cuits are used. From hundreds of MHz up to 1011 Hz 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 (telecommunication, 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 − 1016 ] Hz) we are in the geometrical
optics field and permittivity is determined by reflection and transmission measurements. Molecular chemists master the upper end of the microwaves and the infrared
since their wavelengths are comparable to the size of the molecules. Above 1016 Hz
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).
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 microwaves (0.3-3 GHz), and the static fields theories are being used in many applications, 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 measurements, 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 measured quantities is no longer linear for reasonably sized sample holders and more
complex set-ups have to be implemented. For broad band measurements, reflection 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 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 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 application 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 theoretical 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 representation has a clear advantage above the other three as we will show in the next
section.
26
Chapter 3. Experimental design
Figure 3.1: Equivalent representations of a sequence of dielectric layers traversed by TEM waves.
3.2. Different representations for a coaxial transmission line
3.2
27
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).
28
Chapter 3. Experimental design
2-port network: Scattering matrix representation
Electrical engineers use the scattering matrix ([S]) representation of 2-port networks 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 interaction, 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 equations. 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 developed 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).
3.2
Different representations for a coaxial transmission line
3.2.1
29
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 transmission line equations (C.2) and (C.3) in between two interfaces n − 1 and n, placed
at zn−1 and at zn respectively, can be written as
Vn (z) = Vn+ e−γn (z−zn−1 ) + Vn− e−γn (zn −z) ,
1
[V + e−γn (z−zn−1 − Vn− e−γn (zn −z) ],
In (z) =
Zn n
(3.1)
(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−γn dn
1 −γn dn
Zn e
1
− Z1n
Vn+
Vn−
=
1
1
Zn+1
e−γn+1 dn+1
1
− Zn+1
e−γn+1 dn+1
+
Vn+1
+
Vn+1
, (3.3)
or in its compact form as
R
ML
n V n = M n+1 V n+1 ,
where
Vn=
Vn+
Vn−
(3.4)
,
(3.5)
and
ML
n
=
e−γn dn
1 −γn dn
Zn e
1
− Z1n
and
MR
n
=
1
1
Zn
e−γn dn
− Z1n e−γn dn
. (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:
−1
V 1 = (M L
1)
"N −1
Y
n=2
#
L −1
MR
MR
n (M n )
NV N.
(3.7)
Now, we can group its elements into three main sections with four interfaces, see
Fig. 3.2.
30
Chapter 3. Experimental design
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
L −1
P = MR
p (M p ) ,
L =
where p stands for the sample holder
−1
(M L
1)
"n=p−1
Y
L −1
MR
n (M n )
n=2

n=N
Y−1
R = 
n=p+1

−1
R
R
(M L
n) Mn MN,
#
,
(3.8)
(3.9)
(3.10)
and equation (3.7) can now be rewritten as
V 1 = LP RV N .
(3.11)
But, V 1 and V N 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
1
S22
= LP R
S12
0
,
(3.12)
= ĹṔ Ŕ
S21
0
.
(3.13)
and when it does from Port 2
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
0
image of the transmission line. So L corresponds to the sections between Port 2
0
and the sample holder, represented now by P . And the sections from the sample
0
0
holder to Port 1 are included in R . Notice that P = P , 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 transmission 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 measurements 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, BakerJavis 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,
32
Chapter 3. Experimental design
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
a
e*
b
e
e
0
0
dp
Ref. plane 1
Ref. plane 2
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+γp dp + e−γp dp
Zp (e+γp dp − e−γp dp )
1
+γp dp − e−γp dp )
e+γp dp + e−γp dp
Zp (e
1
1
Υ
,
(3.14)
1
1
0
Z0 − Z0
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.
3.2
Different representations for a coaxial transmission line
33
Working out the matrices product of equation (3.14) we obtain,
1
Γ
e+γp dp
=
4
"
Z0
2(1 + e−2γp dp ) + ( Z
+
p
Z0
−
(Z
p
2(1 + e−2γp dp )
Z
( Zp0 −
Zp
Z0 )(1
Z0
− (Z
p
Z0
Zp )(1
−
Zp
−2γp dp )
Z0 )(1 − e
e−2γp dp )
− e−2γp dp )
+
Zp
Z0 )(1
− e−2γp dp )
#
Υ
0
,
(3.15)
some more handling and naming ζ = e−γp dp and r to the local reflection coefficient,
given by equation (C.18), we can rewrite equation (3.15) as
1
Γ
1
=
ζ(1 − r2 )
1 − r2 ζ 2 r(ζ 2 − 1)
r(1 − ζ 2 ) ζ 2 − r2
Υ
0
.
(3.16)
And now it is easy to go back to the Transmission Line formulation as it is possible
to express the reflection and transmission as
Γ =
Υ =
r(1 − ζ 2 )
,
1 − r2 ζ 2
ζ(1 − r2 )
.
1 − r2 ζ 2
(3.17)
(3.18)
These are the global reflection and transmission coefficients given by equations (C.30) and (C.31) for this specific configuration.
Having expressed the reflection and transmission of a TEM wave along a transmission line as equation (3.16) will prove very useful in the coming sections, where
we study the occurrence of resonant frequencies and we prove that Γ and Υ are
independent measurements.
Resonant frequencies
Broad frequency band measurements of standing waves present destructive inter00
ference, at resonant frequencies. For low-loss materials (εr < 10−2 εr 0 ) of constant
permittivity over the frequency range, the off-diagonal terms of the matrix that
relate the input and output voltages, equation (3.16), are zero, since ζ 2 = 1 at
those frequencies, and therefore Γ = 0 and Υ = ±1.
The implications that arise from these frequencies are many, even for high-loss
materials, but we are too early in this thesis to give an extensive explanation about
them. Here, we simply introduce the concept of resonant frequencies for low-loss
34
Chapter 3. Experimental design
materials as they are a recurrent topic in the coming chapters. In the next chapter,
we see how the measurement accuracy is poorer around the resonant frequencies
(Subsection 4.2.2). In Chapter 5 we explain how it is possible to compute the
permittivity from these frequencies for special cases, see Subsection 5.2.1, a well
known method from interference optics for single frequency measurements. We
also discuss the deterioration of the accuracy in the reconstructed permittivity at
the vicinity of these frequencies depending on the method used for reconstruction
in Section 5.4.
3.3
Γ and Υ: independent measurements
To prove that Γ and Υ are indeed independent measurements we follow a similar
procedure as used by (Claerbout, 1968). While he used it to prove that for the
acoustic equivalent of a shorted transmission line the reflection and transmission
response of a lossless line are not independent, we use it to prove that Γ and Υ are
independent in absence of a perfect reflection at one side of the transmission line.
In essence, in both, the seismic and the electromagnetic case, the product of k
layer matrices can be written as,
V+
V−
1
1
= k
ζ
F (ζ)
G(ζ)
ζ 2k G(1/ζ) ζ 2k F (1/ζ)
V+
V−
,
(3.19)
k
where V + and V − stand for the down going and up going fields. And the determinant of the propagation matrix is
det = [F (ζ)F (1/ζ) − G(ζ)G(1/ζ)] = 1.
(3.20)
By taking the up and down going wave fields at either side of the last interfaces
according to equation (3.11),
+ + V
1
V
Υ
=
and
=
.
(3.21)
V− 1
Γ
V− k
0
Equation (3.21) is just another representation of equation (3.11) and the reflection
and transmission responses are found as,
1
F (ζ)Υ
ζk
Γ = ζ k G(1/ζ),
1 =
(3.22)
(3.23)
and in view of equation (3.20) no relation between F (ζ) and G(1/ζ) can be found
and therefore, Υ and Γ are independent.
3.4. Conclusions
3.4
35
Conclusions
In this chapter, we have justified our experimental choices and theoretical representation. We are interested in the study of the permittivity of sands partially
saturated, in the meters to the decimeter range of the spectrum. We also want to
achieve those saturations by flowing liquids through the sample. For those reasons
we found it most convenient to use a customized coaxial transmission line, whose
technical characteristics are presented in the next chapter.
Coaxial transmission lines have been treated differently by different authors and we
have developed a new combined representation to do it, the Propagation Matrices
representation. It consists of rewriting the Transmission Line representation in
matrix form, and relating the S-parameters to the total reflection and transmission
of the line seen as a 2-port network. It simplifies the understanding and computation
of TEM propagation through transmission lines.
This combined representation has many advantages as we show in the coming
chapters, but as a start, let us point to the fact that it has allowed us to prove the
independence of the reflection and transmission coefficients.
36
Chapter 3. Experimental design
In theory, there is no difference between theory and practice.
But, in practice, there is.
Jan L.A. van de Snepscheut (1953-1994)
Chapter 4
Tool for accurate permittivity
measurements
In this chapter we present the tool we designed and used for measuring the permittivity of sands, its measurement characteristics, the calibration procedure and the
comparison of the measurements with the calibrated forward model.
4.1
Introduction
We are interested in the electromagnetic behavior of sandy environments in a broad
frequency range, and as we saw in the previous chapter, the most suitable set-up is
a coaxial transmission line. Our tool is, therefore, designed as such a line, for which
all four S-parameters can be measured. This is the first time, to our knowledge,
that all four S-parameters of a coaxial transmission line are modelled so accurately
with transmission line theory.
We present the tool and its measurement characteristics in the next section and
then, its forward model is introduced in Section 4.3. To obtain such an accurate
model we had to make several calibration steps, as we show in Subsection 4.3.2,
where we also comment on several model assumptions. In Section 4.4 we show a
comparison of the model with measurements, together with a study of the sensitivity of the model, and we predict expected accuracies for the reconstruction of
permittivity from measurements.
37
38
Chapter 4. Tool for accurate permittivity measurements
Coaxial Transmission Line
TUL.1
SH
TUL.2
TUR.2
C
TUR.1
fluid outlet
fluid inlet
Transition Unit
Sample Holder
Transition Unit
Ref. plane 2
Ref. plane 1
Figure 4.1: Coaxial Transmission line probe for permittivity measurements of porous media
and its sectioning. SH: sample holder, TU: transition unit, L: to the left of SH, R: to the
right of SH, C: connector.
Fluid distributor unit
1 Carved fluid distributor
grey: teflon
white: carved fluid path
2 Inlet into carved path
3 Fluid inlets/outlets
4 Metallic outer ring and inner pin
TOP VIEW
3
1
3
2
3
4
3
1
4
SIDE VIEW
3
Figure 4.2: Fluid distributors units.
3
4
4.2. Technical design and measurement characteristics
4.2
39
The Tool: technical design
and measurement characteristics
The technical description of the tool is presented in Subsection 4.2.1. However,
before presenting the results that support this design, in Section 4.3, we first discuss
the measurement characteristics of the tool, in Subsection 4.2.2, to be able to
separate model errors from measurement errors.
4.2.1
Technical Design
The tool was designed such that it is possible to determine the full S-parameter
matrix of the system (measuring both the transmission and reflection coefficients
of a sample is crucial in the determination of the permittivity). The geometry and
size of the probe ensure that only the TEM mode propagates along the line, and,
therefore, it can be described with Transmission Line Theory. However, by simple
modifications the sample holder can be transformed into a circular waveguide,
similar to the one described in Taherian et al. (1991) and it is possible to determine
the permittivity of rigid cores with a full wave description, see Section B.3. The
sample holder was chosen to be of a representative volume (10 cm long, 3 cm in
outer diameter and 0.9 cm in inner diameter) and allowing for fluid flow. It is gold
plated to ensure low losses of energy in the line conductors, and its several parts
can be characterized separately for accurate modelling.
The probe (Figure 4.1) consists of three main sections; two transitions units (TU)
and the sample holder (SH). Both transition units are again, composed of three
sections: a conical part, a cylindrical part and a fluid distributor. They can be dismounted, enabling separate measurements of the two transition units together for
high accuracy calibration measurements. The transition units are Teflon filled and
long enough to prevent any higher order mode generated from reaching the measurement plane. The conical part of the transition unit eliminates the impedance
jump between the cable connection and the sample holder such that the generated
higher order modes are negligible. When the line is completely filled with Teflon
the impedance throughout is very close to 50 Ω.
The fluid distributors can be connected to four inlets and four outlets for fluids.
Figure 4.2 describes their features. The fluid enters the sample through the inlets located on the sides of the gold plated outer ring and into the carved teflon
distributors, appropriate filters are placed on top. The carved fluid path ensures a
reasonably homogeneous flow through the sample.
40
4.2.2
Chapter 4. Tool for accurate permittivity measurements
Measurement Characteristics
The probe can be connected to the two ports of an S-parameter test set, and
the full S-parameter matrix of the network is measured with a Network Analyzer, controlled by a PC. We use an HP 8357A Network Analyzer together with
a 85046A S-Parameter set, so that the frequency range at which we can operate
goes from 300 KHz to 3 GHz. However, depending on the material, the effective
lower limit of this range can be in the order of several hundreds of MHz.
Any influence from the cables that connect the tool to the S-parameter test set is
compensated for by doing a Full 2-port calibration, so that the measurement plane
is moved to the connection of the cables to the tool, see Figure 4.1. For a full 2port calibration, reflection measurements for an open, a load and a short connector
on both cables are done, as well as, thru-measurements and isolation (HP, 1993).
Once the cables are calibrated, the three main bodies of the probe are assembled.
First one transition unit to the sample holder, that is then filled with the material in
study and finally the other transition unit. In case of flow experiments, convenient
filters are placed at the interfaces between the sample holder and the transition
units. These three pieces are tightly screwed together. And the cables are screwed
to the end connectors at both transition units. These are very delicate processes
and a good connection is essential to obtain accurate measurements.
The tool components were design such that the transition units could be compensated by doing a ”user defined” calibration (HP, 1993). The process is equivalent
to the calibration of the cables but the standards were specially designed and constructed to match the transition units characteristics. This calibration attains the
best results since it removes any undesirable effects from the units. Unfortunately
it was impossible to obtain a proper short circuit of the transition units and we
could not calibrate them experimentally.
Since the experimental calibration of the transition units failed, we then fined tuned
the forward model of the tool with multiple measurements of the different components. We refer to it as the calibration of the model. Note, that it is not an
experimental compensation of the transition units but a calibration of the model
with measurements.
4.2. Technical design and measurement characteristics
41
Precision of the measurements
From a statistical analysis on one hundred measurements for different materials,
we determined that the measurements are very stable and their precision can be
improved by stacking, if needed. For our study, single measurements are accurate
enough, as they have three significant figures. This is clear from Figure 4.3 where,
as an example, the standard deviation of 10 groups of 2 measurements each, have
been plotted for the real and imaginary parts of the reflection (S22 ) and transmission
(S21 ) when the sample holder is filled with air. For the whole frequency range the
difference between any two transmission measurements is always lower than 3·10−3 ,
and lower than 10−3 for reflection. This leads to a true dynamic range of 50-60
dB.
It is useful to look at the relative precision of these measurements in amplitude and
phase. By doing so, we are able to derive important measurement characteristics.
In Figure 4.4 we have plotted the relative precision of reflection (S22 ) and transmission (S21 ) when the sample holder is filled with air. From the figure it is clear
that the phase is always more accurate than the amplitude but that below certain
frequencies, 300 MHz for air, it is not well determined, specially for reflection data.
This corresponds to a wavelength ten times larger than the sample holder. The
accuracy in the amplitude of the measurements depends on whether the measurement is a reflection or a transmission coefficient. For transmission there is a slow
linear increase of inaccuracy with frequency but it is maintained below 5 · 10−3 .
On the other hand the accuracy of a reflection coefficient varies significantly and
not linearly with the frequency. The inaccuracy increases on both the high and
low limit of the frequency spectrum as well as in the vicinity of the resonant frequency, 1.5 GHz for air. This enhanced inaccuracy around resonant frequencies is
observed in other materials. In between these peaks the inaccuracy is maintained
below 10−3 . The explanation and occurrences of these resonant frequencies is due
to the destructive interference introduced in Section 3.2.2. The inaccuracy around
resonant frequencies increases because the amplitude of the signal is very small and
the Network Analyzer has troubles determining its phase. This can also happen
with the transmission coefficient of lossy materials, as the signal damps out above
certain frequencies. For amplitudes around 4 · 10−3 the error can be of the order
or 30% and increase up to 50% for half those amplitudes.
42
Chapter 4. Tool for accurate permittivity measurements
ℜ
0
10
−2
10
10
22
std(S )
−2
−4
−4
10
10
−6
10
−6
0.5
1
1.5
2
2.5
3
0
10
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
Frequency (GHz)
3
0
10
10
−2
−2
10
10
12
std(S )
ℑ
0
10
−4
−4
10
10
−6
10
−6
0.5
1
1.5
2
2.5
Frequency (GHz)
3
10
Figure 4.3: Standard deviation for 10 groups of 2 measurements for the real and imaginary
parts of S22 and S21
relative precision of S
Phase
0.02
0.008
0.01
0.006
0
0.004
−0.01
0.002
0
0
12
relative precision of S
22
Amplitude
0.01
1
2
3
0.01
−0.02
0
1
2
3
0.02
0.008
0.01
0.006
0
0.004
−0.01
0.002
0
0
1
2
frequency (GHz)
3
−0.02
0
1
2
frequency (GHz)
3
Figure 4.4: Relative precision for a group of 2 measurements. In amplitude and phase of
S22 and S21 for an air sample.
4.3. Forward Model and Calibration
43
Number of points
The Network Analyzer HP 8357A can perform from single point frequency measurements up to 1601 points. The measurement and computing time increases
with increasing number of points. We decided to use the maximum number of
points offered by the apparatus to obtain a high density of permittivity values over
the spectrum. That way we expect to detect any significant event. If a particular frequency region is of special interest, the frequency range can be narrowed,
increasing the frequency points density in that region, in order to obtain a more
precise description.
4.3
Forward Model and Calibration
To model the S-parameters of the probe we used the standard transmission line
theory for a multi-section line presented in Appendix C. And for the calibration we
used multiple measurements of the different components that form the tool, to fine
tune the forward model.
4.3.1
Forward Model
Transmission line theory can be applied when only the TEM mode is propagating.
This is ensured by operating the line below the cut-off frequency of the first order
mode, T M01 mode (see Appendix B). It will propagate when
√
λcutoff ≥ 2 εr (a − b),
where a and b are the outer and inner radius of the conductors that form the line.
For our sample holder this will occur when it is filled with a material whose relative
permittivity is higher than 25 and only for very high frequencies. For water, at
25◦ C, this mode propagates above 1.6 GHz (λ ≈ 2 cm) enhancing the inaccuracy
of the model. However, for most sandy samples, whose relative permittivity values
are generally smaller than 25, the propagation of only the TEM mode is ensured.
In Appendix D, it has been proven that the diagonal scattering parameters are the
same when measured at the reference planes of a perfectly symmetric line with a
homogenous filling material. They are equal to the global reflection coefficient ΓN
including all internal reflection and transmission effects. The off-diagonal elements
of the S-parameter matrix are always equal to each other and to the global transmission coefficient ΥN , including all internal reflection and transmission effects.
Our probe is not symmetrical (see Figure 4.1) and therefore the diagonal Scattering parameters are not equal to each other, but they are equal to the corresponding
global reflection coefficients measured at the corresponding port. Then,
44
Chapter 4. Tool for accurate permittivity measurements
S11 S12
S21 S22
=
Γ11
Υ21
Υ12
Γ22
,
(4.1)
where the global reflection and transmission coefficients are given by the recursive
expressions (C.30) and (C.31), and the subindices refer to the departing and measuring planes 1 or 2 (see Figure 4.1). From those expressions and from the ones for
the propagation constant γn , (C.4), the impedance Zn (C.8) and the transmission
line parameters for a coaxial transmission line, equations (C.10) to (C.13), we can
see that our model is governed by
{an , bn , dn }
{µ∗n , ε∗n }
w = 2πf
the dimensions of the sections that form the line;
outer radius, inner radius and the length of the sections.
the electromagnetic properties of the materials filling
the sections of the line; the complex magnetic permeability
and the complex electric permittivity.
the radial frequency of operation.
Direct comparison of this model with actual measurements reveal significant differences between both. These differences are due to several reasons and are explained
in detail in the next subsection. So, although theoretically everything is known
except for the permittivity of the material in the sample holder, in practice, it is
necessary to perform many different measurements on cables, connectors, transition
units and the whole probe to calibrate the tool. In Figure 4.5 we have plotted the
real and imaginary parts of the measured [S msr ] reflection [S22 ] and transmission
[S21 ] when the sample holder is filled with air, together with the modelled parameters [S mod ] and their difference [S msr − S mod ]. The model is qualitatively valid, but
quantitatively it is out of phase and significantly different from the measurements.
To reduce this difference, we spent quite some effort in the calibration of each
component and the final tuning of the model. We show this calibration together
with several model assumptions, and their justification, in the next subsection.
4.3.2
Calibration
The calibration of the tool consists in fine tuning the parameters that control the
forward model. Comparing the measurements to the model we were able to find the
optimum parameters that describe the behavior of our probe. For that purpose, two
main test materials were used: air and Teflon. In the comparison of measurements
and model we looked at all the reflection and transmission coefficients in their
complex form, real-imaginary or amplitude-phase. We assumed that the conical
sections of the transition units could be represented by a single cylinder and that
the hollow fluid distributors could be modelled as solid teflon disks. We then,
followed a step by step procedure, looking at every component and parameter that
4.3. Forward Model and Calibration
45
ℜ
0.5
0.5
0
0
−0.5
−0.5
S
22
1
−1
0.5
1
1.5
2
2.5
3
−1
1
1
0.5
0.5
0
0
−0.5
−0.5
12
S
ℑ
1
−1
0.5
1
1.5
2
2.5
Frequency (GHz)
3
−1
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
Frequency (GHz)
3
Figure 4.5: Measured (solid line) and modelled (dashed line) reflection and transmission
coefficients for an air sample, and their difference (dotted line) .
could improve our model. We started with the simplest model and geometrical
data taken from the technical specifications of the probe, changing one parameter
at a time. Let us first justify the model assumptions.
Assumptions of the calibrated forward model
In the forward model we modelled the conical sections of the transition units as a
single cylinder and the fluid distributors, although formed by a hollow teflon disk,
are represented by a solid teflon disk. Here, we justify these assumptions.
1. Modelling the conical sections of the transition units: The transition
units of the probe have a conical section to eliminate the impedance jump
between the cable connection and the sample holder such that hardly any
energy is converted to higher order modes. But on a circuit model a cone
has to be replaced by series of cylinders that will mimic the behavior of the
cone. It was proven that only one cylinder of the same length of the cone
and a proper radius was needed to model the cone. Adding more cylinders
did not improve the accuracy of the model. The improvement caused by the
46
Chapter 4. Tool for accurate permittivity measurements
replacement of the cone by 100 cylinders was smaller than the measurement
accuracy and therefore there is no need to use a more complicated model, so
that the cone is treated as one single cylinder.
2. Fluid distributors: Between the transition units and the sample holder there
are fluid distributors (Figure 4.2), which ensure a more homogeneous flow
into the sample. These distributors are Teflon made, but they are not solid
cylinders and fluid conductors have been carved into them. By performing
measurements with the actual fluid distributors filled with tap water and
comparing them to the model with solid Teflon cylinders in place of the
distributors we could check the validity of this assumption. In Figure 4.6 we
have plotted the amplitude and phase of the transmission coefficient S12 of a
sample of tap water with hollow fluid distributors and the model (with solid
disks) for the same sample. It is clear that the assumption is valid, at least
up to the cut-off frequency of the TEM mode for water: 1.6 GHz.
Amplitude
Phase
1
0
exp: fluid distributors
mod: solid disks
0.9
−10
0.8
−20
0.7
−30
S12
0.6
0.5
−40
0.4
−50
0.3
−60
0.2
−70
0.1
0
0
1
2
Frequency (GHz)
3
−80
0
1
2
Frequency (GHz)
3
Figure 4.6: Measured transmission coefficients for a tap water sample. The solid line represents measurements done with the fluid distributors, and the dotted line to the modelled
coefficients with the solid disks in the place of the actual fluid distributors.
4.3. Forward Model and Calibration
47
Calibration steps
Here we list the calibration steps, and in Figure 4.7 we show the improvement of
each step by plotting the absolute difference between the model and the measurement for every step when the probe is filled with air. Since the initial difference
between the model and the measured data is bigger for transmission than for reflection, the advantage of this calibration shows more clearly in the transmission
data.
1. Fitting geometrical parameters: Most geometrical parameters (length and
radii of the sections shown in Figure 4.1) of the probe can be measured
independently and others have to be extracted from their technical specifications or from independent measurements. However, these may vary due
to construction, or compression when the probe is mounted, or expansionscontractions from temperature changes. Optimizing for these parameters
proved to be crucial, and resulted in a general reduction in both reflection
and transmission differences. Their values are listed in Table 4.1
Table 4.1:
Geometrical
parameters
for the tool.
[mm]
dn
an
bn
TU1.1
57.5
8.9
2.7
TU1.2
46
15
4.5
SH
100
15
4.5
TU2.2
46
15
4.5
TU2.1
57.5
8.9
2.7
C
18.3
2.7
0.8
Although the transition units can be modelled as symmetric their geometrical
parameters vary some milliliters from their technical specifications.
2. Permittivity of Teflon: Over the working frequency range of the Network
Analyzer, the relative permittivity of Teflon is reported in literature to be
in between 2 and 2.1. The minimum difference between the model and the
measurement was found at 2.05. We have tried using a measured permittivity
for Teflon per frequency, but it did not improve the accuracy of the model.
The deviations of the measured permittivity for Teflon from the mean value
of 2.05 are so small that the model is insensitive to them.
3. Losses: a lossless ideal line has a unitary scattering matrix (equation (D.7)).
So, by plotting the sum of the squares of the reflection and transmission
coefficient we can check how ideal our probe is.
In Figure 4.8 we have plotted this sum (solid line) for the transition units
connected together and for the whole tool, with air as sample. It is clear
that for both cases there is a linear deviation with frequency. The loss of
amplitude due to the skin effect has been plotted as a dashed line.
48
Chapter 4. Tool for accurate permittivity measurements
ℜ
ℑ
0.5
0
0
S22
0.5
−0.5
0.5
1
1.5
2
2.5
3
−0.5
0.5
0
0
S21
0.5
−0.5
0.5
1
1.5
2
2.5
Frequency (GHz)
3
−0.5
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
Frequency (GHz)
3
Figure 4.7: Difference between the measured and modelled reflection and transmission
coefficients for an air sample, at the different steps of the calibration of the tool. Before
calibration (solid line), step 1 (dashed line), step 2 + 3 (dotted line).
Tool: AIR
only Transition Units
1
0.95
0.95
0.9
0.9
0.85
0.85
0.8
0.8
0.75
0.75
|S22|2 + |S21|2
1
exp
with Rc
with δ
0.7
0.5
1
1.5
2
2.5
Frequency (GHz)
3
0.7
0.5
1
1.5
2
2.5
Frequency (GHz)
3
Figure 4.8: Unitary property check for the transition units connected together (tool without
sample holder), right figure, and for the whole tool when the sample holder is filled with
air, left figure. Experimental data (solid line), including conduction losses (dashed line),
including a loss δ (dotted line).
4.4. Experimental Results
49
As we can see in the figure, conduction losses alone, cannot account for the
deviation from unity. However, its linear trend and its consistency, whether
the sample holder is measured or not, suggested that we could model it as a
δ loss in the exponential terms of the conduction along the transition units,
transforming equation (C.23) with
c0
γ̂Transition Units = e−j ω [(εr
Teflon )1/2 −jδ
],
(4.2)
The model with a loss (δ = 1.5 · 10−3 ) has been plotted in Figure 4.8 as a
dotted line. For both cases, whole tool or only transition units, it accounts
very well for the total loss trend present in the measurements, without considering conduction losses. We have tried to include experimental losses into
the model in different ways and this has been the most successful one.
4.4
Experimental Results
In the previous sections we have described the probe, its forward model and the
calibration steps, needed to reduce the difference between the model and the measurements. In this section we present two experimental results on materials of
known permittivity to show the validity of the calibrated model. We also report on
the sensitivity of this model.
4.4.1
Results on materials of known permittivity
Air
Figure 4.9 shows that the difference between the calibrated model and the measurement has been reduced significantly. This is clear, if we compare it to the
difference between the non calibrated model and the same measurement, shown in
Figure 4.5. This improvement is due to the calibration of the model itself. Even
though the maximum difference is bigger than the precision of the measurements
by one order of magnitude, the obtained accuracy is to our knowledge still better
than reported in the literature.
On average, the model differs from the measurements, 2% for reflection and 1%
for transmission, for both real and imaginary parts, although at some frequencies
it is three times as much.
50
Chapter 4. Tool for accurate permittivity measurements
ℜ
ℑ
1
0.5
0.5
0
0
−0.5
−0.5
S22
1
−1
0.5
1
1.5
2
2.5
3
−1
1
0.5
0.5
0
0
−0.5
−0.5
S21
1
−1
0.5 1 1.5 2 2.5
Frequency (GHz)
3
−1
0.5
1
1.5
2
2.5
3
0.5 1 1.5 2 2.5
Frequency (GHz)
3
Figure 4.9: Measured (solid line), reflection and transmission coefficients for an air sample,
its calibrated model (dashed line), and their difference (dotted line).
Ethanol
In Figure 4.10 we have plotted the reflection and transmission coefficients for measured data of ethanol (99.9% pure), the model for these data and their difference.
The permittivity of ethanol is well defined from its Debye parameters (Gemert,
1972), but it cannot be used as calibration material since its permittivity is not
standardized and small temperature and purity changes affect it. Still, we can
compare model and measurements qualitatively.
The transmission is very well modelled, although the loss of signal above 1.5 GHz
increases the inaccuracy in the phase of the measured data. The measured and
modelled reflections are slightly out of phase. Nevertheless, varying the Debye permittivity of ethanol in 1% results in a noticeable increase in the difference between
data and model results over the whole frequency range. We will use this tech-
4.4. Experimental Results
51
ℜ
ℑ
1
0.5
0.5
0
0
−0.5
−0.5
S22
1
−1
0.5
1
1.5
2
2.5
3
−1
1
0.5
0.5
0
0
−0.5
−0.5
S21
1
−1
0.5 1 1.5 2 2.5
Frequency (GHz)
3
−1
0.5
1
1.5
2
2.5
3
0.5 1 1.5 2 2.5
Frequency (GHz)
3
Figure 4.10: Measured (solid line) Reflection and Transmission coefficients for an ethanol
(99.9% pure) sample, its calibrated model (dashed line), and their difference (dotted line).
nique in the coming section to estimate expected errors in the reconstruction of
permittivity with the presented forward model.
4.4.2
Sensitivity of the model
To study the sensitivity of our model, we performed experiments with numerically modelled and measured data. It is possible to determine the accuracy in the
permittivity if the difference between the model and the measurements is known.
Conclusions can be drawn about the maximum error of the possible reconstructed
permittivity from the accuracy of the measurement itself and the differences between the measurement and model results as a function of permittivity. In Figure 4.11, we show the result of this analysis. The plot shows the maximum error
in permittivity, as a function of normalized wavelength, that gives a change in the
model result, which is just observable such that it leads to an increased difference
52
Chapter 4. Tool for accurate permittivity measurements
between data and model result above an uncertainty threshold. For the uncertainty threshold we use the maximum error in the data, which leads to the worst
result scenario. It is plotted as a function of normalized wavelength, because the
resonances at large wavelengths are expected to prohibit the accurate permittivity
reconstruction ability. These resonances are related to the ratio of wavelength and
sample holder length, rather than frequency.
relative error in permittivity %
50
% error
40
30
20
10
0
0
10
20
30
40
50
60
70
5
% error
4
3
2
1
0
0
3
6
λ/L
9
12
15
sample holder
Figure 4.11: Maximum error in electric permittivity, which does not lead to observable
model error as a function of normalized wavelength.
With an air filled sample, the normalized wavelength of 60 is equivalent to a frequency of 50 MHz. It is clear that if we would allow a maximum error of 5% in the
reconstructed permittivity, we cannot use the set up below 250 MHz for air. Of
course the lower bound of the usable frequency goes down proportional with the
inverse square root of the permittivity of the material filling the sample holder. So
for most soil samples, varying in permittivity from 4 to 25 (dry to wet), the lowest
usable frequency would be 125 MHz and 50 MHz, respectively. These limits are confirmed by experimental data in Chapter 5. For lower frequencies, other techniques
4.4. Experimental Results
53
should be used. Robust information on the permittivity from higher frequencies
could be used to stretch this lower limit, but it will only be an extrapolation. For
frequencies higher than twice the lowest usable frequency the permittivity can be
reconstructed with an error below 1%. The smallest error that we are able to
achieve is 0.1%, which is due to the limited accuracy of the measurements.
Sample length
This type of study enables us to forecast sample length requirements for optimum
measurements. For example, we can determine the shortest sample holder we can
use if we want to ensure a 1% accuracy on a certain material. This is done by
plotting the frequency versus the sample holder length for a certain accuracy.
3
600
2.5
500
Frequency limit (MHz)
Frequency limit (GHz)
In Figure 4.12 we have represented the curve for a minimum accuracy of 1% for an
air sample. If we were going to use a 3 cm sample holder the measurements will
be 1% accurate for frequencies above 2 GHz, whereas if we use a 30 cm sample
holder we could use most of the spectrum, from 200 MHz onwards.
2
1.5
1
0.5
0
0
400
300
200
100
5
10
15
Lsample holder (cm)
20
0
0
50
Lsample
holder
100
(cm)
Figure 4.12: Frequency limit depending on the sample length for an air sample to ensure
a maximun error of 1%.
54
Chapter 4. Tool for accurate permittivity measurements
When performing dynamic flow experiments it is important to keep the samples as
short as possible, but as we can see from Figure 4.12 for permittivity purposes we
would like to have them as long as possible. With this technique we can reach a
compromise to satisfy both requirements.
4.5
Conclusions
We have constructed a coaxial transmission line for accurate measurements of permittivity from 300MHz to 3GHz. The forward model representing the reflection and
transmission along the line is in very good agreement with the measured data after
a thorough calibration. It has been a hierarchal tuning and it involved calibrating
the length and radii of every section forming the line, finding the most suitable
permittivity for Teflon in the proper frequency range, modelling the losses present
along the line and corroborating the assumptions made in the model in relation
with the conical sections and the fluid distributors.
We have shown that relative changes in the permittivity in the order of 1% can be
detected over a wide frequency band up to 3 GHz, while the lowest usable frequency
depends on the permittivity of the material filling the sample holder.
Let us now summarize the most important technical and measurement characteristics of the presented tool:
Characteristics of the tool
Technical
Measurements
Coaxial Transmission Line
allows Fluid Flow
Transmission Line
Theory applies
True dynamic range
of 50-60 dB
4 S-parameters measured
Transmission more reliable
than Reflection
a careful calibration is needed to obtain accurate results
Results? Why, man, I have gotten lots of results! If I find
10,000 ways something won’t work, I haven’t failed. I am not
discouraged, because every wrong attempt discarded is often a
step forward.
Thomas Edison (1847-1931)
Chapter 5
Reconstruction methods for
permittivity from measured
S-parameters
In this chapter, we present and compare analytical and optimizing reconstruction
methods of materials of known EM properties. We also show in detail the occurrence of resonant frequencies, and their implications.
5.1
Introduction
The reconstruction of permittivity from measured S-parameters is a broad and complex problem. In the high-frequency regime, the relation between the EM properties
and the measured quantities is highly non-linear, making the reconstruction cumbersome and unstable. Many researches have tackled this problem and solved it in
different ways. There are two main streams: Analytical and Optimization methods.
Analytical methods
The electrical permittivity is computed from analytical expressions involving the
S-parameters of the sample holder only. Although this chapter is dedicated to
the measurement of the electrical permittivity of porous media, the nature of the
problem allows to solve also for the magnetic permeability. So we will present the
solution of both properties and the inherent problems associated with it.
There are three ways to obtain the reflection and transmission of the sample holder
only. First, if a commercial fixture is used, then moving the reference planes from
55
56
Chapter 5.
Reconstruction Methods
the test ports to the sample interface is a matter of removing the phase delay, e.g.
Rau & Wharton (1982), since the empty line characteristic impedance matches the
impedance of the instrument’s test ports. However, reported accuracy problems
arise from the uncertainty of the sample location and not every sample is suitable
for these type of lines. Second, it is possible to compute the permittivity and
permeability from the measurements at the test ports via analytical expressions with
the new technique we have developed, see Subsection 5.2.2. And third, you can
remove the contribution of unwanted transition sections by compensation methods.
Kruppa & Sodomsky (1971), Freeman et al. (1979), Shen (1985) and Chew (1991)
apply basically the same technique. They compensate for the transition sections by
a tedious experimental calibration, which is also prone to additional measurement
errors. We discuss this method in Subsection 5.2.4.
Once the S-parameters of the sample holder only are obtained different reconstruction formulas can be used. Nicolson & Ross (1970) and Weir (1974) combined
the equations for the scattering parameters such that the system of equations
could decouple. But its solution is divergent for low-loss materials at frequencies
corresponding to integer multiples of one-half wavelength in the sample. Stuchly
& Matuszewsky (1978) obtained two explicit equations for the permittivity by a
slightly different derivation. They are unstable for low-loss materials at frequencies
corresponding to integer multiples of one-half wavelength in the sample. Palaith
& Chang (1983) analyzed all three equations emphasizing in the inherent errors
they produced and narrowing them down by a mapping method. Ligthart (1983)
presented a method for shorted line measurements where the scattering equations
for the permittivity were solved over a calculated uncertainty region and the results
were then averaged, but they suffer from the same problems when low-loss materials are considered. Finally, Baker-Javis et al. (1990) minimized the instability of
the Nicolson-Ross-Weir equations by considering only non-magnetic materials and
an iterative procedure. All the methods consider only the ideal case presented in
Subsection 3.2.2 and assume that the S-parameters have been properly compensated, but the procedure is not trivial. In Subsection 5.2.3 we show how they all
correspond to the same method, and therefore suffer from the same instabilities.
Optimization methods
The EM properties of the sample can also be computed via an optimization procedure, minimizing a cost function involving the measured and modelled S-parameters
of the tool where the material is placed. No compensation is needed.
Belhadj-Tahar et al. (1990) and Taherian et al. (1991) applied this method to
a coaxial-circular waveguide. The first used a gradient technique and the second
modified Newton method. Their sample sizes are relatively small and the method
results are inaccurate for low-loss samples as well as for low frequencies since the
5.1. Introduction
57
wavelength becomes too large compared to the sample size. Nguyen (1998) used a
Nelder-Mead Simplex method on shorted reflection data of a coaxial waveguide.
The possibilities are as many as the drawbacks. It will be ideal to use the most
suitable method for each specific material, but that is almost unfeasible. Therefore,
we compare analytical and optimization reconstructions of different materials to
choose the best way to proceed.
5.1.1
Chapter outline
First, we present the analytical methods in Section 5.2, where we start off by studying the occurrence of resonant frequencies in our tool (Subsection 5.2.1). Resonant
methods are usually set aside when broad band measurements are done. However,
it is very favorable to study their occurrence in detail, specially for low-loss materials whose permittivity is constant in our frequency range of interest, and to
by-pass some inaccuracies of the reconstruction methods. Then we present the
Propagation Matrices Method (PM) in Subsection 5.2.2. We have developed this
technique that, in essence, is a method for moving the reference planes from the
measurement planes to the interfaces of the sample holder, taking into account
impedance discontinuities among sections and losses along the line, and applying
an analytical expression to compute the permittivity of the sample. Writing the
solution of the propagation matrices for the ideal case (considering only the sample
holder), in Subsection 5.2.3, enables us to prove how all the published analytical
expressions are in fact, the same method. And in Subsection 5.2.4 we discuss the
validity of the application of the compensation methods (CO) for our tool and frequency range. Finishing the analytical methods section with Subsection 5.2.5 where
we compare the performance of the propagation matrices and the compensation
methods. Then we move forward to optimization methods (OP), in Section 5.3,
where we minimize different cost functions with a modified Newton optimization
routine. In Section 5.4 we compare all the different solutions and state a preference.
Finally, in Section 5.5 we conclude the chapter.
5.1.2
Experimental considerations
All the methods presented in this chapter, have been tested with measured and modelled data for materials of known permittivity. The modelled data were calculated
with the calibrated forward model from the previous chapter and the corresponding
permittivity. We will refer to it as [S mod ], and it comprises all four S-parameters
of the tool when the sample holder is filled with a particular material. The model
takes the measurement planes at the connections with the cables, just as in a real
measurement.
58
Chapter 5.
Reconstruction Methods
The measured data are actual measurements of the tool when the sample holder
is filled with a certain material. We refer to it as [S exp ] since it is obtained experimentally. To acquire the four S-parameters that form [S exp ], the material of study
is placed in the sample holder and the tool is assembled together and connected
to the Network Analyzer (NA) via the calibrated cables, as explained in Subsection 4.2.2. Then each pair of measurements [Γ, Υ] are recorded by the computer
at the corresponding port to form the full matrix. The NA sweeps 1601 points over
the specified frequency range.
The measurements present in this chapter were done using the whole range available
at the HP 8357A NA, from 300 KHz to 3 GHz, to be able to study the performance
of the methods in the entire spectrum.
As tests materials we differentiated between non-frequency and frequency dependent materials to check the ability of the different methods in reconstructing distinct
permittivity profiles. On one hand, materials that maintain a constant permittivity
over the broad frequency range used in these experiments, also present a low-loss.
To represent these type of materials we chose air and alumina. Air has a well known
relative permittivity of 1, that allows for a quantitative comparison of results. Alumina is a composite of aluminum oxide that presents a relatively high permittivity
of around 9. On the other hand, frequency dependent materials are also highly dispersive. To represent these we chose ethanol, a polar liquid, whose permittivity is
defined by Debye’s formula, given in Appendix E, and its Debye parameters. These
parameters depend on temperature and purity. Reported Debye parameters are in
many cases incomplete. Landolt-Börnstein (1996) only lists the static parameter
for different temperatures. The only complete set that was close in temperature
conditions to our experiment (23◦ C) is the one measured by Van Gemert (1972)
(24◦ C) but he does not report on the purity of his samples and his parameters have
significant errors. Therefore, an absolute comparison with standard values cannot
be performed, but we can corroborate if our results are in agreement with the accepted model for polar liquids or not. In order to do so, once the permittivity of
the sample was reconstructed per frequency with the appropriate method, we fitted
a Debye model to the result. Obtaining the Debye parameters from optimizing the
following cost function
costDebye =
Pnf
∗
∗
i=1 |(εr )rec − (εr )Debye |
,
Pnf
∗
i=1 |(εr )Debye |
(5.1)
where nf is the total number of frequencies, (ε∗r )rec is the reconstructed permittivity and (ε∗r )Debye is the Debye permittivity. In this manner, we fit a frequency
dependent model to the permittivity of ethanol computed without any frequency
model, and as a reference we refer to the values measured by van Gemert (1972).
5.2. Analytical Methods
59
Along this chapter the structure of the sections, devoted to the PM, CO and OP
reconstructions, is very similar. First the method is presented and then it is tested
in two extreme situations; the first one with a material of low-loss and low contrast,
air, whose permittivity is constant, and the second one with a material of high loss
and frequency dependent permittivity, ethanol. To test and to be able to weight
the performance of the individual methods presented in this chapter, we compare
the reconstructed permittivity with that of the sample. Their accuracy is derived
from this comparison.
5.2
Analytical Methods
In this section we have included all the existing methods to compute the electrical
permittivity from analytical expressions. It is structured as follows:
In Subsection 5.2.1 we introduce the method of the Resonant Frequencies that allows us to calculate the real part of the permittivity from the resonances of the line.
This is the first time, to our knowledge, that this technique is applied in a transmission line, as it is usually reserved for resonant cavities. In Subsection 5.2.2 we
present the method we have developed to compute the permittivity and permeability from the S-parameters measured at the test ports. This is done via an analytical
expression. In Subsection 5.2.3 we show how all existing analytical expressions for
the computation of permittivity are in fact, the same.
The compensation method, presented in Subsection 5.2.4, suffers from instabilities
at many frequency points over our spectrum and it is necessary to use a modelled
compensation for our tool. Finally, in Subsection 5.2.5 we compare the reconstruction capabilities of the presented methods.
5.2.1
Resonant frequencies: permittivity implications
Resonant methods are very accurate in reconstructing the permittivity of a sample
at fixed frequency points. For each frequency the dimensions of the resonator have
to be changed, making broad band measurements almost impossible. They also
require small samples, specially for high-loss materials.
0
00
In Chapter 3 we saw that for low-loss materials (εr < εr /100) of constant permittivity over the frequency range, the reflection coefficient goes to zero at resonant
frequencies, since ζ 2 = 1. At those frequencies, the accuracy of the reflection
coefficient is worse than at any other part of the spectrum, see Section 4.2.2. This
obstructs the reconstruction of permittivity from broad band measurements, but
we can extract valuable information from the occurrence of these resonances.
60
Chapter 5.
Reconstruction Methods
We introduced ζ, in Subsection 3.2.2, as ζ = e−γp dp , while the propagation constant
√
of non-magnetic materials, γ, can be written as γ = j cw0 ε∗r , see Section 2.4.2, so
that the resonant frequencies occur when
4πfk p ∗ εr dp = 2kπ
<
c0
for
k = 1, 2 . . .
(5.2)
where dp is the length of the sample. They are, thus, periodic and depend on
the length of the sample holder and the permittivity of the material. Since ε∗r is
a complex quantity and can depend on the frequency, let us distinguish between
different cases:
Low-loss non-frequency dependent materials
For materials whose relative permittivity is not a function of frequency and it is
0
real; ε∗r = εr , the reflection coefficient will go to zero at periodic intervals and
its permittivity over the whole frequency range can be computed from these single
frequency points as
k c0
εr =
2 fk dp
0
2
with
fk = kf
for
k = 1, 2 . . .
(5.3)
In Figure 5.1 we have plotted the amplitude of the measured S11 for an air sample
and for an alumina sample, placed in our tool (solid line), together with the model
(dotted line) for those materials considering only the sample holder (ideal case, see
Subsection 3.2.2).
From the figure, it is clear, that the resonances of the tool depend only on the
material placed in the sample holder. The measured reflection of the tool goes to
zero at the same frequencies as the model considering only the sample holder.
Note that the measurements at the reference planes (solid line) are not very different
from the model for the sample holder only, so that the effect of the necessary
transition units is minimized. We think, that these amplitude differences are caused
by losses occurring at the connections between the different sections that form the
tool.
For these samples, resonant frequencies, fk /k and computed permittivities are listed
in Table 5.1
5.2. Analytical Methods
61
Air
0.5
Tool measurement
Sample Holder model
0.4
|S11|
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Alumina
1
0.8
|S11|
0.6
0.4
0.2
0
0
0.5
1
1.5
frequency (GHz)
Figure 5.1: Measured (full tool: solid line) and modelled (ideal case: dotted line) |S11 |
for Air (top) and Alumina (bottom) samples.
fk (GHz)
Table 5.1: Resonant frequencies, period and computed permittivity for an
Air and Alumina sample.
1.51
2.95
mean values
0.50
1.01
1.52
2.04
2.54
mean values
fk /k (MHz)
Air
1510
1475
1492 ± 18
Alumina
500
505
507
510
508
505 ± 5
0
Computed εr
0.98
1.03
1.00 ± 0.03
9.0
8.8
8.7
8.6
8.7
8.8 ± 0.2
62
Chapter 5.
Reconstruction Methods
The resonances are periodic within 1% error for both air and alumina. This indicates
that their permittivities are not dependent on frequency over the frequency range of
interest and that the reconstructed permittivities are valid over the whole frequency
range. This can also be checked by the comparison between the measured and the
modelled reflection coefficient plotted in Figure 5.1. S11 has been modelled with
the computed mean permittivity, that present a standard deviation of 1% for air
and alumina. The modelled resonant frequencies coincide with the measured ones,
validating the computed permittivity.
High-loss non-frequency dependent materials
We couldn’t find any material with a high loss and a constant permittivity in our
frequency range of interest. However, it is interesting to take a look at the modelled
reflection of such a material, see Figure 5.2.
ε* = 4 − 0.2j
0.35
0.3
0.25
|S11|
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
frequency (GHz)
2
2.5
3
Figure 5.2: Modeled (ideal case) |S11 | for a sample of constant permittivity ε∗ = 4 − 0.2j.
5.2. Analytical Methods
63
The hypothetical material has a loss of about 5% of the real permittivity value,
ε∗ = 4 − 0.2j. But the reflection coefficient still presents periodic minima, and the
√
computed < ε∗r from these minima is 3.99. The presence of a loss results in an
inaccurate calculation of the real part of permittivity, but if the loss is not very big,
√
the value computed for < ε∗r is a good approximation of ε,r .
The presence of a constant small loss in the material doesn’t change the location
of the minima, it only varies the amplitude of these; decreasing amplitude with
increasing frequency. So if a reflection pattern of periodic minima of decreasing
amplitude is encountered, we can be certain that the material’s permittivity is
constant over the frequency range and that it’s loss is small but not negligible.
Frequency dependent materials
For these materials, the mentioned method of computing permittivity from resonant frequencies for the whole frequency range is not valid since their occurrence
is not periodic. It can still be used to obtain the real part of permittivity at certain
frequency points from the broad band measurement, where the reflection coefficient
has a minimum. This could be handy since the accuracy of the measurements is
not so good in the vicinity of these minima so that the reconstruction of permittivity around them can suffer from high inaccuracy, and via resonant methods we
can determine quite accurately the value of the real part of permittivity for those
resonant frequencies.
To compute the loss of the material from resonant frequencies is not as straight
forward as calculating the real part of permittivity for non-lossy materials. Anyhow,
there are no materials with high losses that maintain it constant over the frequency
range so that we would not gain anything from computing the loss on a single
frequency point. We refer the reader to von Hippel (1954) for further insight.
5.2.2
The propagation matrices method
We introduced the notation needed for this method in Chapter 3, Subsection 3.2.1.
We mentioned there that we represent both reflection coefficients [S11 , S22 ] with
the term Γ and both transmission coefficients [S12 , S21 ] by Υ. This enables us to
write only one set of equations but we have to keep in mind that there are two sets
of measurements, [S11 , S12 ] and [S22 , S21 ], one pair at each port, and therefore,
there will be two sets of solutions.
Equation (3.12) can now be written as
1
Γ
= LP R
Υ
0
.
(5.4)
64
Chapter 5.
Reconstruction Methods
The EM properties of the material under study are contained within the P matrix
that can be calculated from equations (3.6) and (3.8)
P=
1
2
Zp (e+γp dp − e−γp dp )
e+γp dp + e−γp dp
e+γp dp + e−γp dp
− e−γp dp )
1
+γp dp
Zp (e
.
(5.5)
Now, we can rewrite equation (5.4) into a more convenient form as
A
B
1
Γ
=P
C
D
,
C
D
(5.6)
where
A
B
−1
=L
and
=R
Υ
0
.
(5.7)
Substituting P into equation (5.6) and eliminating the exponential terms, we find
a solution for the impedance Zp of the sample under test as,
A2 − C 2
.
(5.8)
B 2 − D2
If, instead, the impedance terms are eliminated, a solution in terms of the exponentials is found
Zp2 =
e+γp dp + e−γp dp
AB + CD
= cosh (γp dp ) =
.
2
AD + BC
We hence find and expression for the propagation factor as,
AB+CD
)
acosh( AD+BC
.
γp = ±
dp
For simplicity, we will further refer to the quotient
β=
AB + CD
.
AD + BC
AB+CD
AD+BC
(5.9)
(5.10)
as β, hence:
(5.11)
The electric permittivity and magnetic permeability of the sample are related to
0
the impedance and propagation factor via equation (C.21). The conditions εr > 0,
00
0
00
εr ≥ 0, µ > 0 and µ ≥ 0 are physical constraints, cf. equation (2.64), and the
sign for γp is determined, cf. equation (2.66), and so is the sign for the acosh
function.
5.2. Analytical Methods
65
We find two coupled equations for ε∗r and µ∗r
s
Zp
µ∗r
=
∗
εr
Z0
p
c0 γp
ε∗r µ∗r =
jω
the impedance method,
(5.12)
the propagation method,
(5.13)
with Zp and γp defined in equations (5.8) and (5.10), respectively.
In Maxwell’s theory ε∗r and µ∗r appear coupled in characteristic combinations and
therefore, any method trying to measure the permittivity will also be able to measure
the permeability, specially in their coupled forms. Equations (5.12) and (5.13) are
a perfect example. The first one represents the impedance of the medium and the
second one the inverse of the velocity at which EM waves propagate. The coupling
can be a drawback since the accurate determination of both properties is not always
possible.
Zp and γp are a combination of the measured reflection and transmission coefficients
at the measurement planes. The propagation matrices method effectively moves
the measurement planes to the interfaces of the sample holder, taking into account
phase and impedance changes, as well as losses along the line.
If we divide and multiply equations (5.12) and (5.13) we find the solutions for
µ∗r and ε∗r
µ∗r
=
ε∗r =
c0 Zp γp
,
jw
Z0
c0 Z0 γp
.
jw
Zp
(5.14)
(5.15)
We have to remember that we always obtain a pair of solutions depending on the
pair of measurements used in their calculation.
Let us reconstruct µ∗r and ε∗r for an air sample.
Magnetic permeability and electric permittivity of air
To test the validity of the Propagation Matrices method, we first apply it to modelled data. The relative permittivity and permeability of air are real and 1 for the
whole frequency range.
66
Chapter 5.
Reconstruction Methods
*
*
εr
µr
1.02
1.04
mod
rec
1.02
0.98
1
ℜ
1
0
1
2
3
0
−3
20
1
2
3
−3
x 10
5
0
10
−5
5
−10
0
−15
ℑ
15
x 10
−5
0
1
2
frequency GHz
3
−20
0
1
2
frequency GHz
3
Figure 5.3: Modelled and reconstructed electrical permittivity and magnetic permeability
of Air.
In Figure 5.3 we have plotted the modelled and the reconstructed EM properties
for an air sample. The method reconstructed the modelled data perfectly well,
for both real and imaginary parts, except at the resonant frequencies where the
reconstructed data diverges.
Let us now see how it works with experimental data. In Figure 5.4 we have plotted
the standard and the reconstructed EM properties for an air sample. This time,
the reconstructed data diverges significantly from the true value of 1, for both real
and imaginary parts, specially at the resonant frequencies.
5.2. Analytical Methods
67
*
*
εr
µr
1.5
2.5
std
rec
2
ℜ
1
1.5
0.5
0
0
1
1
2
3
0.5
0.5
0
1
2
3
2
1.5
0
ℑ
1
0.5
−0.5
0
−1
0
1
2
frequency GHz
3
−0.5
0
1
2
frequency GHz
3
Figure 5.4: Standard and reconstructed electrical permittivity and magnetic permeability
of Air.
The propagation matrices method works perfectly well with modelled data but fails
to accurately reconstruct the EM properties of the sample from true data. For an
explanation, we have to look at the modelled and experimental Zp and cosh (γp dp )
for air, see Figure 5.5.
The modelled Zp and cosh (γp dp ) are very stable and this makes the method work
perfectly with modelled data. But the experimental Zp suffers from instability
while cosh (γp dp ) does not. This Zp instability affects the reconstruction of the
permeability and permittivity. From the expressions (5.8) and (5.9), it is clear, that
Zp is more prone to be unstable than cosh (γp dp ).
However, cosh (γp dp ) is not exempt of problems. For materials with no or small
loss, the imaginary part of cosh (γp dp ) is close to zero which causes the imaginary
part of the acosh(β) to be poorly determined. In Figure 5.6 we have plotted the
real and imaginary parts of the experimentally determined cosh (γp dp ) for an air
68
Chapter 5.
4
2.5
x 10
Z
Reconstruction Methods
cosh(γ d )
p
p
p
1
from exp
from mod
0.9
2
0.8
amplitude
0.7
0.6
1.5
0.5
1
0.4
0.3
0.5
0.2
0.1
0
0
1
2
Frequency (GHz)
3
0
0
1
2
Frequency (GHz)
3
Figure 5.5: Experimental and reconstructed Zp and cosh (γp dp ) for an Air sample.
sample (left hand side of the figure) together with its acosh, in the center of the
figure. It is clear that =[acosh(β)] suffers from computational instabilities, but it
can be amended by applying a careful unwrapping.
In the right hand side of Figure 5.6 we have plotted the real part of acosh(β) and
its linearized imaginary part. Two techniques for linearization can be applied. If
=(β) is of the order of measurement accuracy it could be set to zero and then the
instabilities disappear, although its unwrapping still has to be amended, according
to the sign choice given by equation (2.66). In case =(β) is bigger than measurement accuracy, as it is for certain frequencies for air, we cannot just set a linear
imaginary part of acosh(β), but instead we take its absolute value and again amend
its unwrapping.
5.2. Analytical Methods
69
β
acosh(β)
1
ℜ
0.5
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
−0.5
−1
1
2
3
0.05
ℑ
acosh(β)
0.1
0
1
2
3
0
8
8
6
6
4
4
2
2
1
2
3
0
0
−0.05
1
2
3
Frequency (GHz)
−2
experimental
1
2
3
Frequency (GHz)
0
−2
linearized
1
2
3
Frequency (GHz)
Figure 5.6: Experimental β for an Air sample (left hand side), its acosh in the center, and
in the right hand side the same acosh with the imaginary part linearized.
Zp is experimentally unstable and β = cosh (γp dp ) can produce numerical instabilities for low loss materials. Therefore, any solution including Zp will suffer from
instability, making the method less accurate, and any solution including cosh (γp dp )
should be revised in the case of low loss materials, and amended if necessary. Since
the propagation matrices method reconstructs the permittivity and permeability of
a sample from both, Zp and cosh (γp dp ), it does not do it accurately. However, we
deal with non-magnetic sands and, in the next section, we eliminate Zp from the
solution, making it more robust. In that sense, we compute the velocity of propagation from equation (5.13) and assume the magnetic permeability to be that of
free space so we can compute the permittivity.
70
Chapter 5.
Reconstruction Methods
Non-magnetic materials
For non-magnetic materials, the relative magnetic permeability is 1, and then, two
different solutions for the permittivity can be derived from equations (5.12) and
(5.13)
Zp
Z0
2
ε∗r
=
ε∗r
c0 γp 2
=
jω
the impedance method,
(5.16)
the propagation method,
(5.17)
We have already seen, that the Zp method will be very unstable and we will only
use the propagation method. However, it is important to explicitly write these two
solutions, as we will show in the coming section, Subsection 5.2.3, that all the
analytical solutions published till now, are these same two methods. But first, let
us see how it works on actual experimental data
Non-frequency dependent materials: AIR
In Figure 5.7 we have plotted the standard and the reconstructed electrical permittivity for an air sample with the PM propagation method, for both pairs of
measurements, [S11 , S12 ] and [S22 , S21 ] .
The reconstructed permittivity presents three significant features: around 500 MHz
on both the real and the imaginary parts, and below that frequency, the accuracy
of the [S22 , S21 ] pair worsens. Below 250 MHz both solutions start to diverge. And
at 1.5 GHz there is a clear jump. These features were already predicted from the
measurements characteristics in Section 4.4.2, although the experimental accuracy
threshold of 1% is lower (500 MHz) than predicted (650 MHz). From 500 MHz to
3 GHz the error of the real and imaginary parts of both solutions is always between
±1%, as predicted in Chapter 4. The 5% threshold is set below 50 MHz and the
2% around 250 MHz .
A result within 1% for air, for such a broad band, is a great achievement. It is also
a confirmation of the accurate reconstructions we expect for materials of unknown
permittivity, since the air result is a measure for the errors introduced by the tool.
5.2. Analytical Methods
71
ε*
r
1.04
1.02
ε
,
1
0.98
0.96
0.94
0.5
1
1.5
2
2.5
3
0
−0.01
ε
,,
−0.02
−0.03
std
PM rec11,12
PM rec
−0.04
22,21
−0.05
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 5.7: Standard and PM reconstructed permittivity for an Air sample.
Frequency dependent materials: ETHANOL
Applying the same method to ethanol (99.9% pure), a polar liquid, whose permittivity varies along our frequency range, we find the reconstructed permittivity (dashed
line) plotted in Figure 5.8 for the pair [S11 , S12 ] and Figure 5.9 for [S22 , S21 ]. The
fitted Debye model is also shown (dotted line).
The reconstructed permittivities are very well captured by the frequency dependent
Debye model. They present three significant features: around 300 MHz, the imaginary part of εrec deviates from the Debye fitted model. Below 50 MHz the solution
starts to diverge. And at high-frequencies, above 1.5 GHz the influence of noise
is clear. These frequency limits correspond to the ones predicted in Section 4.4.2
for ethanol. Above 1.5 GHz the signal is no longer transmitted through the sample
and the Network Analyzer has difficulties determining the phase of the transmission
coefficient, boosting up the inaccuracies in the reconstructed permittivity, however,
the general trend is captured.
72
Chapter 5.
Reconstruction Methods
*
εr
30
PM rec
11,12
Debye fit
ε
,
20
10
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0
ε
,,
−5
−10
−15
Figure 5.8: PM reconstructed permittivity of ethanol (dashed line), for the pair [S11 , S12 ],
and the fitted Debye model (dotted line).
*
εr
30
PM rec
22,21
Debye fit
ε
,
20
10
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0
ε
,,
−5
−10
−15
Figure 5.9: PM reconstructed permittivity of ethanol (dashed line), for the pair [S22 , S21 ],
and the fitted Debye model (dotted line).
5.2. Analytical Methods
73
The fitted Debye model and the reconstructed permittivities differ less than 1%
between 300 MHz and 2 GHz. If the Debye parameters are changed by 1% this
difference increases between 2 and 3 times. The fitted Debye parameters are listed
in Table 5.2.
EthanolGemert
Ethanolfit
11,12
Ethanolfit
22,21
T (◦ C)
24
23
23
εs
25 ± 1
25.2
26
ε∞
4.6 ± 0.7
4.6
4.6
fr (M Hz)
900 ± 60
866
834
σdc (µS/m)
0.8 ± 0.1∗
6
27
Table 5.2: Experimental and fitted Debye parameters. Data for ethanol taken from van
Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ.
The fitted Debye parameters have been optimized to the reconstructed permittivities.
Note that the fitted parameters are within the error bounds of the van Gemert
values. The conductivity given as a van Gemert parameter was, in fact, measured
in the laboratory. The fitted conductivity is orders the magnitude bigger, but this
enables the Debye model to capture the low frequency behavior of the imaginary
part of the reconstructed permittivity.
We have shown that the combination of the propagation matrices, PM and the
propagation method reconstructs satisfactorily the permittivity of non-frequency
and frequency dependent non-magnetic materials without a-priori information. It
shows frequency thresholds predicted from measurement characteristics. It is simple
to program and very fast. We can learn more about it in the next Subsection 5.2.3,
where we will simplify the problem to the ideal case to prove that all other analytical
solutions are equivalent, and therefore, suffer from the same problems.
5.2.3
Analytical solutions for the ideal case
As mentioned in the introduction of this chapter, many researchers published different analytical solutions. They all involve the knowledge of the S-parameters at
the interfaces of the sample holder. That is, they all were derived for the ideal case
presented in Chapter 3. To prove that in fact they are all the same solution let us
first write the expressions for the Zp -γp methods. We can find A, B, C and D
from equation (5.7) for the ideal case and then rewrite equations (5.16) and (5.17)
as
74
Chapter 5.
(1 − Γ)2 − Υ2
(1 + Γ)2 − Υ2
2
c20
1 + Υ2 − Γ2
∗
εr = − 2 2 acosh
ω dp
2Υ
ε∗r =
Reconstruction Methods
the Zp method,
(5.18)
the γp method.
(5.19)
The impedance and the propagation reconstruction methods for an ideal line, equation (5.18) and equation (5.19) are already equivalent to those derived by Stuchly
& Matuszewsky in (Stuchly & Matuszewsky, 1978). They claimed that the propagation equation is ambiguous for certain sample lengths, but we do not see such
ambiguity since the acosh of a complex number z is well defined as
√
√
acosh(z) = ln z + z + 1 z − 1 .
(5.20)
and physical constraints determine the sign for the complex logarithm, as expressed
for equation (5.9). At resonance, where z = ±1, the logarithm is zero and the
permittivity will not be accurately determined. This occurs with low-loss materials
when almost no reflection and total transmission occur at resonant frequencies.
Palaith and Chang (Palaith & Chang, 1983) comprise a comparison of three methods, their 1/Z 2 method is the first equation present in Stuchly and Matuszewsky
(Stuchly & Matuszewsky, 1978) and of course the impedance method. The second
is their so-called K 2 method that they claim to be new, while it is equal to the
second equation from Stuchly and Matuszewsky (Stuchly & Matuszewsky, 1978)
and the same as the propagation method, since they write:
−γp dp
ζ=e
1
=
2Υ
2
1+Υ −Γ
2
±
h
2
1+Υ −Γ
2 2
2
− 4Υ
i1/2 .
(5.21)
To find γp we would take the natural logarithm of the right hand side which can
then be written as a acosh in virtue of equation (5.20) ending up with the same
expression for ε∗r as equation (5.19). As a third solution, Palaith & Chang (Palaith
& Chang, 1983) analyze the Nicolson-Ross-Weir method, published by Nicolson
and Ross in (Nicolson & Ross, 1970) and by Weir (Weir, 1974) for both magnetic
permeability and electrical permittivity. We restrict ourselves to the solution for
the permittivity. Their equation is as follows:
ε∗r
c0 1 − r
ln(ζ),
=j
ωdp 1 + r
(5.22)
where r is the local reflection coefficient and ζ is expressed in terms of Γ and Υ
by equation (5.21), and again we obtain the propagation method. Note that the
5.2. Analytical Methods
75
expression of equation (5.22) is somewhat misleading because the local reflection
coefficient in the right-hand
side contains the unknown complex permittivity of the
√ ∗ h 1−r i
sample under test: εr = 1+r , so that, in fact, it is not another equation but the
same as the square root of equation (5.19). These methods are obviously the same
and they naturally all suffer from the same problems. They are not well behaved
for low-loss materials, especially at frequencies corresponding to integer multiples
of one-half wavelength in the sample, see Stuchly & Matuszewsky (Stuchly &
Matuszewsky, 1978), or in other words at resonant frequencies. Ligthart (Ligthart,
1983) and Baker-Javis et al. (Baker-Javis et al. , 1990) have tried to by-pass this
ill behavior in two different ways but using the same equations. Ligthart (Ligthart,
1983) presented a method for shorted line measurements where the scattering
equations for the permittivity were solved over a calculated uncertainty region and
the results were then averaged, but he could not avoid the low-loss problem. Finally,
Baker-Javis et al. (Baker-Javis et al. , 1990) minimized the instability by an iterative
procedure, without truly solving the problem.
5.2.4
Compensation Parameters
The compensation method has been used by several authors. Kruppa & Sodomsky
(1971) published the system of equations needed and Freeman et al. (1979),
Shen (1985) and Chew (1991) applied it, respectively, to two commercial fixtures
and a large customized cell. It is based on a Scattering Matrix representation
(Section 3.2) of the coaxial transmission line, dividing it into three major sections
each represented by its own scattering matrix.
This method accounts for experimental errors introduced by the transition units.
The S-parameters of the sample holder only are obtained from the S-parameters
measured at the reference planes, by a cumbersome set of operations. Six short
circuit measurement and a through measurement of only the transition units connected together are needed to compensate and to characterize the scattering matrix
of both transition units.
Figure 5.10 is an example of this method applied to modelled data of an air sample.
On the left hand side, we have plotted the amplitude and phase of both reflection
and transmission of the whole tool. In the middle of the figure, the two plots
correspond to the compensated parameters, or in other words, the S-parameters
of the sample holder only. The amplitudes remain unchanged while the phases are
different. This is easy to understand if we consider that this is a modelled case in
which both transition units behave ideally and therefore, the compensation is only
a phase move-out.
76
Chapter 5.
Smod
r
sample holder
1
1
0.8
2
0.8
12
|S | and |S |
ε,
Smod
tool
S11
S12
0.6
1
0.2
0.2
11
0.4
1
2
1.5
0.6
0.4
0
Reconstruction Methods
3
0
0.5
1
2
3
0
1
2
3
θ(S11) and θ(S12) (°)
,,
εr
0
100
100
0
0
−0.02
−0.04
−0.06
−100
−100
1
2
3
Frequency (GHz)
−0.08
1
2
3
Frequency (GHz)
−0.1
1
2
3
Frequency (GHz)
mod
Figure 5.10: Modelled S-parameters of the whole tool (Stool
) and of only the sample
mod
holder (Ssampleholder ) for an air sample and the computed permittivity.
In the right hand side of Figure 5.10 we find the reconstructed permittivity from
the compensated parameters with the propagation method. The real part of the
permittivity is well detemined, while the imaginary part is well determined except
for two frequency points. These frequencies are related to resonances. In the
compensation equations there is one term obtained as the inverse of a subtraction
between two short-circuited measurements. These measurements introduce new
resonances from the teflon filling of the tool, that are not present in the data
measurements, and propagate along the compensation.
For actual measurements this method doesn’t behave that well in our spectrum,
precisely due to the propagation of the mentioned resonances. In Figure 5.11 we
have plotted the amplitude and phase of the S-parameters of the whole tool at
the left hand side, and in the center the S-parameters of only the sample holder
exp
compensated with actual measurements, Ssampleholder
. While, in the right hand
exp
mix
side S
has been compensated with modelled data, Ssampleholder
.
5.2. Analytical Methods
77
Sexp
Sexp
tool
sample holder
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.8
12
|S | and |S |
1
S11
S12
11
0.6
0
θ(S11) and θ(S12) (°)
Smix
sample holder
1
2
3
0
1
2
3
0
100
100
100
0
0
0
−100
−100
−100
1
2
3
Frequency (GHz)
1
2
3
frequency (GHz)
1
2
3
1
2
3
frequency (GHz)
exp
) and of only the sample
Figure 5.11: Experimental S-parameters of the whole tool (Stool
exp
mix
) compenholder (Ssampleholder ) compensated with experimental data and (Ssampleholder
sated with modelled data.
From the figure it is clear that the compensated parameters with true measurements
suffer from instability, with major deviance at resonant frequencies. Reconstructing
permittivity from these parameters will not succeed. However, if S exp is compensated with modelled measurements the instability is reduced to only two frequency
points. This technique involves only compensating for the phase move-out from
the reference planes of measurement to the sample holder, or in other words, the
travel distance along the transition units.
Previous authors successfully applied this method to their tools because they were
air filled and their spectrum was such that the resonances did not affect the calibration since they fallen out of it. Chew et al. (1991) applied it in a broadband
modifying the compensation in the high frequency range by using modelled short
circuited reflections.
78
Chapter 5.
Reconstruction Methods
The compensation is unstable for broad frequency measurements and it has to be
modelled to succeed in the reconstruction of the permittivity. As an example, we
discuss the reconstructions for air and ethanol.
Non-Frequency dependent materials: AIR
In Figure 5.12 we have plotted the reconstructed permittivity from the modelled
compensated experimental S-parameters (right hand side of Figure 5.11), here
indicated as CO.
This reconstruction presents the same features observed in the PM reconstruction.
The same frequency limits and accuracies hold. And their major differences are the
instability points already foreseen in the modelled S-parameters example. It is very
similar to that found with the Propagation Matrices method since, in essence, they
are very similar methods.
*
εr
1.04
1.02
ε
,
1
0.98
0.96
0.94
0.5
1
1.5
2
2.5
3
0
−0.01
ε
,,
−0.02
−0.03
std
CO rec11,12
CO rec22,21
−0.04
−0.05
0.5
1
1.5
Frequency (GHz)
2
2.5
Figure 5.12: Standard and CO reconstructed permittivity for an Air sample.
3
5.2. Analytical Methods
79
Frequency dependent materials: ETHANOL
Applying the same compensation procedure to the ethanol measured S-parameters,
as that employed with air, the reconstructed permittivities (dashed line) plotted
in Fig. 5.13 for the pair [S11 , S12 ] and Fig. 5.14 for [S22 , S21 ], show again the
same features as the ones reconstructed with the propagation matrices method, in
Section 5.2.2, although the difference between the fitted Debye model (dotted line)
and the reconstructed ε∗r is worse.
The two peaks present in the air permittivity reconstruction due to resonances in
the compensation do not show in the reconstruction of the permittivity of ethanol
because their effect is of smaller amplitude due to the very small transmission above
1.5 GHz.
With this method, however, the fitted Debye model and the reconstructed permittivities differ less than 1% between 800 MHz and 1.5 GHz for the real part and
less than 2% for the imaginary part. The fitted Debye parameters are listed in
Table 5.3.
EthanolGemert
Ethanolfit
11,12
Ethanolfit
22,21
T (◦ C)
24
23
23
εs
25 ± 1
25.3
24.9
ε∞
4.6 ± 0.7
4.6
4.6
fr (M Hz)
900 ± 60
906
926
σdc (µS/m)
0.8 ± 0.1∗
7600
8400
Table 5.3: Experimental and fitted Debye parameters. Data for ethanol taken from van
Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ.
The Optimized Debye parameters have been fitted to the CO reconstructed permittivities.
Note that the fitted parameters are within the error bounds of the van Gemert
values. The conductivity given as a van Gemert parameters was, in fact, measured
in the laboratory. The fitted conductivity is orders the magnitude bigger, but this
enables the Debye model to capture the low frequency behavior of the imaginary
part of the reconstructed permittivity.
80
Chapter 5.
Reconstruction Methods
*
εr
30
CO rec
11,12
Debye fit
ε
,
20
10
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0
ε
,,
−5
−10
−15
Figure 5.13: CO reconstructed permittivity of ethanol (dashed line), for the pair [S11 , S12 ],
and the fitted Debye model (dotted line).
*
εr
30
CO rec
22,21
Debye fit
ε
,
20
10
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0
ε
,,
−5
−10
−15
Figure 5.14: CO reconstructed permittivity of ethanol (dashed line), for the pair [S22 , S21 ],
and the fitted Debye model (dotted line).
5.2. Analytical Methods
5.2.5
81
Propagation Matrices vs Compensation Parameters
The difference between the PM and the CO reconstructed permittivities, for both
air and ethanol and both their real and imaginary parts, above their 2% accuracy
thresholds (250 MHz for air and 50 MHz for ethanol) is of the order of measurement
accuracy (10−3 ). This reinforces the similarity of the results of these methods for
an almost ideal tool as ours. If the tool had not been designed as a match load and
big impedance contrasts were to be present amongst the different sections that
form the transition units the results from the Propagation Matrices method will
differ from the ones computed with the modelled compensation slightly more, since
the latter does not correct for them, and its reconstruction will deteriorate.
Ideally, the CO method with experimental compensation would, in principle, be
the most suitable one as it characterizes experimentally the true electromagnetic
behavior of the transition units. However, apart from being a tedious task, since it
includes dismantling and connecting the different sections of the tool many times,
which is prone to experimental errors, it also limits the frequency range of the
measurements. This limitation, as we have seen, comes from the fact that the
experimental compensation suffers significantly from resonances and, instead, a
modelled compensation must be performed. Therefore, the PM method is more
suitable to correct for the effect of the transition units, as it accounts for both
impedance and phase changes. In addition to this, the fit of the Debye model to
the CO reconstructed permittivities for ethanol is worse than that of the PM.
In Section 5.4 we will compare the analytical and optimized reconstructions, but
since, the PM method provides a better reconstruction than the CO method, we
will take the PM method as representative for analytical solutions.
We can already conclude that for broadband permittivity reconstruction from
S-parameters measurements via analytical expressions a good and accurate model
of the tool in use is mandatory. Any of the existing methods involve certain modelling of the transition units connecting the sample holder to the Network Analyzer,
and therefore, the calibration of the model to true measurements is of great importance. The deviations between the tool measurements and the model limit the
accuracy of the reconstruction.
82
5.3
Chapter 5.
Reconstruction Methods
Optimization methods
This technique does not use analytical expressions to reconstruct the permittivity, instead, it minimizes a cost function involving the measured and modelled
S-parameters of the tool. The attainment of the S-parameters of the sample holder
only is not an issue, but the convergence of the method can become one. They
are necessary if only reflection or transmission is measured, since the analytical
methods require always a combination of the two.
The optimization methods can be very varied in cost functions to minimize and
the minimizing procedure is limited to non-linear techniques. Belhadj-Tahar et
al. (1990) and Taherian et al. (1991) applied this method to a coaxial-circular
waveguide. The first applied a gradient technique to an L2 norm, and the second,
a modified Newton method to an L1 norm. Nguyen (1998) used a Nelder-Mead
Simplex method on shorted reflection data of a coaxial waveguide.
We have used a Quasi-Newton algorithm implemented in the optimization toolbox
of Matlab, (1996), as fminunc. It is an unconstrained nonlinear optimization routine that saves computing time by approximating the Hessian with an appropriate
updating technique. It is limited to real numbers and we by-passed the problem by
splitting the output of the cost functions to their real and imaginary components.
The tolerance was set to measurement accuracy and computing time varied with
the cost functions.
The optimization is done per frequency and over the full spectrum. An initial guess
of the permittivity of the sample, for the first frequency point, is given and S mod
is computed with the forward model. By minimizing a specific cost function, the
optimized permittivity is calculated and used as initial guess for the next frequency
point. This is repeated until the permittivity of the sample has been reconstructed
for the whole frequency range.
We have tested different cost functions to study their performance. With modelled
data all of the tested functions reproduced the permittivity introduced in the model,
but with true measurements they varied in their performance. We tried normalized and non-normalized cost functions. We only show the results obtained with
normalized cost functions, which showed a better performance. We have made a
summary of these in Table 5.4.
Both cost functions involving products of the S-parameters diverge for high-loss
materials whose permittivity is a function of frequency. And their performance
with low-loss materials with constant permittivity is worse than that of L1 and L2
norm.
5.3. Optimization methods
83
Table 5.4: Optimized cost functions and summary of performance.
Methods based on the L1 and L2 norm succeed in reconstructing the permittivity
of air but have to be weighted when ethanol is considered. L2 fails for frequencies
above 1.5 GHz. This weighting is necessary because of the small transmission
present in highly lossy materials above certain frequencies and the sole reliability
in reflection is not enough for the optimization to converge. However, a careful
weighting has to be done because if the transmissions are emphasized too much
the solution does not converge either. The weighting consists in finding the proper
reflection/transmission ratio that will produce a convergent optimization.
We also tested the performance of these same cost function but using only one
pair of reflection and transmission measurements instead of the four of them. We
obtained the same behavior, but the accuracy was slightly worse. It is then preferable to use the complete Scattering matrix with L1 norm as the best suitable cost
function to optimize, i.e. first row of Table 5.4. Its results are presented in the
next section.
84
Chapter 5.
Reconstruction Methods
ε*
r
1.04
1.02
ε
,
1
0.98
0.96
0.94
0.5
1
1.5
2
2.5
3
0
−0.01
ε
,,
−0.02
−0.03
std
OP rec
−0.04
−0.05
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 5.15: Standard and OP reconstructed (dotted line) permittivity for an Air sample.
ε*
r
30
OP rec
Debye fit
ε
,
20
10
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0
ε
,,
−5
−10
−15
Figure 5.16: OP reconstructed permittivity of ethanol (dashed line), and the fitted Debye
model (dotted line).
5.3. Optimization methods
85
Non-frequency dependent materials: AIR
In Figure 5.15 we have plotted the OP reconstructed electrical permittivity for
an air sample. As with the analytical methods, the OP reconstruction worsens
below 500 MHz on both the real and the imaginary parts and below 250 MHz the
solution starts to diverge. This time, the resonant frequency, at 1.5 GHz, is not
seen in the OP reconstructed permittivity. From 500 MHz to 3 GHz the accuracy
of the solution is mostly of the order of 1% for the real part, but this time, the
imaginary part is within measurement accuracy. This result is very similar to that
obtained with the analytical methods, slightly worse for the real part and better for
the imaginary part.
Frequency dependent materials: ETHANOL
With ethanol the optimized reconstruction is slightly different to that obtained
with the analytical methods. In Figure 5.16 we have plotted the OP reconstructed
electrical permittivity for this material. The solution holds up very well in the entire
frequency range and only below 10 MHz it starts to diverge. At high-frequencies,
above 2 GHz, the influence of noise is clear, and at the very end of the spectrum
0
a few points are off for εr .
The fitted Debye model and the reconstructed permittivities differ less than 1%
between 300 MHz and 1.5 GHz for the real part and from 500 MHz to 1.5 GHz for
the imaginary part. The fitted Debye parameters are listed in Table 5.5.
EthanolGemert
Ethanolf it
T (◦ C)
24
23
εs
25 ± 1
24.7
ε∞
4.6 ± 0.7
4.4
fr (M Hz)
900 ± 60
922
σdc (µS/m)
0.8 ± 0.1∗
700
Table 5.5: Experimental and fitted Debye parameters. Data for ethanol taken from van
Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ.
The fitted Debye parameters have been optimized to the OP reconstructed permittivities.
Again, the fitted parameters are within the error bounds of the van Gemert values, and the low frequency behavior of the imaginary part of the reconstructed
permittivity is captured by the conductivity.
86
Chapter 5.
Reconstruction Methods
ε*
r
1.04
1.02
ε
,
1
0.98
0.96
0.5
1
1.5
2
2.5
3
0
−0.01
ε
,,
−0.02
−0.03
std
PM
OP
−0.04
−0.05
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 5.17: PM and OP reconstructed permittivity for an Air sample.
|ε* −ε* |/|ε* |
rec
fit
rec
0.05
PM
OP
0.04
ℜ
0.03
0.02
0.01
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
Frequency (GHz)
2
2.5
3
0.05
0.04
ℑ
0.03
0.02
0.01
0
Figure 5.18: Differences between the PM and OP reconstructed permittivities for Ethanol
and their respective fitted Debye models.
5.4. Analytical vs Optimization
5.4
87
Analytical vs Optimization
Along this chapter we have presented the different methods and their independent
reconstructions. It is now time to compare the reconstruction of permittivity via
analytical and optimization methods.
There are three ways to analytically reconstruct the permittivity of a sample from
measured S-parameters. We presented them together with experimental results in
Section 5.2. The determination of permittivity from resonant frequencies is limited
to low-loss materials that present a constant permittivity over the frequency range
and therefore we will not compare it to any other reconstruction method. We
propose it as an alternative for the reconstruction of low-loss materials.
In Subsection 5.2.5 we already stated our preference for the Propagation Matrices
method above the Compensation Parameters. Although, we showed that the results
difference was of the order of measurement accuracy, but for ethanol, the fit of the
Debye model performed better on the PM reconstructed permittivities.
The analytical reconstructions solve for two permittivities, depending on which pair
of measurements is used. For simplicity, in this section we only show the results of
a single pair, as the results do not vary significantly if the other pair is chosen.
The optimization methods are diverse in cost functions and in Section 5.3 we ruled
out several of them and chose the L1 norm as the best possible.
In Figure 5.17 and Figure 5.18 we have plotted the reconstructed permittivities for
air and ethanol, with the Propagation Matrices method and that from optimization.
For both materials, the optimization reconstruction (OP rec: dashed line) is slightly
better than the Propagation Matrices reconstruction (PM rec: dotted line). The
two main accuracy thresholds, 2% and 5%, vary for reconstruction methods, materials and real and imaginary parts of permittivity.
The air reconstruction is peculiar, in its real part, because of the jump present
at 1.5 GHz where the PM reconstruction goes from overestimation at smaller frequencies to underestimation, while the OP reconstruction stays overestimating its
value. However, both reconstructions lie within ±1%, and the presence of the
jump confirms the resonance of air at that frequency. For the imaginary parts, the
optimization is definitely superior to the propagation matrices method. It is clear
that both methods exceed 5% accuracy in the real part below 50 MHz and around
250 MHz they exceed the 2% threshold. Ethanol follows a less complicated trend,
and both solutions are well captured by the Debye model. It is not conclusive
whether the propagation matrices reconstruction is better than the optimization
one or viceversa.
88
Chapter 5.
Reconstruction Methods
We can then conclude that any method is valid for the reconstruction of permittivities, whether they are low-loss and constant over the frequency range or highly
lossy and frequency dependent. We could, maybe, argue that the optimization
reconstruction is slightly better in its accuracy and that its solution is more stable
for low frequencies.
It would be, then, most adequate to use both methods to obtain results in a broader
spectrum. However, for the rest of this thesis we only use the propagation matrices
method because the optimization is very time consuming and it is necessary to
tune the cost function to each particular material, as the weighting needed for its
convergence depends in the ratio of the amplitudes of reflection and transmission.
On the contrary, the propagation matrices method is very fast and easy to program
and it always gives the same solution given the same data.
5.5
Conclusions
Along the chapter we have presented and compared the different reconstruction
methods for permittivity from measured S-parameters. We already evaluated their
individual performance with different materials and compared the three main methods presented. This section is intended as the closing summary of the chapter,
although it is not in order of appearance of the sections but in an order more in
accordance to the natural progression of the research.
As we have seen, there are two ways to obtain the permittivity of a sample from
its measured S-parameters: analytically or via an optimization. Both methods
determine the permittivity at each frequency independent of a predefined model of
the frequency dependence of the permittivity.
Traditionally, obtaining the frequency dependent permittivity of a material via analytical expressions was done in two steps. First, the measured S-parameters of
the transmission line were moved from the measurement planes to the interfaces
of the sample holder. Secondly, they were introduced into analytical expressions to
compute the permittivity. The nature of these expressions allow to reconstruct one
permittivity from each pair of reflection/transmission coefficients, and in the case,
the full scattering matrix is measured, two solutions can be computed.
There are two ways to move the measured S-parameters of the transmission line
from the measurement planes to the interfaces of the sample holder. If it is an ideal
tool a simple phase correction is sufficient, see Rau & Wharton (1982), but in practice this technique does not take into account errors present in true experimental
set-ups. Compensation techniques were developed to experimentally account for
them, it is cumbersome and very time consuming, and as we have shown in Subsection 5.2.4 it cannot be applied for broad band measurements. In fact, it must
5.5. Conclusions
89
be replaced by a modelled compensation in order to obtain reasonable results.
Complementary, in this chapter, we have developed a new method, the propagation matrices (5.2.2). While it moves the measured S-parameters of the transmission line from the measurement planes to the interfaces of the sample holder taking
into account impedance jumps between sections and phase changes, the nature of
the solution technique allows to directly find analytical expressions to compute both
permittivity and magnetic permeability. It is based on a simple concept and it is
easily programmable.
The analytical expressions, equations (5.12) and (5.13), found from the propagation
matrices approach, depend on Zp and cosh (γp dp ), two different combinations of
the corrected S-parameters. The instability of Zp is responsible for the considerable
errors encountered in the reconstruction of permittivity and permeability of air.
Therefore, since we are interested in non-magnetic soils, we restrict ourselves to
the reconstruction of permittivity from the propagation method (equation (5.17)).
It is a successful approach and we are able to reconstruct, both real and imaginary
parts, of the relative permittivity of air within ±1% error and obtain for ethanol, a
permittivity that fits the Debye model. For low-loss materials, =(acosh(β)) needs
special attention.
This approach also enabled us to show that all the existing analytical expressions are
the same (Subsection 5.2). For simplicity we only considered the solutions proposed
for permittivity reconstruction, but it is also valid for permeability. Writing Zp and
cosh (γp dp ) for the ideal case, so that no correction is needed and they are a direct
combination of the reflection and transmission coefficients at the interfaces of the
sample holder (see equations (5.18) and (5.19)), showed that they were equivalent to the equations proposed by Stuchly & Matuszewsky (1978), and later by
Palaith & Chang (1983). By further mathematical considerations we also showed
their equivalence to the Nicolson-Ross-Weir equation. So, 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,
because they are essentially the same methods. It is only now, that we have showed
it. We also showed the superiority of the methods based in the cosh (γp dp ) parameter above the unstable ones based on Zp (see for example Stuchly & Matuszewsky
(1978)).
The results obtained with the propagation matrices and with the modelled compensation parameters are comparable because of our almost ideal tool (see Subsection 5.2.5), but their different corrections and the better fit of the Debye model
to the PM reconstructed permittivity of ethanol (PM accounts for impedance and
phase changes and CO only for the latter) make the PM method superior.
The accuracy thresholds predicted in Chapter 4 are confirmed experimentally by
the PM and CO solutions. For air, below 50 MHz the difference between the
90
Chapter 5.
Reconstruction Methods
true permittivity of air and the reconstructed one is bigger than 5%. And the 2%
threshold is located around 250 MHz. From 500 MHz to 3 GHz the solution is
always between ±1% of the true data. Around the resonant frequency of 1.5 GHz
there is a clear deterioration of the solution. For ethanol, the comparison cannot
be quantitative but must be qualitative, and we have shown that a Debye model
can be fitted to the reconstructed permittivity. The obtained parameter lie within
the accuracy interval given by van Gemert (1972).
The fact that these methods present inaccuracies around resonant frequencies for
low-loss materials, encouraged us to introduce the resonant frequencies method
(5.2.1), within the analytical methods. To our knowledge this is the first time
that it is applied for a transmission line and not for a resonator. For materials of
constant ε∗r and with a low loss, it is possible to compute its permittivity for the
whole frequency range from a single resonant frequency or a set of them. Their
periodicity give a measure of the constancy of the property, and the amplitude of
the reflection coefficient at those points indicate the presence or absence of loss.
This method, then, overcomes the problems present in other analytical solutions
with low-loss materials.
When the reconstruction of the permittivity of a sample is not done from analytical
expressions, it must be reconstructed via an optimization procedure. In Section 5.3
we proved that the normalized L1 norm is the best cost-function to minimize (of
the ones we considered) and that the results obtained with it are slightly better than
those computed with the PM method, specially in the lower limit of the spectrum.
It would be, then, most adequate to use both methods to obtain results in a broader
spectrum. However, for the rest of this thesis we only use the propagation matrices
method because the optimization is very time consuming.
Based on the results obtained with the propagation matrices method for air, we
expect to reconstruct permittivities within ±1% error from 3λ/L up to 3 GHz,
while the lower limit can be relaxed by using optimization. However, for lossy
materials, this upper limit is conditioned by its loss, as the transmission gets smaller
the accuracy of the phase of the measurements deteriorates taking its toll in the
reconstructed permittivity, that clearly presents noisy behavior. However, it stills
captures its trend, and it can be amended with noise reduction techniques.
All our science, measured against reality, is primitive and
childlike — and yet it is the most precious thing we have.
Albert Einstein (1879-1955)
Chapter 6
Permittivity states of
mixed-phase; two and three
component sands
This chapter is dedicated to the study of the permittivity of dry, partially and fully
saturated sands, and how the distribution of saturation affects the EM properties.
6.1
Introduction
In Chapter 4 we saw that the probe used to measure the permittivity of our samples
allows for fluid flow through them, and in Chapter 5 we studied different reconstruction techniques for the permittivity of calibration samples. It is time to put
our tool to work and investigate its performance in the measurement of samples at
different saturations. After all, the main interest for field applications, is to find out
the fluid content of the soil from permittivity measurements. To do this, we compare our measurements with existing Mixing Laws, see Appendix E. Furthermore,
since our tool allows for fluid flow, it also enables us to study different saturation
distributions and their influence.
The sample can be saturated in different ways, fluid flowing perpendicular or parallel
to the wave’s direction of propagation and at any angle in between, or, as most
researchers do, it can be saturated outside the tool and then filling the sample holder
with it. All these saturation techniques generate different saturation distributions
in the sample even when the average saturation is the same, but how this influences
the permittivity measurements has not been studied yet.
91
92
Chapter 6. Permittivity states of mixed-phase
In this chapter we compare the different saturation techniques and we show how a
simple 2-layer model can account for the case in which the flow is perpendicular to
the direction of propagation of the waves, and we can find the fluid front position
from the measured S-parameters.
It is now important to clarify two troublesome terms, homogeneous and heterogeneous. They come from ancient Greek and literally mean of the same kind and
other kind. These nouns and their adjectival and adverbial forms are widely used in
this chapter in particular, and in science in general, with a slightly altered meaning,
or should I say scale? Science is renown for its relativeness and these terms are not
exceptions to the rule.
A certain volume of sand with fluid in its pores is certainly heterogenous at the
microscale but we can refer to it as homogenous in the mesoscale if the fluid
is homogeneously distributed in that volume. The sample as a whole turns into
an entity with a homogenous average saturation, porosity, permeability and even
permittivity. What happens when the fluid is not evenly distributed, for example
in the case of layering? In this chapter, we will then, be referring to two different
heterogeneities, one at the microscale, pore to pore, and the other one at the
mesoscale, differently saturated sections of the sample.
The correct use of the terms is intrinsically linked to the scale of interest and to
the physical property under study. A single porosity might be representative of a
certain volume of sand but the distribution of saturation may be so that an average
saturation does not correctly describe the sample. Therefore, it is important to
place the terms in the proper context to correctly understand its meaning.
In the specific case of referring to the permittivity of a sample it is important to
remember that the forward model, used along this thesis, considers the sample
holder as a whole and the Propagation Matrices method reconstructs a single effective permittivity for each pair of reflection/transmission parameters, even when
the sample holder is filled with two or more materials. As we will see in the coming
sections the reconstructed permittivities are sensitive to certain heterogeneities.
6.1.1
Chapter Outline
In Subsection 6.2.2 we give a detailed description of the samples, the experimental
procedures for the sample preparation and saturation and the S-parameters measurements. And in Subsection 6.2.3 we present the permittivities of three samples
saturated in different ways. The permittivities are computed from the measured
S-parameters with the Propagation Matrices method presented in Chapter 5.
In Section 6.3 we show the advantages of averaging the two permittivities obtained
with the mentioned method over averaging the S-parameters and in Section 6.4 we
6.2. Saturation technique and permittivities
93
model the case in which the flow is perpendicular to the direction of propagation
of the waves with a simple 2-layer model. In Section 6.5 we study the anomalies
present in the reconstructed permittivities of the 2-layer samples, whereas in Section 6.6 we compare our measurements with existing Mixing Laws, and finally, we
conclude this chapter with Section 6.7.
6.2
Saturation technique and permittivities
To study the effect of fluid flow on the permittivity of the samples under investigation, we performed three sets of experiments for comparison. One in which the flow
was assumed to be perpendicular to the propagation of the TEM waves, another
in which it was parallel, and a third one in which the saturation was homogenously
distributed over the sample, for control purposes. For each case we measured 6
steps, from a fully dry sample to a fully saturated one. These are explained in
detail in Subsection 6.2.2, while in Subsection 6.2.3 we present the reconstructed
permittivities and their comparison in Subsection 6.2.4. Let us first give a detailed
description of the samples.
6.2.1
Samples
All the samples were from loose quartz grains with grain sizes in the range
[112 − 280]µm. Their grain size distribution is shown in Figure 6.1. Two thirds
of the grains have radii of 160 or 180 µm. This narrow distribution of radii ensures homogeneous packing and a fairly close range of porosities among different
samples.
Grain Size Distribution of Quartz
0.5
0.45
0.4
Mass Ratio
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
100
120
140
160
180
200
Grain Size (µm)
220
240
260
280
Figure 6.1: Grain size distribution for the quartz grains that form the samples.
94
Chapter 6. Permittivity states of mixed-phase
The quartz grains were dried in an oven for two days at 70◦ C to ensure that no
water was present at the beginning of each experiment.
The sample holder was filled with these grains in different ways, as explained in
the next section, and the fluid used to flow into the sample was distilled water.
The temperature in the lab remained in the range [21 − 23]◦ C throughout all the
experiments. The porosities of the samples were computed from mass-volume
relationships and their inherent error is of the order of 0.5%.
6.2.2
Saturation technique
In the three flow experiments, the filling of the sample holder with the dry grains
differed. Since the resulting porosities were not equal for all the samples, we not
only looked at water saturation (Sw ) in volume ratio, but also at the mass ratio
water/quartz (mw /mq ) to be able to compare results. The measurements are very
sensitive to sample preparation.
No-Flow: saturation homogeneously distributed
To obtain a homogeneously distributed saturation, we added the desired amount
of water to a certain weight of quartz grains and stirred it well until no visible
lumps were present. Then, the sample holder was filled up, from above, with the
mixtures and the final porosity and saturation of the samples were computed from
mass measurements. Figure 6.2 represents these samples.
Figure 6.2: No-Flow: saturation homogeneously distributed.
6.2. Saturation technique and permittivities
95
Filling the sample holder with already made partially saturated samples resulted in
samples with different porosities, and although we tried to ensure a homogenous
distribution of water throughout the sample the filling of the sample holder could
be responsible for some layering and/or trapped air.
One could argue that with this type of set-up and due to gravity, the distribution
of water in the sample varies with time. However, the sample is only 10 cm long
and the time it takes from the moment of sample preparation to the measurement
of the S-parameters is in the order of 10 minutes. Whereas gravity will take over
the capillary forces in matter of several hours.
Flow perpendicular to the direction of propagation: Gravity drainage
All throughout this experiment, the sample holder is maintained perpendicular to
the ground. Figure 6.3 shows a schematic representation of this procedure.
The sample holder was first filled with water and then dry quartz grains were
carefully added so that the first measurement was done on a fully saturated sample
(SW ). During the whole filling process the tool is under vibration to help the
compaction of the grains and to remove possible trapped air. This technique
resulted in the lowest porosity for all three experimental sets, 40%. Later we let
the water flow out with the help of gravity (top to bottom), in steps. When no
more water flows out of the sample, we assume the sample to have a residual
water saturation (Sr ). Note that this is just a way of naming the saturation of this
step and that it is not the true residual water saturation. It refers to the obtained
saturation at the last step of a gravity drainage. To dry up the sample (S0 ) we
connected a vacuum pump to the fluid outlets of the sample holder and let it pump
for 24 hours.
The amount of water that came out was monitored by a balance and because of the
geometry of the set-up, the tubes connecting the sample holder to the water balance
were always full, so that saturation determination was very simple and precise. At
every step we monitored the changes in the S-parameters and the measurement
was taken when there was no noticeable change in them, sometimes having to wait
for a few minutes. At the end, we had determined the S-parameters of the sample
in several stages of saturation, from fully dry to fully saturated.
This gravity drainage is possible with enough height between the sample and the
balance. In our case, this was of 1.25 meters.
This is a very porous sample with most grains being of the same size, and although
a sharp interface between the residual and fully saturated regions is a simplistic
assumption, we can think of the fluid front as being perpendicular to the direction
of propagation of the TEM waves.
96
Chapter 6. Permittivity states of mixed-phase
Figure 6.3: Flow perpendicular to the direction of propagation.
We also tried to perform imbibition experiments, but those resulted in misleading
results. During the imbibition, the grains compacted and a fine layer of water
formed on top of the sample.
Flow parallel to the direction of propagation
First, the sample holder is put in an upright position and filled with the dry grains.
By this procedure we obtained a porosity of 43%. Then, the sample holder is turned
and maintained parallel to the ground and the fluid is injected step by step via a
controlled syringe until it overflows. Figure 6.4 shows this procedure. The amount
of water that is injected into the sample is measured with the controlled syringe,
but there is uncertainty to it because of trapped air in the tubes connecting the
sample holder with the in-flow system. It was not possible to maintain a constant
filling of the tubes and therefore the amount of water in the sample holder was not
accurately known.
In this type of saturation it is hard to know the distribution of the fluid front, and
with the flow inwards a dry sample there is risk of grain replacement and further
compaction.
6.2. Saturation technique and permittivities
Figure 6.4: Flow parallel to the direction of propagation.
97
With these three different techniques we saturated very similar samples. In the next
subsection, we show results for all cases and compare them. We should keep in mind
that the permittivity reconstruction with the Propagation Matrices method yielded
two permittivities, one computed with the pair [S11 , S12 ] and another with the pair
[S22 , S21 ] (Subsection 5.2.2). From now on, we will refer to these parameters as
ε∗12 and ε∗21 so that it is clear from which S-parameters they have been calculated.
6.2.3
Permittivities for the differently saturated samples
For each technique, six different saturations have been reached, from fully dry to
fully saturated. For the no-flow case, the saturation is assumed to be homogeneously distributed, but for the other two cases, it is not and therefore, the value
given corresponds to an effective saturation. Furthermore, as mentioned earlier,
the porosities are not equal for all samples due to the experimental procedure, and
in order to be able to compare the results it is more convenient to refer to the
gravimetric moisture content, mw /mq (water/quartz mass ratio). Unfortunately it
was not possible to obtain samples with equivalent ratios and the comparison is
more qualitative than quantitative.
Each subsection corresponds to a saturation technique, and in each of them, we list
the measurement steps together with the porosity of the sample and their effective
water saturation, in volume and mass ratio. Errors are stated when significant. We
also show ε∗12 and ε∗21 for each step.
98
Chapter 6. Permittivity states of mixed-phase
Homogeneously mixed
30
6
ε
,
20
5
10
4
3
2
1
0
0.5
1
1.5
2
2.5
0
1
2
3
−1
4
−2
5
ε
,,
3
−3
6
−4
−5
0.5
1
1.5
2
Frequency (GHz)
2.5
3
Figure 6.5: Complex permittivity for six samples homogeneously saturated. The dashed
line corresponds to ε∗12 and the dotted line to ε∗21 .
6.2. Saturation technique and permittivities
99
No-Flow: saturation homogeneously distributed
Having to saturate the sample outside of the sample holder, results in samples with
slightly different porosities.
Table 6.1: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±0.5) and
water-to-sand mass ratio (∆(mw /mq ) =
±0.2) for the six samples mixed outside
the sample holder. All quantities are
given in %.
Step
φ
Sw
mw /mq
1
2
3
4
5
6
43
45
45
46
45
42
0
16
32
47
66
100
0
5
10
15
21
28
Figure 6.6 presents the real and imaginary component of both ε∗12 and ε∗21 . A
blown up version of this figure (Figure 6.8) can be found in pages 104-105. The
real parts are more or less constant for the whole frequency range and their values
increase with increasing water content. The imaginary parts show a linear increase
with frequency and the loss enlarges with water content. Their trend is less smooth
than their real counterparts, and they clearly diverge for low frequencies due to the
low frequency limit of the measurement technique.
In Chapter 5 we saw that when the samples were fully homogeneous (air and
ethanol) ε∗12 and ε∗21 differed within measurement accuracy, and, as we will see,
their similarity can be used to estimate the homogeneity of a sample.
In this case, we have tried to homogeneously distribute the water saturation
throughout the sample, and from the results, it is clear that the dry sand is very
close to this assumption and could be considered as a homogenous sample with an
effective permittivity of around 2.5 − 0.03j. As the water content increases, the
difference between the ε∗12 and ε∗21 also increases, being maximum for the last two
steps. However, this difference is still acceptable and always smaller than 2% above
1 GHz.
The permittivity of step 5 shows two clear deviations from the constant trend of
0
00
both εr and εr around 500 MHz. As we will see in the coming sections, the
reconstructed permittivities present synchronized anomalies when the samples are
composed of 2 or more layers, and we think that these deviations are indicators of
layering and due to the sample preparation. This effect is more pronounced in the
00
anomalies of the otherwise smooth εr . In a smaller scale this is also present in step
4 around 1 GHz.
100
Chapter 6. Permittivity states of mixed-phase
Flow perpendicular to the direction of propagation
30
6
20
ε
,
5
4
10
3
2
1
0.5
1
1.5
2
2.5
3
0
1
2
−1
3
−2
4
5
ε
,,
0
6
−3
−4
−5
0.5
1
1.5
2
Frequency (GHz)
2.5
3
Figure 6.6: Complex permittivity for the six steps of a flowing experiment when the
flow is perpendicular to the direction of propagation of the TEM mode. The dashed line
corresponds to ε∗12 and the dotted line to ε∗21 .
6.2. Saturation technique and permittivities
101
Gravity Drainage: Flow perpendicular to the direction of propagation
In this case, the sample is the same all throughout the experiment, and therefore,
the porosity remains constant. It is also, the lowest porosity obtained with the three
different methods.
Table 6.2: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±0.5) and
water to sand mass ratio (∆(mw /mq ) =
±0.2) for the six flowing steps, when the
flow is perpendicular to the direction of
propagation of the waves. All quantities
are given in %.
Step
φ
Sw
mw /mq
1
2
3
4
5
6
40
40
40
40
40
40
0
27
53
67
83
100
0
7
13
17
21
25
Figure ?? presents the real and imaginary components of both ε∗12 and ε∗21 . A
blown up version of this figure (Figure 6.9) can be found in pages 105-106. The
reconstructed permittivities, are certainly not as smooth as the ones computed when
the sample was homogenously saturated, except, of course for the dry, residual water
and fully saturated steps (1, 2 and 6). For all the other steps, the real parts are
more or less constant for frequencies above 1 GHz and their value increases with
increasing water content, but below 1 GHz they present anomalies. The imaginary
parts show a linear increase with frequency and the loss increases with water content
but their trend is certainly more erratic than their real counterparts. The anomalies
on the real part are also present in the imaginary part.
Again, as the water content increases, the difference between the two computed
permittivities also increases, being maximum for the fifth step. The fully saturated
sand behaves as the homogenously mixed sample from the previous experiment.
Above 1 GHz the difference in the real part of permittivity is still acceptable and
smaller than 2%, but the imaginary parts differ considerably.
102
Chapter 6. Permittivity states of mixed-phase
Flow parallel to the direction of propagation
30
6
5
4
10
3
2
ε,
20
1
0.5
1
1.5
2
2.5
3
0
1
−2
2
−4
5
63
45
4
ε
,,
0
−6
−8
−10
0.5
1
1.5
2
Frequency (GHz)
2.5
3
Figure 6.7: Complex permittivity for the six steps of a flowing experiment when the flow
is parallel to the direction of propagation of the TEM mode. The dashed line corresponds
to ε∗12 and the dotted line to ε∗21 .
6.2. Saturation technique and permittivities
103
Flow parallel to the direction of propagation
As in the previous flowing experiment, the sample is the same all throughout the
experiment, and therefore, the porosity remains constant.
Table 6.3: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±7) and
water to sand mass ratio (∆(mw /mq ) =
±2) for the six flowing steps, when the
flow is parallel to the direction of propagation of the waves. All quantities are
given in %.
Step
φ
Sw
mw /mq
1
2
3
4
5
6
43
43
43
43
43
43
0
7
36
54
72
100
0
2
11
16
21
29
The reconstructed permittivities ε∗12 and ε∗21 for this experiment are presented in
Figure 6.7. A blown up version of this figure (Figure 6.10) can be found in pages
108-109. They behave differently than those computed in the two previous experiments, but still, they increase with increasing water content. The real parts are no
longer constant but present a growing tendency with frequency, except for the fully
dry and saturated samples.
The permittivity of the fully saturated sample is consistently higher than that of its
equivalent counterparts and the difference in porosity cannot account for it. From
experimental experience, we think that this is due to the saturation technique used.
As we already mentioned, flowing into a dry sample allows for grain repositioning
and compaction that could result in a small empty chamber at the top of the
sample holder where more water could be stored explaining the high value of the
permittivity.
The anomalies show up during a broader part of the spectrum and the imaginary
parts are not consistent, since the apparent loss in partially saturated samples is
greater than the fully saturated sample. This time, the difference in the real part of
permittivity is only smaller than 4% for frequencies above 2 GHz, but the imaginary
parts differ considerably. As we will see in the coming sections, the anomalies
present in the previous flow experiment can be explained with a layered model.
We think that the anomalies present in this experiment is also due to layering.
However, this layering cannot be modelled with homogeneous cross sections and
therefore cannot be included in our model. Since, we cannot make a thorough study
of these samples we will not use it for comparison with the other two techniques.
ε,
Homogeneously mixed
15
5
10
4
3
1
Chapter 6. Permittivity states of mixed-phase
2
5
3
2.5
2
1.5
1
0.5
0
104
25
6
20
1
2
−0.5
3
−1
4
−1.5
ε
,,
−2
5
−2.5
−3
6
−3.5
6.2. Saturation technique and permittivities
0
−4
−4.5
−5
0.5
1
1.5
Frequency (GHz)
2
2.5
3
105
Figure 6.8: Complex permittivity for six samples homogeneously saturated. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 .
106
Flow perpendicular to the direction of propagation
25
6
20
5
15
ε,
10
3
5
2
1
0
0.5
1
1.5
2
2.5
3
Chapter 6. Permittivity states of mixed-phase
4
1
2
ε
,,
−0.5
−1
3
−1.5
4
−2
5
−2.5
6
−3
−3.5
6.2. Saturation technique and permittivities
0
−4
−4.5
−5
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.9: Complex permittivity for the six steps of a flowing experiment when the flow is perpendicular to the direction of propagation of
the TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 .
107
108
Flow parallel to the direction of propagation
30
6
25
5
4
15
3
10
2
5
1
0
0.5
1
1.5
2
2.5
3
Chapter 6. Permittivity states of mixed-phase
ε,
20
1
−1
2
−2
3
−3
6
4
ε,,
−4
5
−5
−6
−7
6.2. Saturation technique and permittivities
0
−8
−9
−10
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.10: Complex permittivity for the six steps of a flowing experiment when the flow is parallel to the direction of propagation of the
TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 .
109
110
6.2.4
Chapter 6. Permittivity states of mixed-phase
Comparison and interpretation of results
Here, we compare only the results of the first two sets of experiments, because we
ruled out the last one since the saturation technique provided samples that violated
the model assumptions (homogeneity in the cross-section). All the samples consisted of quartz grains with the same grain size distribution, and although saturated
in different ways, their permittivities are comparable.
The first step corresponds to the fully dry state. The porosities of the two samples
differ by 2% and their permittivities by 5%. The sample with lowest porosity
has a higher permittivity, since the quartz to air ratio is higher. ε∗12 and ε∗21 for
both samples are within experimental error and the dry sands can be treated as
homogenous.
0
As water is added the results are consistent, higher values of εr and greater losses for
higher water content. From steps 2 to 4 the water to quartz ratio is always bigger
for the sample subject to gravitational drainage than for those homogenously mixed
outside of the sample holder, and so are their permittivities. Step 5 has the same
mw /mq for both experiments, however, the permittivity of the sample with 40%
porosity (gravity drainage) is 2% bigger than the one with 45% (homogeneously
mixed). This is related to the different ratios of quartz/air/water they posses. When
the samples are fully saturated (step 6), their permittivities are again comparable,
and they are higher for the one with higher water to quartz ratio.
In principle, when all possible water has been drained with the aid of gravity, the
remaining residual water will be evenly distributed along the sample and it can be
treated as homogeneous (step 2 of the experiment in which the flow is perpendicular
to the direction of propagation of the TEM waves).
Only steps 3 to 5 of the sample are subject to gravitational drainage and present
remarkable anomalies, whereas steps 1, 2 and 6 do not present these anomalies.
Furthermore, ε∗12 and ε∗21 are within experimental errors for steps 1 and 2 and
differ by less than 2% for the sixth step. It is thus clear that the heterogeneous
samples can be identified from the reconstructed permittivity. The consistency
of the anomalies in all heterogeneous samples, and the distortion of ε∗ for the
case in which the fluid front is parallel to the direction of propagation, points to
heterogeneity. Furthermore, the fact that we are able to reproduce the anomalies
by modelling the sample as a two layer sample (see Section 6.4) encourages this
explanation. In Section 6.5 we study their occurrence and distribution.
The interpretation of the results would not be complete without the attempt to
model these results with known mixing models, but we have preferred to do so at
the end of the chapter, when all the needed information is available.
6.3. Averaging ε∗12 and ε∗21 over averaging the S-parameters
6.3
111
Averaging ε∗12 and ε∗21 over averaging
the S-parameters
Along the previous subsections we have presented results on both ε∗12 and ε∗21 .
Their advantage is clear when dealing with heterogeneous samples, as the more
inhomogeneous a sample is the more different they are. However, two permittivities
complicate the analysis and comparison, so for simplicity we have decided to average
them out per frequency. Even more so, when it is possible to do it without loosing
heterogeneity information.
Obtaining a single effective ε∗r per frequency can be done in two different ways,
averaging the permittivities into a single value per frequency or reducing the 4
S-parameters into 2, averaging both reflections and transmissions into single parameters, also per frequency, and invert to obtain a single permittivity. Reducing
the two pairs of reflection and transmission data into one single pair and then invert, has to be done on a perfectly symmetric tool, otherwise it would not be valid,
because the reflections would be different due to the asymmetry of the tool. For
our tool, we have to compensate for the connector that enables the connection
with the cable from Port 2 (see Chapter 4), a simple phase correction takes it into
account.
In an ideal case of a perfectly homogeneous sample and with the compensated
symmetric tool, both reflections and transmissions should be equal. For our tool
this is true within experimental errors when the sample holder is filled with a homogeneous material. Then, averaging the S-parameters per frequency yields the
same result as if two permittivities are computed from the 4 S-parameters and then
averaged. However, the samples we are interested in, always have a certain degree
of heterogeneity, and it is logical that the average of the S-parameters will involve
a loss of information about it. To corroborate it, we have performed these two
averaging approaches on the No-Flow experiment where the samples were assumed
to be homogeneous. With the obtained permittivity we have reconstructed the data
and looked at the similarities between the measured and reconstructed data.
We show results only for step 4 of the experiment, but in all cases the results
were similar. Averaging the measured S-parameters and then, reconstructing the
permittivity (ε∗s ) always resulted in permittivity values between 2% and 4% bigger
than those computed from the averaged ε∗12 and ε∗21 (ε∗m ), so that both the real
part and the loss were greater, see Figure 6.11. The phase of the reconstructed
S-parameters was equally well modelled but the amplitude of reflection and transmission was underestimated by ε∗s , see Figure 6.12.
112
Chapter 6. Permittivity states of mixed-phase
11
ε
,
10
9
8
0.5
1
1.5
2
2.5
3
0
εm
ε
12
ε
21
ε
ε,,
−0.5
−1
s
−1.5
−2
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.11: Averaged complex permittivities for step 4 of the no-flow experiment. ε∗12
and ε∗21 are represented by dotted lines, their mean ε∗m by the solid line, and the dashed
line corresponds to the permittivity computed from the averaged S-parameters.
e
es
Figure 6.12: Amplitude of the measured and reconstructed S11 and S12 , for step 4 of the
no-flow experiment. The solid line corresponds to the measured parameters, the dotted
line to the reconstructed parameters from ε∗m and the dashed line to the reconstructed
parameters from ε∗s
6.4. 2-layer samples
113
From now on, we will show a single permittivity per sample/water content, that is,
the average of ε∗12 and ε∗21 , i.e. ε∗m . As we have seen, it is a better representation
than ε∗s , plus the anomalies due to heterogeneities do not disappear and a single
effective parameter represents the sample.
In the next section, we model the response of the sample in which the flow is
perpendicular to the direction of propagation of the TEM as a 2-layer sample,
obtaining the thickness of the layers in two different ways.
6.4
2-layer samples
For the case in which the flow is perpendicular to the direction of propagation of the
TEM mode (Figure 6.3) we can picture the sample from steps 3 to 5, as consisting
of two layers, one fully saturated and the other one with residual water saturation
and assume each layer to be homogeneous. If we knew the height of these two
layers we could model the S-parameters corresponding to such a configuration and
test the validity of the 2-layer assumption. Figure 6.13 shows this model. Although,
it is a very simple approach, the model yields a good approximation.
Figure 6.13: Model for a sample in
which the flow is perpendicular to the direction of propagation of the TEM mode.
SW is the saturation of the fully saturated layer and equal to 100% (step 6 in
Table 6.2) and Sr is the saturation of the residual water saturated layer and equal
to 27% (step 2 in Table 6.2). H is the total height of the sample, 10 cm, and h
is the height of the fully saturated layer. To find out h, there are many different
methods, but we will limit ourselves to two. The first one is based on an effective
saturation assumption and the second one is a phase approximation.
114
Chapter 6. Permittivity states of mixed-phase
6.4.1
hSw from saturation
If we assume an effective water saturation (Sw ) for the whole sample and a homogeneously distributed porosity, then, equation (6.1) must hold, and since all the
other parameters are known hSw can be obtained.
(6.1)
Sw · H = Sr · (H − hSw ) + SW · hSw
6.4.2
hθw from phase
In this case, we do not make any assumptions about the saturation of the sample,
but instead, we take the phase of the transmission coefficients to be linear and suppose that the contribution of multiples can be neglected. Now, θw is the measured
phase of Υ (θ(Υ)) for any step from 2 to 6 of Table 6.2, θW is θ(Υ) of the fully
saturated sand (step 6) and θr is θ(Υ) of the residual saturated sand (step 2).
θw · H = θr · (H − hθw ) + θW · hθw
6.4.3
(6.2)
Results
Solving equations (6.1) and (6.2) we found the different heights listed in Table 6.4.
The heights are given in cm, and the ones computed from the saturation assumption, have an error of ±0.1. For those computed from the phase approximation it
is not possible to estimate the error a-priori.
Table 6.4: Heights [cm] of the fully saturated sand layer for the five flowing steps,
from Sr at 2 to SW at 6, when the flow
is perpendicular to the direction of propagation of the waves.
Step
h Sw
hθw
2
3
4
5
6
0
3.5
5.5
7.7
10
0
3.9
6.0
7.9
10
With the heights from Table 6.4, and the reconstructed permittivities for the fully
and the residual water saturated sands we modelled the sample as two layers and
generated computed S-parameters for such a model. Comparing the measured
S-parameters and the synthesized ones we found that hθw is a better approximation
to the true height of the saturated layer than hSw .
6.4. 2-layer samples
115
In Figure 6.14 we have plotted the real and imaginary parts of S11 and S12 for
step 4. The solid line corresponds to the measured data, the dotted line to the
modelled data taking hSw , and the dashed line to the modelled data taking hθw .
From the figure it is clear that using hθw results in a closer fit to the measured
data. Specially, for the transmission, where the modelled data with hSw is clearly
out of phase. These results are consistent in steps 3 and 5.
Then, the assumption that a sample, where a two-phase flow (air-water) is occurring
and a fluid front present, can be treated with an effective saturation underestimates
the position of the fluid front. In contrast, the linear approximation to the phase
of the transmission coefficient yields a good estimation. However, if the error in
the saturation was of the order of 5% (10 times bigger than the computed error)
the estimates of the layer height would be comparable. The error difference is
considerable, but the experimental conditions were not optimum, and, we can only
positively affirm that the phase approximation is more reliable.
ℜ
ℑ
1
0.5
0.5
0
0
−0.5
−0.5
S11
1
−1
1
0.5
1
1.5
2
2.5
3
meas
mod h(Sw)
mod h(θ )
−1
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
Frequency (GHz)
3
1
w
0.5
0
0
−0.5
−0.5
S12
0.5
−1
0.5
1
1.5
2
2.5
Frequency (GHz)
3
−1
Figure 6.14: Real and imaginary parts of S11 and S12 for step 4 of the experiment in
which the flow is perpendicular to the direction of propagation of the TEM. The solid line
corresponds to the measured data, the dotted line to the modelled data taking hSw and
the dashed line to the modelled data taking hθw
116
Chapter 6. Permittivity states of mixed-phase
From the modelled S-parameters with hSθw we can reconstruct a modelled effective
permittivity (dashed line) and compare it to the one reconstructed from the measured data (solid line), corroborating the good approximation of the height of the
saturated layer from the phase of the transmission coefficient (Figure 6.15). Note,
that the reconstructed modelled permittivity also has the anomalies present in the
permittivity reconstructed from measured data.
In Figure 6.15 we have plotted the reconstructed permittivities for step 5 of the gravitational drainage experiment from the measured S-parameters [S exp ] (solid line)
together with the reconstructed permittivities from the synthesized S-parameters
].
with a two layer model in the sample holder, [Shmod
] and [Shmod
S
θ
w
w
Gravity Drainage Experiment
25
ε,
20
15
10
0.5
1
1.5
2
2.5
3
0
ε,,
−1
−2
meas
mod h(Sw)
mod h(θw)
−3
−4
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.15: Averaged complex permittivities for step 5 of the experiment in which the
flow is perpendicular to the direction of propagation of the TEM. The solid line corresponds
to the measured data, the dotted line to the modelled data taking hSw and the dashed line
to the modelled data taking hθw .
6.5. Anomalies of layered samples
117
It is clear that the best solution is given from taking hθw instead of hSw . The real
parts coincide except for the regions around the anomalies present in ε∗hθ . These
w
anomalies match those of ε∗rec below 500 MHz, but are more numerous. In fact,
there seems to be a relation between the height of the fully saturated layer and the
number and amplitude of anomalies present. We think that the sharpness of the
interface introduced in the model could be responsible for the clearer appearance of
anomalies, while in the experimental case the interface between fully and residual
water saturated sand is more gradual. In the next section we will test this hypothesis. The imaginary parts are not so well captured by the 2-layer model because
the more frequent anomalies mask the data. The results obtained for steps 3 and
4 are comparable.
6.5
Study of the anomalies present in reconstructed
permittivities of layered samples
In Subsection 6.2.3 we saw how the reconstructed permittivities of layered samples
present anomalies in their real and imaginary parts, steps 3 to 5 of Figure 6.9.
However, the ε∗rec of the same sample when the saturation was homogeneously
distributed, steps 1, 2 and 6 of Figure 6.9 were smooth. This pointed out the fact
that those anomalies could be originated from the inhomogeneity of the sample.
Moreover, in Section 6.4 by modelling the sample with 2 layers, one fully saturated
and another with residual water saturation, we were able to reconstruct the measured S-parameters and their associated permittivities, anomalies included. In this
section we want to investigate further these anomalies and the parameters that
influence them.
To study these anomalies we have worked with modelled data and we have created
3 sets of numerical experiments with different conditions. In all three sets we have
used an approximation to the permittivities for the residual (ε∗r ) and fully water
(ε∗W ) saturated quartz sand (steps 2 and 6 of Figure 6.9). Over the frequency
range of operation, the real parts can be considered constant and the imaginary
parts can be approximated with a linear dependence on frequency (f ). Their values
are
ε∗r = 5 − j · 0.83 · 10−10 · f
ε∗W = 21.5 − j · 9.17 · 10−10 · f
In the first place, we study the effect of the length of the sample holder, reconstructing the permittivities of samples of 10, 20 and 30 cm long, for four different
volume fractions of the 2 different materials. All the samples studied are shown
in Figure 6.16, the grey layers correspond to ε∗W and the white ones to ε∗r , the
118
Chapter 6. Permittivity states of mixed-phase
numbers on the left reflect the volume fraction of material with permittivity ε∗W to
that of ε∗r . Figure 6.17 shows the reconstructed real permittivity for these samples.
0
Note that the scale of the εr axis is very big.
Figure 6.16: Modelled samples for different sizes of sample holder.
2/10
100
H = 10 cm
H = 20 cm
H = 30 cm
50
4/10
0
50
6/10
0
50
8/10
0
50
0
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.17: Real permittivity for four different volume fractions in three sample holders
of different length.
6.5. Anomalies of layered samples
119
For constant volume fraction, the anomalies are ordered, they occur at lower frequencies for longer sample holders, and move to higher frequencies and they expand
their size for broader frequency spans for shorter sample holders. As the volume
fraction of the wet quartz increases so do the amplitude of the anomalies. This indicates that the contrast in the permittivity of the layers has a role in the amplitude
of the anomalies, and although, the sample holder length influences the frequency
span and location, making the sample holder longer doesn’t make the anomalies
disappear.
Secondly, we investigate the role of the position of the layer. Figure 6.18 shows
the modelled samples we used, they consist of a 2 cm long wet layer at different
locations in a 10 cm sample holder. The rest of the sample holder is filled with
residual water saturated sand. This time, we show the amplitude and phase of the
synthesized S-parameters for such samples in Figure 6.19.
Figure 6.18: Modelled samples for different positions of the wet layer. The
numbers correspond to the position of
the layer in cm from the left hand side
of the sample holder.
The phase of the transmission coefficients are equal, and so are their amplitudes
for equivalent samples, 0 and 8, and 2 and 6. Whereas the amplitudes and phases
of the reflection coefficient are certainly different. Note that for the first 100-200
MHz all coefficients are equal. Of course, when reconstructing the permittivity
from these S-parameters, the anomalies are distinct for distinct positions of the
wet layer, but, in the low frequency limit they all have the same permittivity and
as frequency increases they oscillate around a constant value of permittivity lower
than that of the low frequency limit. We discuss this further at the end of this
section.
The position of the wet layer certainly matters and a single effective permittivity
cannot capture the EM response of these samples, as it would produce the same
S-parameters for the six samples.
120
Chapter 6. Permittivity states of mixed-phase
Amplitude
Phase
1
4
0.8
2
S11
0.6
0
0.4
−2
0.2
0
0.5
1
1.5
2
2.5
3
1
−4
0.5
1
1.5
2
2.5
3
0.5 1 1.5 2 2.5
Frequency (GHz)
3
4
0.8
2
S12
0.6
0.4
0.2
0
0
0
2
4
6
8
0.5 1 1.5 2 2.5
Frequency (GHz)
−2
3
−4
Figure 6.19: Modelled S-parameters for samples with the same volume fraction
wet/residual, but with the wet layer at different positions (in cm from the left of the
left hand side of the sample holder).
The last numerical experiment focuses in the width of the layers, and consists in
maintaining the volume fraction of wet/residual constant for a 10 cm sample, and
evenly distributing the initial 2 cm wet layer in smaller and smaller layers throughout
the sample holder. Figure 6.20 shows the samples studied.
The synthesized S-parameters, Figure 6.21, are different for the three first samples
and equal for the last three, implying that the width and distribution of the layers
play and important role in whether the samples can be considered equivalent from
an EM point of view, or not. Again, for the first 100-200 MHz the S-parameters
are equal for all samples, but this time, the phases of the transmission coefficients
are slightly different for the first three samples.
6.5. Anomalies of layered samples
121
Figure 6.20: Modelled samples for different distributions of wet layer.
Amplitude
Phase
1
4
0.8
2
S11
0.6
0
0.4
−2
0.2
0
0.5
1
1.5
2
2.5
3
1
S12
0.4
0.2
0
0.5
1
1.5
2
2.5
3
0.5 1 1.5 2 2.5
Frequency (GHz)
3
4
0.8
0.6
−4
2
1
3
7
15
31
63
0.5 1 1.5 2 2.5
Frequency (GHz)
0
−2
3
−4
Figure 6.21: Modelled S-parameters for samples with the same volume fraction
wet/residual, but with diminishing wet layer width and increasing number of layers (in
the legend).
122
Chapter 6. Permittivity states of mixed-phase
Let us now take a look at the reconstructed permittivities of the last two experiments. The volume fraction of wet sand to that of residual water saturated sand
is constant and equal to 0.2. Figure 6.22 shows ε∗ for all six samples in which the
width of the wet layer is maintained constant but its position changes.
10
0
2
4
6
8
ε,
9
8
arim
crim
7
6
0.5
1
1.5
2
2.5
3
0
ε,,
−0.5
crim
arim
−1
−1.5
−2
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.22: Reconstructed permittivities from modelled S-parameters for samples with
the same volume fraction wet/residual, but with the wet layer at different positions (in cm
from the left of the left hand side of the sample holder).
They are all different but it is interesting to note that at the lowest frequency they
all have the same value, corresponding to the static prediction of the arithmetic
mean, εarim , equation (E.3). Above 1.5 GHz, the real parts of the permittivity
all tend to the value predicted by the CRIM (Complex Refractive Index Model)
model, equation (E.19) with c = 0.5. The arithmetic mean and the CRIM model
are explained in detail in Appendix E, but for convenience, we list their expressions
for this particular example.
6.5. Anomalies of layered samples
123
ε∗arim = fr ε∗r + fW ε∗W ,
(ε∗crim )0.5
=
fr (ε∗r )0.5
+
(6.3)
fW (ε∗W )0.5 ,
(6.4)
where fr is the volume fraction of material with permittivity ε∗r (residual water
saturated) and fW that of material ε∗W (fully saturated).
Above 1.5 GHz, the real parts of the permittivity all tend to the value predicted
ε∗crim , with a maximum standard deviation is 3%. The imaginary parts are unfortunately more erratic. At 1.5 GHz, the wavelength corresponding to the permittivity
of the mixture given by the CRIM model (λcrim ) is approximately four times the
width of the wet layer.
Figure 6.23 shows ε∗ for all six samples in which the volume fraction is maintained
constant, but the width of the wet layer is reduced, and the thinner but more
numerous wet layers are distributed along the sample.
10
ε,
9
arim
8
crim
7
6
0.5
1
1.5
2
2.5
3
0
ε
,,
−0.5
−1
−1.5
−2
1:
3:
7:
15
31
63
crim
arim
20.0
6.67
2.86
: 1.33
: 0.64
: 0.32
0.5
1
1.5
Frequency (GHz)
2
2.5
3
Figure 6.23: Reconstructed permittivities from modelled S-parameters for samples with
the same volume fraction wet/residual, but with diminishing wet layer width and increasing
number of layers (odd sequence) [number of wet layers: width of wet layers].
124
Chapter 6. Permittivity states of mixed-phase
This time, as the width of the inclusions decreases, both real and imaginary parts
of the permittivity, tend to ε∗arim . Only the first two samples at high frequencies
tend to ε∗CRIM . When the layer width is 2.86 mm the standard deviation of the real
part of ε∗ is about 2% for the first half of the spectrum and increases up to 6% at
the high frequency end. When they are 1.33 mm it is already smaller than 0.5%
for the first half and around 1% for the second half. The imaginary parts deviate
slightly more than their real counterparts but we can consider the sample with 15
equally spaced layers of 1.33 mm wide to be homogeneous and with a permittivity
given by ε∗arim .
The fact that this experiment considers an odd sequence of layers (the change in
width ratios is not constant) makes it difficult to find the relation between layer
width and the frequency at which ε∗arim changes to ε∗crim . In figure 6.24 we have
plotted the permittivities of a series of samples equivalent to those of figure 6.21
but with an even sequence of layers, such that their width ratio is always a factor
of two.
Effective permittivities for layered systems
12
1:20.0
2:10.0
4:5.00
8:2.50
16:1.25
32:0.62
11
10
ε,
r
9
8
7
6
5 8
10
10
9
10
10
10
11
frequency [Hz]
Figure 6.24: Reconstructed permittivities from modelled S-parameters for samples with
the same volume fraction wet/residual, diminishing wet layer width and increasing number
of layers (even sequence). In the legend [number of wet layers: width of wet layers].
6.5. Anomalies of layered samples
125
For this specific case, the frequency at which ε∗arim changes to ε∗crim , ft corresponds
to approximately
wet
ft = fc · Nlayers
+1
with
fc = 1.1GHz,
(6.5)
wet ) is related to the width of these layers,
and since the number of wet layers (Nlayers
2
as dlayer = N wet
[cm], the transition frequency can be rewritten as
layers
ft = fc ·
2
dlayer
+1 .
(6.6)
The transition frequency, ft depends on the width of the layers and on fc that is
determined by the volume fraction and permittivities of the two components. For
very narrow layers this frequency is very high but finite.
Note that for frequencies smaller than ft the values of permittivity values correspond
precisely to ε∗arim while for frequencies bigger than ft they oscillate around ε∗crim .
Local electrodynamic equilibrium
Rio & Whitaker (2000a) constrained the validity of using one-equation models for
the dynamic electric and magnetic fields in a two component mixture with equations (E.11), (E.12) and (E.13). They derived them using the method of volume
averaging. In essence, and according to del Rio and Whitaker, when a mixture violates these constrains it cannot be represented by single effective electromagnetic
properties, and a Two-Equation Model must be used (Rio & Whitaker, 2000b).
Moreover, if the mixture can be represented by a One-Equation Model, then its
effective permittivity is given by the arithmetic mean, equation (E.3).
fh fi (εh − εi )2
1,
(fh εh + fi εi )(εh + εi )
fh fi (µh − µi )2
1,
(fh µh + fi µi )(µh + µi )
fh fi (σh − σi )(εh − εi )
1.
(fh σh + fi σi )(εh + εi )
(6.7)
(6.8)
(6.9)
The samples used in this modelled experiments do not obey the Local Electrodynamic Equilibrium. Since their condition for permittivity, equation (E.11), is in
this particular case 0.2, certainly not very small compared to 1, and yet a single
permittivity is a very good representation of the sample. The other two conditions
are met for non-magnetic (µh = µi = µ0 ) and low-loss materials (σh ≈ σi ≈ 0).
126
Chapter 6. Permittivity states of mixed-phase
In the case of thin layers evenly distributed in the sample holder the permittivity
is given by the arithmetic mean, as del Rio & Whitaker (2000a) claim, but for
high enough frequencies and a 2 cm wet layer located at different positions within
a water residual saturated layer it is given by the CRIM model, equation (E.19),
contradicting their conclusions.
From these modelled experiments we can conclude that the anomalies present in the
reconstructed permittivities of layered samples are dependent on the permittivity
contrast between the different layers, the distribution of these layers and the length
of the sample holder. However, this dependency is not trivial. It is important to
note that the distribution of the layers has a strong impact in the permittivity, and,
while at low frequencies they all tend to the arithmetic mean, at higher frequencies
it depends on the electrical length of the layers.
At this point, we are ready to interpret the reconstructed permittivities, of the two
valid saturation techniques, from an effective medium point of view.
6.6
Interpretation
Mixing Laws relate the effective permittivity of a mixture in terms of the permittivities of each constituent and their volume fraction. The literature on the
effective properties of materials and mixtures is very vast (Appendix E) and their
applications range from the design of new composites for electronic components
to the estimation of water content of porous materials (biological tissues, wood,
bedrock . . . ).
Estimating the fluid content of a rock core or soil sample from measured permittivities is yet a rather imprecise task. The permittivity depends on the soil structure,
texture, composition, porosity, fluid nature, content and distribution, temperature,
and the mixing models only consider the permittivity of the components of the
mixture and their volume fractions. Moreover, the most widely used Mixing Laws
have been developed under the consideration of static or quasi-static fields but are
applied in the high frequency range of the spectrum.
Different researchers have tried to find the best mixing law for different types of soils
and rocks. Topp (1980) found an empirical relation for frequencies between 1 MHz
and 1 GHz, and its has been extensively used. Shutko & Reutov (1982) found the
Complex Refractive Index Method to give a good approximation to the permittivity
of different soils in the range [1-10] GHz and Bruggeman-Hanai at 10 GHz as
a good prediction for soils with low moisture content. Hallikainen et al. (1985)
pointed out that although soil texture may not affect the permittivity of dry soils, it
6.6. Interpretation
127
certainly does so for wetted soils, specially at the lower end of the frequency range.
Dobson et al. (1985) found that both the Four Component Model and a Power
Law with a coefficient c = 0.65, are capable of describing the complex dielectric
permittivity above 1.45 GHz for different soils (sand, silt and clay). According to
Chelidze & Gueguen (1999b) the Bruggeman-Hanai model with L = 1/3 works
fine for the Megahertz region of the spectrum and different rocks. Finally, Seleznev
et al. (2004) studied the permittivity of partially saturated carbonate rocks, from
300 KHz to 3 GHz, finding that the best fit was that of the CRIM model.
In this section, we compare our results for the homogeneously mixed and gravity
drainage experiments with existing Mixing Laws. We also show that even if the Local Electrodynamic Equilibrium is not satisfied, again, a single effective permittivity
can represent the sample and not necessarily by the arithmetic mean.
Our samples have 2 components and two phases for steps 1 and 6, dry and fully
saturated sands, and 3 components and three phases in the intermediate saturations, quartz grains, air and water. For convenience, we have listed the parameters
used in this section.
ε∗q
ε∗a
ε∗w
φ
Sw
Sr
SW
6.6.1
permittivity of quartz
permittivity of air
permittivity of water
porosity of the sample
saturation of the sample
residual water saturation
full saturation
4.65 (Parkhomenko, 1967)
1
Debye model for water (Section E.2.2)
Tables 6.1 and 6.2
Tables 6.1 and 6.2
27%
100%
Dry sand
The dry samples were obtained in different ways and possess slightly different porosities and permittivities. Their frequency dependence is negligible and we can represent them with their mean. To model their permittivities we have used the relation
proposed by Dobson et al. (1985) and given by equation (E.22), and the reinterpretation of the Claussius-Mossotti model, equation (E.23).
According to Dobson et al. (1985) the permittivity of dry soils (εs ) can be computed
from their density (ρs ), given in [gr/cm3 ], via
εs = (1 + 0.44ρs )2 − 0.062.
(6.10)
128
Chapter 6. Permittivity states of mixed-phase
It is surprising to relate a unitless property to the square of a density, but these are
empirical fittings to experimental data on different dry soils. We think it is more
appropriate to compute the permittivity from the following reinterpretation of the
Clausius-Mossotti relation, equation (E.1), given in Section E.2.1, and rewritten
here for simplicity
ε=
1 + 23 N α
ε0 .
1 − 13 N α
(6.11)
Since this expression has been successfully used to compute the permittivity of
different gases from known polarizabilities, we wanted to test it for mixtures. For
non-interactive materials we can compute α of each material from its permittivity
and equation (6.11), and by superposition we can find the permittivity of the dry
sand mixture as
3 + 2[Nq αq + Na αa ]
,
(6.12)
ε∗cm =
3 − [Nq αq + Na αa ]
where the subscript a refers to the air (αa = 0), and the subscript q refers to
the quartz. According to Parkhomenko (1967) the permittivity of quartz over the
frequency range of interest is 4.65.
The permittivities computed with the different models are listed in Table 6.5
Table 6.5:
Relevant
data for two dry sands
samples, including reconstructed and modelled
permittivities.
φ
ρs
ε∗rec
ε∗Dobson
ε∗cm
ε∗mod
Homogeneously Mixed
Gravity Drainage
43 %
1.47 gr/cm3
2.49 - 0.02j
2.68
2.36
2.47 - 0.005j
40 %
1.60 gr/cm3
2.64 - 0.03j
2.88
2.47
2.57 - 0.005j
The empirical Dobson model overestimates by around 8% the reconstructed values
of permittivity for these two samples. Dobson’s equation is in fact a fit to different
dry soil types and dependent on experimental conditions. The other empirical
relation, by Shutko, equation (E.21) gives even higher predictions, and we have not
included it here.
The Claussius-Mossotti model, ε∗cm , underestimates the permittivity a 5%. However, if the ”dry” sand contained a very small quantity of water (Sw = 1%), double
the accuracy of the water saturation, the underestimation of the Claussius-Mossotti
model could be explained. Using the CRIM equation to model the dry sand with
such a small moisture content the result is ε∗mod , and not only is the error in the
estimation of the permittivity considerably smaller, but also the loss is present in
6.6. Interpretation
129
the result, although it is underestimated. For this particular case
ε∗mod = ε∗cm (1 − φSw ) + ε∗w φSw .
(6.13)
Overall, we consider the Claussius-Mossotti reinterpretation to give the most adequate estimate for the permittivity of dry sands. If these sands were not perfectly
dry, as it could have absorb moisture from the environment when filling the sample
holder in the case of the homogeneously mixed samples, or in case of the gravity
drainage the sample was dried with a vacuum pump during 24 hours and total
absence of moisture cannot be ensured, their permittivity is very well captured by
the Claussius-Mossotti reinterpretation together with the CRIM model, assuming a
very small quantity of water.
6.6.2
Fully saturated sand
The 6th step of the Homogeneously Mixed and Gravity Drainage experiments correspond to fully saturated samples. That is, they are 2 component mixtures: quartz
grains and water, occupying the total porosity of each sample.
In the modelling of the fully saturated sands we have tried the CRIM model,
(ε∗crim )0.5 = (ε∗q )0.5 (1 − φ) + (ε∗w )0.5 φ
(6.14)
the Power Law model with c = 0.65 proposed by Dobson et al. (1985),
(ε∗Dobson )0.65 = (ε∗q )0.65 (1 − φ) + (ε∗w )0.65 φ
(6.15)
the Maxwell-Garnett formula given by equation E.15, and the symmetric Bruggeman
formula, equation E.17.
The results on both samples were similar. The best fit is given by the CRIM model,
however, it overestimates the permittivities by 10%. All the others overestimate
or underestimate its value by even greater margins. A smaller exponent in the
Power Law model would result in a more accurate result. This can be observed in
Figures (6.25) to (6.28) in the next section.
In view of these results, in the next subsection we will compare the reconstructed
permittivities at different saturations with the Power Law model and different exponents.
130
Chapter 6. Permittivity states of mixed-phase
Local electrodynamic equilibrium
In Section 6.4 we showed how our modelled data did not obey the Local Electrodynamic Equilibrium proposed by del Rio & Whitaker (2000a) and nevertheless
could be represented by a single permittivity. Do these fully saturated sands also
contradict del Rio & Whitaker theory? They are two component and their volume
ratios result in a Local Electrodynamic Equilibrium of around 0.5, which is not very
small compared to 1, and therefore, this mixture does not obey the condition on
permittivity given by del Rio & Whitaker (2000a). Moreover, and in contradiction
to their theoretical results, the reconstruction of its permittivity gave a solid value
throughout the whole frequency range not equal to the arithmetic mean, which is
60% higher.
This experimental result together with the synthesized example of Section 6.4 suggest that the derived conditions by del Rio & Whitaker (2000a) for mixtures are
not confirmed by our numerical or experimental results. Our numerical results are
confirmed by the experimental results; one-equation models can be used up to
high frequencies for two component mixtures and the high frequency limit for the
effective relative permittivity is the CRIM model.
6.6.3
Partially saturated sands
For both experiments, the saturation varies from fully dry to fully saturated in 6
steps. We have modelled the results with the Power Law model for 3 components
(c3 ) and different exponents, c = [0.3, 0.4, 0.5, 0.6] (equation (6.16)). They read
(ε∗c3 )c = (ε∗q )c (1 − φ) + (ε∗q )c Sw φ + (ε∗a )c (1 − Sw )φ,
(6.16)
Note that the Power Law model changes name with changing exponent, c = 0.5
corresponds to the CRIM model, c = 0.65 to the coefficient found by Dobson et
al. (1985) and c = 1/3 to the Looyenga equation.
For the Gravity Drainage Experiment, we have also modelled the partially saturated
sands with the CRIM model for 2 components, taking as components the CRIM
model for each layer. Then, the permittivity of the two layer samples is modelled
as
0.5
(ε∗ )2layers
= (ε∗r )0.5 (H − h) + (ε∗W )0.5 h,
(6.17)
where ε∗W is the permittivity of the fully saturated layer, step 6 in Table 6.2, h is its
height (Table 6.4), ε∗r is the permittivity of the residual water saturated layer, step
2 in Table 6.2, and H is the total height of the sample (10 cm). The permittivities
6.6. Interpretation
131
of both layers have been also modelled with the CRIM model. For step 2 (the
residual water saturated sand) it is given by equation (6.16) taking Sw = 1, and for
step 6 (the fully saturated sand) Sw = 0.27. In this way, we introduce geometry in
the CRIM model for steps 3 to 5 of the Gravity Drainage experiment, as the two
layer model presented in Section 6.4.
Figures 6.25 to 6.28 show the real and imaginary parts of the reconstructed permittivity versus saturation, at 500 MHz, 1, 2 and 3 GHz for the homogenously mixed
experiment. They also include the Power Law model with different coefficients,
equation (6.16). For all frequencies, the best model compared to the data depends
0
00
on the saturation and on ε or ε .
00
0
The 3-component modelling of ε requires higher coefficients than ε . In both real
and imaginary parts of the permittivity, there is a clear jump around saturations
of 50%, where the best fitting model changes exponent, from a lower value to a
higher one.
0
ε below 50% saturation is very well captured by the Power Law model with a coefficient equal to 0.3. Step 4 of the Homogeneously Mixed Experiment (Sw = 47%)
is in a transition zone between the Power Models with coefficients 0.3 and 0.4. For
higher saturations, the model with exponent equal to 0.4 gives the best fit. This
00
behavior is also found in ε , but now, the jump is from a coefficient equal to 0.5 to
0.6. The real part of the permittivity is always better captured than its imaginary
counterpart. However, as frequency increases the imaginary part of permittivity is
better captured by the model.
Although it is not shown, when comparing the reconstructed permittivities of the
Gravity Drainage experiment with the different Power Law models, the same behavior as that of the Homogeneously Mixed samples is found. However, the fitting to
the permittivity of the Homogenously Mixed samples is better than to those subject
0
to Gravity Drainage. In the first case, the error of the model for ε is smaller than
1% for all partial saturations, except for the transition saturation that it is 3%. For
00
ε the maximum difference is of 0.1. The accuracy regarding the Gravity Drainage
samples is twice as big, although sufficiently small.
It is possible to obtain a better fit to the Gravity Drainage samples if the 2 layer
model (equation (6.17)) is chosen. Figures 6.29 to 6.32 show the real and imaginary
parts of the reconstructed permittivity versus saturation, at 500 MHz, 1, 2 and 3
GHz for the gravity drainage experiment and this model.
0
The error of the fit for ε is smaller than 1% and the fit to the imaginary part
results improves significantly. Note that with this model, the exponent used for
the real and imaginary parts is the same. From our results, we can coclude, that
the CRIM model including geometry, gives the best fit to the permittivities of the
Gravity Drainage Experiment samples.
132
Chapter 6. Permittivity states of mixed-phase
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
−3
0
1
GHz
ε0.5
rec
0.3
(εc3)
(εc3)0.4
(ε )0.5
c3
0.6
(εc3)
0.5
Sw
1
Figure 6.25: Reconstructed permittivities at 500 MHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with
coefficients 0.3, 0.4, 0.5, and 0.6.
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
1
−3
0
GHz
ε0.5
rec
(εc3)0.3
(ε )0.4
c3
(ε )0.5
c3
(ε )0.6
c3
0.5
Sw
1
Figure 6.26: Reconstructed permittivities at 1 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with
coefficients 0.3, 0.4, 0.5, and 0.6.
6.6. Interpretation
133
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
−3
0
1
ε0.5 GHz
rec
(ε )0.3
c3
(ε )0.4
c3
(ε )0.5
c3
0.6
(εc3)
0.5
Sw
1
Figure 6.27: Reconstructed permittivities at 2 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with
coefficients 0.3, 0.4, 0.5, and 0.6.
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
1
−3
0
GHz
ε0.5
rec
(εc3)0.3
(ε )0.4
c3
(ε )0.5
c3
(ε )0.6
c3
0.5
Sw
1
Figure 6.28: Reconstructed permittivities at 3 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with
coefficients 0.3, 0.4, 0.5, and 0.6.
134
Chapter 6. Permittivity states of mixed-phase
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
−2.5
ε0.5 GHz
rec
)0.5
(ε*
crim 2 layers
0
0
0.5
Sw
−3
0
1
0.5
Sw
1
Figure 6.29: Reconstructed permittivities at 500 MHz for the Gravity Drainage experiment
versus water saturation (Sw ) and the CRIM model of equation (6.17)
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
1
−3
0
ε1 GHz
rec
*
0.5
(εcrim)2 layers
0.5
Sw
1
Figure 6.30: Reconstructed permittivities at 1 GHz for the Gravity Drainage experiment
versus water saturation (Sw ) and the CRIM model of equation (6.17)
6.6. Interpretation
135
0
25
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
−3
0
1
ε2 GHz
rec
*
0.5
(εcrim)2 layers
0.5
Sw
1
Figure 6.31: Reconstructed permittivities at 2 GHz for the Gravity Drainage experiment
versus water saturation (Sw ) and the CRIM model of equation (6.17)
25
0
−0.5
20
−1
ε,
ε,,
15
−1.5
10
−2
5
0
0
−2.5
0.5
Sw
1
−3
0
ε3 GHz
rec
*
0.5
(εcrim)2 layers
0.5
Sw
1
Figure 6.32: Reconstructed permittivities at 3 GHz for the Gravity Drainage experiment
versus water saturation (Sw ) and the CRIM model of equation (6.17)
136
Chapter 6. Permittivity states of mixed-phase
Other researchers have found also the Power Law model to be the best model
describing the electric properties of different soils. Dobson et al. (Dobson et al. ,
1985) found a coefficient c = 0.65 for different soils and frequencies above 1.45
GHz, and Seleznev et al. (2004) found the CRIM model to give the the best fit to
the permittivity of partially homogeneously saturated carbonate rocks, in our same
frequency range.
6.7
Conclusions
In this chapter, we have studied the effect of fluid flow through a sand sample in
relation to its permittivity. Here we summarize the most relevant conclusions of our
study. Despite the simplicity of the experiments used in this study, and the lack
of thorough fluid flow study, the experiments properly illustrate the influence of
the fluid distribution in the permittivity. And although the assumptions about this
fluid distribution have been very simplistic, the reconstructed permittivities were
properly modelled with these assumptions.
We showed that the saturation technique has a clear effect on the reconstructed
permittivity and that extra care is needed for proper measurements. If the sample is
initially dry and then saturated with the flow parallel to the direction of propagation
of the TEM waves, the control on the saturation is very poor and there is a risk of
compaction of the grains, leading to a higher content of liquid. This extra saturation
together with the fact that we cannot model these types of samples, because they
have non homogeneous cross sections, made us rule out this saturation technique.
The other two techniques are superior. The liquid content is easily controlled and
accurately determined, and the reconstructed permittivities are consistent. Their
difference rests on the saturation distribution and sample properties. While the
technique that mixes the sand outside the sample holder with a certain amount
of liquid ensures a homogeneous distribution (HM), except for minor disturbances,
the method in which the liquid is let to drain with the aid of gravity (GD) results
in distinctive 2-layer samples, one with residual saturation and the other one fully
saturated. However, all the samples of the HM were different and possessed different porosities. With this methodology, it is very difficult to obtain samples of the
same porosity, plus it is very time consuming. Whereas the sample subject to GD
was always the same, with constant porosity, it is a considerably faster experiment
and it is easier to determine the permittivity at more saturation levels.
The reconstructed permittivities (ε∗12 and ε∗21 ) for perfectly homogenous materials
(air and ethanol) are equal within measurement accuracy, while for heterogenous
materials they are different. How different depends on the scale and distribution
of the heterogeneity and the permittivity contrast, the higher the contrast and the
scale, the higher the difference. Of course, there is also a frequency dependence.
6.7. Conclusions
137
The permittivities of the samples of the HM experiment, are equal within 2%
above 1 GHz, and therefore we can consider them homogeneous. The real parts
of permittivities of the GD experiment are also equal within 2% above 1 GHz, but
their imaginary parts difference is slightly higher. The frequencies below 1 Ghz are
in an interesting region, because that is the region where the anomalies due to
heterogeneities are dominant.
0
00
The ε of partially saturated sands does not depend strongly on frequency, and ε
shows an increase with increasing frequency. While the steps corresponding to the
dry, residual saturation and fully saturated sands of the GD experiment (steps 1,
2 and 6) have equivalent ε∗12 and ε∗21 , also below 1 GHz, the steps corresponding
to 2-layer samples (steps 3, 4 and 5) present anomalies in that frequency region.
These anomalies are present only in the samples in which a 2-layer assumption of
the saturation distribution is reasonable, and to a smaller extension in steps 4 and
5 of the HM experiment, but not in the rest of the samples with a homogeneous
distribution. Their further study in Section 6.5 corroborated the fact that these
anomalies are indeed result of heterogeneities in the sample.
I think of the similarity between ε∗12 and ε∗21 as a measure for heterogeneity in the
small scale, and the anomalies present in both of them as a measure for a larger scale
of heterogeneity. For partially saturated sands, the small scale heterogeneities have
a small impact on their permittivities (resemblance within 2%) and therefore can be
represented by a single parameter. Moreover, if the permittivities have anomalies,
due to layer heterogeneities, and they are substituted by a single ε∗ , they do not
disappear. We can, then, represent a sample with a single permittivity loosing only
redundant information on the small scale heterogeneities that we already know of,
since we deal with a 3 component random mixture. To do so, it is best to average
ε∗12 and ε∗21 per frequency, and not to average the S-parameters and then compute
a single permittivity (Section 6.3). Anyhow, we recommend the reconstruction of
both ε∗12 and ε∗21 in order to check these assumptions, and then, if the small scale
heterogeneities does not have a big impact, average them out and work with a
single parameter.
The sample subject to gravity drainage 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, equation (6.2). The error made if the width is computed from effective
saturation assumptions, equation (6.1), is considerable. In addition, in Section 6.4
we showed how the reconstructed permittivities from the 2-layer model (modelled
samples with layer widths computed from the phase of the transmission coefficients,
and permittivities of the two layers from the reconstructed permittivity of the fully
and residual water saturated samples) were equivalent to the measured permittivities of the real experiment. Nevertheless, the anomalies of the modelled samples
were more pronounced and extended for a broader frequency range. We think that
this is because the interface of the modelled samples represents a sudden change
138
Chapter 6. Permittivity states of mixed-phase
of properties, while, the change in the real samples is more gradual, leading to
smoother anomalies.
Being able to accurately determine the width of the fully saturated layer implies
that we can use the tool to monitor the movement of the fluid front, opening new
doors in the investigation of fluid flow in porous media by means of electromagnetic
measurements. Nguyen et al. (1996) published an equivalent method, at a bigger
scale, to find the capillary transition zone from ground penetrating radar data.
In Section 6.5 we studied the anomalies present in the reconstructed permittivities
of 2-layer samples, with modelled experiments. We showed how the anomalies
depend on the permittivity contrast between the different layers, the width of these
layers and the length of the sample holder. In addition, the distribution of the
layers has a strong impact on the permittivity, and, whereas at low frequencies all
sample permittivities tend to the arithmetic mean, no matter their distribution, at
higher frequencies it depends on the electrical length of the layers. For a single
thick fully saturated layer, the permittivity corresponds to that of the CRIM model
above 1.5 GHz, but if that layer is divided into sufficiently thin layers, the volume
fraction kept constant, and the layers are evenly distributed, then the permittivity
corresponds to the arithmetic mean in our frequency range.
There is, therefore, a transition frequency that depends on the width of the layers,
the volume fraction and the permittivities of the two components. For very thin
layers this frequency is very high but finite. The amount of parameters that influence
the anomalies (layer width, permittivity contrast and layer distribution) make it very
difficult to draw an absolute conclusion and more investigation is necessary.
These results can contribute to the improvement of direct and inverse modelling.
Most modern modelling problems are so complicated that they can be solved only
using numerical techniques. These can be either local or global techniques. In
both cases, the medium needs to be discretized and the optimal discretization of
a piecewise continuous medium is an unsolved problem for all wave problems. The
optimal average for the partial equations that describe these fields, is unknown. Different researches have applied different techniques. In seismics, Muir et al. (1992)
modelled elastic fields across irregular interfaces with the harmonic average, while
Moczo et al. (2002), used both harmonic and arithmetic averaging of the elastic
parameters in a finite difference modelling scheme. Our results suggest that the
arithmetic averaging should be used for low frequency fields and the CRIM model
for high frequencies.
The most important class of problems for modelling is, of course, when the strong
heterogeneities call for grid refinements, which usually introduce spurious scattering at the connection of the fine grid to the coarser grid (Falk, 1998). He reduced
the spurious scattering using harmonic averages of the shear modulus and arithmetic averages of the mass density, for velocities he used the harmonic average.
6.7. Conclusions
139
Examples of these heterogeneities are rough surfaces either at the Earth surface or
in the subsurface with strong electric parameter contrasts. Schneeberger (2003)
successfully accounted for small scale heterogeneities (due to a rough surface) in a
single effective parameter, using the Complex Refractive Index Model for modelling
surface emission in radiometry measurements.
For local modelling methods, apart from averaging the medium parameters across
boundaries of discontinuities in the medium parameters, the minimization of the
computational domain is of great importance to minimize computational cost, especially for three-dimensional modelling. Since the advent of very efficient boundary
conditions, known as Perfectly Matched Layers (PML’s, Berenger (1994)), many
other attempts have been reported to optimize these PML’s and the optimal type
of layers would be the one represented by a sequence that allows transmission and
is reflection free (Hoop et al. , 2002).The study reported in this thesis can increase
our understanding of how to construct such PML’s.
In the interpretation of the reconstructed permittivities for the HM and GD experiments in terms of Mixing Laws (Section 6.6), we saw that a slight difference
in porosity has an effect in the value of the permittivity of dry sands (higher for
lower porosities). And that the empirical relations between the permittivities and
the sands densities, given by Dobson et al. (1985) and Shutko & Reutov (1982),
overestimate their value while the reinterpretation of the Clausius-Mossotti model
(equation (E.23)) underestimates it. While, a very small quantity of water could
account for this underestimation no logical explanation could be found for the overestimation of the empirical models, besides their self-empirical nature. Due to this,
we find that the dry sands should be modelled with the reinterpretation of the
Clausius-Mossotti model in opposition to the empirical models.
The best mixing formula for partially and fully homogenously saturated sands
(HM experiment and steps 2 and 6 of the GD experiment) has proved to be a Power
Law equation. Different exponents are needed depending on the saturation level
0
and if we are interested in the real or imaginary parts of the permittivity. The ε of
these samples, is in very good agreement (error of 1%) with the Power Law model
[c = 0.3 for Sw < 50% and c = 0.4 for Sw > 50%]. This is not in contradiction
with other researchers results, where variety is dominant (see Section E.5). In fact,
the fit of the Power Law model to the reconstructed permittivities is better than
most published data, but our samples were equal in composition. A broader range
of sands would certainly lead to lower accuracies, since the dispersion of coefficients
will be bigger.
For the 2-layer samples (steps 3 to 5 of the GD experiment) the error made by
the Power Law model is larger. However, we could still model them in that manner if we allowed for a lower precision (error of 2%). In that sense, the partially
saturated sands, whose saturation is clearly distributed in 2-layers, could be considered homogenous. The validity of this approximation decreases with increasing
140
Chapter 6. Permittivity states of mixed-phase
permittivity contrast. Instead, we can use the CRIM model to model the 2 layer
samples (equation 6.17) and improve considerably the agreement between the measurements and the model results. This approach includes geometry in its modelling
since first the permittivities of each layer is computed with the CRIM model and
then the permittivity of the combined 2 layer sample is calculated with the same
model and the volume fractions are given by the heights of the layers.
Del Rio & Whitaker (2000a) derived constrain conditions for the validity of using
effective parameters (one-equation models) for the dynamic electromagnetic properties of mixtures. They define the Local Electrodynamic Equilibrium, given by
equations (E.11), (E.12) and (E.13), and they claim that if the constrains are met
the effective permittivity will be that corresponding to the arithmetic mean, which
is true in the low frequency approximation. In this chapter, we have given two
examples that contradict their statements: the modelled samples of Section 6.5
(fSW = 0.2 and fSr = 0.8) and the real fully saturated samples, (fSW = 0.4 and
fSr = 0.6).
Both samples do not meet the Local Electrodynamic Equilibrium constraint and
yet, can be represented by an effective permittivity. In the case of the modelled
samples, this depends on the frequency and the distribution of the fully saturated
layer. Moreover, the saturation distribution affects the value of this ε∗ , when the wet
layers are thin enough and evenly distributed, the effective permittivity corresponds
to the arithmetic mean, but if it is a single thick layer, then its value is given by the
CRIM model for high enough frequencies, whereas, if the sample is fully saturated,
then it is better modelled by the Power Law 0.4 . These experimental and modelled
results contradict the derived conditions by del Rio & Whitaker (2000a) for the
validity of using effective parameters (one-equation models) for the electromagnetic
properties of mixtures.
Along this thesis we have supported our choice of tool and reconstruction method,
and throughout this chapter we have presented the characteristics of different saturation techniques and their corresponding permittivities. It is now time to explain
the best experimental methodology according to our studies.
6.7.1
Recommended experimental methodology
In the determination of the effective permittivity of dry, partially or fully saturated
soils we recommend the use of a Gravity Drainage experiment. The experimental
advantages of the GD experiment over the HM mixed are evident:
• The same sample can be measured at many saturation degrees.
6.7. Conclusions
141
• It allows for two phase flow, and although in this thesis we have presented
results with air and water as the two flowing phases, in principle, they could
be any other liquids or gases (limitations given by fluid viscosity and soil
permeability)
• Extra information about the movement of the effective fluid front along the
sample can be easily found very accurately.
• Depending on the permittivity contrast between the 2-layers and the desired
accuracy, these samples can be modelled as homogenous and the results are
analogous to true homogeneous samples. So that, they can be used in the
study of the effective properties of soils at different saturations.
• A simple 2-layer model together with the CRIM model account extremely
well for the permittivity of these samples.
A Gravity Drainage experiment is, in principle, always possible but it is only appropriate when the permittivity contrast is small so that the loss of accuracy when
treating the sample as homogenous is not large. If the sample cannot be considered
to have an effective permittivity, then, this experiment is suitable for the study of
2-layer samples, alone. In the study of effective properties of samples with large
permittivity contrasts, the samples must be saturated outside the sample holder.
In view of our experience, the sand(SW )/sand(Sr ) contrast (21/5) is in the safe
area while a 2 layer sample consisting of one layer of water and another of air
is not (80/1). Yet, most sands have a small contrast between the fully and the
residually saturated soils and we expect that this technique can be applied for most
cases leading to a better understanding of the relations between fluid saturation
and permittivity.
142
Chapter 6. Permittivity states of mixed-phase
We never stop investigating. We are never satisfied that we
know enough to get by. Every question we answer leads on to
another question. This has become the greatest survival trick
of our species.
Desmond Morris (1928- )
Chapter 7
Conclusions and
recommendations
In this thesis we have shown the successful design of an accurate tool to investigate
the electrodynamic response of soils. We have also shown that their response is
very much dependent on the distribution of components of the sands, and that
heterogeneities as small as λ/100 can be detected. This challenges existing mixing
laws, that only consider the components volume fractions and permittivities, and
when EM fields can be considered quasi-static. We have also proposed a new
method to treat the interfaces present in the soils. This will enhance existing
models of the subsurface.
The tool 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. This design is most suited to measure loose sand samples.
The same tool can be modified to measure solid cores. It would be appropriate to
expand this study to lower frequencies and use the same tool as a capacitor.
We also introduced a new combined representation of the measurements, the Propagation Matrices Representation. It allowed us to prove the independence of the
reflection and transmission coefficients, as well as, to develop a novel analytical
inversion of the measurements. Moreover, it provides a representation in which the
sections interact by matrix multiplication making its mathematical treatment and
physical understanding simpler than existing methods.
The forward model representing the reflection and transmission along the line is in
very good agreement with the measured data after a profound calibration. We have
shown that relative changes in the permittivity in the order of 1% can be detected
over a wide frequency band up to 3 GHz, while the lowest usable frequency depends
on the permittivity of the material filling the sample holder.
143
144
Chapter 7. Conclusions and recommendations
We compared the different inversion techniques to obtain the complex permittivity
(ε∗ ) from the measured S-parameters. All techniques determine ε∗ per frequency
point without a pre-defined model of the frequency dependency of permittivity. The
Compensation techniques have several disadvantages. They can be applied only in
a narrow frequency band. They are very time consuming since they require many
measurements, which increases the experimental errors. On the other hand, the
optimization method gives very good results. However, an appropriate weighting
is needed, which is unknown a-priori. We also presented our new method, the
propagation matrices, it allows to find explicit expressions to compute both complex
electric permittivity and magnetic permeability. It is based on a simple concept and
it is easily programmed.
The analytical expressions found from the propagation matrices approach, depend
on two different combinations of the corrected S-parameters. The one related to
the sample’s impedance is unstable and since we are interested in non-magnetic
sands, we restrict ourselves to the reconstruction of permittivity from the other
method, which is related to the propagation factor of the sample. We successfully
reconstructed the relative permittivity of air within ±1% error. And we could fit
a Debye model to the reconstructed permittivity of ethanol, within the same error
bound.
The formulation of the measurements with the propagation matrices method allowed us to prove that all existing analytical expressions are, in fact, the same. 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.
Our tool, a customized coaxial transmission line together with the propagation
matrices inversion, is able to reconstruct permittivities within ±1% error from 3λ/L
up to 3 GHz, while the lower limit can be relaxed by using optimization. However,
for lossy materials, this upper limit is conditioned by its loss. As the amplitude
of the transmission gets smaller, the accuracy of the phase of the measurements
deteriorates taking its toll on the reconstructed permittivity that clearly presents
noisy behavior. Nevertheless, it stills captures its trend, and it can be amended with
noise reduction techniques. To improve the knowledge over the measurements it is
desirable to include error propagation in the reconstruction of permittivity.
We also studied the effect of fluid flow through a sand sample in relation to its
permittivity. We showed how the saturation technique has a clear effect in the
reconstructed permittivity and that extra care is needed for proper measurements.
Depending on the goal of the measurement it is best to homogeneously saturate
the sample outside the sample holder or to perform a gravity drainage to change the
saturation. The second technique has the advantage that the same sample can be
measured at different saturation levels. However, the saturation distribution results
in distinctive 2-layer samples, whose reconstructed permittivity exhibit anomalies
145
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.
Since we are able to accurately determine the width of the layers present in a sample
subject to gravity drainage, we can use the tool to monitor the movement of the
fluid front, opening new doors in the investigation of fluid flow in porous media by
means of electromagnetic measurements.
With the aid of synthetic experiments, we showed how the anomalies depend on the
permittivity contrast between the different layers, the distribution of these layers
and the length of the sample holder. In addition to this, the distribution of the
layers has a strong impact on the permittivity. The amount of parameters that
influence the anomalies (layer width, permittivity contrast and layer distribution)
make it very difficult to draw a quantitative conclusion and more investigation is
necessary. This should be done together with an analytical study of layered systems,
which, thanks to the propagation matrices representation can be easily done.
When the size of the heterogeneities is smaller that λ/100 (quasi-static field approximation is valid), the permittivity of all samples composed of two effective materials
corresponds to the arithmetic mean, while for higher frequencies the distribution of
the materials has a direct effect on the value of the permittivity. In case they form
a 2-layer sample, the permittivity of the sample will be given by the harmonic mean
of the permittivities (velocities) of the components. As mentioned in Section 6.7
these results can contribute to the improvement of direct and inverse modelling
and increase the understanding of how to construct PML’s.
We found that dry sands should be modelled with the reinterpretation of the
Clausius-Mossotti model, in opposition to the empirical models. Their real permittivity is constant over the frequency range as well as their imaginary part that
is very small. The real part of the permittivity of partially saturated sands does
not depend strongly on frequency, while its imaginary part shows an increase with
increasing frequency.
The best mixing formula for partially and fully homogenously saturated sands has
proved to be a Power Law equation. Different exponents are needed depending on
the saturation level and whether we are interested in the real or imaginary parts of
the permittivity. We think that our results on multilayered samples can explain the
variety of exponents encountered in this thesis and in the existing literature. We
saw that the permittivity of a layered sample was given by the arithmetic mean for
frequencies smaller than a transition frequency, and above it, it oscillated around
that of the CRIM model.
The Power Law is also able to model the permittivity of 2-layer samples, but the
error it makes is larger than when it is used for modelling homogenous samples.
If a lower precision is allowed (error of 2%), the partially saturated sands, whose
146
Chapter 7. Conclusions and recommendations
saturation is clearly distributed in 2-layers, can be considered homogenous. The
validity of this approximation decreases with increasing permittivity contrast. If we
are interested in their heterogeneity, geometry can be included in the modelling
using the CRIM model. The permittivities of both layers are also modelled with
the CRIM model, and their volume fractions are determined by the height of the
layers. It gives the best fit to the permittivities of the Gravity Drainage Experiment
samples.
We presented numeric and experimental examples that contradict the theory derived
by del Rio & Whitaker (2000a). Our experiments did not meet the constraints
for the Local Electrodynamic Equilibrium, and yet the samples were adequately
represented by an effective permittivity. Furthermore, εef f is not necessarily the
arithmetic mean of the permittivities of the components, as stated by del Rio &
Whitaker.
Based on our results we can conclude that sands can be represented by a single
value of permittivity. However, the effective parameter depends very strongly on the
constituents, volume fractions and distribution within the sand, and these must be
taken into account. It should also be considered that the permittivity of sands can
vary with frequency, not because of the frequency dependence of the permittivities
of its constituents, but because of their geometrical distribution.
Appendix A
General solution for the wave
equation in polar coordinates
This appendix presents the general solution for the Maxwell wave equations in a
cylindrical confined region, the symmetry of this structure recommends the use of
polar coordinates.
A.1
Polar coordinates
In polar coordinates a point P in space is represented by three coordinates r, φ,
and z, according to Fig. (A.1)
x1 = r cos φ,
(A.1)
x2 = r sin φ,
(A.2)
x3 = z,
(A.3)
and the three mutually perpendicular base vectors being {r̂, φ̂, ẑ} of unit length
each. A vector G = {Gr , Gφ , Gz } can be written as
G = Gr r̂ + Gφ φ̂ + Gz ẑ.
147
(A.4)
148
Appendix A.
General solution for the wave equation in polar coordinates
Figure A.1: The polar coordinate system
Expressing the curl (∇×) and divergence (∇·) of a vector G and the Laplacian of
a scalar ψ in polar coordinates will help us later on.
1
1
∂r (rGr ) + ∂φ Gφ + ∂z Gz ,
r
r 1
∇×G =
∂φ Gz − ∂z Gφ r̂ + [∂z Gφ − ∂r Gz ] φ̂
r
1
+ [∂r (rGφ ) − ∂φ Gr ] ẑ,
r
1
1
∇2 ψ =
∂r (r∂r ψ) + 2 ∂φ ∂φ ψ + ∂z ∂z ψ.
r
r
∇·G =
A.2
(A.5)
(A.6)
(A.7)
General solution for the wave equation
The source free wave equations for the electric and the magnetic field, equations (2.69) and (2.70), have the same shape and may be written in terms of
Ĝ, standing either for the electric field, Ê, or the magnetic field, Ĥ.
∇2 Ĝ − γ̂ 2 Ĝ = 0,
(A.8)
it can be solved for a general component F̂i , where the subscript i refers to the
r-radial, φ-angular or z-component, by separation of variables.
Let Ĝi be written as the product of three independent functions, each depending
only on one variable, i.e. R(r) for the radial, Φ(φ) for the angular and Z(z) for
the z-component.
Ĝi (r, φ, z) = R(r)Φ(φ)Z(z).
(A.9)
A.2. General solution for the wave equation
149
We solve the homogeneous equation introducing this expression into equation (A.8)
and doing the appropriate separation (the sign of the separation constants is chosen
so that the solution has physical meaning), lead to the following separate differential
equations and solutions, for each variable
∂z ∂z Z − γ̂z2 Z = 0 −→ Z = c1 exp(γ̂z z) + c2 exp(−γ̂z z), (A.10)
∂φ ∂φ Φ + ν 2 Φ = 0 −→ Φ = c3 exp(jνφ) + c4 exp(−jνφ), (A.11)
r2 ∂r ∂r R + r∂r R + (k̂c2 r2 − ν 2 )R = 0 −→ R = c5 Jν (k̂c r) + c6 Nν (k̂c r), (A.12)
where
γˆz and ν 2 are the separation constants,
k̂c2 = γˆz 2 − γ̂ 2 ,
Jν is the Bessel function of the first kind of order ν,
Nν is the Bessel function of the second kind of order ν.
The separation constants and the coefficients of the linear combination of solutions
can be determined imposing initial and boundary conditions. Properties concerning
the Bessel functions can be found in (Abramowitz & Stegun, 1972).
For k̂c2 = 0 the wave will propagate with the propagation constant of the medium
γ̂ (i.e. γˆz 2 = γ̂ 2 ). But for this particular value of k̂c the differential equation on
r (A.12) is degenerate, so that the solution given is not valid and it has to be solved
separately. This case gives rise to the TEM mode and will be solved explicitly in
the Subsection B.2.1.
For imaginary values of γ̂, Z(z) is the sum of two waves, one travelling in the
positive z-direction, exp(−γˆz z), and the other one in the negative z-direction,
exp(γˆz z). At time t = 0, when the source is switched, only one wave travelling
in the positive z-direction is present, but when it encounters a discontinuity it will
then be reflected, therefore, there will be waves propagating in both z-directions.
The coefficients c1 and c2 depend, then, on the amplitude of the exciting source
and the reflections that take place within the guide. For the general solution we
will take Z(z) = exp(γˆz z), where γˆz is determined from
γˆz 2 = k̂c2 + γ̂ 2 ,
(A.13)
and k̂c will come from imposing boundary conditions to the radial solution, and γ̂
depends on the medium properties and is given by equation (2.61).
Imposing the continuity of the field and its derivate with respect to the angular
coordinate
Φ(0) − Φ(2π) = 0
and
(∂φ Φ)(0) − (∂φ Φ)(2π) = 0
150
Appendix A.
General solution for the wave equation in polar coordinates
it is found that the separation constant ν must be an integer, and taking the
convention of the right-hand (i.e. clockwise rotation implies positive z-direction
propagation) we will take positive integers
ν = m = 0, 1, 2, . . .
The boundary condition to be imposed on the radial component of the solution is
that the tangential component of the electric field is zero along a perfect conducting
wall. This depends on the geometry and characteristics of the guide, so we will
leave them for later.
The solutions (for the non-degenerate case) can then be written as the product of
Z(z), Φ(φ) and R(r)
ĝim (r, φ, z) = Bm [c1 Jm (k̂c r) + c2 Nm (k̂c r)] exp(jmφ) exp(−γˆz z),
(A.14)
where Bm are the coefficients of the expansion that can be determined from orthogonality as
Z Z
[ĝim ĝi∗m ]dS = 1,
(A.15)
with the integration expanded over the entire cross section of the waveguide.
The total field Ĝi is then the sum over m of ĝim
Ĝi (r, φ, z) =
∞
X
ĝim (r, φ, z)
m=0
where ĝim is given by the equation (A.14).
To find the specific solutions of propagating waves along any cylindrical structure
we have to impose boundary conditions to Maxwell’s equations, but now written in
polar coordinates.
Appendix B
Waveguides
The solution of Maxwell equations for a coaxial waveguide can be found in this
Appendix, together with cut-off frequencies for the TE and TM modes
Electromagnetic waves propagate along waveguides in specific patterns. To find
these, Maxwell’s equations have to be solved, subject to the proper boundary conditions, and in a convenient coordinate system. In this appendix we solved them
for the specific case of a coaxial waveguide. We also propose a variation of the
tool, presented in this thesis, for the permittivity determination of solid rock cores,
see Section B.3.
B.1
Maxwell’s equations in polar coordinates
Maxwell’s equations (2.55) and (2.56) can be written splitted into their polar components, see appendix A.
1
− ∂φ Ĥz − ∂z Ĥφ + η̂ Êr = −Jˆre ,
r
i
h
− ∂z Ĥr − ∂r Ĥz + η̂ Êφ = −Jˆφe ,
i
1h
− ∂r (rĤφ ) − ∂φ Ĥr , + η̂ Êz = −Jˆze
r
151
(B.1)
(B.2)
(B.3)
152
Appendix B.
1
∂φ Êz − ∂z Êφ + ζ̂Hr = −K̂re ,
r
i
h
∂z Êr − ∂r Êz + ζ̂ Ĥφ = −K̂φe ,
i
1h
∂r (rÊφ ) − ∂φ Êr . + ζ̂ Ĥz = −K̂ze
r
Waveguides
(B.4)
(B.5)
(B.6)
We will now consider the case of TM and TE modes in each configuration (circular
and coaxial waveguides). On each mode the z-components of either the magnetic
(for TM mode) or the electric (for TE mode) field is zero and all the other components can be written in terms of the non-zero z-component field. The TEM mode
will be derived in the Coaxial Waveguide section, as it can only propagate in that
configuration of the two studied here.
As the only coefficients and separation constant to be determined are those of the
radial component. We can simplify the problem writing the fields as
Ê(r, φ, z) = ê(r) exp(jmφ) exp(−γˆz z),
(B.7)
Ĥ(r, φ, z) = ĥ(r) exp(jmφ) exp(−γˆz z),
(B.8)
where for each mode we have
TM
TE
êT M = {êr , êφ , êz }
ĥT M = {ĥr , ĥφ , 0}
êT E = {êr , êφ , 0}
ĥT E = {ĥr , ĥφ , ĥz }
where the z-components of each mode correspond to a Fourier expansion of the
form
ζ̂(r) =
∞
X
ĝzm (r)
with
ĝzm (r) = Bm [c1 Jm (k̂c r) + c2 Nm (k̂c r)], (B.9)
m=0
again the coefficients of the expansion Bm can be determined from orthogonality,
equation (A.15).
Introducing these expressions for the electric and the magnetic fields into Maxwell’s
equations (B.1)-(B.6) new systems of equations are obtained for the TE and TM
modes.
B.1. Maxwell’s equations in polar coordinates
B.1.1
153
Transverse magnetic: TM
Taking into account that the longitudinal component of the magnetic field ĥz = 0,
we will write the differential equation for the longitudinal component of the electric
field êz and all the other electromagnetic field components can be related to it. An
electric source current in the z-direction, Jˆze , will give rise to such a field. However
the equations are presented and solved in a source-free domain for simplicity. The
system of equations that give rise to this mode reads
1
k̂ 2
1
∂r ∂r êz + ∂r êz + ∂φ ∂φ êz + k̂c2 êz = − c Jˆze ,
r
r
η̂
êr =−
ĥφ =−
η̂
k̂c2
γˆz
k̂c2
∂r êz ,
∂r êz ,
η̂ jm
∂φ êz ,
k̂c2 r
γˆz jm
∂r êz .
ĥr = −
k̂c2 r
êφ =
(B.10)
(B.11)
(B.12)
The electromagnetic field components of the transverse magnetic mode will be
determined imposing the appropriate boundary conditions to equation (B.9).
B.1.2
Transverse electric: TE
For this mode the longitudinal component of the electric field êz = 0 and this time
a magnetic source current in the z-direction, K̂ze , gives rise to this mode. Similarly
to the previous case
1
1
k̂ 2
∂r ∂r ĥz + ∂r ĥz + ∂φ ∂φ ĥz + k̂c2 ĥz = − c K̂ze ,
r
r
ζ̂
êφ =
ĥr =−
ζ̂
k̂c2
γˆz
k̂c2
∂r ĥz ,
∂r ĥz ,
ζ̂ jm
∂φ ĥz ,
k̂c2 r
γˆz jm
∂φ ĥz .
ĥφ = −
k̂c2 r
êr = −
(B.13)
(B.14)
(B.15)
The solution to equation (B.13) is given for a general case by equation (B.9) and
imposing the appropriate boundary conditions, the electromagnetic field components of the transverse electric mode will be determined.
154
Appendix B.
Waveguides
Figure B.1: Electric and
Magnetic fields for a TEM
wave propagating along a
coaxial waveguide.
B.2
Coaxial Waveguide
Along this type of configuration (see Fig. (B.1)) the principal mode TEM, together
with the TM and TE, can propagate. Let us first consider the TEM mode
B.2.1
Transverse electromagnetic: TEM
This mode has only two non-zero components, Êr and Ĥφ , and it will propagate
when the waveguide is exited with a longitudinal source volume density of electric
current, Jˆze . We have already seen that for k̂c2 = 0 the differential equation on
r (A.12) is degenerate, so that it has to be solved separately. However the other
differential equation on z (A.10) is still valid and γˆz 2 = γ̂ 2 . So that the fields can
be written as
Êr (r, φ, z) = êr (r, φ) exp(−γ̂z),
(B.16)
Ĥφ (r, φ, z) = ĥφ (r, φ) exp(−γ̂z).
(B.17)
Maxwell’s equations, in this particular case, read
−γ̂ ĥφ + η̂êr = 0,
1
+ ∂r Ĥφ = Jˆze ,
r
−γ̂êr + ζ̂ ĥφ = 0,
∂φ Êr = 0.
(B.18)
(B.19)
(B.20)
(B.21)
B.2. Coaxial Waveguide
155
Equations (B.19) and (B.21) show the non dependence of the fields with the φcoordinate, and then
êr = êr (r)
and
ĥφ = ĥφ (r),
and from equation (B.19) and taking into consideration that the current Iˆ is related
to Jˆe according to
Iˆ =
Z
e
Jˆ · dl,
ĥφ can be expressed in terms of Iˆ as
ĥφ (r) =
Iˆ
,
2πr
and êr can be derived from equation (B.18). Finally the expression for the electric
and magnetic fields that compose the TEM mode propagating along a coaxial
waveguide read
Êr
" #1/2
Iˆ
ζ̂
exp(−γˆz z),
=
η̂
2πr
Ĥφ =
Iˆ
exp(−γˆz z),
2πr
(B.22)
(B.23)
h i1/2
Iˆ
where we recognize η̂ζ̂
2π as the initial source potential V0 , which is a customary
notation. And now the fields can be written as
V̂0
exp(−γ̂z),
r
1/2
η̂
V̂0
=
exp(−γ̂z).
r
ζ̂
Êr =
(B.24)
Ĥφ
(B.25)
So that the electric and magnetic fields are perpendicular to each other and to the
direction of propagation z, see Fig. (B.1). Their propagation factor is γ̂, that can be
complex (lossy materials) and therefore the mode can be attenuated and dispersed,
but they have no cutoff frequency as they can propagate for any frequency.
For a perfect mode (no longitudinal component) to exist the walls of the waveguide
should be perfect conductors, but in reality this is not the case and the metals have
156
Appendix B.
Waveguides
a high, but finite, conductivity, so that the electric field has a slight longitudinal
component (êm
z 6= 0), and the signal is slightly attenuated as it propagates.
B.2.2
TM
For this mode the non-zero longitudinal component of the electric field is
êzmn (r, φ) = [c1 Jm (k̂cn r) + c2 Nm (k̂cn r)] exp(jmφ).
(B.26)
The second kind Bessel function Nm goes to infinity when r = 0, but in a coaxial
waveguide it is within the inner conductor, so that there are no problems of finiteness
and Nn is an admissible solution (c2 6= 0). Since êzmn must vanish at r = a and
r = b,
−
Nm (k̂cn b)
Nm (k̂cn a)
c1
=
,
=
c2
Jm (k̂cn a)
Jm (k̂cn b)
(B.27)
where k̂cn is given from the non-vanishing roots of
Nm (k̂cn a)Jm (k̂cn b) − Nm (k̂cn b)Jm (k̂cn a) = 0.
(B.28)
Making use of the properties of the Bessel functions of first and second kind (see
Abramowitz & Stegun (1972)) and assuming asymptotic behavior, the approximate
solution for k̂cn is found to be
k̂mn =
πn
a−b
with n = 1, 2, . . .
As the system of equations (B.27) is undetermined we can choose c1 and c2 to be
c1 = Nm (k̂cn a)
and
c2 = −Jm (k̂cn a).
(B.29)
We can now write the expressions for the fields that form this mode in terms of a
0
function ψmn (r) and its differential ψmn
(r)
ψmn (r) = Nm (k̂cn a)Jm (k̂cn r) − Jm (k̂cn a)Nm (k̂cn r),
0
ψmn
(r)
=
0
Nm (k̂cn a)Jm
(k̂cn r)
−
0
Jm (k̂cn a)Nm
(k̂cn r),
(B.30)
(B.31)
where
0
Nm
(k̂cn r) =
"
0
Jm
(k̂cn r) =
"
m
k̂cn r
m
k̂cn r
#
Nm (k̂cn r) − Nm+1 (k̂cn r) ,
#
Jm (k̂cn r) − Jm+1 (k̂cn r) ,
(B.32)
(B.33)
B.2. Coaxial Waveguide
157
so that the field components of this mode read
êzmn =
ψmn (r) exp(jmφ)
Êz =
∞ X
∞
X
TM
Xmn
êzmn exp (−γ̂zn z),
∞
∞ X
X
TM
Xmn
êrmn exp (−γ̂zn z),
∞ X
∞
X
TM
Xmn
êφmn exp (−γ̂zn z),
∞ X
∞
X
TM
Xmn
ĥrmn exp (−γ̂zn z),
∞ X
∞
X
TM
Xmn
ĥφmn exp (−γ̂zn z),
m=0 n=1
êrmn =
êφmn =−
−
γ̂zn
k̂cn
0
ψmn
(r) exp(jmφ)
Êr =
m=0 n=1
jm γ̂zn
ψmn (r) exp(jmφ)
r k̂c2
Êφ =
m=0 n=1
n
ĥrmn =
ĥφmn =
jm η̂
ψmn (r) exp(jmφ)
r k̂cn
−
η̂
k̂cn
0
ψmn
(r) exp(jmφ)
Ĥr =
m=0 n=1
Ĥφ =
m=0 n=1
(B.34)
(B.35)
(B.36)
(B.37)
(B.38)
TM
where the coefficients of the expansion Xmn
are determined from equation (A.15)
and read
TM
Xmn
=√
1
1
(m)
π k̂c χm
where
(m)
χm
= a2 [Jm (k̂cn a)Nm+1 (k̂cn a) − Nm (k̂cn a)Jm+1 (k̂cn a)]2
− b2 [Jm (k̂cn a)Nm+1 (k̂cn b) − Nm (k̂cn a)Jm+1 (k̂cn b)]2 .
(B.39)
(B.40)
Propagation Properties
From equation (A.13) we can see that the propagation of this mode depends on the
material it propagates through and the frequency at which it does. If γˆz is purely
imaginary there will be no attenuation, but in case it has a real part (αz ) it will
get attenuated in the z-direction by the factor exp(−αz z), the frequency at which
this change occurs is known as cutoff frequency, because below this frequency the
mode is evanescent. We need to distinguish between media with and without loss.
158
Appendix B.
Waveguides
• Lossless medium: the propagation factor becomes a real number
γ̂ 2 = −w2 εµ = −w2 /c2 and a cutoff frequency wn can be found for each
n-th root
k̂cn =
πn
w2
πn
= (γ̂zn )2 + 2 → wcn =
c.
a−b
c
a−b
(B.41)
• Medium with losses: the propagation factor γ̂ has already a real and an
imaginary part, so that γˆz also has a real part, no matter what value k̂c
takes. For these media the modes behave as attenuated waves for all
frequencies.
B.2.3
TE
For this mode the non-zero longitudinal component of the magnetic field is
ĥzmn (r, φ) = [c1 Jm (k̂cn r) + c2 Nm (k̂cn r)] exp(jmφ).
(B.42)
Since êφmn must vanish at r = a and r = b, and êφmn ∝ ∂r ĥzmn we have
−
c1
N 0 (k̂c b)
N 0 (k̂c a)
= m n = m n ,
0 (k̂ a)
0 (k̂ b)
c2
Jm
Jm
cn
cn
(B.43)
where k̂cn is given from the non-vanishing roots of
0
0
0
0
Nm
(k̂cn a)Jm
(k̂cn b) − Nm
(k̂cn b)Jm
(k̂cn a) = 0.
(B.44)
The approximate solution for k̂cn is found to be
k̂c,m1 =
k̂c,mn =
2
for n = 1,
a+b
π(n − 1)
for n = 2, 3, . . .
a−b
Now c1 and c2 are chosen
0
c1 = N m
(k̂cn a)
and
0
c2 = −Jm
(k̂cn a)
(B.45)
So that the fields that form this mode are written in terms of the function ϕmn (r)
and its differential ϕ0mn (r)
0
0
ϕmn (r) = k̂cn [Nm
(k̂cn a)Jm (k̂cn r) − Jm
(k̂cn a)Nm (k̂cn r)]
ϕ0mn (r)
=
0
k̂c2n [Nm (k̂cn a)Jm
(k̂cn r)
−
0
Jm (k̂cn a)Nm
(k̂cn r)]
(B.46)
(B.47)
B.2. Coaxial Waveguide
159
0 (k̂ a) and J 0 (k̂ a) are equal to equations (B.32) and (B.33) when
where Nm
cn
m cn
r = a.
so that the field components of this mode read
ĥzmn =
ϕmn (r) exp(jmφ)
Ĥz =
∞ X
∞
X
TE
Xmn
ĥzmn exp (−γ̂zn z),
∞ X
∞
X
TE
ĥrmn exp (−γ̂zn z),
Xmn
∞ X
∞
X
TE
Xmn
ĥφmn exp (−γ̂zn z),
∞ X
∞
X
TE
Xmn
êrmn exp (−γ̂zn z),
∞ X
∞
X
TE
Xmn
êφmn exp (−γ̂zn z),
m=0 n=1
ĥrmn =
ĥφmn =−
−
ζ̂ 2
γ̂zn k̂c2n
ϕ0mn (r) exp(jmφ)
Ĥr =
m=0 n=1
ζ̂ 2
jm
ϕmn (r) exp(jmφ)
r γ̂zn k̂c2
Ĥφ =
m=0 n=1
n
êrmn =
−
jm ζ̂ 2
ϕmn (r) exp(jmφ)
r k̂c2
m=0 n=1
n
ζ̂ 2
êφmn =
Êr =
k̂c2n
ϕ0mn (r) exp(jmφ)
Êφ =
m=0 n=1
(B.48)
(B.49)
(B.50)
(B.51)
(B.52)
TE
where the coefficients of the expansion Xmn
are determined from equation (A.15)
and read
1 1
TE
Xmn
= √ (e)
π χm
(e)
χm
=
a2 −
−
b2 −
m2
!
k̂c2n
!
m2
k̂c2n
where
(B.53)
[Jm (k̂cn a)Nm+1 (k̂cn a) − Nm (k̂cn a)Jm+1 (k̂cn a)]2 −
[Jm (k̂cn a)Nm+1 (k̂cn b) − Nm (k̂cn a)Jm+1 (k̂cn b)]2 .
Propagation Properties
Following the same reasoning as in the case of TM modes we find a cutoff frequency
when they propagate along a coaxial waveguide filled with a lossless medium, and
it reads
160
Appendix B.
wc,m1 =
wc,mn =
B.3
2
c
for n = 1,
a+b
π(n − 1)
c
for n = 2, 3, . . .
a−b
Waveguides
(B.54)
(B.55)
Coaxial-Circular Waveguide
Belhadj-Tahar et al. (1990) and Taherian et al. (1991) proposed a coaxial-circular
waveguide for permittivity measurements of solid rock cores. It consists of coaxial
transition units and a circular sample holder to host the core. This has an advantage
over performing these types of measurements with a coaxial sample holder, as
the core doesn’t need drilling and the flow through the sample will not have a
preferential path. Moreover air gaps between the central pin and the core are
avoided.
The tool presented in this thesis, see Chapter 4, can be modified to work as a
coaxial-circular waveguide. It is relatively easy and consists in removing the central
conductor of the sample holder and adjusting the ones of the transition units.
Figure B.2 shows a diagram for this modified tool.
Figure B.2: Modified tool: coaxial-circular waveguide
B.3. Coaxial-Circular Waveguide
161
To reconstruct the permittivity of a rock core from measured S-parameters in this
type of configuration, a full-wave description of the waveguide is needed. The fields
in the coaxial sections will propagate as the TEM, TE and TM modes presented
previously in this Appendix, and in the circular section similar equations can be
obtained for the TM mode, the only one generated when going from a coaxial into
a circular section. Then the total fields at the reference planes can be computed
numerically and via an optimization routine the permittivity can be reconstructed.
162
Appendix B.
Waveguides
Appendix C
Transmission Lines
This appendix contains the equations of wave propagation through a transmission
line, and more explicitly through a coaxial configuration
TEM transmission lines are a specific group of waveguides. They are composed
of two or more conductors whose cross section perpendicular to the direction of
propagation is constant (i.e. two parallel wires, coaxial cable, micro-strip line). In
such a configuration, as we showed in Appendix B, TEM waves (transverse electromagnetic; both electric and magnetic fields lie entirely in the plane perpendicular
to the direction of propagation) are able to propagate. Other modes, TE (transverse electric; the electric field lies in the plane perpendicular to the direction of
propagation) and TM (transverse magnetic; the magnetic field lies entirely in the
plane perpendicular to the direction of propagation) are also excited, but the line
is usually operated below the cutoff frequency of these modes so they do not propagate. In this situation, it is possible to reformulate Maxwell’s waves equations in
terms of circuit parameters and voltage and current waves, see Section C.1.
C.1
Transmission Line Equations
In Appendix B we show the existence of a purely transversal electromagnetic mode
that can propagate along a coaxial waveguide. It can be proven, in a similar way
(solving Maxwell’s equations subject to the boundary condition that the tangential
component of the electric field vanishes on the surface of the conductors), that
this type of waves may propagate in uniform lines, consisting of any combination of
conductors and dielectrics where the cross-section of the line is constant (parallel
wires, coaxial cables, etc).
163
164
Appendix C.
Figure C.1:
paths
Transmission Lines
Integration
But Maxwell’s equations applied to the specific case of transmission lines can be
ˆ as they can be written
expressed in terms of voltage V̂ and current I,
Z
Z
ˆ
Ĥ · dl,
(C.1)
V̂ =
Ê · dl
and
I=
Ct
C0
where Ct is an integration path transverse to z and C0 a closed contour, see
Fig. (C.1). The equivalent to the wave equations for a TEM mode propagating
along a coaxial waveguide, equations (2.69) and (2.70), in terms of voltage and
current are
∂z ∂z V̂ − γ̂ 2 V̂ = 0,
∂z ∂z Iˆ − γ̂ 2 Iˆ = 0,
(C.2)
(C.3)
R
when the line has been exited by an initial voltage, V̂0 = Ct Ê(r, 0) · dl, and γ̂ is
again the propagation factor, but has to be written in terms of the transmission
line parameters.
C.1.1
Propagation factor (γ̂) of a transmission line
The propagation factor of general media, with conductivity σ̂, permittivity ε̂, permeability µ̂ and taking s = jw reads
q
γ̂ = η̂ ζ̂
where
η̂ = σ̂ + jwε̂
and
ζ̂ = jwµ̂,
and its transmission counterpart can be written as
γ̂ =
p
(G + jwC)(R + jwL),
(C.4)
C.1. Transmission Line Equations
165
where
G is the conductance and equivalent to σ̂ [S/m]
C is the capacitance and equivalent to ε̂ [F/m]
R is the resistance due to the non perfect conductors [Ω/m]
L is the inductance and equivalent to µ̂ [H/m]
The resistance R is determined by the finite conductivity σc of the non perfect
conductors and the skin depth δ. A resistivity Rs can be defined as
r
1
2
Rs =
where
δ=
,
δσc
wµσc
and thus R = Rs /(circumference of the line).
These parameters are all defined per unit length and depend only on the geometrical
aspects of the line and the filling material. Explicit formulas for their calculation
can be found in the literature (see, for example, Karmel et al. (1998)).
The solutions to the transmission line equations (C.2) and (C.3) are
V̂ (z) = V̂0+ e−γ̂z + V̂0− e+γ̂z ,
ˆ
I(z)
= Iˆ0+ e−γ̂z + Iˆ0− e+γ̂z .
(C.5)
(C.6)
The voltage and the current waves consist of two waves, one travelling to the right
and the other one to the left. The complex numbers V̂0+ , Iˆ0+ , V̂0− and Iˆ0− are not
independent of each other as they must obey the transmission line equations, (C.2)
and (C.3) and thus
V̂0−
V̂0+
=
−
= Z0 ,
Iˆ+
Iˆ−
0
(C.7)
0
where Zc is the characteristic impedance (Ω) of the transmission line. And it is
a function of the material electromagnetic properties, µ̂, ε̂, σ̂ and σc , and the
cross-sectional geometry of the line
Zc =
s
R + jwL
.
G + jwC
(C.8)
166
Appendix C.
Transmission Lines
The impedance at any position is defined as the ratio between the voltage and
current on the line
Z(z) =
V̂ (z)
,
ˆ
I(z)
(C.9)
and the admittance is the inverse of the impedance Y (z) = 1/Z(z).
Transmission Line Parameters for a coaxial transmission line
For a coaxial transmission line with inner radius b and outer radius a these
parameters become
L =
C =
G =
R =
C.2
a
µ̂
ln
,
2π
b
2π ε̂
,
ln(a/b)
2πσ̂
,
ln(a/b)
1 1 1
Rs .
+
2π a b
(C.10)
(C.11)
(C.12)
(C.13)
Local Reflection and Transmission coefficients
Consider an interface n located at zn , the medium on the left has impedance Zn
and on the right Zn+1 . Part of an incident wave on the interface will be reflected
and part will be transmitted, this is represented in Fig. (C.2).
Figure C.2:
Incident,
reflected and transmitted
waves on the nth interface
C.2. Local Reflection and Transmission coefficients
167
A local reflection coefficient and a local transmission coefficient can be defined as
the ratios of the reflected and transmitted components of the voltage wave to the
incident component.
rn± =
t±
n =
V̂n∓
,
V̂n±
#±1
"
±
Vn+1
Vn±
(C.14)
.
(C.15)
In order to find these ratios we need to impose boundary conditions to the voltage
and current wave expressions. For zn = 0, these expressions are
z<0
z>0
The voltage and current waves on each side of the interface are given by
V̂n (z) = V̂n+ e−γ̂n z + V̂n− e+γ̂n z
+
V̂n+1 (z) = V̂n+1
e−γ̂n+1 z
Iˆn (z) = Iˆn+ e−γ̂n z + Iˆn− e+γ̂n z
+
Iˆn+1 (z) = Iˆn+1
e−γ̂n+1 z
a local impedance can be defined as
Zn =
V̂n+
Iˆn+
−
= − V̂ˆn−
Zn+1 =
In
+
V̂n+1
+
Iˆ
n+1
V̂ −
n+1
= − ˆ−
In+1
and then the waves can be rewritten as
+
V̂n+1 (z) = V̂n+1
e−γ̂n+1 z
V̂n (z) = V̂n+ e−γ̂n z + V̂n− e+γ̂n z
Iˆn (z) =
1
Zn
h
V̂n+ e−γ̂n z − V̂n− e+γ̂n z
i
Iˆn+1 (z) =
+
1
−γ̂n+1 z
Zn+1 V̂n+1 e
These expressions under the boundary conditions, that the fields must be continuous
at the interface
lim V̂n+1 = lim V̂n ,
(C.16)
lim Iˆn+1 = lim Iˆn ,
(C.17)
z↓0
z↓0
z↑0
z↑0
168
Appendix C.
Transmission Lines
lead to the equations for the local reflection and transmission coefficients
C.2.1
rn+ =
Zn+1 − Zn
,
Zn+1 + Zn
(C.18)
t+
n =
2Zn+1
.
Zn+1 + Zn
(C.19)
The terminated transmission line
Assume a transmission line of constant impedance Zc which is terminated at a
distance z = L from the origin by a load impedance ZL . Its reflection coefficient
at the interface L, where the line is terminated, depends thus on the impedance of
the load and of the line, according to
rL =
ZL − Zc
,
ZL + Zc
(C.20)
so for different types of load we have:
• Short-circuit load ZL = 0 and RL = −1.
• Open-circuit load ZL = ∞ and RL = 1.
• Matched load ZL = Zc and RL = 0.
C.3
Line parameters for an ideal section
For ideal lines (resistance due to the conductors that form the line is negligible) the
propagation constant and the impedance of each section can be expressed in terms
of the electromagnetic properties of the materials filling it
γ̂n2 = −ω 2 ε̂∗n µ̂∗n
and
Zn = g
s
µ̂∗n
,
ε̂∗n
(C.21)
where g is a geometry factor that depends on the geometry of the line. For a
coaxial transmission line g = ln(a/b)
2π .
C.4. Global Reflection and Transmission coefficients
169
The local reflection coefficient can be rewritten as
r
rn = r
µ̂∗n+1
ε̂∗n+1
µ̂∗n+1
ε̂∗n+1
−
+
q
q
µ̂∗n
ε̂∗n
.
(C.22)
µ̂∗n
ε̂∗n
If the materials are non magnetic, µ̂∗ = µ0 these equations simplify to
ω2 ∗
ε ,
c20 r,n
Z
p 0∗
where
εr,n
q
p ∗
εr,n − ε∗r,n+1
q
.
p ∗
εr,n + ε∗r,n+1
γ̂n2 = −
Zn =
rn =
C.4
(C.23)
Z0 = g
r
µ0
,
ε0
(C.24)
(C.25)
Reflection and Transmission coefficients
for a multi-section transmission line
Let us now consider a multi-section transmission line consisting of N sections, see
Fig. (C.3). The sections are determined by abrupt changes in impedance at the
interfaces between them. At z = zs < 0 the line is driven by a source and at
z = zN = L it is terminated by a load impedance ZL .
Figure C.3: Multi-section transmission line
170
Appendix C.
Transmission Lines
If Γn is the local reflection coefficient at the nth interface the voltage and current
waves on the left hand side (zn−1 ≤ z ≤ zn ) of the nth interface can be written as
i
h
V̂n (z) = V̂n+ e−γ̂n z + Γn e+γ̂n (z−2zn ) ,
i
1 + h −γ̂n z
Iˆn (z) =
V̂n e
− Γn e+γ̂n (z−2zn ) ,
Zn
(C.26)
(C.27)
and on the right hand side (zn ≥ z ≥ zn+1 )
h
i
+
e−γ̂n+1 z + Γn+1 e+γ̂n+1 (z−2zn+1 ) ,
V̂n+1 (z) = V̂n+1
h
i
1
Iˆn+1 (z) =
V̂n+ −γ̂n+1 z − Γn+1 e+γ̂n+1 (z−2zn+1 ) .
Zn+1
(C.28)
(C.29)
It is possible to derive a recursive expression for the multiple reflection coefficient
Γn and for the multiple transmission coefficient Υn if the continuity of the fields
across the nth interface, equations (C.16) and (C.17) are taken into account
Γn =
rn + Γn+1 e−2γn+1 dn+1
,
1 + rn Γn+1 e−2γn+1 dn+1
(C.30)
Υn =
n Y
(C.31)
k=1
P
n
1 + Γk
e− k=1 γk dk ,
−2γ
d
k+1
k+1
1 + Γk+1 e
where rn is the local reflection coefficient given by equation (C.18) and dn+1 =
zn+1 − zn is the length of the n + 1 section.
Appendix D
Scattering Matrix
Here, we show the explicit relations between the total reflection and transmission
coefficients, found in the previous Appendix, with the S-parameters. We also show
the unitary property of the Scattering Matrix for a lossless line
2-port networks are described by electrical engineers via three different matrices,
Impedance [Z], Admittance [Y ] and the Scattering Matrix [S]. The impedance
matrix relates reflected voltage to incident current and viceversa in the case of
admittance representation.
The scattering matrix relates reflected voltage to incident voltage on circuit components with two distinct ports or connections, V − = SV + , see Figure (D.1). And
the entries of the matrix, the S-parameters are related to the reflection/transmission
response of the electrical components forming the network and their interaction.
And therefore, they are the favorite representation for reflection/transmission problems.
This voltage relation can be written as
−
V01
−
V02
=
S11 S12
S21 S22
Figure D.1: 2-Port Network.
171
+
V01
+
V02
.
(D.1)
172
Appendix D.
Scattering Matrix
So that, the diagonal elements of [S], Sii are given by
Sii =
V0i−
,
V0i+
(D.2)
it is then clear that Sii is the reflection coefficient at the ith port, when every other
port is terminated in a matched load.
The off diagonal terms, Sij , i 6= j are given by
Sij =
V0i−
V0j+
j 6= i,
(D.3)
and Sij , i 6= j is the transmission coefficient from the jth port to the ith port, when
every other port is terminated in a matched load.
D.1
S-parameters and the total reflection and transmission
coefficients of a transmission line
Assuming a perfectly symmetric and homogeneous line, [S] is symmetric and their
components are related to the total reflection, Γn , and total transmission, Υn ,
coefficients of a transmission line, given by equations (C.30) and (C.31), as follows
S11 = S22 =
V1−
= Γ1 e2γ1 zr ,
V1+
(D.4)
where zr is the position where the voltage ratio is measured. If this reference plane
is taken at the origin of the line (zr = 0) then
S11 = S22 = Γ1 .
(D.5)
For the off-diagonal terms and taking the reference plane of measurement at the
origin or end of the line, we have
S12 = S21 = ΥN .
(D.6)
D.2. Unitary property of the Scattering Matrix
D.2
173
Unitary property of the Scattering Matrix
For a lossless ideal line it can be proven that [S] is unitary (Karmel et al. , 1998),
and the S-parameters obey the following properties
2
X
i=1
2
X
|Sik |2 = 1,
∗
Sik Sir
= 0,
i=1
k = 1, 2
k, r = 1, 2
(D.7)
k 6= r.
(D.8)
For lossless material filling the line, these properties can be used to determine how
close a line is to ideal.
174
Appendix D.
Scattering Matrix
Appendix E
Permittivity models
This appendix gives a short introduction into the theoretical models that try to
explain the EM behavior of materials and mixtures. It is not extensive, for further
insight and full derivations we refer to the cited literature.
E.1
Introduction
The electromagnetic behavior of materials has been, is and will continue to be
a very broad research topic, for obvious reasons. Along history, many different
authors have been involved in the production of theoretical and empirical models
that could explain the experimental evidence.
The first capacitor was constructed by Cunaeus and Musschenbroek as far back
as 1745, and it became very popular for a variety of experimental purposes under
the name of Leyden jar. However, it was Faraday in 1837, who published the first
numerical measurements on dielectrics, a term introduced by him (Böttcher, 1952).
The beginning of the systematic investigation of the dielectric properties can be
established with the works of Mossotti (1850) and Clausius (1879), as they attempted 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 ε and the
volume fraction occupied by the conducting particles (Section E.2.1).
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 ClausiusMossotti theory to take into account the permanent moments of the molecules.
His theory, later extended by Onsager (1936) and Kirkwood (1939), linked the
175
176
Appendix E.
Permittivity models
dispersion of ε∗ 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
significantly from Debye’s theory of relaxation. Cole & Cole (1941) pioneered
the first approach to interpret the non-Debye relaxation of materials by means
of a superposition of different relaxation times and Jonscher (1983) rebates that
the relaxation behavior at molecular level is intrinsically non-Debye due to the
cooperative molecular motions, the ”many-body interactions” approach.
All these researchers modelled the macroscopic behavior of dielectric materials composed 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.
A permittivity mixture model expresses ε∗ of the mixture in terms of the permittivities of each constituent and their volume fraction. The literature on the effective
properties of materials and mixtures is very vast and we have decided to include in
this appendix just a few of the most relevant ones. Nevertheless, you can find comprehensive reviews by Wang & Schmugge (1980), Shutko & Reutov (1982), Dobson
et al. (1985), Chelidze & Guguen (1999a), Sihvola (1999) and Choy (1999), among
others.
This appendix is divided into four sections, the first three contain different models
grouped according to their characteristics. Section E.2 presents two classical models
to compute the permittivity of materials from known parameters, Section E.3 deals
with existing bounds for the permittivity of mixtures and Section E.4 contains the
most widely used mixing formulas. The last section summarizes some of the most
important experimental results.
E.2
Classical approaches to pure materials
This section contains the classical models for gases, due to Clausius-Mossoti, and
Debye’s model for polar liquids.
E.2. Classical approaches to pure materials
E.2.1
177
Clausius-Mossotti formula
This famous formula is derived from the idealization of a spherical particle with
polarizability α embedded in a homogeneous background of permittivity ε0 . The
permittivity of N of these particles is given by:
ε=
1 + 32 N α
ε0 .
1 − 13 N α
(E.1)
It is also known as the Lorenz-Lorentz equation, since it was independently formulated for the optical range by Lorentz in The Netherlands and Lorenz in Denmark
(Hippel, 1954). Note that the permittivity has a singularity at N α = 3, the
Clausius-Mossotti catastrophe and it is due to the cavity model they used.
E.2.2
Debye’s model
As we have mentioned, Debye (1929) developed a model for the complex permittivity of a spherical polar molecule relaxing in a viscous medium, based on the
assumption of a single relaxation time for all molecules, and it is described by the
following equation
ε∗r = ε∞ +
σdc
εs − ε∞
−j
,
1 + jωτr
ωε0
(E.2)
where j is the imaginary unity, ω is the angular frequency [1/s], ε0 is the permittivity
of free space, ε∞ is the permittivity in the limit ω → ∞ and εs when ω → 0,
τr = 1/(2πfr ) is the characteristic relaxation time given in [s], and σdc is the direct
current conductivity of the material [S/m]. Although not explicitly mentioned in
the formula, these parameters are temperature dependent.
In Chapter 5 we showed how the reconstructed permittivities per frequency for
ethanol, a polar liquid, could be fitted with a Debye model very accurately. Here,
we show the Debye parameters for distilled water
WaterHasted
Waterfit
T (◦ C)
20.6
22
εs
79.9
77.4
ε∞
4.22
4.81
fr (GHz)
17
19.4
σdc (µS/m)
0.6 ± 0.1∗
0.06
Table E.1: Debye parameters for distilled water from literature (Hasted, 1973) and the
fit to our measurements. ∗ The conductivity shown for WaterHasted was measured with a
conductivity meter in-situ.
178
Appendix E.
E.3
Permittivity models
Bounds for the effective permittivity of mixtures
Several authors limit the effective permittivity value to lie between bounds, based
on general assumptions, rather than finding an explicit solution for the effective
permittivity.
We present equations for two-component mixtures where εi and εh are the permittivities of the inclusion and host, respectively, and fi and fh are their corresponding
volume fractions.
E.3.1
Geometric bounds
Figure E.1: Geometrical limits of a mixture of 2 materials.
The geometries in which 2 materials can be mixed is unlimited, but the most
extreme cases are when the materials are distributed as shown in Figure E.1. Then
the effective permittivity of any mixture cannot exceed the upper or lower bounds
given by the arithmetic and harmonic means
εmax
ef f = fi εi + (1 − fi )εh ,
εi εh
.
εmin
ef f =
fi εh + (1 − fi )εi
(E.3)
(E.4)
In electromagnetics, these bounds are also known as the Wiener bounds. For N
components they read
εmax
ef f =
1
εmin
ef f
i=N
X
=
fi εi
i=1
i=N
X
i=1
fi
.
εi
(E.5)
(E.6)
E.3. Bounds for the effective permittivity of mixtures
E.3.2
179
Hashin-Shtrikman bounds
For macroscopically homogeneous and isotropic composites, Hashin & Shtrikman
(1962), derived bounds for the magnetic permeability of an N-component mixture
based on energy considerations. Accordingly, for the permittivity
εmin
ef f = ε1 +
A1
A1
1 − 3ε
1
where
A1 =
εmax
ef f = ε2 +
A2
A2
1 − 3ε
2
where
A2 =
N
X
fi
+
1
3ε1
fi
+
1
3ε2
1
i=2 εi −ε1
N
−1
X
1
i=1 εi −ε2
,
(E.7)
,
(E.8)
where ε1 corresponds to the minimum permittivity found in the N components,
and ε2 to the maximum.
For a two component mixture, these bounds reduce to the Maxwell-Garnett model,
where ε1ef f corresponds to the component i acting as inclusion into the host component h and ε2ef f interchanges the roles of inclusion/host.
Ri→h
= εh 1 + 3fi
1 − fi Ri→h
Rh→i
= εh 1 + 3fi
1 + (1 − fi )Rh→i
ε1ef f
ε2ef f
E.3.3
εi − εh
, (E.9)
2εh + εi
εh − εi
. (E.10)
=
2εi + εh
where
Ri→h =
where
Rh→i
Local Electrodynamic Equilibrium constrains
Rio & Whitaker (2000a) constrained the validity of using one-equation models
for the dynamic electric and magnetic fields in a two component mixture with
equations (E.11), (E.12) and (E.13). They derived these constrains using the
method of volume averaging.
fh fi (εh − εi )2
1,
(fh εh + fi εi )(εh + εi )
fh fi (µh − µi )2
1,
(fh µh + fi µi )(µh + µi )
fh fi (σh − σi )(εh − εi )
1.
(fh σh + fi σi )(εh + εi )
(E.11)
(E.12)
(E.13)
180
Appendix E.
Permittivity models
In essence, and according to del Rio and Whitaker, when a mixture violates the
constrains given by equations (E.11), (E.12) and (E.13) it cannot be represented
by single effective electromagnetic properties, and a Two-Equation Model must be
used (Rio & Whitaker, 2000b). Moreover, if the mixture can be represented by
a One-Equation Model, then its effective permittivity is given by the arithmetic
mean, equation (E.3).
In Chapter 6 we presented numeric and experimental examples where these bounds
are not met and yet the samples were adequately represented by an effective permittivity that is not necessarily the arithmetic mean of the permittivities of the
components.
E.4
Mixing Laws
This section gathers the most commonly used mixing laws. Traditionally they are
given in implicit form, but here we also present their explicit solution and we only
state the solution for spherical inclusions.
Most mixing formulae originate from solving the Laplace equation excluding the
wave-propagation properties of EM fields. They are a static approach and, according to Sihvola (1999), can be considered quasi-static when the wavelength of the
field divided by 2π is bigger than the size of the inclusion. But this is only an
approximation. There are extensions of the static mixing formulae for time-varying
fields but they are hardly used in geophysical applications, while the use of the
static formulae is very wide spread.
E.4.1
Maxwell-Garnett
The Maxwell-Garnett mixing rule consist in rewriting the microscopic terms of the
Clausius-Mossotti equation (E.1) in terms of the permittivities of the host and the
inclusion. It reads
m
X εi − εh
εef f − εh
fi
=
,
εef f + 2εh
εi + 2εh
(E.14)
i=1
and it is also known as the Rayleigh mixing formula. In its explicit form for two
components
εef f
= εh 1 + 3fi
Ri→h
,
1 − fi Ri→h
(E.15)
E.4. Mixing Laws
181
−εh
is the Rayleigh reflection coefficient of a small sphere in the
where Ri→h = εεii+2ε
h
quasistatic limit.
It is anti-symmetric, if the roles of host and inclusion are interchanged the result is
different. This approach of treating one of the components as host medium, and
the inclusion as a perturbation against the background has been widely used and
there are many Maxwell-Garnett type of mixing formulae (Sihvola, 1999).
Equation (E.15) is equal to equation (E.9) and if the host and inclusion components
are interchanged it is equal to equation (E.10). The Hashin-Shtrikman bounds for
a 2-component mixture correspond to the Maxwell-Garnett model.
E.4.2
Bruggeman
Bruggeman (1935) derived several mixing rules depending on the geometry of the
mixtures. His most famous contribution is a symmetrical formula where, in contrast
to the Maxwell-Garnett type of solutions, the effective medium itself is considered
as the background against which polarizations are measured. It assumes that the
components are homogeneous and randomly distributed, and that the size of the
components is very small compared to the size of the mixture. It is also known as
the Böttcher equation (Böttcher, 1952). For m components with permittivities εi
it reads
m
X
i=1
fi
εi − εef f
= 0,
εi + 2εef f
(E.16)
and its explicit solution for two components is
εef f
r
εh εi
= F + F2 +
2
where
F =
1
[(3fi − 1)εi + (3fh − 1)εh ] .
4
(E.17)
For dilute mixtures (fi 1), Maxwell Garnett and Bruggeman, predict the same
results, since, up to the first order in fi their formulas are the same.
E.4.3
Bruggeman-Hanai
Bruggeman also derived a non-symmetrical formula later modified by Hanai (1936)
and generalized by Sen et al. (1981)
182
Appendix E.
Permittivity models
εef f − εi εh L
= 1 − fi ,
εh − εi εef f
(E.18)
where L is a depolarization factor to be fitted with core measurements. In the
original formula by Bruggeman L = 1/3.
E.4.4
Power Law
This series of formulae relate a certain power of the permittivity of the mixture to
the same power of the permittivities of the components weighted by their volume
fractions.
εcef f =
m
X
fi εci ,
(E.19)
i=1
when c = ±1 equation (E.19) describes the geometric bounds given in Subsection E.3.1. Birchak et al. (1974) used c = 1/2, and then equation (E.19) is
commonly known as the Complex Refractive Index Method (CRIM), which is a
harmonic mean of velocities, or the arithmetic mean of slownesses. By contrast,
Looyenga (1965) used c = 1/3.
E.4.5
Four-Component Dielectric mixing Model
This model was first proposed by Dobson et al. (1985) based on a previous model
given by de Loor (1963). It is based on plate-shaped inclusions dispersed in a host
medium and reads
εef f =
3εs + 2ff w (εf w − εs ) + 2fbw (εbw − εs ) + 2fa (εa − εs )
,
s
− 1) + fa ( εεas − 1)
3 + ff w ( εεfsw − 1) + fbw ( εεbw
(E.20)
where the subscripts bw, f w, a and s refer to bound water, free water, air and soil,
respectively. They found that εbw = 35 − j · 15 yielded a good fit to the measured
data.
This is an interesting approach because it treats the bound water separately from
the free water. However, it is very difficult to determine the amount of bound water
present in the system and its permittivity.
E.5. Results
E.4.6
183
Dry Sand
Dry sand is a two-component mixture and the influence of texture and grain size
seem negligible. According to Shutko & Reutov (1982) the permittivity of dry soils
(εs ) can be computed from their density (ρs ), given in [gr/cm3 ], via
εs = (1 +
ρs 2
) ,
2
(E.21)
whereas Dobson et al. (1985) found a slightly different relation
εs = (1 + 0.44ρs )2 − 0.062.
(E.22)
It is surprising to relate a unitless property to the square of a density, but these are
empirical fittings to experimental data on different dry soils. We think it is more
appropiate to compute the permittivity from the following reinterpretation of the
Clausius-Mossotti relation
Clausius-Mossotti reinterpretation
Since equation (E.1) has been successfully used to compute the permittivity of different gases from known polarizabilities, we wanted to test it for mixtures. For noninteractive materials we can compute α of each material from its permittivity and
equation (E.1). By superposition we can find the permittivity of the m-materials
mixture as
P
Nj αj
3+2 m
Pmj=1
εef f =
(E.23)
3 − j=1 Nj αj
The results presented in Subsection 6.6.1 support this model.
E.5
Results
Different researchers have compared measured permittivities of different soils at
different water contents and temperatures to existing mixing models. In this section,
we summarize existing results but it is a complex problem and, in general, they are
only qualitative results.
Shutko & Reutov (1982) found the Refractive Index Method to give a good approximation to the permittivity of different soils in the range [1-10] GHz and BruggemanHanai at 10 GHz is a good prediction for soils with low moisture content. Hallikainen
et al. (1985) pointed that although soil texture may not affect the permittivity of
184
Appendix E.
Permittivity models
dry soils, it certainly does for wetted soils, specially at the lower end of the frequency range. Dobson et al. (1985) found that, both, the Four Component Model
and a Power Law with a coefficient c = 0.65, were capable of describing the complex dielectric permittivity above 1.45 GHz for different soils. And, according to
Chelidze & Gueguen (1999b) the Bruggeman-Hanai model with L = 1/3 works
fine for the Megahertz region of the spectrum and different rocks. Finally, Seleznev
et al. (2004) studied the permittivity of partially saturated carbonate rocks, from
300 KHz to 3 GHz, finding that the best fit was that of the CRIM model.
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Bibliography
Summary
As geophysicists we are interested in the imaging capabilities of electromagnetic
(EM) waves. Our 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.
Sands are one of the most common materials found in the subsurface. They can
be dry, partially or fully saturated. Their electromagnetic response depends mainly
on, porosity, saturation, and the electromagnetic properties of the sand grains, the
saturating fluid and air. It is possible to obtain the saturation of the material from
its EM properties through the Mixing Laws. However, there are many of these
laws and the experimental evidence does not favor any one in particular. They only
consider volume fractions and electromagnetic properties of the constituents of the
sand, while the geometrical characteristics of the saturation distribution and the
heterogeneities present in the sand are not taken into account.
This thesis tests whether the geometry of the saturation distribution within the sand
and the possible heterogeneities present in it have an impact in its electromagnetic
properties. In order to do so, we have designed and constructed a customized tool
in which fluid flow is allowed through the sample. Much of this thesis deals with the
design and calibration of the tool and the reconstruction of the EM properties of the
sample. We review various experimental designs for measuring the electromagnetic
properties of soil samples, and construct a customized coaxial transmission line
whose reflection and transmission coefficients can be determined in the frequency
range from 300MHz to 3GHz. The electromagnetic properties of the material
are obtained from the inversion of these coefficients, the so-called reconstruction
of the EM properties. The forward model that describes the propagation of EM
waves along the line, needs to be calibrated to be in very good agreement with the
measured data.
191
192
Summary
We introduce a new notation (Propagation Matrices) to describe the transmission
lines. It provides a representation that simplifies the understanding of the interactions between the different sections that configure these tools. And enables us to
prove the independence of the transmission and reflection measurements, and to
develop a novel analytical inversion of the measurements.
The electromagnetic properties that determine the propagation of EM waves are the
magnetic permeability and the electric permittivity. Inverting for both properties
results in relatively big errors, and since most sands are non-magnetic materials we
restrict our investigation to the study of the permittivity alone.
We analyze and compare existing reconstruction techniques with the novel one
based in the Propagation Matrices and find it the most convenient. We successfully reconstruct the relative permittivity of air within ±1% error. And we could
fit a Debye model to the reconstructed permittivity of ethanol, within the same
error bound. The inversion obtained with the Propagation Matrices representation
identifies all existing inversions equal to either one of the two fundamental and independent solutions. This explains the inaccurate reconstructions of the permittivity
of low-loss materials, indistinctively of the expression used.
Near the end of the thesis, we investigate the effect of the distribution of constituents with measured and modelled experiments. With the aid of numerical
experiments, we show that a distribution of layers has a strong impact on the
permittivity, and that heterogeneities as small as λ/100 are detectable. We also
studied the effect of fluid flow through a sand sample in relation to its permittivity.
We showed how the saturation technique has a clear effect in the reconstructed
permittivity and that extra care is needed for proper measurements. Depending
on the goal of the measurement it is best to homogeneously saturate the sample
outside the sample holder or to perform a gravity drainage to change the saturation.
The second technique has the advantage that the same sample can be measured at
different saturation levels. However, the saturation distribution results in distinctive 2-layer samples, whose reconstructed 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 and
then, the tool can be used to monitor the movement of the fluid front.
And finally, we 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. The Power Law is also able to model the permittivity of 2-layer samples, but
the error it makes is larger than when it is used for modelling homogenous samples.
If a lower precision is allowed, the partially saturated sands, whose saturation is
clearly distributed in 2-layers, can be considered homogenous. The validity of
this approximation decreases with increasing permittivity contrast. For the first
193
time in literature, we present a technique to include the geometrical distribution
of saturation in a Power Law equation. With this technique the modelling of the
permittivity of 2-layer samples is very accurate.
Based on our results we can conclude that sands can be represented by a single
value of permittivity. However, the effective parameter depends very strongly on the
constituents, volume fractions and distribution within the sand, and these must be
taken into account. It should also be considered that the permittivity of sands can
vary with frequency, not because of the frequency dependence of the permittivities
of its constituents, but because of their geometrical distribution.
Ainhoa G. Gorriti
194
Summary
Samenvatting
Elektrische karakterisering van zanden met heterogene saturatieverdelingen
We zijn als geofysici genteresseerd in de afbeeldingmogelijkheden van elektromagnetische (EM) golven. Ons doel is een beeld te maken van de ondergrond uit de
responsie van de Aarde op de in de ondergrond aangebrachte EM verstoringen. Het
is dan van essentieel belang de reactie van de materialen, die vaak in de aardse
ondergrond worden aangetroffen, op de gebruikte EM velden te weten.
Zand is een van de meest voortkomende ondergrondse materialen; het kan droog
zijn of gedeeltelijk of geheel met water verzadigd. De elektromagnetische responsie
wordt voornamelijk bepaald door de porositeit en de elektromagnetische eigenschappen van de zandkorrels, de vloeistoffen en lucht die de holle ruimte vullen.Het is
mogelijk de verzadigingsgraad van het materiaal te bepalen uit de EM eigenschappen via Mengregels. Er zijn echter veel van deze mengregels en experimentele resulaten lijken geen enkele regel te onderbouwen. De mengregels beschouwen slechts
de volume fracties and de elektromagnetische eigenschappen van de verschillende
componenten van het zand/vloeistof/lucht mengsel, terwijl de geometrische karakteristiek van de verzadigingsverdeling en de in het zand aanwezige heterogeniteit
niet worden meegenomen.
Dit proefschrift bekijkt in hoeverre de geometrie van de verzadigingsverdeling in het
zand en de mogelijk aanwezige heterogeniteiten van invloed zijn op de elektromagnetische eigenschappen van het geheel. Hiervoor hebben wij een speciaal instrument
ontworpen en gebouwd waarin vloeistofstroming door het monster mogelijk is. Een
belangrijk deel van dit proefschrift behandeld het ontwerp en de kalibrering van
het instrument en een rekenwijze om de elektromagnetische eigenschappen van het
monster te reconstrueren uit de metingen. Een aantal experimentele ontwerpen,
om elektromagnetische eigenschappen van grondmonster te meten, wordt onder de
loep genomen om uiteindelijk te kiezen voor de coaxiale transmissielijn, waarvan
de reflectie- en transmissiecofficinten kunnen worden bepaald in de frequentieband
van 300MHz tot 3GHz. De elektromagnetische eigenschappen van het materiaal
worden verkregen door invertering voor deze cofficinten, de zogenaamde reconstructie van de EM eigenschappen. Het voorwaartse model dat de propagatie van
195
196
Sumenvatting
de EM golven langs de lijn beschrijft, moet worden gekalibreerd om tot een zeer
nauwkeurige beschrijving van de meetgegevens te komen.
We introduceren een nieuwe notatie (Propagatie Matrices) om de transmissielijnen
te beschrijven. Deze representatie vergemakkelijkt het begrip van de interacties
tussen de verschillende delen van het totale instrument. Het stelt ons in staat
te bewijzen dat de reflectie- en transmissiemetingen onafhankelijk van elkaar zijn,
alsook om een nieuwe analytische inversie van de meetgegevens te ontwikkelen.
De elektrische permittiviteit en de magnetische permeabiliteit zijn de twee elektromagnetische eigenschappen die de propagatie van EM golven bepalen. Inversie voor
beide eigenschappen resulteert in relatief grote fouten, en omdat de meeste zanden
niet-magnetische materialen bevatten hebben we onze onderzoekingen tot alleen
de permittiviteit beperkt. Bestaande reconstructietechnieken worden geanalyseerd
en vergeleken met de nieuw ontwikkelde techniek gebaseerd op de Propagatie Matrices, welke de meest geschikte blijkt. De relatieve permittiviteit van lucht wordt
binnen een foutmarge van 1
Tegen het eind van het proefschrift onderzoeken we het effect van de verdeling van
de bestanddelen aan de hand van fysische en gemodelleerde experimenten. Met
behulp van de numerieke experimenten tonen we aan dat de verdeling van laagjes een grote invloed hebben op de permittiviteit en dat heterogeniteiten tot een
grootte van een honderdste golflengte detecteerbaar zijn. We bestuderen ook het
effect van vloeistofstroming door het zandmonster op de permittiviteit. De manier
van satureren heeft een effect en hieraan dient aandacht te worden gegeven bij het
uitvoeren van dergelijke experimenten. Afhankelijk van het doel van het experiment zijn de volgende twee procedures het meest aan te bevelen: ofwel wordt het
zand met een vooraf bepaalde hoeveelheid vloeistof gemengd tot een homogeen
mengsel, ofwel wordt volledig met vloeistof verzadigd zand in de monsterhouder
geplaatst waarna met behulp van zwaartekracht de verzadigingsgraad wordt verlaagd onder gecontroleerde omstandigheden. De tweede manier heeft als voordeel
dat de porositeit van het monster ongewijzigd blijft bij alle verzadiginggraden. Het
nadeel is echter dat de verzadigingsverdeling duidelijk in twee-laags monsters resulteert, waarvan de gereconstrueerde permittiviteit uitschieters vertoont vanwege het
grensvlak. Dit soort monsters kunnen zeer nauwkeurig door twee-laags modellen
worden beschreven indien de dikte van één van de twee lagen uit de fase van de
transmissiemeting wordt bepaald, waarna het instrument gebruikt kan worden om
de verplaatsing van het vloeistof grensvlak kan worden gevolgd.
Uiteindelijk vinden we op basis van onze experimenten dat de machtsregel, van
alle bestaande regels, de beste mengregel is voor, gedeeltelijk en geheel, homogeen
verzadigde zanden. Helaas zijn verschillende exponenten nodig bij verschillende
verzadigingsgraden en worden zowel het rele als het imaginaire deel met dezelfde
macht voldoende nauwkeurig beschreven. De machtsregel kan ook worden gebruikt
bij het homogeniseren van twee lagen, maar dan is de nauwkeurigheid minder dan
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bij homogeen gemengde monster met dezelfde verzadigingsgraad. Dit homogeniseren van gelaagde monsters is mogelijk indien deze verminderde nauwkeurigheid
nog voldoende is. Deze verminderde nauwkeurigheid neemt nog verder af met
toenemend contrast tussen de twee lagen. Dit is de eerste maal dat in de literatuur
de geometrische verdeling van de verzadiging is meegenomen in een machtsregel.
Met deze techniek kan de permittiviteit van twee-laags monsters zeer nauwkeurig
worden gemodelleerd.
We concluderen op basis van onze resultaten dat zanden door een enkele permittivteitwaarde kunnen worden gerepresenteerd. De waarde van deze effectieve parameter hangt echter sterk af van de bestanddelen, volume fracties en de verdelingen
in het zand en deze moeten worden meegenomen. Men dient zich te realiseren dat
de permittiviteit van zand frequentieafhankelijk kan zijn, niet vanwege de frequentieafhankelijkheid van de bestanddelen maar vanwege de geometrische verdeling
ervan.
Ainhoa G. Gorriti
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Sumenvatting
No son todos los que están, ni están todos los que son
Spanish saying
Acknowledgements
Finally we get to the part of the writing that most enjoy most, the acknowledgements. If you got here after reading all the previous chapters, congratulations! If,
on the contrary, this is the first page you looked for, well . . . congratulations too,
there is very little book left.
Writing a book takes a lot of effort and can be anything but an individual enterprise.
Much of the effort came from myself but this thesis would not be what it is without
the help of so many others.
First of all, I would like to thank Hans Bruining and Evert Slob for giving me the
opportunity of doing this research. All of the committee members for helping me
improving this text considerably. Prof. Fokkema for his constant motivation and
support, and for his contagious enthusiasm. And my supervisor, Evert Slob, for
giving me freedom and confidence in my research. Evert was there whenever I
needed him, he knew when to push and when to relax, working with him cannot
be dull. I am sure that wherever I go I will miss him. Hartelijke bedank, jefe!
The experimental set-up and the experiments present in this thesis were done in
the Dietz Laboratory. They would not have been possible without all the technical
team. Impossible is not in their dictionary. Andre Hoving, Peter de Vrede, Leo
Vogt and Karel Heller I want to thank you for your invaluable help and specially
for having those smiles and laughs ready at any moment of the day.
I have spent 5 years in the lab and running back and forth to the Mijnbouw. Many
people have come and go. I shared great and not so great moments with many.
Thank you all! Specially Nikolai for being a wonderful office mate and WillemJan for helping me out with the fluid flow through porous media. Marco, Quoc,
Mohammed, Roald and Jeroen for your different point of views. And Pacelli for
those evening talks.
Coffee breaks played a very important role as depressurises and problem solvers. I
am grateful for the martian surrealism, the political activism, the endless resources
of Google, and the vending machine. Antonio, Cas, Tanguy, Aletta and Sevgi see
you all in Mars and neighboring planets!
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200
Acknowledgements
I thought that at this point I would not need to thank my friends and family. It
would be understandable not to have any friends or family left after the last year
when I turned off the switch of the real world and crawled into my cocoon. But
hey! they supported me a great deal and deserve to be here. You are too many, so
I will keep it simple. I thank you all for the scientific discussions, the great days out
in the mountains, dinners and nights out ”de baretos”. The losers, the numerous
spanish mafia and its Leiden branch. And those back in el patio de vecinas. I am
truly grateful to my second family, I+D and los Filipiris, because without them
life would be so much boring. And my first family, abuelos, tios y primos, por su
gratificante alegria y saber vivir.
This list would not be complete without my parents. They introduced me to
skepticism and wonder and that is one of the most important things in my life, the
roots of my scientific character and a great source of fun. Thanks for believing
in me, for your unconditional support and for taking care of me when I needed it.
To my brother, for being ready for a caustic chat at any time and for his technical
support from the other side of the globe. And at last to Antonio, thank you for
being there when Murphy was ruling my world.
Muchı́simas gracias a todos,
About the author
Ainhoa González Gorriti was born in Madrid, Spain,
on the 6th of June 1974. During her childhood her
family moved around the country and overseas, until
they settled down back in Madrid. There, she finished
highschool in 1992 and joined the faculty of Physics
at the Universidad Complutense. Thanks to a scholarship, the year of ’95 she studied at the University
of Montana, and in the summer of ’97 she enjoyed
an internship at Norsk Hydro. Finally, in 1999 she
graduated. That year, she moved to Delft where she
joined the Dietz Laboratory, Faculty of Civil Engineering and Geosciences (CITG), department of Geotechnology. She investigates the electromagnetic characterization of geophysical materials. The work carried
out during the last years forms the basis of this thesis.
Currently, she works as a postdoctoral researcher in
the same field. Apart from geophysics and science in
general, she likes to travel all over the world, specially
if it involves going up and down mountain ranges.
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About the author
Publications
Peer-reviewed articles
Gorriti, A.G. & Slob, E. C. 2004. A new Tool for accurate S-parameters measurements and permittivity reconstruction, Submitted for publication: Trans. of
Geoscience and Remote Sensing
Gorriti, A.G. & Slob, E. C. 2004. Synthesis of all known analytical permittivity reconstruction techniques of non-magnetic materials from reflection and transmission
measurements, Submitted for publication: Geoscience and Remote Sensing Letters
Gorriti, A.G. & Slob, E. C. 2004. Comparison of the different reconstruction techniques of permittivity form S-parameters, Submitted for publication: Trans. of
Geoscience and Remote Sensing
Proceedings and symposium abstracts
Gorriti, A.G. & Slob, E. C. 2004. Permittivity Measurements and Effective Medium
Considerations, Proc. 10th International Conference on Ground Penetrating Radar
21-24 June 2004, Delft, pages 747-750.
Gorriti, A.G. & Slob, E. C. 2003. Analytical Determination of Permittivity from
Measured S-parameters, Proc. 65th EAGE Conference & Exhibition. 2-5 June
2003, Stavanger, paper F27.
Gorriti, A.G., Slob, E. C. & Bruining, J. 2002. A Coaxial Transmission Line for
Accurate Permittivity measurements from 300KHz to 3GHz, Proc. 8th EEGS-ES
Conference, 8-12 September 2002, Aveiro, pages 547-550.
Gorriti, A.G., Slob, E. C. & Bruining, J. 2002. Accurate Reconstruction of Permittivity from Coaxial Transmission Line Measurements, Proc. 8th EEGS-ES Conference, 8-12 September 2002, Aveiro, pages 551-555.
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