MASTER'S THESIS
Sensus Aquae in Ferrum ac Air
Characterization of Electrical Properties of a Moisture Measurement System
Christian A. M. Karlsson
Master of Science in Engineering Technology
Space Engineering
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
Sensus Aquae in Ferrum ac Air
Characterization of Electrical Properties of a Moisture Measurement System
Christian A.M. Karlsson
Luleå University of Technology
Department of Computer Science, Electrical & Space Engineering
Division of EISLAB
2011-08-24
I
Cover illustration:
Images from left to right
LKAB Kiruna aerial picture (2008), photo taken by LKAB
LKAB mine Kirunavaara, cross-section of mine, image produced by LKAB
Iron ore from Kirunavaara mine (2010), photo by C. Karlsson
The first CEO of Kirunavaara and Loussavaara mine, Hjalmar Lundbohm, drawing published by LKAB
A pile of magnetite concentrate (2010), photo taken by C. Karlsson (assisted by Rolf Schröter)
LKAB KK4 lightened, photo by LKAB
Pellets in a gloved hand, photo by LKAB
II
III
ABSTRACT
Fast precise moisture measurement in mineral process plants, has long been a difficult
problem. In this master thesis a foundation for a new way of measurement using transmission
line theory, is laid out. Macroscopic electro-magnetic properties of the measured sample can
be linked to those of the individual constituents of the sample. Thus one is believed to be able
to do a precise estimate of the amount of moisture in a sample. In our case the samples are
moist magnetite concentrate.
The characterization of sample macroscopic electro-magnetic properties is done by using a
coaxial cell, performing frequency spectra measurement. In this master thesis, the electrical
behavior that links the measured data to the macroscopic electro-magnetic properties is
examined. It exists two setups, one inductive and one capacitive.
The DC resistivity of the moist magnetite concentrate is determined and some interesting
phenomena, suggested to be self potential were discovered. This phenomenon is in general
known from geochemistry of mineral rich clays. This made the measurements of resistivity
tricky to perform, some measurements were achieved and a new relation between moisture of
the moist magnetite concentrate and resistivity is presented.
Studies have been performed on the history of attempts of solving this problem, as well as
current relevant research. Reference samples have been prepared in a geotechnical laboratory.
Impedance frequency behavior of the measurement equipment is measured and presented,
with different curves for different moisture content. From these measurements it is obvious
that the moisture affects the impedance of the measurement equipment and thus it is possible
to use this method to measure moisture content, if one can determine the electrical model
which describes the measurement system.
A few equivalent circuits of transmission lines have been discussed, and one was chosen to
represent the purely coaxial part of the coaxial cell. Other parts have been modeled with other
elements of AC theory.
A new electrical model for the coaxial probe is proposed, based on LC resonances. Some
Comsol FEM simulations have been performed on the coaxial cell, as well as on electric
property sensors. Ideas to designs of electrical sensors are discussed. It is mainly the geometry
which has been investigated.
First it was proposed that a phenomenon of resonance occurs at the interface of the bottom
and coaxial part of the cell, LC resonance. The resonance effect has later been investigated
and identified as quarter wavelength resonance, due to the length of the measurement cell and
change of wave velocity due to wave propagation in MUT.
Simulations of proposed circuit equivalents in Orcad with PSpice have given some correlation
with measured data for dry and non-conducting samples.
There is still work to be done, to complete an accurate mathematical model of the system.
IV
V
PREFACE
This master thesis covers some of the first trembling steps toward the solution of measuring
moisture content in the magnetic granular media, moist magnetite concentrate, also containing
some different minerals which add further complexity to the problem of electrical
measurements. This problem has engaged engineers and scientists for a few decades.
The recipient of the result is the mining company LKAB which is interested in the technique
for the refinement of their pelletization process. Also some electronic companies have shown
some interest to future developments for a wider production of measurement units. The reason
for high precision moisture measurements at LKAB is the effect it puts on the formation of
pellets. Pellets are round balls with a certain prerequisites of roundness and diameter. When
shaping the pellets, the moisture content of the moist magnetite concentrate affect the quality
and the shaping of the pellets. Thus having the ability to easily monitor the moisture content
of moist magnetite concentrate, would increase productivity and improve quality of pellets.
The idea is to find the moisture content by using one or two electric property mixing
formulas, the Maxwell-Garnet formula and / or the Bruggeman formula. These two formulas
describe the relation between electric properties and constituent volume fractions, thus one
can deduce the different volume fractions by measuring the electric properties.
In this master thesis we will not go into calculations with mixing formulas, but focusing on
the measurements of the electric properties of the material under test (MUT). The electric
properties which describe the media are electric permittivity [F/m], magnetic permeability
[H/m] and resistivity [Ωm]. Later these properties can be used together with the already
known properties of each separate component which then allows us to extract the moisture
content from one or both of the mixing formulas.
Christian Karlsson
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ACKNOWLEDGEMENT
I would like to thank my supervisor Torbjörn Löfqvist for his support, inspiration and
devotion to the project. Again Torbjörn Löfqvist and Johan Borg for extremely important
laboratory assistance without which the project would have been far more time consuming.
Lars-Göran Westerberg (TFMM) for his discussion around Comsol Multiphysics modelling
and Sverker Fredriksson (TFMM) for refinement of my report writing technique, through a
few courses.
Many people at EISLAB LTU has brought understanding to the problem and given valuable
help in theoretical issues. The following peoples are appreciated for these contributions
Torbjörn Löfqvist, Johan Borg, Åke Wisten, Johan Carlsson and Kalevi Hyyppä. Mikael
Larsmark provided help by constructing electrode plates for the resistivity measurement
equipment.
Dr. Per-Erik Martinsson (Project Manager for Process IT) for his interest and support for the
project SAFA. Thanks to Dr. Bertil Pålsson (TKG) for access to the geotechnical laboratory.
Magnus Westerstrand (TKG) for discussions on process water characteristics and self
potential. For test samples and introduction to the refining process by LKAB and Åsa
Partapuoli. Discussions of details on the process between grinding iron ore and the pellets
formation, as well as the characteristics of the magnetite concentrate and pellets by Seija
Forsmo.
Also many thanks to the people who have supported me throughout my master studies, those
who have made me determined to finish my master and my family for their support.
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TERMS, DEFINITIONS AND ABBREVIATED TERMS
AC
B
BUW
CCPL
Concentrate
DC
Dry Case
E
EISLAB
E-M
GPR
Green
Pellet
HUT
LTU
M
M-G
MUT
PEEC
SAFA
SDSP
TDEM
TKG
VNA
Wet Case
WikiUN
Alternating Current
Bruggeman
Bauhaus-University Weimar
Creative Commons Public License
Concentrated mineral powder
Direct Current
Material measurement where the Rp parameter is high enough to be neglected in
the model
Electric
Embedded Internet System Laboratory
Electro-Magnetic
Ground Penetrating Radar
Non-Sintered Pellet
Helsinki University of Technology
Luleå University of Technology
Magnetic
Maxwell-Garnett
Material Under Test
Partial Element Equivalent Circuit (method)
Sensus Aquae en Ferrum ac Air (project name)
Soil Dielectrics Spectroscopy Probe
Time Domain Electro Magnetic Measurement
Department of Chemical Engineering and Geosciences
Vector Network Analyzer
Material measurement where the Rp parameter is low enough to be
considered a part of the circuit.
Wiki User Name
X
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CONTENTS
Abstract .................................................................................................................................... IV
Preface ...................................................................................................................................... VI
Acknowledgement ................................................................................................................. VIII
Terms, Definitions and Abbreviated Terms .............................................................................. X
Contents ...................................................................................................................................XII
Figures ................................................................................................................................... XIII
1
2
3
4
5
Introduction ......................................................................................................................... 1
1.1
The Objective of the Project ........................................................................................ 1
1.2
The Objective of this Master Thesis ............................................................................ 1
Literature Study ................................................................................................................... 2
2.1
A Short Introduction to the LKAB Process of Interest ............................................... 2
2.2
General Properties of Magnetite Concentrate and Pellet............................................. 3
2.3
A Short Introduction to the Measurement Environment ............................................. 4
2.4
Summary of LKAB´s Earlier Tested Measurement Techniques ................................. 4
2.5
Other Research Groups ................................................................................................ 6
2.6
Possible Measurement Techniques .............................................................................. 7
2.7
Suitable Setups for Magnetic and Conductive Materials ............................................ 9
2.8
Electric Equivalent Models ......................................................................................... 9
Lab Work and Measurements ............................................................................................ 11
3.1
Preparation of Moist Magnetite Concentrate Samples .............................................. 11
3.2
Resistivity (DC) ......................................................................................................... 12
3.3
Permittivity, Permeability and Parasitic Resistance (AC) ......................................... 15
Why Comsol Modeling ..................................................................................................... 20
4.1
Coaxial Cell – Inductive and Capacitive Setup ......................................................... 20
4.2
Capacitive Surface Sensors ....................................................................................... 22
Calculations and Electric Modeling .................................................................................. 25
5.1
Evaluation of Models and Motivation for Chosen Model ......................................... 25
5.2
Improvements in Calculation after A. Saremi Work ................................................. 30
5.3
Calculations Using the Saremi Circuit Model - Inductive Setup .............................. 31
5.4
Simple Model Analyses ............................................................................................. 35
5.5
Calculation Using Electric Circuit Proposed by C. Karlsson .................................... 41
5.6
Resonance Analysis ................................................................................................... 48
6
Discussion of Results and Final Conclusions .................................................................... 50
7
Discussion of Future Work ................................................................................................ 51
8
References ......................................................................................................................... 53
Appendix A – Geometric and Material Constants ................................................................... 55
XII
Appendix B – Preparation of Moist Magnetite Concentrate .................................................... 56
Appendix C – Resistivity Measurements ................................................................................. 57
Appendix D – Mathematical System - Saremi Model ............................................................. 58
Appendix E – Inductive Mathematical System - Karlsson Model ........................................... 59
Appendix F – Capacitive Mathematical System - Karlsson Model ......................................... 61
Appendix G – Matlab Functions - Cost and Zload Functions.................................................. 62
Appendix H – Matlab Script – Inductive Optimizer ................................................................ 63
Appendix I – Matlab Script – PLOT Results ........................................................................... 66
Appendix J – Matlab Script – Constants .................................................................................. 68
Appendix K – Matlab Script – Data Loader ............................................................................ 69
FIGURES
Figure 1: A flow diagram of a part of the LKAB process. ......................................................... 2
Figure 2: A picture of a sandcastle, illustrating the properties that granular material gains from
moisture. This picture has been released in to the public domain, at pdphoto.org. ................... 3
Figure 3: A pile of dry magnetite (left) and a pile of 11 % wet sample pile (right). .................. 4
Figure 4: Schematic picture of linking coaxial elements, licensed use under CCPL (CC BYSA 3.0) by WikiUN Qianchq [30; 31]. .................................................................................... 10
Figure 5: Industrial mixer, photo by T. Löfqvist ...................................................................... 11
Figure 6: Table of ideal moisture content and actual moisture content. .................................. 12
Figure 7: In the back there is a white bucket with moist sample of magnetite concentrate. In
the front there is a resistivity measurement setup, consisting of a plastic pipe, with outside
covered in aluminium foil. In the bottom of the pipe, there is a circular copper surface. The
surface constituting an electrode connected to a wire exiting the bottom of the setup. A similar
circular surface can be thread into the pipe from above, thus connecting through the sample
put in beforehand to the surface in the bottom and finally constitute a connected system. One
can do measurements between the two cables exiting the setup. The wooden plate in the
bottom is for stability of measurement setup. To the left, an abandoned measurement device, a
general resistance meter. .......................................................................................................... 13
Figure 8: Schematic picture illustrating the geometric dependence of the resistivity
measurement setup, licensed use under CCPL by WikiUN Omegatron [30]. ......................... 14
Figure 9: DC resistivity versus actual moisture content. The red arrow indicates that the value
at zero percent moisture, most probably can be considered infinite, as the measurement
equipment reached its limit. The actual characteristic is believed to be of the type one over
fraction, according to this diagram though, from two percent to eleven percent an almost
linear behaviour. ....................................................................................................................... 15
Figure 10: Measurement setup with coaxial cell and VNA. The coaxial cell is connected only
through one VNA port. The wooden box contains a calibration set for calibration of the VNA.
.................................................................................................................................................. 16
Figure 11: Real part impedance spectra for inductive setup translated from the measured S11
parameters. ............................................................................................................................... 18
XIII
Figure 12: Imaginary part impedance spectra for inductive setup determined from the
measured S11 parameters. ........................................................................................................ 18
Figure 13: Real part impedance spectra for capacitive setup determined from the measured
S11 parameters. ........................................................................................................................ 19
Figure 14: Imaginary part impedance spectra for capacitive setup determined from the
measured S11 parameters. ........................................................................................................ 19
Figure 15: Cross-Section of coaxial cell. ................................................................................. 20
Figure 16: 3D Model of Inductive Coaxial Cell. ..................................................................... 21
Figure 17: 3D Model of Capacitive Coaxial Cell. ................................................................... 21
Figure 18: Geometry of Capsense1. ......................................................................................... 22
Figure 19: Graphical E-Field Representation of Capsense1. ................................................... 22
Figure 20: Graphical E-Field Representation of Capsense2. ................................................... 23
Figure 21: Graphical E-Field Representation of Capsense3. ................................................... 24
Figure 22: Graphical E-Field Representation of Capsense4. ................................................... 24
Figure 23: L-type equivalent circuit with correction equivalent. ............................................. 27
Figure 24: T-type equivalent circuit with correction equivalent. ............................................. 27
Figure 25: PI-type equivalent circuit with correction equivalent. ............................................ 28
Figure 26: Saremi equivalent circuit with correction equivalent. ............................................ 28
Figure 27: Semi-lumped semi-distributed equivalent model with split correction lid and
bottom....................................................................................................................................... 29
Figure 28: Karlsson proposed inductive equivalent circuit with correction equivalent. .......... 29
Figure 29: C. Karlsson proposed capacitive equivalent circuit with correction equivalent. .... 30
Figure 30: Real µr provided by optimization of Saremi model for different water content. ... 33
Figure 31: Imaginary µr provided by optimization of Saremi model for different water
content. ..................................................................................................................................... 33
Figure 32: Conductivity provided by optimization of Saremi model for different water
content. ..................................................................................................................................... 34
Figure 33: Error surface representing the impedance error for the optimized Saremi model
compared to the measured impedance...................................................................................... 34
Figure 34: µr provided by simple inductive model for air. ...................................................... 36
Figure 35: µr provided by simple inductive model for water. ................................................. 36
Figure 36: µr provided by simple inductive model for dry magnetite concentrate. ................. 38
Figure 37: µr provided by simple inductive model for 2% moisture. ...................................... 38
Figure 38: µr provided by simple inductive model for 4% moisture. ...................................... 39
Figure 39: µr provided by simple inductive model for 6% moisture. ...................................... 39
Figure 40: µr provided by simple inductive model for 7% moisture. ...................................... 40
Figure 41: µr provided by simple inductive model for 8% moisture. ...................................... 40
Figure 42: Measured equivalent impedance for air (theoretical value by simple model),
Matlab Calculated Impedance Using C. Karlsson Circuit. ...................................................... 42
XIV
Figure 43: Measured equivalent impedance for de-ionized water (mQ).................................. 42
Figure 44: Measured equivalent impedance for dry magnetite concentrate. ........................... 43
Figure 45: Orcad air equivalent with constant maximum magnitude resistance. .................... 44
Figure 46: Orcad plot of equivalent impedance for air. ........................................................... 44
Figure 47: Orcad water equivalent with constant maximum magnitude resistance. ................ 45
Figure 48: Orcad plot of equivalent impedance for water........................................................ 45
Figure 49: Matlab calculation of the air case. .......................................................................... 46
Figure 50: Matlab calculation of the water case. ..................................................................... 47
Figure 51: Matlab calculation of the dry case. ......................................................................... 47
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1
INTRODUCTION
This master thesis covers measurement of E-M properties of magnetic and conductive
granular material, moist iron ore concentrate. The reason to measure E-M properties is that it
forms the foundation for determining moisture content.
1.1
The Objective of the Project
The objective of the project is to do precise moisture measurement of moist iron ore
concentrate. Having the ability to easily monitor the moisture content of moist iron
concentrate, would increase productivity, improve the quality of iron ore pellets and make
production cheaper.
The vision of this project is to develop an in-line moisture sensor unit, capable of accurate
precise measurement of moisture down to 0.1 % precision. To achieve this, it is necessary to
know the density and temperature of the MUT, if the sample is not in a controlled
environment. Hence for characterization of measurement equipment and MUT, we stay in a
controlled environment, the laboratory. Later when field measurements are performed, the
effect of temperature and density deviations has to be incorporated.
1.2
The Objective of this Master Thesis
The objective of this master thesis is to develop models for determination of E-M properties,
which constitutes a basis for a new measurement technique for determining moisture content.
The master thesis shall also work as documentation for further research and development, as
well as describe the work performed during the master thesis period.
The E-M properties which describe the characteristics of material under test (MUT) are the
electric permittivity, εr [F/m], magnetic permeability, µr [H/m], and resistivity ρ [Ωm]. Later
these properties can be used to find the moisture content of the MUT.
For the extraction of the E-M properties εr and µr, the intention is to use a single or two
different near-field antennas / sensors. Probably one electric and one magnetic sensor, needs
to be designed, if a combination is not possible. The E-M properties can be retrieved from the
expressions of capacitance and inductance. Then the value of εr and µr can be determined due
to the following relationship [1], in equation 1.1:
(1.1)
Where C [F] is the capacitance obtained from the equivalent models when the model has been
applied to the moisture measurement values of impedance. C0 [F], the capacitance according
to the model when air (or preferably vacuum) constitute the MUT.
The work is based on an earlier attempt to determine moisture content, from which the results
brought some doubt. It was preferable to find a better lumped electrical equivalent model for
the measurement cell and make new measurements to verify the model. This cell is supposed
to be evaluated by reflection measurements which could be related to impedance and at the
end to capacitance and inductance, through the electrical equivalent model. Further literature
studies and discussions have brought a deeper understanding and have resulted in some ideas,
which will be presented throughout this master thesis.
1
2
LITERATURE STUDY
Studies have focused on different techniques to measure moisture or E-M properties. The
analyses have covered what the physical and chemical properties are of the magnetite
concentrate used in the moist filter cake at LKAB? Other tested methods for measuring
moisture in moist magnetite concentrate for LKAB? Which other research groups exists?
Which apparatus has been used for E-M measurements on materials? Which setups can be
used for measurements on magnetic and conductive materials? Which electric models are
there? What work has earlier been performed within the project at LTU?
2.1
A Short Introduction to the LKAB Process of Interest
The LKAB process that is of interest for the understanding of the problem is the balling
process and the preparation of the moist magnetite concentrate to become a green pellet, the
wet process. Magnetite is a specific type of iron ore, below these terms will come in context.
For the balling process it is very important to control the moisture content of the moist
magnetite concentrate. First the iron ore is ground to suitable size for pelletizing, cleaned by
magnetic separation and flotation. The resulting slurry is mixed with different additives and
put through press filters to reduce water content, resulting in a moist filter cake suitable for
balling. The moist filter cake is stored in a storage silo, where it is further transported from
and mixed with a binder and again put in a storage silo. From there portioned to a balling
drum, where “seeds” starts the balling process and the balls slowly grows, exemplified by
rolling a snowball. The described part of the process can be seen in Figure 1.
Figure 1: A flow diagram of a part of the LKAB process.
2
If one wish to understand the reasons behind the interest of moisture content of the moist filter
cake (or our lab equivalent, moist magnetite concentrate) one could think of the properties of
sand castles depending of sand moisture content (Figure 2), sand being another granular
material. Also magnetic effects operate in this material, causing attraction and repulsion
between particles.
Figure 2: A picture of a sandcastle, illustrating the
properties that granular material gains from moisture.
This picture has been released in to the public
domain, at pdphoto.org.
A selection is made by screening the green pellets to correct size [2]. Green pellets that has
exceeded the determined size, goes to a new grounding station and are inserted into the
balling drum as seeds again. Smaller green pellets also go back to the drum as seeds. Those
green pellets that have passed screening are loaded to the pelletizing machine for drying and
sintering, at about 250º C and 1250º C respectively.
It is of course during the wet process which LKAB would like to perform online moisture
measurements and more specifically between the filters and balling drum (red square in
Figure 1). Today these measurements are performed manually once per work shift; a sample
is taken, weighed, dried and weighed again, this will provide the amount of moisture as
percent mass [3].
2.2
General Properties of Magnetite Concentrate and Pellet
Magnetite is an iron oxide containing, FeO·Fe2O3 (Fe3O4). The magnetite concentrate are
finely ground iron ore. The fineness of the magnetite concentrate is roughly 80% - 45 µm in
diameter (meaning that 80% of the magnetite granules are 45 µm or less in diameter),
controlled by screening. Pelletizing is done using external binders, where the most common
one is bentonite clay, commonly sodium activated bentonite clay. Typical composition of
magnetite concentrate is 71 % Fe, 23 % Fe2+ and 0.6 % SiO2. Additives are about 0.5 %
bentonite binder and grounded olivine, (Mg,Fe)2SiO4. Our laboratory equivalent however
contains as close to pure dry magnetite concentrate as it ever becomes in the LKAB process.
For mathematical modeling our approach is to assume a clean matrix of magnetite based on
spherical particles.
The water mass concentration of the moist filter cakes used today is around 8 - 9 %. At this
concentration it provides a suitable base for the agglomeration process. Too low water content
3
will have a to slow growth or none at all, too high will either grow too fast and / or collapse
due to insufficient binding properties. In modeling the moist magnetite concentrate will be
modeled as a fixed matrix of magnetite with fixed inclusions.
The accepted green pellets have a diameter around 10 mm with some small margin.
Figure 3: A pile of dry magnetite (left) and a pile of 11 % wet sample pile (right).
2.3
A Short Introduction to the Measurement Environment
The MUT (moist magnetic concentrate) is due to its fineness and magnetic properties a dirty
material and thus tends to smear down most things in its environment. The following
characteristic phenomena occur in the moist magnetite concentrate; conduction, galvanic
corrosion, self potential and it acts abrasively. When looking at the moist filter cake, one can
add seasonal change of the salinity of the process water (as well as the moisture abundant in
the MUT.)
2.4
Summary of LKAB´s Earlier Tested Measurement Techniques
Throughout the years many methods of measuring moisture content have been tested [3], the
currently used measurement technique is the manual weigh-dry-weigh test, which provides an
accurate result. This process is time consuming, why LKAB would wish to have an in-line
and real-time measurement technique.
Tested methods which are not related to the measurement technique we try to introduce will
be mentioned. Methods related to our measurement technique will be deeper discussed with
emphasize on their implications for our measurement technique.
Unrelated Techniques Tested
First the weigh-dry-weigh method has been the target for development and refinement, but
still this method suffers from difference in sample collection, heating may induce chemical
reactions changing mass relations, the method is time consuming and does not allow in-line or
real-time measurements.
Different general commercial moisture measurements units have been tested without any
stable results, mainly due to the complex nature of the moist magnetite concentrate. Chemical
analysis of moisture dependence of chemical reactions resulting in a pressure proportional to
the moisture contents has been tested; the pitfall of this test seems to have been its minor
sample volume.
4
Multivariate analysis has been tested; but has been considered in minor further interest to this
project. The problems with the technique are the stability of the method when it is exerted to
external disturbance and that it is a system that needs training by estimating moisture from
pre-known moisture samples. Radiometric measurement technique has been tested where the
observation only could be done for long term variation over the period of a day. Therefore has
been considered being a non sufficient measurement technique.
Related Techniques Tested
Within the microwave area, an instrument called Hydro-Probe II has been tested, by
submersing it into the moist magnetite concentrate. The general opinion then was that the
equipment did not work as the moist magnetite concentrate was absorbing around 90 % of the
signal. The signal used was 300 MHz to 1.2 GHz, either this experiment shows that the
commercial techniques for moisture measurement does not work for moist magnetite
concentrate, due to its complex nature (if the instrument was not developed having the
complex nature in mind) or it shows that EM waves has trouble of penetrating the MUT,
especially in the microwave range.
At a seminar of moisture measurements with microwaves in subject [3] it was stated that
microwaves could not be used as a measurement technique as moist magnetite concentrate
absorbs the microwaves. Also commercial companies have stated that this is the case, that
microwaves cannot be used as measurement technique due to the major absorption of signal
power. Measurements of optical reflection have been tested. Optical techniques in general
require massive maintenance, calibration and are very sensitive to external disturbance. Only
if a method can find some frequency where all disturbances are extremely low compared to
the response of the variable representing the moisture content this method has usability.
Electric conductivity has been investigated as an interesting method, by measuring the
conductivity between two electrodes; one could determine the moisture content. In the process
industry this method is not believed to be of sufficient stability due to the many parameters
from which the moisture measurement depends on. The measurement depends on the grain
size, process water salinity, moist magnetite concentrate temperature, geometry, compression
and moisture content. Another experiment confirms that the resistivity is heavily dependent of
moisture content [4]. Resistivity is the reciprocal of conductivity.
Short on the Apparent Demands of the Measurement Equipment
When analyzing earlier evaluated measurements it is apparent that





Accuracy - The accuracy should be around 0.1%
Tolerant - The measurement should be stable towards environmental changes /
process disturbance
Instant - Preferred to be real-time measurements
Independent - Should have minimal maintenance
Representative - Propose a value representing a whole batch
These are quite tough demands under the circumstances for in-line measurements. However
the method presented in this master thesis is believed to cope with them.
Untested Techniques and Our Role in the Continuation
From studies of earlier tested techniques it obvious that one will need to utilize a
multivariable solution as all methods suffer from other environmental effects. The author
5
believes that either submerged measurements with sample contact will have the most
controllable and/or measurable environment or a macro sized capacitance and/or inductance
and/or resistivity measurement module. Examples of non tested methods (2005) are heat
conduction, density, measurements with ultrasound and infrasound, pressure posed to static
obstacles, nucleus spin resonance (Proposed by the company Omicron) and other chemical or
heating related methods.
There are also many untried measurement techniques related to electromagnetic and/or
frequency spectra measurement. These methods are mentioned as radio waves (transmission),
dielectric property measurements, electromagnetic AC measurements, inductive methods,
capacitive methods; at this point it seem a bit messy as all these seems to be different parts of
the same kind of measurement technique, just that they are mathematically- and model-wise
separated.
Some short conclusions when it comes to electromagnetic measurements. One has tested
transmission of microwaves submerged, optical surface reflections of microwaves (outside
media) and electrical resistivity measurements submerged. Where the transmission and optical
reflection techniques has been discarded due to lack of quality. The resistivity has shown to
be sensitive to moisture content, but faces uncertainties due to large number of dependent
parameters.
The method proposed by T. Löfqvist and C. Karlsson is to utilize a model which measures the
AC spectra of capacitive, inductive and conductive behavior of the moist magnetite
concentrate. By large means it is reasonable to believe that a commercial product must also be
able to measure the temperature and resistivity of process water and dry magnetite or utilize
some kind of calibration at least month wise. At first the ambition would be to prove the
legality of this method. A big strength is that this method utilizes many parameters which by
themselves are moisture dependent. If one could determine these parameters, there would be
many parameters to use in moisture estimation. Only one of the parameters is sensitive for all
constituents, that is capacitance and one parameter is sensitive for the magnetite that is
inductance, the actual material dependent variables are the electric permittivity (dielectric
constant) and magnetic permeability (magnetization constant). Together these two will allow
us to determine the amount of magnetic particles (magnetite grains) from air and water, with
that as input we can determine the amount of particles with higher dielectric constant from
those that have less. That is water from air that in the end allows us to estimate the water
content. This estimation is done using a two phase mixing formula causing an input to a three
phase mixing formula, allowing the water content to be determined. Another interesting
property we might find from knowing the permeability is the dry density or mass volume of a
moist powder.
2.5
Other Research Groups
A book containing an overview of theory and techniques in the field of E-M aquametry have
been written by K. Kupfer at BUW [5]. One co-writer of the book is A. Sihvola at HUT
whose research is focused to the area of material characterization of matter with high
dielectric loss.
An Italian group involved in planetary research and sub soil measurements has performed
measurements with a toroidal probe. This was a step in the direction of building an instrument
intended for a mission to Mars to perform moisture / ice content measurements in Martian
soil. The Martian soil connects with our experiments due to its rich iron content.
6
There are also many other people around the world which has brought contributions to and
been working on the problems of moisture measurements, as well as measuring other material
properties using the electromagnetic properties.
Short information of A. Sihvola’s Work and Concerning M-G and B
A. Sihvola has been working with aquametry [6; 7], with the main target of building model
systems for material with high dielectric losses. In his work he has studied the geometry
effects of host and inclusions, where the host is the main constituent and inclusions are
secondary constituents. Mainly it is the formulas by M-G and B, which have resulted in more
knowledge for the effects in using mixing formulas. He is the author of the book,
Electromagnetic Mixing Formulas and Applications [7]. A. Sihvola together with L. Jylhä has
also developed a differential formula for effective permittivity [8] which combines the effects
of the M-G and B [7] formulas, thus they have achieved a formula that are valid for quite high
permittivity contrasts which seems to be the case of these measurements. Initially the
ambition will be to prove the electric behavior of the measurement equipment.
Short Information on the Work Performed by the Italian Group
The group consists of many members of different disciplines; astrophysics, electronics,
geology, computer and system engineering. They have been working with several techniques
to discover the E-M parameters of the sand of Mars, as SDSP, GPR and TDEM. The reason
why they use the many techniques is that the environment at Mars is varied and they have no
knowledge of any parameter in advance. On earth one knows some parameters and can
calibrate for them in calculations.
2.6
Possible Measurement Techniques
Studies were performed on summarizing documents and articles evaluating different
instruments; Venkatesh and Raghavan (2005) [9], the NIST standard document [10] by
Baker-Jarvis et al. (2005) and the two ASTM standard documents D 7449 (2008) [11] and D
5568 (2009) [12]. The NIST document is the most valuable of the documents.
Literature studies and discussions have come up with a few different ideas to setups using
capacitance and inductance to determine the E-M constants. The different methods are based
on reflection measurements or reflection-transmission measurements. The reflection methods
will be discussed further in subchapters to identify problems or advantages of each method,
using facts from article studies.
Reflection Measurements
 Open-ended coaxial measurements
 Toroidal probe
 Open and closed coaxial cell
 Micro-strip sensor
Reflection-Transmission Measurements
 2-port coaxial cell
 2-port transformer
Earlier unsuccessful experiments on plate capacitors, N. Örlander, and a technique developed
in an earlier report by A. Saremi, a version of reflection measurements of an inductive
7
(closed) / capacitive (open) coaxial cell. This open/closed technique will be further developed
by the author later in this master thesis.
Open-Ended Coaxial Measurements
This measurement technique is based on the change of fringing field capacitance due to
change of material measured, in example compared to air. The simplest interpretation is this
by Jaspard and Nadi [13], in equation 2.2. First the interpretation of measurement as
admittance, in equation 2.1:
(2.1)
Where Y11 [1/Ω or S] correspond to the admittance of the system using only one port of the
VNA. Y0 [1/Ω or S], is the characteristic admittance of the VNA cable, in our case (1/50) S.
The S11 parameter is a scattering parameter which says something of the reflection caused in
the macroscopic measurement equipment. The indexes in all cases refer to the first element of
a matrix, where the index represents a port number; in this case only port one. Thus looking at
the admittance due to the reflection of a pulse sent at port one, observed at port one.
(2.2)
G [1/Ω or S] is the conductance (inverted impedance) due to leak currents between the center
conductor and the outer cylinder. The angular frequency ω [rad/s], Cfc [F] capacitance
regarding the full length of coaxial cable, from the VNA cable connector to the cut surface
used as sensor and Cfm [F] being the full material dependent fringing field capacitance.
Simulations in Comsol have not been able to confirm the validity of these, which could have
been caused by bad conditions for boundary values in the simulation or that the model is
insufficient. Most probably this is due to adaptations of the model, which has not been
implemented, dependency of other geometrical factors. There is also a more advanced model
for determining the complex εr, described in the book Microwave Electronics: Measurement
and Materials Characterization [14]. Even though the title of the book refers to microwaves,
this is somewhat applicable for RF as well.
This technique / theory might be needed for the construction of a surface sensor for the
measurement module, but probably cannot be used to characterize the material properties in
our case, as the response of the fringing field capacitance should be quite small and it will
only measure the capacitive property within the measured volume. This was realized from
Comsol FEM simulations and reading articles.
Also from simulations and articles it seems to be most sensitive at the core, where current are
injected and there are a certain area of the flange shield which gives optimal fringing field
response. Also letting the core penetrate deeper and letting the flange stay at the surface may
have a good impact on the fringing field capacitance. Important contributions to this area
which the author has studied follows; Zheng and Smith (1991) [15], Moreau and Aziz (1993)
[16], Gasvenor (1993) [17], Baker-Jarvis (1994) [18], Aimoto and Matsumoto (1996) [19],
Folgerö and Tjomsland (1996) [20], Jaspard and Nadi (2001) [13], Hagl et al (2003) [21],
Cheng et al. (2006) [22], Ellison and Moreau (2006,2008) [23; 24], MacLaughlin and
Robertsson (2007) [25] and Oppel et al (2008) [26].
8
Toroidal Probe
As described above this has been tested earlier by an Italian group. The toroidal probe seems
to be a good measurement technique, but the trick is to get the material into the toroid. The
solution of the Italian group is to use an air winded toroid, which would allow the toroid to be
pressed down into the moist magnetite concentrate. Then the toroid is filled with moist
magnetite concentrate and as the inductance is only dependent of the material characteristics
of the material inside the toroid windings, it is easy to determine the measured volume.
The group used a constant volume of Martian soil and the sensitivity should be really good
due to the concentrated fields. This geometry may be a good geometry for the magnetic
sensor to be used for in-line measurements. No experiments, modeling or deeper literature
studies have been done yet, on this method.
Open and Closed Coaxial Cell
An earlier report by A. Saremi [27] has been written on the coaxial cell, but some of the
modeling is questionable. Some new measurements, analysis and simulations will bring
further light on the subject. The general overview sources contained the best written material
found for this measurement technique, as well as previous measurements performed at
EISLAB [27] and newly acquired knowledge from new measurements and modeling. See
chapter 5 for further information.
Micro-Strip Sensor
A micro-strip sensor is believed to be one of the sensors to be used for an in-line solution.
Together with the magnetic toroid / loop / transformer it would constitute a full laboratory
solution for the measurement of εr and µr. The drawback associated with this type of sensor is
its surface sensitivity and that it will not represent a whole batch properly. To solve these
issues, one solution could be to have a lot of measurement units well distributed and
submersed in the moist magnetite concentrate. This would allow a mean value representation
of the moist magnetite concentrate moisture content and decrease the risk of moist concentrate
surface anomalies.
Interesting articles discussing this type of sensor are published by Avitabile et al. (2001) [28],
interdigital dielectrometry by Guggenberg and Zaretsky (1995) [29] and Sonnet Application
Note, SAN-206A (2006) [30], for design of RFID sensors. RFID sensors are probably useful
sensors to use for material measurements in the radio frequency spectra, even though the
measurement in itself has nothing to do with RFID.
2.7
Suitable Setups for Magnetic and Conductive Materials
When determining moisture in magnetic and conductive media, it is important to use both
inductive and capacitive technique. For the material characterization it seems to be suitable to
use the open and closed coaxial cell. This will allow measurement of a constant volume,
which will simplify the verification of the measurement. When measuring in-line, the pair of a
micro-strip sensor (mainly capacitive) and a magnetic sensor (mainly inductive) will probably
constitute the laboratory measurement setup.
2.8
Electric Equivalent Models
There are two types of electrical equivalent models which can be used to model a
transmission line [1]. It is the lumped parameter model and the distributed parameter model.
In this sense we either see the transmission line as a RLC circuit or that the element is made
9
up of infinitesimal elements of RLC circuits. The modeling which has been performed on the
coaxial cell is of the lumped parameter type.
When it comes to distributed parameters, there is a common model for the impedance of an
element of a transmission line [1] (characteristic impedance). There is a special formula to
relate the characteristic impedance to the impedance seen by a VNA, through different
segments of different lengths for lossy lines. A schematic sketch in Figure 4 describe the
different variables in equation 2.3.
Figure 4: Schematic picture of linking coaxial elements, licensed
use under CCPL (CC BY-SA 3.0) by WikiUN Qianchq [30; 31].
(2.3)
(2.4)
Here the Zi+1 [Ω] the complex impedance at any point on the line, looking towards the load.
The Zi [Ω] are the complex impedance at the distance li [m] towards the load. Z0,i [Ω] the
characteristic impedance of the coaxial line, linking point (i+1) with (i). γi [1/m] is the
propagation constant (segment i), αi [dB/m] the attenuation constant (segment i) and βi
[rad/m] the phase constant (segment i). The expressions for matched, short and open refers to
special conditions regarding the impedance of the load, where one can see the impedance at
Z2 by using Z0 = Z0,1 and γ = γ1. However in our case using the coaxial cell, the aluminum
bottom will not be represented by the short condition as resistance and inductance exists.
Using the plastic lid we might be able to use the condition, open, provided that one consider
that the plastic lid constitute another coaxial element and the fringing field capacitance are
small enough. The fringing field capacitance is the capacitance due to the escaping field in the
open or semi-open case.
The continued work within this master thesis will be with the lumped parameter model.
Lumped parameter models for two conductor transmission lines can be modeled in three
different ways according to literature; L-type, T-type and PI-Type [1] p.477 and 533. The
variation of these makes it plausible that other combinations of the components could be used
as well.
The lumped parameter model proposed by A. Saremi is another one [27], see chapter 5.3.
10
3
3.1
LAB WORK AND MEASUREMENTS
Preparation of Moist Magnetite Concentrate Samples
Purpose
The sample preparation of moist magnetite concentrate was performed in the laboratory of the
TKG department. The samples will provide both a basis for calibration and test. For full
liability these measurements should be repeated a few times with newly prepared moisture
samples.
Equipment
The equipment used was 9 plastic buckets with lid, scale, an industrial mixer (depicted in
Figure 5), measuring cylinder, de-ionized water and bags with dry magnetite concentrate.
Execution
Weighing empty bucket with lid, weighing half-filled bucket with lid and calculating the
amount of dry magnetite and the amount of water that should be added to give each moistened
sample its right moisture content 0, 2, 4, 6, 7, 8, 9, 10, 11 mass percent. Due to inaccuracy of
the amount of water added to the samples (equation 3.1), also actual percentage has been
calculated (equation 3.2).
(3.1)
(3.2)
Where mm is the mass of magnetite, mw is the mass of water, fw is the mass fraction of water
and the mass fraction of magnetite are of course fm=1-fw.
Figure 5: Industrial mixer, photo by T. Löfqvist
11
Results and Conclusions
The resulting moisture contents deviates quite a bit from the intended, thus using separate
buckets measuring dry magnetite and water, will make the percentage more precise and
constitute a better testing frame. The actual moisture contents are presented in the table in
Figure 6 below.
Moisture %m Real %m
2
2,044
4
3,054
6
6,061
7
7,278
8
8,311
9
9,373
10
10,43
11
11,44
Figure 6: Table of ideal
moisture content and actual
moisture content.
3.2
Resistivity (DC)
Purpose
The purpose of determining resistivity was to model parasitic resistance in the model of the
impedance for calculation of permittivity / permeability dependence.
Equipment
The setup (depicted in Figure 7) consisted of a cylindrical plastic pipe with a circular plate in
the bottom and a height adjustable circular plate in the top. Both plates connected to wires.
The measurement apparatus were a regular resistance measurement unit and a fine tunable DC
voltage measurement unit with programmable current.
12
Execution
Figure 7: In the back there is a white bucket with moist sample of magnetite concentrate. In
the front there is a resistivity measurement setup, consisting of a plastic pipe, with outside
covered in aluminium foil. In the bottom of the pipe, there is a circular copper surface. The
surface constituting an electrode connected to a wire exiting the bottom of the setup. A
similar circular surface can be thread into the pipe from above, thus connecting through the
sample put in beforehand to the surface in the bottom and finally constitute a connected
system. One can do measurements between the two cables exiting the setup. The wooden
plate in the bottom is for stability of measurement setup. To the left, an abandoned
measurement device, a general resistance meter.
The measurement equipment was connected to the wires of the measurement apparatus. First
DC Resistance was measured with a resistance meter, it was noticed that the DC resistance
was changing rapidly, this seemed strange. Second voltage was measured with the same
equipment and no voltage source attached, still the measure equipment showed a voltage
across the plates. Decision to use constant current measurement equipment instead was made,
and then one could make measurements of voltage for positive and negative current. Followed
by calculation of resistance and taking the mean value of them to find the most accurate
resistance. For each sample measurement, also the height of the cylindrical bulk mass sample
was noted. The resistance was then converted to resistivity and conductivity by a geometric
dependence [1] p.166 using equation 3.3. A schematic picture in Figure 8 depicts the different
variables.
(3.3)
13
Figure 8: Schematic picture illustrating
the geometric dependence of the
resistivity measurement setup,
licensed use under CCPL by WikiUN
Omegatron [30].
Where ρ [Ωm] is the resistivity, R [Ω] the resistance measured on the sample length l [m] and
using electrodes of area A [m2].
Results and Conclusions
The measurement data and results can be found in appendix C and results in the following
figure, Figure 9. The effect of generating a voltage, suggested to be self potential, is very
interesting and will certainly affect electrical measurements on the moist magnetite
concentrate. At least for DC measurements and could be another interesting phenomenon of
study further for other reasons or maybe for measurement of moisture with DC levels. This is
a problem which mainly occurs during DC measurements, should not interfere too much in
the AC model.
Analysis of the measured data show that resistivity is approximately reversely proportional to
the moisture content expressed as a fraction, at least within the interval 0-12 % moisture. Only
doubts would be around 2-4 %, but as our area of interest is mainly around 8 % the results can
be considered sufficient. The bad results around 2-4 % could also somehow be related to the
measurements, as the voltage was fluctuating. The recording of values was done when the
fluctuations slowed down a bit. One could issue that it should be done after a certain time, but
satisfactory results could be presented with the current method.
14
Figure 9: DC resistivity versus actual moisture content. The red arrow indicates that
the value at zero percent moisture, most probably can be considered infinite, as the
measurement equipment reached its limit. The actual characteristic is believed to be of
the type one over fraction, according to this diagram though, from two percent to
eleven percent an almost linear behaviour.
The measurements have been performed in a laboratory environment where the temperature
has been at room temperature and the packing density of the MUT has been as low as possible
but still so that the amount of macroscopic air pockets are at a minimum. Measuring higher
moisture percentage gets more troublesome as air pockets are unavoidable due to the
agglomeration process. Despite this effect it is obvious from the measurement data that the
conductivity is very high, around 10 % moisture. The measurement of dry magnetite
concentrate gave the same voltage value as it did when the two electrodes were in free space.
This indicates that if moisture content goes towards zero, the resistivity goes toward infinity.
This implies that there are two models needed to model each setup of equipment one with
parasitic resistance and one without, if it is not possible to use a combined model which
satisfies both cases.
3.3
Permittivity, Permeability and Parasitic Resistance (AC)
Purpose
By determining the E-M properties of the moist magnetite concentrate it is possible to
determine moisture content through the mixing formulas Maxwell-Garnett and Bruggeman.
Thus this measurement is the core of this project to accurately determine the E-M properties.
For accuracy it is also very important that we understand the effects in the MUT and the
measurement cell. Which covers the most of the time spent in this master thesis work. The
first objective is to measure the complex and frequency dependent S11 parameters. From there
one can calculate the E-M properties.
15
Equipment
Equipment used was a VNA and the measurements cell, depicted in Figure 10. The
measurement cell drawing, dimensions and more pictures are to be found in appendix A. The
measurement cell can be used in two setups, inductive (for magnetic properties) using
metallic bottom and capacitive (for electric properties) using plastic bottom. Thus we end up
with two equations and two unknowns for each measurement. The cables to the VNA were 50
Ω cables.
Figure 10: Measurement setup with coaxial cell
and VNA. The coaxial cell is connected only
through one VNA port. The wooden box contains
a calibration set for calibration of the VNA.
Execution
Scripts for GPIB communication were used to make ten measurements for each sample, and a
statistical analysis is possible to determine the variance and other statistical measures, when a
satisfactory model has been found. The script acquires the frequency used and the complex
S11 parameters from the VNA as a matrix of the size of frequency times ten. The matrices are
stored in Matlab files (.mat). The S11 parameter could be translated to the impedance seen at
the connection between 50 Ω VNA cable and the connector-lid system of the measurement
cell. According to equation 3.4 below:
(3.4)
Zeq is the impedance seen by the VNA and correspondingly the impedance matched with the
equivalent circuit discussed in chapter 5, at the time of publishing believed to be best
described by the model proposed by C. Karlsson. Γ represents the reflection coefficient [1]
16
p.442. In our case Z0 is 50 Ω (characteristic impedance of VNA cable) and reflection Γ is
equal to the S11 scattering parameter (one VNA port measurement).
Measurements were done on various moisture contents 0 %, 2 %, 6 %, 7 %, 8 %, 9 %, 10 %
and 11 %. Where the intervals are shorter around 8 % to give better accuracy at the interval
where it is intended to determine the moisture content in the end. Data from moistening
process could be found in appendix B. Some measurements have been done on regular water,
de-ionized water and air.
Results, Modifications and Conclusions
The values from initial measurements on air had a low correlation with model values; this was
believed to be caused by the VNA, that it measures impedance best around 50 Ω. Then air
measurements are supposed to be badly represented by measurement data. By physical means,
this was believed to be caused by lack of parasitic current (associated to the parasitic
resistance). For example, air has very low conductivity, as well as it causes a rather low
inductance, compared to water of a factor of approximately 80.
These conditions would result in a loss of the parasitic resistance in the model by A. Saremi
as the resistance gets very high and thereby can be neglected. This case also occurs when
measuring de-ionized water or dry magnetite concentrate, which also has a high resistivity
(low conductivity). Later analysis has disproved this idea, the bad correlation were probably
caused by bad data treatment and/or insufficient modeling. For further analysis, see chapter
5.5.
Some modifications to the measurement device have been done. The connection between
connector and center rod was first a coiled spring, as it was feared that this spring somehow
would affect the impedance behavior of the circuit, it was replaced by a needle-hole
connection, where the needle has a bow of feather material to tighten the connection
(commercial component). It seem like the effects on the impedance was minor, but the
connection between the conductors was greatly enhanced, which allows measurements to be
performed with ease. Also the center rod has been polished.
Measurements on samples of moist magnetite concentrate clearly show that the system cannot
only be modeled by an inductance or a capacitance alone. In the case with metal bottom
(inductive), first we will assume that all current goes through the conductor and therefore the
capacitance is not part of the model. Then if there is no capacitance behavior, it is clear that
the interaction between the leak current associated with the parasitic resistance and the current
that passes through the material dependent inductor interact to form a non-linear inductive
behavior. This will be discussed further in the discussion of chapter 5, Calculation and
Electrical Modeling. New model proposes that there should be capacitance in the inductive
model as well.
The first peak of the curves is probably the peak of the quarter wave resonance, which has a
decreasing frequency and magnitude, for increasing moisture content. In the imaginary
inductive case (Figure 12) the quarter wave resonance is more dominant than the other effects
and hence the nice asymptote. There is some confusion of the curve corresponding to 6 and 7
% in all graphs. The effects in the capacitive setup are still not identified, but are believed to
follow the theory as strictly as the inductive case.
17
Figure 11: Real part impedance spectra for inductive setup translated from
the measured S11 parameters.
Figure 12: Imaginary part impedance spectra for inductive setup determined
from the measured S11 parameters.
18
Figure 13: Real part impedance spectra for capacitive setup determined from
the measured S11 parameters.
Figure 14: Imaginary part impedance spectra for capacitive setup
determined from the measured S11 parameters.
19
4
WHY COMSOL MODELING
To get a greater understanding of capacitance and inductance contributions, as well as having
a geometrical view of the fields allows us to develop better instruments or make our
equivalent models better. The equivalent models might need correction from fringing field
effects resonance phenomenon or similar.
Several Comsol models have been developed, some models of the inductive and capacitive
setup. A few versions of microstrip capacitive sensors have been evaluated. There has been no
breakthrough with any sensor so far, but one can see that the fringing field capacitance of the
capacitive coaxial set up should be negligible, if the surrounding media are air. This depends
on the length of the coaxial cell, combined with the spread and extended field in the media air
(outside the coaxial cell), which has a fairly lower permittivity than does the magnetite
concentrate and water. If one would immerse it into the moist magnetite concentrate this
would probably not be the case, concerning open ended probing.
4.1
Coaxial Cell – Inductive and Capacitive Setup
A graphical representation of the coaxial cell presented in Figure 15 below:
Figure 15: Cross-Section of coaxial cell.
For the inductive cell in Figure 16, we can see that the electric field is strongest along the
strictly coaxial part; the measurement cell excluded the lid and bottom. The electric field
weakens when approaching the edges of the coaxial cell. The edges in this case are the
inductive short (metal bottom) and the lid with connection to coaxial cable and eventually the
VNA. Especially magneto static or quasi static simulation of this setup could be of further
value to understand such things as end effects.
The end effects observed, means that the capacitance calculated in the inductive case
according to the latest equivalent model would be slightly larger than the actual one. For the
capacitive setup in Figure 17 the capacitance of the last part, the coaxial segment where the
plastic lid constitute dielectric will have an increased capacitance in reality. First due to the
fringing field effects and second due to the plastic layer attached outside the cylinder (part of
the plastic bottom) which further will increase effects of the fringing field. Due to the length
20
of the coaxial part we will assume that the irregularities are negligible. Thus we will neglect
fringing field effects and loss of capacitance due to the inductive short. Later these effects
might need more study, when one considers more accuracy and sensitivity for measurements.
Figure 16: 3D Model of Inductive Coaxial Cell.
Figure 17: 3D Model of Capacitive Coaxial Cell.
21
In Figure 17 the outer discs (blue), tell that there is no electric potential outside the
measurement cell (as expected). The dotted lines (red) show the electric field and the arrows,
pointy structure (red), show the orientation of the field inside the cylinder. The internal disk,
rainbow colored, shows the magnitude of the electric potential.
At this point the models does not offer more than further geometrical understanding of the
problem, the models must be more sophisticated to give more information. The further need
off modeling will rather be towards the in-line measuring sensors, than on the coaxial probe.
4.2
Capacitive Surface Sensors
Some different surface sensor geometries have been simulated, offering some insight to the
distribution of electric field and electric energy density. The geometries simulated are of a 2D
symmetrical one, that means that the view should be rotated 360° around the left edge or r=0.
After this the sensor becomes somewhat of a circular disc.
Figure 18: Geometry of Capsense1.
Figure 19: Graphical E-Field
Representation of Capsense1.
During these simulations it was discovered that long grounding planes will enhance the
sensitivity of the sensor, an approximate of twice the diameter of the centre electrode seems to
be a good dimension. In the picture above there are two smaller rings as grounding plane, this
is not efficient. Instead if one could use the second ring as a guard, meaning putting the same
voltage as to the centre electrode, but without direct connection. Thus the electric fields that
generate the capacitance will not bulge at the ends. This will give a more centre aligned field.
22
Figure 20: Graphical E-Field
Representation of Capsense2.
Another thing to address could be the length of the electric field lines; it is generally known
that capacitance is proportional to electrode area and reversely proportional to the distance
between the electrode areas. What this really means is that the distance between the electrode
areas causes the field lines to travel a further distance before decoupling in ground. Thus short
field lines are field lines with a great impact on capacitance. If one sticks with the surface
sensor, one would like to have a maximal area of the centre electrode, because this is where
the sensitivity is at its highest. A maximal area would be achieved by a sphere, where one
could experiment with different radius. When area are increased one will need higher power
to maintain the energy density, as the current are more spread out on the surface.
The authors recommendation is to use some kind of parallel or semi parallel electrode
solution, this to maximize the capacitance measurability and thereby accuracy of
measurements. One possibility would be to use geometry similar to an old thread roll or timeglass with an extended waist, thus one might avoid problems with the moist magnetite
concentrate not entering the cavity of measurements. This kind of geometry would belong
with the parallel plate type, be of cylindrical symmetry and possibly have conical plates. The
plates would also have a hole in the middle to allow cables to the outer electrode.
23
Figure 21: Graphical E-Field
Representation of Capsense3.
Another interesting thing was discovered, maybe not too surprising, when using multiple
circles with increasing radius. Fed with the same feed (voltage), it has a superposition effect
(Figure 22). This means it has a very good electric energy density; however the capacitive
system also becomes very complex. If one would utilize this system one would need some
advanced algorithm calculating the capacitance seen by each electrode ring, thus using the
different capacitances for as good value as possible.
Figure 22: Graphical E-Field Representation
of Capsense4.
24
5
5.1
CALCULATIONS AND ELECTRIC MODELING
Evaluation of Models and Motivation for Chosen Model
The first electrically equivalent models investigated consist of two parts, the first part are the
correction part for correction between the coaxial theory (the material dependent part) and the
VNA.
The second and most important part is the coaxial theory determined part [1]. The coaxial part
is the part which allows extraction of the E-M material characteristics. The first part of the
circuit analysis has been inspired by A. Saremis work.
The latest models consist of the material dependent part and the modeling of the bottom part;
it seems that any resistance or inductance of the lid causes no larger effects of overall
impedance. In the end it could off course affect the accuracy of the E-M parameters; this is a
question for the accuracy analysis once the method has been verified in general.
The behavior of these E-M characteristics is not known, thus to generally validate the model it
is necessary to first of all well know the different characteristics and the models counterpart,
the parameters. If modeling is not fully correct these extracted E-M parameters will be wrong.
This however does not imply that the moisture content from mixing formulas would be
wrong, as the E-M properties are determined the same way for all measurements, the relation
between composite parameter (containing different properties) still could cause accurate
results for moisture. The formulas are valid for different parameters, why not a composite
parameter, but this would require further investigation. However the main idea still is to try to
find the actual E-M parameters, as these might also be useful for other purposes.
First of all, one can analyze the impedance of the whole setup, to see which characteristics
there are. It is also important to keep correct physical measures of the device in equations;
even though one might have better equation correlation with slight offset, otherwise the model
may compensate for formula errors and thereby give wrong answers. This when considering
accurate determination of the individual E-M parameters.
To characterize the connector-lid system (VNA-coaxial cell) we can measure the short
circuited simple system, namely lid and bottom connected. However this cannot directly be
used as a modeling correction for the connector-lid system as there are also the characteristics
of the bottom included. Here we need to separate the effects of the connector-lid and bottom
system. Also it is important to notice that the areas of these surface current spread on the
bottom will be different connecting directly to the lid instead of through the coaxial
cylindrical part, the area is bigger for the latter. Due to area dependence of the surface current,
the parameters Rc and Lc cannot be directly split into lid and bottom, without knowing
anything about the area dependence.
Theoretical studies of the bottom phenomena was not been successful using the Saremi
circuit, it is also suggested that this bottom dependence as part of the single-loop coil would
be material dependent. If so, this would cause a dependence of frequency and moisture. It
would certainly be good to find such a theoretical dependence; probably one can see the
effects of the surface current stream of this connection between the middle conductor and the
bottom as end effects when a conducting rod is connected to a circular disc.
25
The newest idea is to determine the geometrical dependence from measurements on water and
air, then modeling the inductance as µrLG. If that is valid our problem simplifies. Similar
calibrations have been done by others for similar setups.
When choosing an equivalent circuit for moisture determination; it is necessary to combine
the behavior of leakage (G or Rp) with the material dependent inductive behavior. Later it has
been understood that also the capacitance of the coaxial part must be a part of the circuit as
well as the bottom impedance.
The Parameters of Equivalent Circuits for Coaxial Cell
For any two conductor transmission line one can represent the impedance by a lumped
parameter model where the parameters are calculated according to equations 5.1 to 5.4.
Where skin resistance, Rs [Ω], inductance, L [H] (most models) / LM [H] (latest models),
conductance, G [S or 1/Ω], and capacitance C [F] (most models) / CM [F] (latest models).
These parameters however are dependent of the transmission line geometry; hence for coaxial
elements they are as follows [1] (geometrical properties can be found in appendix A):
(5.1)
(5.2)
(5.3)
(5.4)
These parameters were also derived by A. Saremi in [27].
ε [F/m] is the permittivity, µ [H/m] is the permeability, σ [1/(Ωm)] is the conductivity, a [m]
is the core conductor radius, b [m] the inner radius of the cylinder wall, ω [rad/s] the angular
frequency and l [m] is the length of the cylindrical symmetric coaxial segment. The subscripts
c and d stands for conductor and dielectric respectively.
Other parameters are described along with the corresponding models and their development.
In the latest model the bottom resistance, R [Ω], is calculated using the skin-depth, [m], by:
(5.5)
(5.6)
26
Where [Hz] is the frequency and the other variables are the same as described for the
equation above.
If one uses LC resonance theory to model the bottom impedance, the theoretical skinresistance of the bottom needs an adjustment multiplying with a coefficient to get the right
magnitude for the damping of resonances. The bottom inductance might be modeled as
follows in equation 5.7:
(5.7)
Where
[H] is the geometrically dependent inductance and can be
determined for the case when the coaxial cell is empty, containing air or preferably vacuum,
we can find the parameter
inductance of the bottom based on multiplying with the
magnetic permeability.
L-type Circuit
Figure 23: L-type equivalent circuit with correction equivalent.
In the L-type equivalent circuit in Figure 23 it is easy to see that if we short-circuit the
capacitor the leakage would be short-circuited as well, thus we would get a purely inductive
behavior. One should be aware that the characteristics of the material properties also might
interfere with our model, why it is important to model by care and keep addressing the pitfalls of this problem. Minding this, this model does not seem to be the one we are looking for.
T-type Circuit
Figure 24: T-type equivalent circuit with correction equivalent.
If the capacitor is shorted here in Figure 24, the same happen, except the skin resistance and
inductance will be halved. Again this will result in a circuit with only resistance and
inductance. Another way of seeing this case would be to accept C here and realize that the
impedance could also be affected by a capacitance along the coaxial even if it is shorted by
the end. However as the model does not take the opening capacitance into account, so that
27
something can be short-circuited, after all it seems to be a bad model. Keeping these ideas in
mind, continue with the next type of model PI-type circuit in Figure 25.
PI-type Circuit
Figure 25: PI-type equivalent circuit with correction equivalent.
Here the leakage and capacitance divided in two halves, and then if the right capacitor is
short-circuited the effect of half of the capacitance and leakage (G or Rp) will stay in the
model. When starting this analysis one should be aware that this is a step in the direction of
making a semi-lumped semi-distributed model. Because if one appreciate the fact that
capacitance is split in two, it could as well be split in more parts. A distributed model is built
by infinitesimal elements of these equivalent circuits, preferably by the L-type.
Somewhere between the two models we will have a number of lumped models where each
parameter value is divided by the number of the lumped models used, and then we can only
short circuit the last capacitor. The parameters otherwise has the same properties as when only
one lumped model is used. This type of modeling maybe would better reflect the behavior of
the measurement cell. When this interpretation is used, the mathematical system will become
large and bring time consuming analysis.
Thus it is a good idea to analyze the system where we completely can ignore the capacitance,
but still keep the effects of leakage (G or Rp). Thus we will continue working with the Saremi
model when considering optimization, discussed below. If it considered being insufficient for
modeling purpose, we will continue with building a semi-lumped semi-distributed equivalent
circuit. For ease we will build this system by matrices, so that the increase of the number of
lumped models is practically simple.
Saremi Circuit
Figure 26: Saremi equivalent circuit with correction equivalent.
This is the model used in the optimization calculation with somewhat of a success (Figure
26). By previous analysis, this model seemed to be the best model to predict the behavior of
28
the measurement cell. If the analysis is not enhanced using a chain of lumped models by
matrices. Further analysis and earlier knowledge of model simplification suggest that this
model should be replaced by another model.
Thus we will make a jump in the modeling and continue working with an expanded L-type
model. The semi-lumped semi-distributed model are still of interest, maybe for future
analyses.
Semi-Lumped Semi-Distributed Circuit System
Analysis of such a rather complex system is very time consuming, to simplify the work with
the equations and the modeling; it is convenient to choose a matrix approach to the problem.
It exist several sets of parameters to relate the output to the input of a two-port circuit system,
not to be mixed with the ports of the VNA. In this case we would have seen our system as a
two-port system and it will be illustrated in figure 27 below, this if correction parameters for
the lid would be used.
Figure 27: Semi-lumped semi-distributed equivalent model with split correction lid and bottom.
For microwaves the S parameters are good as voltage and current values are not well defined.
One can transform one two port matrix to another for other parameter such as Z, T or G [31].
New Electrical Model Proposed by Karlsson
Figure 28: Karlsson proposed inductive equivalent circuit with correction equivalent.
When analyzing the complex impedance regarding the effects of the correction parameters Rc
and Lc, one could see that subtracting the modeled impedance of the combination of the lid
and bottom from any measured impedance did not change the magnitude of impedance by
much, thus we neglect these effects at the lid position (first) and instead modeling only the
bottom resistance and impedance (last). The bottom skin-resistance is very small, but
according to the LC resonance theory it would affect the impedance much, due to damping
effects of the LC resonance peak which would occur as a result of the parallel connection
between bottom inductance and coaxial capacitance of the coaxial part.
29
In general when considering AC current one would model a conductor as a serially coupled
resistance and inductance. Simple analyses done for dry systems, eg no conduction; air, deionized water and dry magnetite concentrate, resulted in the choice of this model, as by theory
one could build the actual impedance curves and this was done in Orcad with PSpice. The
bottom resistance causes a resonance peak at the real part of impedance as well, more about
this analysis in chapter 5.5.
New in this model are also that the whole material dependent capacitance is kept, this is the
effect of a non ideal system, the inductance and along the coaxial part as well as that of the
lid. Also the capacitance is distributed throughout the system and cannot be short-circuited;
by an ideal coaxial cell it could neither be short-circuited due to the resistance and inductance
of the bottom. Only for a completely ideal conductor case one could totally short the
capacitance, but in such a case probably none of the material parameters we are interested in
would be present.
For the capacitive measurement setup one probably just connects a new segment of coaxial
cell to the inductive version without bottom (in model without bottom inductance). Initially
the plastic bottom effects will be neglected and left for later analysis of model quality.
Figure 29: C. Karlsson proposed capacitive equivalent circuit with correction equivalent.
General for the two models are that the parasitic resistance vanish for the dry case,
corresponding to a case where there are no conductance.
5.2
Improvements in Calculation after A. Saremi Work
A better characterization of the DC behavior of the moist magnetite concentrate was
performed; this provides the model with better guesses or values of the parameter Rp,
depending on which model are in use and if Rp are considered to be frequency dependent or
not. Here it was also discovered that the resistivity of dry magnetite concentrate respond
equally to the resistivity of air which can be considered infinite or at least very large. This was
shown in the experiment by having same values unconnected as connected to the dry sample.
The Saremi circuit optimization modeling was modified to model the magnetic permeability
as a complex unit, according to theory, and guessing both the real and imaginary part together
with the conductivity presents a more valid characterization of the material. The accuracy of
this modeling is highly dependent of the structure and order of segments in the model, the
model needs more work to present accurate results. Representation of modeling error is turned
into a graphical presentation allowing the error to be viewed as a surface spanned by
frequency vector and moisture vector. The data loading is now done from within the folder
easily allowing the whole measurement calculation folder to be copied or moved, still
executable without any modifications.
30
In the new model proposed by C. Karlsson we bring in capacitance into the inductive model
and in general takes full use of the coaxial ac theory. Capacitances also are modeled as a
complex frequency dependent vector which will add further unknowns to the problem.
However in one way this also simplifies our problem a bit, by using this model we exclude
many parts which were not fully correct, as bottom impedance and capacitance of inductive
model. Some initial verification that this model is justified will be discussed in chapter 5.5.
5.3
Calculations Using the Saremi Circuit Model - Inductive Setup
Purpose
The intention with the calculations performed so far is to perform a material model to describe
a spectrum of permeability as separate curves for each moisture sample measured. Also there
could be a frequency dependent conductivity; this will together form a magnetic material
model for the moist magnetite concentrate.
Data
The input data to the mathematical model are the frequency dependent and complex S11
parameters (VNA output) from connector – lid – bottom measurement (correction data).
Performing the same type of measurement on the inductive setup for each moisture sample,
together with the DC resistivity data this provides the data to be used for material modeling.
DC resistivity data based on resistivity from measurements providing guesses for Rp, guesses
are the same for each frequency but changes with the moisture content. S11 parameters are
early transformed into impedance by equation 3.4.
In the modeling case used in this master thesis for this method, even numbered moisture
contents from 2 to 10 % are evaluated in the model. This choice has been done to be able to
compare the output between the measurements made by A. Saremi (even 0-12%) and those
made by C. Karlsson (0, 2, 4, 6, 7, 8, 9, 10 and 11 %).
Then our available data are two correction vectors Rc [Ω] and Lc [H] (determined from
measurement), both of length omega (2000). A vector of parasitic resistance Rp [Ω] (from
measurement in DC and estimated in AC) and a vector Rs [Ω] (theoretical) skin resistance of
length omega. The moisture data are represented by a complex value impedance matrix (2000
x 5).
Mathematical Tools
Calculations are performed in Matlab, full code in appendix G-K. The main mathematical tool
used is the optimization function fminsearch [32; 33; 34]. fminsearch minimizes J the value
its input function, in this case the mean square error of the model.
The input function / cost function is the following:
(5.8)
The cost function provided with, theta = [µre, µim, Rp], the guess parameter and other data
determined from measurements. Other data span correction resistance, correction inductance,
measured impedance and corresponding discrete frequency vector.
(5.9)
31
E [Ω] the actual error of the guessed impedance, Zm [Ω] measured impedance and Zhat [Ω]
modeled impedance.
(5.10)
With included impedance model, where Zc [Ω] is correction, Zp [Ω] parasitic and Zm [Ω]
measured impedances. The measured impedance also holds some theoretical information, as
well as the parasitic impedance which probably also would be determined from
measurements, as it is believed to vary with frequency.
(5.11)
The impedances Ztx [Ω] and Znx [Ω], are only substitutions to make the system more easily
overview able and manageable.
(5.12)
Here Zeq,real [Ω] and Zeq,imag [Ω], are the real part and the imaginary part of the modeled
equivalent impedance.
Preparation and Optimization of Material Properties (µr, Rp)
By equation 3.4 and 5.13 the moisture impedance matrix are transformed into guesses of µr,
based on the much simpler equivalent circuit of only an inductor, having a complex µr
resulting in an equivalent impedance according to equation below:
(5.13)
Then a matrix of size 2000x10, five columns of µre [H/m] real permeability and five columns
of µim [H/m] imaginary permeability is used, value by value together with the corresponding
value from matrix of Rp [Ω] 2000 x 5. Once for each frequency the fminsearch evaluates the
optimal µr and Rp. The error is once again calculated for the optimal values and a surface plot
of the error is made, results of the optimized parameters will be presented next.
32
Presentation of the Results
Figure 30: Real µr provided by optimization of Saremi model for different water
content.
Figure 31: Imaginary µr provided by optimization of Saremi model for different
water content.
33
Figure 32: Conductivity provided by optimization of Saremi model for different
water content.
Figure 33: Error surface representing the impedance error for the optimized
Saremi model compared to the measured impedance.
34
Discussion of the Results
One can see from the data acquired from the optimization of the material parameters that this
optimization suffers from disturbances, most probably from the calculation of the parameters
in the optimization function fminsearch. Strange jumps and non-expected discontinuous
behaviors occur in the parameter spectra. This is probably an effect of many things. First of
all it is believed that the electrical model of the measurement system still is badly
characterized, especially the bottom, which could cause these discontinuities. Second the
small noise on the parameter curves could be related to function-minimum triangulation and
that the number of guesses (or unknowns) makes the system instable.
This model turned out to be of insufficient nature, mathematically weak as non deterministic
model; we know our system shall be deterministic. A decision to analyze the circuit from a
fairly easier point of view, only as a purely inductive component was made, still with
frequency and complex behavior, more of this in chapter 5.4. This decision was made after
identifying that the estimated µr according to this model looks good, compared to the
optimized version.
5.4
Simple Model Analyses
The optimization breakdown, due to mathematical insufficiency, instead a more simple study
was carried out. Then observation of behaviors can be done, even though the actual
magnitudes and numbers will not be applicable in any way.
Starting by analyzing the inductive setup as purely inductive, the division of the complex
inductance to a full complex number, where the real and imaginary parts switch place due to
multiplication by Jω, that is:
(5.14)
Remember the definition of µr (equation 5.13), so that the parts of Zeq [Ω] (here the
impedance of the simple inductive model), will become positive in the impedance equation.
Purpose, data and Mathematical Tools
The purpose of this analysis is to look for special characteristics of the µr curves, which could
guide our design of the equivalent circuit model. The data sets are those of measurement 10
and 11, which are those containing samples of moist magnetite concentrate. In general the
data are of the same kind as those used in optimization explained in chapter 5.3. The
mathematical tool used is Matlab.
Calculations, Results and Discussion
The impedance was converted to inductance, using equation 5.14, further to µr by relations
between inductance and µr, in equation 5.2. The results of the plots performed on µr for
different conditions measured are presented below:
35
Figure 34: µr provided by simple inductive model for air.
Figure 35: µr provided by simple inductive model for water.
36
The air sample measurement provides us with a reasonable µr, however there are some
deviations for low frequency and a little more on the higher frequency, this indicates that
especially the model concerning the reactance are too simple. It is possible to subtract
complex impedance from the measured impedance to make this permeability equal to one as
was expected. However this information is not enough to determine which components there
really are, as they could interact to cause this impedance. If we only know this, one would
suggest that it is the adding of lid inductance which causes this increased of µ r, primarily at
high frequencies. The imaginary part is with noise subtracted zero, all according to theory of
air having a real characteristic permeability. Still there are some modeling issues to solve.
Observing the results from water measurements in Figure 35, a strong resonance appears.
From discussions and calculations, one can see that it is caused by the equivalence of the
electrical length and the quarter wave length.
Recalling the inductive model of Saremi equivalent circuit, where there is no capacitance.
Recalling the coaxial theory and if one reanalyzing the equivalent circuit. All parameters
except the unknown lid inductance are per unit length parameters, but by using the equivalent
circuit with a shorted capacitance we automatically assumes the capacitance not to be a per
unit length parameter, which allows us to short it. However the short (or the metal bottom)
actually does not affect the impedance of the coaxial model, this depends on the inductance
and other impedance elements along the coaxial part. Thus we would have a LC parallel
connection at the end of the equivalent circuit, which could have been causing this LC
resonance. This phenomenon and its consequences for the measurements will be further
investigated in chapter 5.5.
Otherwise the real part of µr is close to one, as expected, as well as that the imaginary part
keeps to zero. If one can assume that the resonance is of the LC type, we could from the
resonance frequency of this graph and the theory behind the LC resonance frequency,
determine the value of inductance for the bottom.
This model is fairly simple, as permeability and permittivity are real. When dealing with
complex permeability and permittivity, one could expect more problems, if they also are
frequency dependent, the system tends to become more theoretically complex.
For the dry magnetite case, we have the same linear / constant basic behavior. Also the
resonance which occurs for all dense media, here it seems to be the same frequency for both
real and imaginary permeability. Same frequency for resonance should imply approximately
the same values of real part and imaginary part. Possibly a leaning real part resonance
indicates a frequency dependent real part. The guess based on the permeability of pure
magnetite are µr=2-3j, which may be bad. This could change when we have made our model
more theoretically complex.
The resonance frequency decrease by increasing moisture content and separation between real
part and imaginary part occurs. The slope of the resonance decreases due to increasing
moisture content. Also some more theoretically complex effects are hidden in the graphs. This
is subject for more study.
37
Figure 36: µr provided by simple inductive model for dry magnetite concentrate.
Figure 37: µr provided by simple inductive model for 2% moisture.
38
Figure 38: µr provided by simple inductive model for 4% moisture.
Figure 39: µr provided by simple inductive model for 6% moisture.
39
Figure 40: µr provided by simple inductive model for 7% moisture.
Figure 41: µr provided by simple inductive model for 8% moisture.
40
The steep slope of the curve is believed to be the effect of a very low parasitic resistance and
could also be affected by changes in relaxation due to increased charge transport. However no
valid conclusions can be drawn from these graphs.
5.5
Calculation Using Electric Circuit Proposed by C. Karlsson
Purpose and Data
In large the purpose of these calculations is the same as for those with the optimization using
the Saremi circuit model. Thus our calculation should first verify the electrical models put up
for the measurement cell. For this purpose one will start by analyzing the dry case due to the
simplicity, and then further complicate the case until the models are thoroughly understood.
When the system is fully known one could start estimating moisture content. Before this one
must fully characterize dry cases and also measurements of wet frequency dependent cases,
that is regular water and moist magnetite concentrate filtering water (process water), either
directly from the mine or laboratory filtered water to make an approximated lab equivalent.
At this point it is possible to use measurements of moisture samples of known moisture
content, using one or two mixing formulas to determine the properties that would be
prescribed by the theory for the moisture samples; if we have too many unknowns we could
also use this data to determine parameters not covered by number of equations. For the latter
case, however one could only verify the model by estimating moisture contents in new
samples. It would be preferred if the model could be completely mathematically proofed.
Measurements 10 and 11, was used for this purpose as they are the most complete sets of data
existing.
Mathematical Tools and Simulation Tools
The S11 parameters have been converted to impedance curves in Matlab and then these are
compared to the curves plotted by Orcad, when the equivalent circuit was built there. The
circuits are designed according to theory from LC resonances.
Measured Data
For air in the coaxial cell, we find that the resonance is outside of our spectra, but we can see
some of the effects of the resonance in the end of our spectra as the imaginary impedance
deflect from its otherwise linear behavior. The de-ionized water spectra, as seen before in
Figure 43, have a very sharp resonance. While the measurements on dry magnetite
concentrate has a slightly more spread resonance.
41
Figure 42: Measured equivalent impedance for air (theoretical value by simple model),
Matlab Calculated Impedance Using C. Karlsson Circuit.
Figure 43: Measured equivalent impedance for de-ionized water (mQ).
42
Figure 44: Measured equivalent impedance for dry magnetite concentrate.
43
Orcad Equivalents and Their Spectra
Orcad Real and Imaginary Impedance [ Ω ]
Figure 45: Orcad air equivalent with constant maximum
magnitude resistance.
Figure 46: Orcad plot of equivalent impedance for air.
So far using frequency dependent resistors has failed, thus the frequency dependent resistors
have been replaced by maximum resistance for each resistor. The resistance is really small
44
and is assumed to have a small impact on our graph. The behavior of the Comsol model looks
similar to the measurement data.
Orcad Real
] Imaginary Impedance [ Ω ]
[ Ωand
Orcad Real and Imaginary Impedance
Figure 47: Orcad water equivalent with constant maximum
magnitude resistance.
Frequency [ MHz ]
Figure 48: Orcad plot of equivalent impedance for water.
Same with water graph, we have similar impedance behavior as for our measurement.
45
Matlab Calculated Impedance Using C. Karlsson Circuit
The latest models for equivalent circuits have been evaluated in Matlab, for comparison with
earlier presented measured data. The following equivalent circuit equation 5.15 has been
used;
(5.15)
Figure 49: Matlab calculation of the air case.
This impedance graph for our theoretical Matlab model for air (Figure 49) well correlate to
the one from Orcad model and less with the simple theoretical model, by comparison with the
actual measurement data (Figure 42) one can see a missing deflection (in upper part of
spectra) of the theoretical data to correlate with measurement data. This missing deflection
might be a sign of missing modeling of lid.
The water (mQ) impedance resonance peak (Figure 50) well correspond to measured data, but
the basic behavior under the resonance might still badly represent the actual circuit. This data
is of course biased as it was used for the modeling; one certainly needs to confirm these
resonant elements towards other MUT.
46
Figure 50: Matlab calculation of the water case.
Figure 51: Matlab calculation of the dry case.
The material values for permittivity and permeability for the dry concentrate, is not yet
determined, thus for our guesses and for the C. Karlsson model do not have a good correlation
or actually none at all (Figure 51). Here more work is needed. Possibly the resonance of the
47
lid will change this behavior or we need to look for other explanations. By pursuing the
capacitive models, we might enhance our chance of understanding the system.
5.6
Resonance Analysis
The frequency spectra of impedance from measurements suggests that the measurement
catches up some resonances, the resonance phenomena analyzed are the LC - resonance and
quarter wavelength resonance. Thus using different models for modeling of LC – resonances
and theory for quarter wave resonance, the results show that the most likely resonance to
occur is the quarter wave resonance. Looking at graphs of LC - resonance models, one can
observe that the phenomena observed does not fully explain those observed at the
measurement spectra.
To overcome these problems with resonances, the most appropriate way, would be to go
towards lower frequencies and/or distributed parameter analyses. Another idea would be some
redesign, where one would risk stepping into another resonance phenomena. In subchapters
each resonance phenomena and its analysis is described in detail.
RLC Resonances in Coaxial Cell
An approach to the modeling problem is to consider the coaxial cell to be a RLC resonant
system. This means that the electrical equivalent circuit of the coaxial cell has parallel and
serially connected inductors and capacitors which might cause resonances. The resistance of
the circuits damp and broadens the base of the resonance peaks. Generally a simple LC
resonator, parallel or serial has a resonant frequency matching equation 5.16 below:
(5.16)
For more complex systems, one might assume a more complex description of the resonant
frequency or as multiple LC resonance contributions to the resonant frequency
If one considers the coaxial cell behavior to be approximated by an RLC system, one
resonance believed to originate from the bottom and another one suspected to come from the
lid connection, the latter has not been modeled yet.
Quarter Wave Length Analysis
The relationship between the quarter wavelength resonance frequency (fr [Hz]) and the
electrical length (l [m]) of the device is as follows:
(5.17)
It is also recommended to stay with a frequency, so that the below relationship will hold:
(5.18)
Thus avoid resonances and allow the use of lumped parameter models.
48
When calculating the quarter wavelength for the corresponding resonance frequencies for
each medium air and water, one can see that the quarter wavelength almost exactly matches
the length of the electrical length of the device. Even though the E-M parameters of the moist
magnetite concentrate are not known, one may assume that they are larger than those for less
dense matter, such as air or vacuum.
There are three roads to pursue; either build smaller sensors for in-line use to "complete" the
measurement with the coaxial cell, look further in the spectra towards lower frequency, go on
with advanced transmission theory or build smaller coaxial cell to avoid quarter wave length
resonances.
49
6
DISCUSSION OF RESULTS AND FINAL CONCLUSIONS
For the dry cases, the equivalent circuit follows theory. The coaxial part can be modeled using
L-type equivalent circuit. The metallic bottom can be modeled as a conductor that is a resistor
and inductor in series and the plastic bottom could probably be modeled as another coaxial
part with plastic as dielectric, which remains to be proved.
Lid effects, seem to be negligible for impedance, but might be necessary if one needs good
accuracy. Based on the knowledge of the system for dry models assuming that the resonance
phenomena originates from parallel LC connection a model for wet case has been proposed.
One for the inductive case and one for the capacitive case, in this sense inductive and
capacitive refer to the end load (bottom) of coaxial cell according to above.
If one can assume that the inductive air inductance carries all geometrical properties and
material dependence μr can be multiplied to acquire inductance for bottom for the case of
media inside coaxial cell. Then the inductive system is fully described with two unknowns,
assumed known resistivity. If the capacitive setup can be modeled as assumed, one would
have two equations and two unknowns for each frequency. Thus the system which is proposed
will assume resistivity constant in frequency spectra but changing with moisture. For more
understanding, please look at chapter 5 where these things are described in more detail.
New measurements of resistivity has given a new picture of the DC resistive behavior of the
moist magnetite concentrate, the dry magnetite concentrate has an almost infinite resistivity.
The more water added the more conductivity and at 11% water the resistivity is very low.
Also a strange behavior of self potential was discovered during the measurements. Self
potential, a concept known from geophysics, caused by charge separation in clay by transport
of charges through the pores of magnetic granular media or flow of mineral rich fluid through
the granular media. Otherwise the calculations of μr have confirmed the theories on complex
μr.
In-line sensing using this method can be constructed in two ways, one way is to build a
segment of the process line as a transmission element and another way is to use small in-line
units, using small inline units will probably imply a use of large amount of units. Ensuring a
representative value for a batch, where moisture content largely differs.
In-line sensing using small immersed sensors will probably consist of two sensors, one
sensing magnetic properties and one sensing electric properties. A few electric geometries
have been analyzed, but a lot of analysis remains to be done before one can develop a
complete in-line sensor.
50
7
DISCUSSION OF FUTURE WORK
It is possible that the possibility of publishing research result from this master thesis in an
article will arise, concerning the measurements of dry samples, meaning samples without
conduction. Presenting the measurement method, equivalent modeling and results from
measurements.
One should analyze the effects of resistivity for AC conditions, if one can use the DC
resistivity or if it is frequency dependent.
Using other methods as Comsol simulations and PEEC simulations one could compare
electrically derived values for different components in the measurement equivalent circuit,
mainly the bottom. It would be favorable to have analytic mathematical equations for these
values.
Research with PEEC modeling is run at LTU under J. Ekman, where a few PhD students
working alongside. This PEEC modeling might be interesting as collaborative project, where
modeling for the air case could be identified and then maybe one could validate bottom
properties.
Also this might be an interesting technique to continue working with towards sensor design of
in-line sensors. The PEEC project has mainly been working with surrounding air media,
however if one could confirm models of air, regularly simple modeling can be adapted to
surrounding material property change. How this adaptations works with PEEC modeling is
unknown to the author.
Measurements of water may improve understanding of the electric model as it has a constant
permeability and a frequency dependent permittivity. Determining the process water
permittivity is a step towards verification of determined permittivity from moisture samples.
When a complete understanding of the dry magnetite E-M properties and of the process water
E-M properties have been established, one can prepare comparison data from mixing
formulas. Combining determined permeability of dry magnetite and determined permeability
of process water in mixing formulas. This should be done for the different actual mass
fractions of moisture, using the prepared moisture samples. This produced set of reference
data will allow evaluation of the validity for our E-M model.
Using a correct resistivity and finely calibrated equipment to eliminate noise, one can have
improved measurements of characteristic impedance. Small noise could appear due to the
damages inside the measurement cylinder (part of coaxial cell), therefore one might reproduce
that cylinder and get it internally polished. One may also again evaluate if some connection
with the bottom could cause any noise or contribution, maybe studying the effect of screw
thread. For this study one may produce extra coaxial cells without these screw threads, using
only frictional connection. Then one can study the effects of geometrical changes and general
stability of model, causing changes in impedance.
When the measurement equipment is improved, the E-M model fully accepted, one can start
to determine the E-M properties of the different moisture samples and compare to the
prepared sample comparison data. If the modeling is correct this should match very well.
When the E-M properties have been fully justified it can be used as a reference for choice of
51
suitable measurement frequency and provide reference values for this frequency. At this point
the project has reached the point of instrumental design for in-line measurements, as well as
presenting a full material model.
When a full material model has been acquired, one can write an article on the material
properties of LKABs moist magnetite concentrates. Here one should present a thorough
analysis of the E-M properties with a stabile foundation in the electric equivalent as well as
the impedance relations and the mixing formulas. Another topic which needs to be attached to
verify the legality of using mixing formulas for identifying the permeability and permittivity is
the E-M analysis with Comsol for the effective E-M parameters based on mixing of grains
and water filled and air filled cavities. The later analysis has been assessed by T. Löfqvist.
When the knowledge of E-M properties has been accumulated one can start to develop an
embedded solution. Two different communication modules have been proposed, RFID
technology suggested by Electrotech in Kalix and the LTU product “Mulle” (acting as a node
for Bluetooth sensor networks). Contact at electro tech is J. Rajala and for “Mulle” at LTU it
is J. Eliasson.
The end product must have a measurement of temperature, resistivity (DC or AC), impedance
and maybe total measurement / sensing volume must be known a priori. Permittivity and
permeability could be determined from an equivalent electrical circuit.
52
8
REFERENCES
[1] Sadiku, M. N. (2001). Elements of Electromagnetics. New York, United States of
America: Oxford University Press.
[2] Forsmo, S. (2007). Influence of Green Pellet Properties on Pelletizing of Magnetite Ore.
Luleå, Norrbotten, Sweden: Luleå University of Technology.
[3] Skott, T. (2005). LKAB: Sammanställning av metoder för fuktmätning av slig (Not
published). Forskning och Utveckling. Kiruna: LKAB.
[4] Turtola, J. (1991). Mätning av resistivitet hos MPC vid olika fukthalter. Meddelande,91104,LKAB.
[5] Löfqvist, T. (2010). Subject Introduction. (C. Karlsson, Interviewer)
[6] Kupfer et al. (2005). Electromagnetic Aquametry. (K. Kupfer, Ed.) Heidelberg, Germany:
Springer-Verlag.
[7] Sihvola, A. (1999). Electromagnetic Mixing Formulas and Applications. Cornwall:
Institution of Electrical Engineers.
[8] Jylhä, L., & Sihvola, A. (2007). DIFFERENTIAL EQUATION FOR THE EFFECTIVE
PERMITTIVITY OF RANDOM MIXTURE OF SPHERES. Electromagnetic Theory
Symposium. 26-28 July. Ottawa, ON, Canada: International URSI Commission B.
[9] Venkatesh, M. S., & Raghavan, G. S. (2005). An overview of dielectric properties
measuring techniques. LE GÉNIE DES BIOSYSTÈMES AU CANADA , Vol.47.
[10] Baker-Jarvis et al. (2005). Measuring the Permittivity and Permeability of Lossy
Materials: Solids, Liquids, Metals, Building Materials and Negative Index Materials. NIST
Technical Note (1536).
[11] ASTM International. (2008). Standard Test Method for Measuring Relative Complex
Permittivity and Relative Magnetic Permeability of Solid Materials at Microwave Frequencies
Using Coaxial Air Line. (ASTM D 7449).
[12] ASTM International. (2009). Standard Test Method for Measuring Relative Complex
Permittivity and Relative Magnetic Permeability of Solid Materials at Microwave Frequencies
Using Waveguide. (ASTM D 5568).
[13] Jaspard, F., & Nadi, M. (2001). OPEN ENDED COAXIAL LINE FOR ELECTRICAL
CARACTERIZATION OF HUMAN BLOOD.
[14] Chen, L., & Varadan, V. K. (2004). Microwave electronics: measurement and materials
characterization. Chichester, England: John Wiley and Sons.
[15] Zheng, H., & Smith, C. E. (1991). Permittivity Measurements Using a Short Open-Ended
Coaxial Line Probe. IEEE MICROWAVE AND GUIDED WAVE LETTERS , vol.1 (no.11),
p.337-339.
[16] Moreau, J. M., & Aziz, R. (1993). Dielectric study of granular media according to the
type of measurement device: coaxial cell or open-ended probe. Meas. Sci. Technol. , vol.4,
p.124-129.
[17] Gasvenor, J. H. (1993). NIST Measurement
Characterization of Materials. Boulder: NIST.
Service
for
Electromagnetic
[18] Baker-Jarvis et al. (1994). Analysis of an Open Ended Coaxial Probe. IEEE Transactions
, Vol.43.
53
[19] Aimoto, A., & Matsumoto, T. (1996). Noninvasive Method for Measuring the Electrical
Properties of Deep Tissues Using an Open-Ended Coaxial Probe. Med.Eng.Phys. , vol.18
(no.8), p.641-646.
[20] Folgerø, K., & Tjomsland, T. (1996). Permittivity measurement of thin liquid layers
using open-ended coaxial probes. Meas. Sci. Technol. , vol.7, p.1164–1173.
[21] Hagl et al. (2003). Sensing Volume of Open-Ended Coaxial Probes for Dielectric
Characterization of Breast Tissue at Microwave Frequencies. IEEE TRANSACTIONS ON
MICROWAVE THEORY AND TECHNIQUES , vol.51 (no.4), p.1194-1206.
[22] Cheng et al. (2006). Evanescent microwave probe study on dielectric properties of
materials. Journal of the European Ceramic Society , Vol.26, p.1801–1805.
[23] Ellison, W.; Moreau, J-M. ;. (2006). Open-Ended Coaxial Probe: Model Limitations (Not
published).
[24] Ellison, W. J.; Moreau, J-M. (2008). Open-Ended Coaxial Probe: Model Limitations.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT , Vol. 57 (no.9).
[25] McLaughlin, B. L., & Robertson, P. A. (2007). Miniature open-ended coaxial probes for
dielectric spectroscopy applications. J. Phys. D: Appl. Phys. , vol.40, p.45-53.
[26] Oppel et al. (2008). Broadband Measurement of Complex Permittivity Using Reflection
Method and Coaxial Probes. RADIOENGINEERING , vol.17 (no.1), p.14-19.
[27] Saremi, A. (2008). Sensus Aquae en Ferrum ac Air - A Prestudy: Electro-Magnetic
Moisture Metering of Magnetite Ore Concentrate (Not published). Eislab. Luleå: Luleå
University of Technology.
[28] Avitabile et al. (2001). A Rugged Active Sensor for Microwave Aquametry. IEEE MTTS Digest , p.2259-2262.
[29] Guggenberg, P. A., & Zaretsky, M. C. (1995). Estimation of one-dimensional complexpermittivity profiles: a feasibility study. Journal of Electrostatics , vol.34, p.263-277.
[30] Sonnet Software Inc. (2006). Using Electromagnetic Analysis for RFID Antenna Design.
Sonnet Application Note (SAN-206A).
[31] Wedelin, B. (1988). KOMPENDIUM - ELKRETSTEORI DEL C - LEDNINGAR och
FILTER. Göteborg, Sweden: Teknologtryck - Chalmers Tekniska Högskola.
[32] Mathworks. (n.d.). fminsearch. Retrieved 2010-10-05, from Mathworks.com:
http://www.mathworks.com/help/techdoc/ref/fminsearch.html
[33] Lagarias et al. (1998). Convergence Properties of the Nelder-Mead Simplex Method in
Low Dimensions. SIAM Journal of Optimization , Vol. 9 (no. 1), p. 112-147.
[34] Mathworks. (n.d.). fminsearch Algorithm. Retrieved 2010-10-05, from Mathworks.com:
http://www.mathworks.com/help/techdoc/math/bsotu2d.html#bsgpq6p-11
[35] Creative Commons. (2008-01-09). Creatice Commons - Attribution - ShareAlike 3.0
Unported. Retrieved 2010-12-28, from Wikimedia Commons - Creative Commons:
http://creativecommons.org/licenses/by-sa/3.0/deed.en
[36] WikiUN:Qianchq. (2010-01-16). Special:ListFiles. Retrieved 2010-12-28, from
Wikipedia: http://en.wikipedia.org/wiki/File:Segments.jpg
[37] Sjöberg, B. (1971). Försök med konduktiv fuktmätare. Utredning, dk nr 28/42,LKAB.
54
APPENDIX A – GEOMETRIC AND MATERIAL CONSTANTS
Coaxial Cell Geometrical Dimensions
a = 0.00495 m
b = 0.027 m
c = 0.03 m
twall = c-b = 0.003 m
ltot= 0.249 m
llid-insert = 0.008 m (Top,bottom and lid-teflon)
lM= ltot- 2llid-insert
lM= 0.233 m (Measured length)
Coax-Cell,
Schematic
Picture
rinner, teflon lid= a
router, teflon lid= 0.0125 m (0.01235)
bottom hole and rod: M8
(Metric 8 mm screw thread)
Coax-Cell, Photo by A. Saremi
rlidprobestick= (thin)~0.001m (+-0.0002)
(med)0.0012 (thick)0.00165
llidprobestick= 0.021 m (from connector Teflon end) (0.022 m)
hlid outside = ~ 0.012 m (+-0.0002)(Short air-line)(Length cal?)
hconnector-teflon = 0.0093m
coaxial connector: square side 0.025m,
cylinder height 0.0218 m, screw thread M3.
Bottom, Aluminium and Plastic
(metric 3 mm)
bottom of coaxial cell can be plastic or metal.
Material Constants and Speed of Light
Lid and coax cell, lid-probe and
core hole
-12
ε0 = 8.85419∙10 (F/m),
μ0 =
(H/m) | (N/A2),
c = 299792458, ε= εr ε0, μ = μ r μ 0
Basic constants:
Property
\Medium
Vacuum
Air
Water
(mQ)
Tap water
Salt
Water
Aluminum
εr
μr
σ (S/m)
1
1
0
1.00059
1.00000037
~0 / 10-16
~80.4
0.999992
~0
~80.4 (f)
0.999992
~10-3
~80.4 (f)
0.999992
~4
~1
1.000022
3.538∙107
Property
\Medium
PTFE
PC
Magnetite
Powder
εr
μr
σ (S/m)
~2.1
~1
~0 / 10-16
~2.9
~1
-
> 1000 (f?)
Guess 2-3j
0.00151
55
APPENDIX B – PREPARATION OF MOIST MAGNETITE CONCENTRATE
Fraction
Magnetite
Powder
w.Container
Empty
Container
Magnetite
Powder
Water
(calc.)
Water
(actual)
Actual
Fraction
0.02
0.04
0.06
0.07
0.08
0.09
0.10
0.11
2968,9
4054,4
3051,4
3911,1
3932,1
3657,3
3463
3721
94,12
93,01
92,98
160,51
160,48
160,32
160,29
160,45
2874,7
8
58,668
97959
60
3961,39
2958,42
3750,59
3771,62
3496,98
3302,71
3560,55
165,057
9167
124,8
188,835
3191
190,87
282,302
4731
294,4
327,966
9565
341,9
345,855
1648
361,7
366,967
7778
384,7
440,067
9775
459,9
0,0204
44
0,03054
2
0,06060
7
0,07278
1
0,08311
6
0,09373
7
0,10432
8
0,11439
The magnetite concentrate used in the moist samples in laboratory is originally as clean as the
LKAB factories can make it.
56
APPENDIX C – RESISTIVITY MEASUREMENTS
Moisture Fraction
0%
2%
4%
6%
7%
7%
8%
9%
10%
11%
0
0.02
0.04
0.06
0.07
0.07
0.08
0.09
0.1
0.11
V(I=-1mA)[V]
V(I=1mA)[V]
l [cm]
R_M- (ohm)
R_M+ (ohm)
-42.11
-7.87
-5.11
-3.2
-3.18
-2.35
-2.6
-1.72
-1.17
-0.64
42.11
6.3
5.18
3.3
3.2
2.34
2.61
1.77
1.18
0.393
6.5
6.0
6.3
6.8
6.8
6.3
6.9
6.0
5.8
5.8
42110
7870
5110
3200
3180
2350
2600
1720
1170
640
42110
6300
5180
3300
3200
2340
2610
1770
1180
393
Res_M+
664,5389
107,7751
83,69203
49,39692
47,90005
37,80682
38,50227
30,02734
20,70851
6,896988
mean_res
664,5389
121,20423
83,126542
48,648486
47,75036
37,887607
38,428508
29,603228
20,620764
9,0643615
Moisture
0%
2%
4%
6%
7%
7%
8%
9%
10%
11%
Res_M664,5389
134,6333
82,56105
47,90005
47,60067
37,96839
38,35475
29,17911
20,53302
11,23174
57
Sigma
0,001505
0,008251
0,01203
0,020556
0,020942
0,026394
0,026022
0,03378
0,048495
0,110322
APPENDIX D – MATHEMATICAL SYSTEM - SAREMI MODEL
Here Zc is correction, Zp parasitic and Zm measured impedances. The measured/optimized
impedance also holds some theoretical information, as well as the parasitic impedance which
also probably are a measured/optimized variable.
Conjugate multiplication and then split in real and imaginary part.
In this case Z0 = VNA-CABLE-RES = 50 Ω and S11 are the complex frequency dependent S11
parameter measured by the VNA.
58
APPENDIX E – INDUCTIVE MATHEMATICAL SYSTEM - KARLSSON MODEL
59
60
APPENDIX F – CAPACITIVE MATHEMATICAL SYSTEM - KARLSSON MODEL
61
APPENDIX G – MATLAB FUNCTIONS - COST AND ZLOAD FUNCTIONS
%*************************************************************************
%**************************costfun****************************************
%*************************************************************************
%Mean square function - With mean square error as output
%Author: Amin Saremi
%Edits by: Christian Karlsson
function J = costfun(theta,R_C,L_C,f,real,img)
constants;
w=2*pi*f;
% define some parameters for measurement cell
skin_depth=1/sqrt((pi*mu_al*sigma_al)*f); % Skin depth
Rs=l/(sigma_al*(2*pi*a*skin_depth))+l/(sigma_al*(2*pi*b*skin_depth)); %AC
resistance (skin resistance)
Z0=real+1i*(img); %putting together Zreal and Zimag to Z0, Z0 is the
measured impedance
mu_real=theta(1);mu_imag=theta(2);Rp=theta(3);%Picking out mu_r and Rp from
theta
%mu_r=mu_real-1i*mu_imag;
Lreal=(l*mu_0*mu_real*log(b/a))/(2*pi); %Calculating the inductance of the
cylinder
Limag=(l*mu_0*mu_imag*log(b/a))/(2*pi); %Calculating the inductance of the
cylinder
%Components of impedance to make the system more easy overviewed
Zt1=Rp*(Rs+w*Limag);
Zt2=Lreal*Rp;
Zn1=Rp+Rs+w*Limag;
real_hat=R_C+((Zt1*Zn1+w^2*Zt2*Lreal)./(Zn1^2+w^2*Lreal^2)); %real part of
Z_hat
img_hat=L_C*w+((w*Zt2*Zn1-w*Zt1*Lreal)./(Zn1^2+w^2*Lreal^2));%imag part of
Z_hat
Z_hat=real_hat+1i*(img_hat); %Z_hat is the calculated impedance.
%**********Mean Square of error*******************************************
E = Z0-Z_hat; %E is the error and Z0 is the measured impedance
J = conj(E)*E; %Scalar product Error squared becomes real
%disp(['r = ', num2str(theta(1)),', a = ',num2str(theta(2)),' , v =
',num2str(theta(3)),', J = ',num2str(J)]);
%*************************************************************************
%*************************************************************************
%**************************ZLOAD******************************************
%*************************************************************************
%% Function for translating the S11 parameter to equivalent impedance of
%Author: Christian Karlsson
%% ZLOAD
function [ ZL ] = zload(S11)
%Zload calculates the ZLoad as a function of frequency
%Zload(S11)
ZL=50.*((1+S11)./(1-S11));
62
APPENDIX H – MATLAB SCRIPT – INDUCTIVE OPTIMIZER
%
%
%
%
Optimization of the inductive case coaxial cell using fminsearch
(inductive_optimizer.m)
Original Author: Amin Saremi
Rewritten and edited by: Christian Karlsson
%% Formalism
%
clear all;
clc;
constants; %loadconstants
loaddata;%f,w,w_m,Zs,Z2,Z4,Z6,Z8,Z10,Z12,
%% Initiations
%
theta=zeros(3,1);
mu_real=zeros(length(w),length(phi));
mu_imag=zeros(length(w),length(phi));
Rp_opt=zeros(length(w),length(phi));
j=zeros(length(w),length(phi));
%% Calculating correction parameters and well known behaviour of cell
(Skinresistance)
%
% Directly from Eq.Circuit of lid Z=R_C+JwL_C
% determine the resistance Rc and inductance Lc of the non-ideal connection
% on top of the cylinder inductive equivalent model (for lid
representation)
disp('Warning about badly conditioned polynomial is expected!, due to
modeling and canceling of unwanted noise.')
L_C=imag(Zs)./w; %Inductance of lid
P=polyfit(f,real(Zs),3); %Stores an 3rd order polynomial which fits
P(f)=real_c
R_C=polyval(P,f); %A nice approximation of serial skin resistance
%% Guessing mu_r values by assuming the circuit of an only inductive
circuit
% with complex mu_r that is a resistor value of wLimag and a inductor value
% of Lreal assuming Rs very small.
Lrguess=imag(Zmeas)./w_m;
Liguess=-real(Zmeas)./w_m; %Minus sign according to definition of positive
impedance and that the reactance part of mu_r will be defined negative
Lguess=[Lrguess,Liguess]; % An length(w)*12 matrix containing guesses of
real and imaginary inductance
mur_guess=((2*pi).*Lguess)./(l*mu_0*log(b/a)); %Already adjusted for
negative reactance
%% Guessing Rp based on resistivity measurements
%
%Fit a 10th order polynomial curve to the dataset of measurements
P1=polyfit(resistivity_data(:,1),resistivity_data(:,2),10); %(moisture
content,resistivity,order of polynomial)
63
%Evaluate the polynomial for the vector phi
res=polyval(P1,phi);
%Assumptions
res_slig=repmat(res,size(w),1); %size(w) x 6
sigma_slig=1./res_slig;
G=(2*pi.*sigma_slig)./log(b/a);
Rp=1./G;
%% Optimization
%% ************** Start of optimization - Information disp ****************
%************************************************************************
disp(' ');
disp('Optimization just started...');
disp('Thanks for your patience!...');
disp(' ');
%% *********************** Start of optimization ************************
%***********************************************************************
%Optimization settings
oldopts = optimset('fminsearch');
newopts = optimset(oldopts,'MaxIter',200);
%newopts = optimset(newopts,'TolFun',1e-3);
%newopts = optimset(newopts,'TolX',1e-3);
tic, %take the execution time for the below code (until instruction toc)
%OPTIMIZATION LOOP
for m=1:length(phi) %Moisture content (2:2:12%)
for n=1:length(w) %frequency start:length(w)
theta(1)=mur_guess(n,m); %real mu_r guess
theta(2)=mur_guess(n,m+length(phi)); %imag mu_r guess
theta(3)=Rp(n,m);
theta_r= fminsearch(@(theta)
costfun(theta,R_C(n),L_C(n),f(n),real(Zmeas(n,m)),imag(Zmeas(n,m))),theta,n
ewopts);
mu_real(n,m)=theta_r(1); %Den optimerade frekvensberoende mu_r_real
mu_imag(n,m)=theta_r(2); %Den optimerade frekvensberoende mu_r_imag
Rp_opt(n,m)=theta_r(3); %Den optimerade frekvensberoende Rp
j(n,m)=
costfun(theta_r,R_C(n),L_C(n),f(n),real(Zmeas(n,m)),imag(Zmeas(n,m)));
end
err=sum(j(:,m))/length(j(:,m));
disp(['mean square error at the frequency range for', num2str(2*m),'%
water content' '= J = ', num2str(err)]);
end;
toc; %Stop the execution time and auto disp "Elapsed time"
figure(1)
surf(j)
64
%% Display
%
% mu_opt=mu_real+1i.*mu_imag; %mu_imag is negative so the condition hold
for mu_opt=mu_real-1i*mu_imag (from guess)
sigma_opt=log(b/a)./(Rp_opt.*(2*pi*l)); %Optimized Conductivity - Sigma
%Calculating optimized impedance for different moistures and plotting
skin_depth=1./sqrt((pi*mu_al*sigma_al).*f); % Skin depth
Rs=l./(sigma_al*(2*pi*a*skin_depth))+l./(sigma_al*(2*pi*b*skin_depth)); %AC
resistance (skin resistance)
%Rbot=log(b/a)./(2*pi*skin_depth*sigma_al);
%Lbot=
clear L;
for m=1:length(phi) %Check so that the model does not deviate
Rp=Rp_opt(:,m);
Lreal=(l*mu_0.*mu_real(:,m).*log(b/a))./(2*pi); %Calculating the
inductance of the cylinder
Limag=(l*mu_0.*mu_imag(:,m).*log(b/a))./(2*pi); %Calculating the
inductance of the cylinder(ev. stoppa in -)
%Components of impedance to make the system more easy overview
Zt1=Rp.*(Rs+w.*Limag);
Zt2=Lreal.*Rp;
Zn1=Rp+Rs+w.*Limag;
real_hat=R_C+((Zt1.*Zn1+w.^2.*Zt2.*Lreal)./(Zn1.^2+w.^2.*Lreal.^2));
%real part of Z_hat
img_hat=L_C.*w+((w.*Zt2.*Zn1w.*Zt1.*Lreal)./(Zn1.^2+w.^2.*Lreal.^2));%imag part of Z_hat
Z_hat=real_hat+1i.*(img_hat); %Z_hat is the calculated impedance.
figure(2)
hold on
plot(f, real_hat,'b') %Plotting optimized mu_r
plot(f, real(Zmeas(:,m)),'r')
figure(3)
hold on
plot(f, img_hat,'b') %Plotting optimized mu_r
plot(f, imag(Zmeas(:,m)),'r')
end
hold off
plot_results
65
APPENDIX I – MATLAB SCRIPT – PLOT RESULTS
% Result plotting function
% Original Author: Amin Saremi
% Editor: Christian Karlsson
%**************************************************************************
%*
Plots of the results
*
%**************************************************************************
threshold=ones(size(w));
%*********** Real Relative Permeability depending on watercontents (real
part)***************
figure
hold on
plot(f, mu_real(:,1),'-b') %Plotting optimized mu_r based on 2% moisture
(frequency dependent)
plot(f, mu_real(:,2),'-g') %Plotting optimized mu_r based on 4% moisture
(frequency dependent)
plot(f, mu_real(:,3),'-y') %Plotting optimized mu_r based on 6% moisture
(frequency dependent)
plot(f, mu_real(:,4),'-m') %Plotting optimized mu_r based on 8% moisture
(frequency dependent)
plot(f, mu_real(:,5),'-c') %Plotting optimized mu_r based on 10% moisture
(frequency dependent)
%plot(f, mu_real(:,6),'-r') %Plotting optimized mu_r based on 12% moisture
(frequency dependent)
plot(f, threshold,'--k') %Plotting threshold
title('Real Relative permeability of the magnetite clay for different water
contents [optimized]');
xlabel('Frequency[hz]');ylabel('Mu_r_,_r_e_a_l');
%*********** Imag Relative Permeability depending on watercontents (imag
part)***************
figure
hold on
plot(f, mu_imag(:,1),'-b') %Plotting optimized mu_r based on 2% moisture
(frequency dependent)
plot(f, mu_imag(:,2),'-g') %Plotting optimized mu_r based on 4% moisture
(frequency dependent)
plot(f, mu_imag(:,3),'-y') %Plotting optimized mu_r based on 6% moisture
(frequency dependent)
plot(f, mu_imag(:,4),'-m') %Plotting optimized mu_r based on 8% moisture
(frequency dependent) %pinkish
plot(f, mu_imag(:,5),'-c') %Plotting optimized mu_r based on 10% moisture
(frequency dependent)%light bluish
%plot(f, mu_imag(:,6),'-r') %Plotting optimized mu_r based on 12% moisture
(frequency dependent)
plot(f, threshold,'--k') %Plotting threshold
title('Imag Relative permeability of the magnetite clay for different water
contents [optimized]');
xlabel('Frequency[hz]');ylabel('Mu_r_,_i_m_a_g');
% %*************Real Relative permeability guesses based on moisture MUT
tests******************
% figure
% hold on
% plot(f, mur_guess(:,1),'-b') %Plotting guess of mu_r based on 2% moisture
(frequency dependent)
% plot(f, mur_guess(:,2),'-g') %Plotting guess of mu_r based on 4% moisture
(frequency dependent)
% plot(f, mur_guess(:,3),'-y') %Plotting guess of mu_r based on 6% moisture
(frequency dependent)
66
% plot(f, mur_guess(:,4),'-m') %Plotting guess of mu_r based on 8% moisture
(frequency dependent)
% plot(f, mur_guess(:,5),'-c') %Plotting guess of mu_r based on 10%
moisture (frequency dependent)
% %plot(f, mur_guess(:,6),'-r') %Plotting guess of mu_r based on 12%
moisture (frequency dependent)
% plot(f, threshold,'--k')%Plotting threshold
% title('Real Relative Permeability Guesses of the Magnetite Clay for
Different Water Contents');
% xlabel('Frequency [hz]');ylabel('Mu_r_,_g_u_e_s_s_,_r_e_a_l');
%
% %*************Imag Relative permeability guesses based on moisture MUT
tests******************
% figure
% hold on
% plot(f, mur_guess(:,1+length(phi)),'-b') %Plotting guess of mu_r based on
2% moisture (frequency dependent)
% plot(f, mur_guess(:,2+length(phi)),'-g') %Plotting guess of mu_r based on
4% moisture (frequency dependent)
% plot(f, mur_guess(:,3+length(phi)),'-y') %Plotting guess of mu_r based on
6% moisture (frequency dependent)
% plot(f, mur_guess(:,4+length(phi)),'-m') %Plotting guess of mu_r based on
8% moisture (frequency dependent)
% plot(f, mur_guess(:,5+length(phi)),'-c') %Plotting guess of mu_r based on
10% moisture (frequency dependent)
% %plot(f, mur_guess(:,6+length(phi)),'-r') %Plotting guess of mu_r based
on 12% moisture (frequency dependent)
% plot(f, threshold,'--k')%Plotting threshold
% title('Imag Relative Permeability Guesses of the Magnetite Clay for
Different Water Contents');
% xlabel('Frequency [hz]');ylabel('Mu_r_,_g_u_e_s_s_,_i_m_a_g');
%*********************Sigma - Conductivity of clay*************************
figure
hold on
plot(f, sigma_opt(:,1),'-b')%Plotting optimised conductivity for 2% water
plot(f, sigma_opt(:,2),'-g')%Plotting optimised conductivity for 4% water
plot(f, sigma_opt(:,3),'-y')%Plotting optimised conductivity for 6% water
plot(f, sigma_opt(:,4),'-m')%Plotting optimised conductivity for 8% water
plot(f, sigma_opt(:,5),'-c')%Plotting optimised conductivity for 10% water
%plot(f, sigma_opt(:,6),'-r')%Plotting optimised conductivity for 12% water
title('Sigma - Conductivity of Magnetite Clay');
xlabel('Frequency[hz]');ylabel('Sigma [S/m]');
67
APPENDIX J – MATLAB SCRIPT – CONSTANTS
% Different constants of our problem both for materials and geometry
% (constants.m)
% Author: Christian Karlsson
% sigma_al=3.54e7; %conductivity of aluminium
% mu_al=1.0141; %Permeability of aluminium
global t_wall t_ptfe eps_0 mu_0 r_ptfe a b c d l eps_air eps_w mu_air mu_w
mu_al sigma_air sigma_w sigma_ws sigma_al
%Basic constants%
eps_0=8.85419E-012; %F/m
mu_0=4*pi*1E-007; %H/m | N/A^2
c=299792458; %
start=15; %index makro
%Geometrical constants%
a=0.00495; %m
b=0.027; %m
d=0.03; %m
%l=0.3; %m (amin value)
%l=0.5; %m (experimentation value)
l=0.249-(1.6e-2); %m (Ck value for inductive) Removed for bottom and lid
insertion
dept=0.008; %m
t_wall=d-b; %m
r_ptfe=0.0125; %m
t_ptfe=0.00785; %m
%Material Specific constants%
%Permitivity
eps_air=1.00059*eps_0;
eps_w=80.4*eps_0;
eps_ptfe=2.1*eps_0;
%Permeability
mu_air=1.00000037*mu_0;
mu_w=0.999992*mu_0;
mu_al=1.000022*mu_0;%Could be complex and frequency dependent Christians
values
%mu_al=1.0141*mu_0;%Could be complex and frequency dependent (Amin values)
mu_ptfe=1*mu_0;
%Conductivity
sigma_air=0; %S/m %Actual: ~1E(-16)
sigma_ptfe=1E-16;
%sigma_w=55*1E-009; %S/m
sigma_w=1E-003; %S/m
sigma_ws=4; %[S/m] for salt water (varies dep. Saltcontent probably ref.
ocean water)
sigma_al=3.538E+007; %S/m %Pure aluminium, changes with type of aluminium
phi=[0.02,0.04,0.06,0.08,0.10]; %moisture content vector
68
APPENDIX K – MATLAB SCRIPT – DATA LOADER
% Data Loader (loaddata.m)
%Author: Christian Karlsson
%%
%
load lidind.mat %Inductive measurement of lid
f=(s(1,:)).';
w=2.*pi.*f;
Sli=(d(1,:)).';
Zli=ZLoad(Sli);
%%
%
load lidcap.mat %cappacitive measurement of lid
Slc=(d(1,:)).';
Zlc=ZLoad(Slc);
%%
%
load airind.mat %
Sai=(d(1,:)).';
Zai=ZLoad(Sai);
%%
%
load aircap.mat %
Sac=(d(1,:)).';
Zac=ZLoad(Sac);
%%
%
load h2oind.mat %
Swi=(d(1,:)).';
Zwi=ZLoad(Swi);
%%
%
load h2ocap.mat %
Swc=(d(1,:)).';
Zwc=ZLoad(Swc);
%%
%
clear s d
69