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A HIGH-SENSITIVITY FLEXIBLE-EXCITATION ELECTRICAL
CAPACITANCE TOMOGRAPHY SYSTEM
A thesis submitted to the University of Manchester
Institute of Science and Technology for the degree of
Doctor of Philosophy
1997
JOSE CARLOS GAMIO ROFFE
Department of Electrical Engineering and Electronics
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LIST OF CONTENTS
ABSTRACT .............................................. vii
DECLARATION ............................................ ix
ACKNOWLEDGEMENTS ........................................ x
CHAPTER 1: Introduction ................................. 1
1.1 OVERVIEW OF ELECTRICAL CAPACITANCE TOMOGRAPHY ...... 1
1.2 MAIN AREAS OF IMPROVEMENT .......................... 3
1.3 AIMS AND OBJECTIVES ................................ 6
1.4 ORGANISATION OF THIS THESIS ........................ 6
CHAPTER 2: Theory of electrical capacitance
Tomography (ECT) ............................. 8
2.1 THE ECT SENSOR AS A SYSTEM OF CHARGED
CONDUCTORS ......................................... 8
2.2 ECT MEASUREMENT STRATEGIES ........................ 12
2.2.1 THE SINGLE-ELECTRODE EXCITATION METHOD ......... 12
2.2.2 QUALITATIVE IMAGE RECONSTRUCTION FOR
SINGLE-ELECTRODE EXCITATION: THE LINEAR
BACK-PROJECTION (LBP) ALGORITHM ................ 14
2.2.2.1 THE SENSITIVITY MAPS ....................... 14
2.2.2.2 THE NORMALISED MEASUREMENTS ................ 16
2.2.2.3 THE WEIGHTED BACK-PROJECTION OPERATION ..... 17
2.2.3 MULTIPLE-ELECTRODE EXCITATION METHODS .......... 17
2.2.3.1 RECONSTRUCTION WITH MULTIPLE-ELECTRODE
EXCITATION ................................. 22
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2.2.3.2 OPTIMAL MULTIPLE-ELECTRODE EXCITATION
PATTERNS ................................... 25
2.3 ECT TRANSDUCERS ................................... 28
2.3.1 THE CHARGE-DISCHARGE CAPACITANCE TRANSDUCER
CIRCUIT ........................................ 29
2.3.2 AC-BASED ECT TRANSDUCERS ....................... 32
2.4 DISCUSSION AND CONCLUSIONS ........................ 35
2.4.1 SINGLE- VS. MULTIPLE-ELECTRODE EXCITATION ...... 35
2.4.2 CHARGE-DISCHARGE VS. AC-BASED TRANSDUCERS ...... 36
CHAPTER 3: Finite-element simulation of single-
electrode and parallel-field excitation ..... 38
3.1 INTRODUCTION ...................................... 39
3.2 FINITE-ELEMENT MODELLING OF ECT SENSORS............ 39
3.3 SINGLE-ELECTRODE EXCITATION ....................... 44
3.4 IDEAL SENSITIVITY MAPS .......................... 45
3.5 PARALLEL-FIELD EXCITATION ......................... 47
3.5.1 PARALLEL-FIELD GENERATION ...................... 47
3.5.2 SENSITIVITY MAPS FOR PARALLEL-FIELD
EXCITATION ..................................... 50
3.6 COMPARISON OF IMAGES OBTAINED USING SINGLE-
ELECTRODE AND PARALLEL-FIELD EXCITATION ........... 52
3.7 DETERMINATION OF MUTUAL CAPACITANCES FROM
PARALLEL-FIELD CHARGE MEASUREMENTS ................ 56
3.8 CONCLUSIONS ....................................... 59
CHAPTER 4: Design of an AC-based multiple-
excitation ECT system ....................... 62
4.1 INTRODUCTION ...................................... 62
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4.2 DESIRED SYSTEM CHARACTERISTICS .................... 64
4.3 GENERAL DESIGN STRATEGY ........................... 65
4.4 THE BASIC DETECTOR CIRCUIT ........................ 67
4.4.1 TRANSFER FUNCTION WITH FREQUENCY-DEPENDENT
OP-AMP GAIN .................................... 68
4.4.2 COMMENTS ON NOISE PERFORMANCE .................. 73
4.4.3 DETAILED CIRCUIT DESIGN ........................ 74
4.4.4 FINAL FREQUENCY RESPONSE ....................... 76
4.4.5 EFFECT OF CHANGES IN STRAY CAPACITANCE ......... 77
4.5 DESIGN OF THE MULTI-EXCITATION ECT TRANSDUCER
CHANNEL ........................................... 80
4.5.1 NOMINAL MEASUREMENT RANGES AND SENSITIVITIES ... 84
4.5.2 EFFECT OF CMOS SWITCH ON RESISTANCE .......... 85
4.6 REFERENCE SINE-WAVE GENERATOR ..................... 91
4.7 SIGNAL CONDITIONING AND DATA CONVERSION SECTION ... 93
4.8 SYSTEM INTEGRATION AND CONSTRUCTION ............... 95
4.9 CALIBRATION OF SYSTEM ELECTRONICS ................ 101
4.9.1 CALIBRATION OF THE EXCITATION VOLTAGE
SOURCES ....................................... 101
4.9.2 DETECTOR CALIBRATION .......................... 103
4.10 EXPERIMENTAL EVALUATION OF THE SYSTEM ........... 106
4.10.1 CAPACITANCE MEASUREMENT COMPARATIVE TEST ..... 107
4.10.2 LINEARITY EVALUATION ......................... 108
4.10.3 INTRINSIC NOISE LEVEL TEST ................... 110
4.10.4 DYNAMIC PERFORMANCE OF THE SYSTEM ............ 112
4.11 IMAGE SAMPLES ................................... 115
4.12 CONCLUSIONS ..................................... 118
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CHAPTER 5: Comparative experimental evaluation of
single- and multiple-electrode excitation
methods..................................... 120
5.1 INTRODUCTION ..................................... 120
5.2 EXPERIMENTS WITH OPTIMUM MULTI-ELECTRODE
EXCITATION ....................................... 121
5.2.1 DETERMINATION OF THE OPTIMUM EXCITATION
VECTORS ....................................... 121
5.2.2 APPLYING THE OPTIMUM EXCITATION VECTORS ....... 122
5.2.3 EXPERIMENTAL PROCEDURE ........................ 127
5.3 EXPERIMENT RESULTS AND DISCUSSION ................ 130
5.3.1 EXPERIMENT No. 1 .............................. 131
5.3.2 EXPERIMENT No. 2 .............................. 136
5.3.3 EXPERIMENT No. 3 .............................. 139
5.3.4 EXPERIMENT No. 4 .............................. 142
5.3.5 EXPERIMENT No. 5 .............................. 145
5.3.6 EXPERIMENT No. 6 .............................. 148
5.4 CONCLUSIONS ...................................... 151
CHAPTER 6: Iterative linear back-projection image-
reconstruction techniques .................. 153
6.1 INTRODUCTION ..................................... 153
6.2 PREVIOUS WORK .................................... 153
6.3 A RECONSTRUCTION ALGORITHM INSPIRED ON FEED-
BACK CONTROL ..................................... 156
6.3.1 FIRST APPROACH: ITERATIVE LBP RECONSTRUCTION .. 156
6.3.2 AN ALGORITHM BASED ON A CONTROL-SYSTEM
ANALOGY ....................................... 163
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6.4 EXPERIMENTAL EVALUATION OF THE
FEED-BACK
ALGORITHM ........................................ 169
6.4.1 RESULTS OF IMAGE RECONSTRUCTION TESTS ......... 169
6.4.2 DISCUSSION .................................... 178
6.5 CONCLUSIONS ...................................... 179
CHAPTER 7: Conclusions and further work ............... 180
7.1 CONCLUSIONS ...................................... 180
7.2 RECOMMENDATIONS FOR FUTURE WORK .................. 182
REFERENCES ............................................ 184
APPENDIX A: System design details and circuit
diagrams .................................. 194
APPENDIX B: Computer programs ......................... 201
B.0 General ......................................... 201
B.1 ECT system monitoring and control programs ...... 201
B.2 Programs used in the optimum excitationexperiments ..................................... 204
B.3 Simulation programs ............................. 205
B.3.1 Programs to calculate the sensitivity mapsfor single-electrode excitation ............... 206
B.3.2 Programs to calculate the sensitivity mapsfor parallel-field excitation ................. 206
B.3.3 Programs to simulate single-electrode-excitation measurements ....................... 207
B.3.4 Programs to simulate parallel-field-excitation measurements ....................... 208
B.3.5 Programs to perform LBP image reconstruction .. 208
B.3.6 Iterative LBP image reconstruction program .... 209
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APPENDIX C: Papers produced as a result of this
work ...................................... 210
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ABSTRACT
Several ways of improving the performance of electrical
capacitance tomography (ECT) systems are presented and
evaluated, including the use of alternative excitation
schemes, more sensitive and less noisy electronics, and more
accurate image reconstruction algorithms.
The design of a new electrical capacitance tomography (ECT)
data acquisition system is presented, having a number of
improvements over the one previously designed at UMIST. The
new system uses AC-based instead of charge-discharge
capacitance transducers, providing an increase in resolution
from 0.26 to 0.025 fF (peak value of noise level) and a
ten-fold improvement in stray-immunity. Thanks to the use of
AC amplifiers before demodulation the problem of drift is
practically eliminated. Phase-sensitive demodulation is
employed in order to be able to discriminate between the
effects of the conductive and capacitive components of the
unknown admittance. Each channel has its own demodulator thus
allowing parallel measurement. The latter, coupled with
high-frequency (500 kHz) operation, results in a potential
acquisition rate of more than 100 frames per second. Finally,
excitation signals can be applied to several electrodes at
the same time, and, thus, the system can be employed to
investigate the possibility of using multi-electrode
excitation patterns.
The use of parallel-field excitation (attempting to mimic
X-ray tomography) is explored employing finite-element
simulation techniques and found to be disadvantageous. It is
shown that, due to fundamental differences in the underlying
physics, an analogy cannot really be established between
X-ray tomography and parallel-field excitation ECT. On the
other hand, the use of optimal or adaptive excitation
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methods, seeking to maximize the visibility or
distinguishability of the permittivity distributions imaged,
were successfully tested against the conventional single-
electrode excitation method, and results are presented for
different permittivity distributions.
Finally, results are presented of an assessment of a new
iterative image reconstruction algorithm based on the quite
singular approach of viewing the reconstruction process as a
feed-back control system.
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DECLARATION
No portion of the work referred to in this thesis has been
submitted in support of an application for another degree or
qualification of this or any other university, or other
institute of learning.
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ACKNOWLEDGEMENTS
I would like to express my appreciation for their help,
encouragement and valuable advice to my supervisor Dr. R C
Waterfall, Professor M S Beck and Dr. W Q Yang.
I also acknowledge the financial support of the Mexican
Petroleum Institute and the National Council for Science and
Technology of Mexico (Conacyt).
Finally, I thank my wife Sara for her patience and support.
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C H A P T E R 1 :
I N T R O D U C T I O N
1.1 OVERVIEW OF ELECTRICAL CAPACITANCE TOMOGRAPHY
Electrical capacitance tomography (or ECT) is one of a
relatively new breed of imaging techniques developed for
industrial process applications, collectively known as
process tomography [1,2]. The aim of all these methods, which
started to evolve in the mid 1980s, is to provide a
non-invasive, non-intrusive means to obtain cross-sectional
images of the interior of process vessels (figure 1.1), which
can be used to control and monitor process operations, or as
a model validation tool in process design.
Fig. 1.1 Process tomography systemFig. 1.1Fig. 1.1
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Although the use of several tomographic modalities has been
explored, including ionising radiation, magnetic resonance
imaging, ultrasonic and optical techniques, electrical
methods based on impedance measurement have generally proved
more suitable for process tomography applications, being
fast, robust and relatively inexpensive. Electrical impedance
tomography (EIT) can be subdivided into resistance,
inductance and capacitance tomography, depending on the
physical quantity being measured.
Fig. 1.2 Electrical capacitance tomography systemFig. 1.2Fig. 1.2
Electrical capacitance tomography is aimed at industrial
processes involving non-conducting materials, or mixtures
where the continuous phase is non-conducting. In a
conventional ECT system (figure 1.2), the sensor takes the
form of a circular array of electrodes placed around an
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insulating pipe and surrounded by a grounded screen (the
latter not shown in the figure for clarity). The data
acquisition unit contains capacitance transducers which are
used to determine the capacitance of all possible electrode
pair combinations, thus producing n(n-1) measurements, where
n is the number of electrodes. By means of a computer and a
suitable algorithm, these data must then be used to
reconstruct an image of the permittivity distribution inside
the sensor, which directly reflects the material
distribution.
The first attempts to do capacitance-based tomography were
carried out more or less at the same time (
1986-88) both at
UMIST [3-5] and the Morgantown Energy Technology Centre (in
the USA) [6-8]. Later on, in 1991, the first real-time ECT
system was developed in a joint project by UMIST, Leeds
University and Schlumberger Cambridge Research [9,10]. ECT
has been applied, at an experimental level, to the on-line
visualisation of gas-oil flows [11,12], as well as imaging
combustion processes [13-17] and fluidised beds [18-20]. For
an excellent review article including numerous applications
and an extensive bibliography (80 references) see [42].
1.2 MAIN AREAS OF IMPROVEMENT
In what follows we identify some of the principal areas
subject to improvement in ECT, which will become the focusand motivation of this work.
The main challenge encountered in the design of an ECT system
comes from the fact that the capacitances to be measured are
extremely small. For example, for a typical 12-electrode
empty sensor the various inter-electrode capacitances range
from 10 to 1,000 femtoFarads, and the expected full-scale
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capacitance changes extend from about 5 to 80 femtoFarads,
for 10 cm long electrodes. Therefore, depending on the
particular permittivity distribution, the system may have to
measure accurately capacitance changes of a few tenths of a
femtoFarad. Furthermore, these very low inter-electrode
capacitances have to be measured in the presence of stray
capacitances to ground several order of magnitude larger
(
100 picoFarads). There is, then, the need for better
high-sensitivity low-noise capacitance transducers, in order
to increase the signal to noise ratio (SNR) and the
resolution of the measurements. With the use of more
sensitive electronics, it would be possible to increase the
spatial and/or axial resolution by employing smaller
electrodes. Alternatively, the higher sensitivity can be
exploited for imaging lean flows.
Another problem area found
Fig. 1.3 Sensitivity mapFig. 1.3Fig. 1.3
for opposite electrodes
i n E C T i s t h a t
inter-electrode capacitances
are much more sensitive to
changes in permittivity near
the electrodes than to
changes occurring near the
centre of the sensor. This
is reflected on the shape of
the sensitivity maps, which
are basically graphs of
dC/d for a specific pair of
electrodes. For instance, infigure 1.3 we can see the
typical sensitivity map for opposite electrodes. It roughly
defines a channel of sensitivity across the sensor area.
However, we can see that the sensitivity is not constant
along this
channel
; the two peaks correspond to the
electrode positions and the sensitivity decreases towards the
centre. As a result of this non-uniform sensitivity effect,
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it is much more difficult to
see
permittivity changes near
the centre than elsewhere in the sensor. The idea of somehow
increasing the sensitivity in the centre to achieve a uniform
sensing area sounds, therefore, very attractive. In the early
stages of this work, it was thought that the problem would be
solved or alleviated if a parallel electric field could be
created inside the sensor by applying specific voltages to
the electrodes [21,22], although, after a thorough
investigation of the matter, it was later found that this was
not entirely the case [23]. There are, however, other types
of multiple-electrode excitation that can be used to improve
the overall sensitivity of EIT systems in general, which have
been already successfully tried for resistance tomography
[30,31,33], but not for ECT.
Let us finally say a word about image reconstruction. To
perform an accurate quantitative reconstruction of the
permittivity distribution inside the sensor from the
capacitance measurements is a very complex task, which
mathematically belongs to the category of inverse problems
and normally involves computationally intensive iterative
procedures based on optimisation. However, by linearising the
problem, a qualitative reconstruction can be done using a
simple and fast algorithm known as linear back-projection
(LBP) [4], adapted from medical X-ray tomography. Because of
its simplicity and speed, virtually all ECT systems use the
LBP algorithm. Nevertheless, the quality of the images
obtained with LBP is rather poor, and there is clearly the
need for better reconstruction methods that can provide bothaccuracy and speed.
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1.3 AIMS AND OBJECTIVES
In view of the considerations presented in the previous
section, the general aims of this work are set as follows:
a) To investigate ways to increase the sensitivity of ECT
systems in the central area of the sensor, particularly
the use of parallel fields and multi-electrode excitation.
b) To develop new capacitance transducers for ECT use, having
more sensitivity and lower noise level than those
currently being employed.
c) To investigate improved reconstruction algorithms based
on iterative LBP methods.
In order to have experimental support, an ECT system will be
designed and built, incorporating the new high-sensitivity
transducers developed.
1.4 ORGANISATION OF THIS THESIS
In chapter 2, the physical situation occurring in an ECT
sensor is described, according to the laws of
electromagnetism. The theoretical implications of using
various different excitation arrangements are presented, and
a comparison is made between them, showing the unsuitabilityof parallel fields. Also, the previous work on ECT system
design in discussed here, and new ways of improving
capacitance transducer sensitivity using AC-based circuits
are proposed.
Chapter 3 presents a critical analysis of the idea of using
parallel-field excitation as a means to increase the
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sensitivity in the centre of the sensor and reduce the
non-uniformity of the sensitivity maps, in an attempt to
mimic X-ray tomography. The analysis is based on finite-
element simulation using the software package PC-OPERA, and
includes a comparison between parallel-field and conventional
single-electrode excitation.
The design and testing of an AC-based high-sensitivity
capacitance transducer is presented in chapter 4. The design
of a multiple-excitation ECT system based on this transducer
is also described.
Chapter 5 presents results obtained with an ECT system built
after the design presented in chapter 4, including the
experimental evaluation of various single- and multiple-
electrode excitation methods.
Chapter 6 discusses the implementation of a new iterative
reconstruction algorithm based on back-projection.
Finally, chapter 7 summarises the main achievements of this
project and gives suggestions about possible areas for future
work.
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C H A P T E R 2 :
THEORY OF ELECTRICAL CAPACITANCE TOMOGRAPHY
2.1 THE ECT SENSOR AS A SYSTEM OF CHARGED CONDUCTORS
In ECT, a number of electrodes are installed around the pipe
or vessel to be imaged, surrounded by a grounded screen. This
is the basic ECT sensor configuration, shown in figure 2.1
for 12 electrodes. The mutual capacitance (defined below) of
the different electrode pairs depends on the permittivity
distribution inside the sensor. When a body is placed in the
sensor there will be a change in the mutual capacitances. The
principle of ECT is to measure the change in mutual
capacitance of the different electrode pair combinations and
then use these measurements to reconstruct an image of the
permittivity distribution in the cross section being
investigated by the sensor. The reconstruction process can be
done with a computer using a simple algorithm known as linear
back-projection (LBP) [4].
Fig. 2.1 Basic 12-electrode ECT sensorFig. 2.1Fig. 2.1
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From a physics point of view, the ECT sensor can be
considered as a special case of a system of charged
conductors separated by a dielectric medium [24-27], the
theory of which was first developed by Maxwell [28]. In our
particular case, with the sensor electrodes acting as the
charged conductors, the electrode charges Qi and the electrode
potentials Vi are related by the following set of linear
equations for an n-electrode sensor
Q1= c11V1+c12V2++c1nVn
Q2= c21V1+c22V2++c2nVn
(2.1) Qn= cn1V1+cn2V2++cnnVn
where the coefficients cii are called the self-capacitance of
electrode i, while the others, cij, with i j, are the mutual
capacitance of electrodes i and j.
Writing equation (2.1) in matrix form we have
Q1 c11 c12 c1n V1 Q2 c21 c22 c2n V2 = (2.2) = Qn cn1 cn2 cnn Vn
or
Q = C V (2.3)
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The matrix C is called the capacitance matrix of the system.
The self and mutual capacitances are sometimes called
coefficients of capacitance and coefficients of
(electrostatic) induction respectively, and in the very old
books they are termed coefficients of capacity and
coefficients of influence. These coefficients depend only on
the geometry and the permittivity distribution of the system,
and have the following important properties:
a) The self-capacitances are always positive.
b) The mutual capacitances are always negative.
c) For every conductor we have
ci1 + ci2 + + c in 0 (2.4)
d) For the mutual capacitances
cij = cji (2.5)
The matrix C is called the capacitance matrix and completely
characterises the system of conductors. C is a non-linear
function of the system geometry and of the permittivity
distribution in the dielectric medium. In our case the
geometry is fixed, so any change in C will be due to a change
in the permittivity distribution.
An ECT sensor (or any system of charged conductors) can also
be modelled using a circuit theory approach, as a network of
component capacitances [29], as illustrated in figure 2.2
using a 4-electrode sensor. To do this, we re-write equation
(2.1) in terms of the voltage difference between the various
electrodes, to put it in the form
Q1= C1V1+C12(V1-V2)+C13(V1-V3)+
Q2= C2V2+C21(V2-V1)+C23(V2-V3)+ (2.6)
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or, in compact form, for electrode i of n
n
Qi = CiVi + Cij(Vi-Vj) (2.7)j=1
(i j)
where the component capacitance between electrode i and
ground is given by
Ci = ci1+ci2+ci3++cin (2.8)
and the inter-electrode component capacitance between
electrodes i and j by
Cij = -cij (i j) (2.9)
Fig. 2.2 Equivalent circuit (based on the componentFig. 2.2Fig. 2.2
capacitances) of a 4-electrode ECT sensor
As mentioned earlier, in ECT we are concerned only with the
inter-electrode capacitances (i.e. the mutual capacitances),
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since this are the ones that depend on the permittivity
distribution inside the sensing area of the sensor. We are
not interested in the component capacitances to ground (i.e.
between the electrodes and the screen) because, due to the
geometry of the sensor (assuming that the inter-electrode
gaps are small), they depend mainly on the permittivity
distribution in the annular region between the electrode ring
and the outer screen, outside the imaging area. We are not
interested in the self-capacitances either because they are
determined by the mutual capacitances and the component
capacitances to ground (see equation 2.8).
Consequently, we are only interested in the mutual
capacitances, only half of which are independent (because of
their reciprocity relationship). So, we can say that all the
information about any change in the permittivity distribution
inside the sensor will be contained in the variations of the
n(n-1) independent mutual capacitances, which form the lower
(or upper) triangular part of the capacitance matrix C.
2.2 ECT MEASUREMENT STRATEGIES
The value of the self and mutual capacitances can be found by
applying known potentials to the sensor electrodes and
measuring the electrode charges. In practice, the
determination on the electrode charges is normally done
indirectly by measuring the electrode currents (Q = i d t), andthe excitation potentials are applied to the electrodes in
the form of a periodic signal of known amplitude.
2.2.1 THE SINGLE-ELECTRODE EXCITATION METHOD
All ECT systems reported in the literature so far, including
those developed earlier at UMIST [10], use single-electrode
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excitation to measure the mutual capacitances, with the
exception of the Morgantown system, which uses a special
bipolar excitation technique [8]. Let us consider a
12-electrode sensor (figure 2.1). Using the single-electrode
excitation method, the mutual capacitances are determined as
follows: First an excitation voltage is applied to electrode
1 while keeping all the others at zero potential and the
charge on electrodes 2 to 12 is measured. According to
equation (2.1), these measurements directly represent c2 1 to
c12 1 . Next, the excitation voltage is applied to electrode 2
while keeping all the others at zero and the charge on
electrodes 3 to 12 is measured, representing c3 2 to c12 2 . This
procedure is repeated, applying voltage to electrode n and
measuring the charge on electrodes (n+1) to 12, until, as a
final step, voltage is applied to electrode 11 and the charge
of electrode 12 is measured. In this way, the 66 independent
mutual capacitance values corresponding to the lower half of
the capacitance matrix are determined (the other 66 being
given by equation (2.5)), requiring 66 electrode charge
measurements.
From a hardware design point of view, single-electrode
excitation has the advantage of requiring only one voltage
source, which can be switched sequentially to the electrode
being used as a source.
The problem with this method is that, because the mutual
capacitances are so small, the electrode charges can also be
very small, and, as a result, the signal-to-noise ratio (SNR)of the measurements tends to be rather poor, even when
low-noise measuring circuits are used. From equation (2.1),
it is clear that, if excitation potentials are applied to
more than one electrode, it is possible to obtain larger
electrode charges, although they would no longer be a direct
measure of any particular inter-electrode capacitance.
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2.2.2 QUALITATIVE IMAGE RECONSTRUCTION FOR SINGLE-ELECTRODE
EXCITATION: THE LINEAR BACK-PROJECTION (LBP) ALGORITHM
In single-electrode excitation ECT systems, image
reconstruction for two-component mixtures is done using the
linear back-projection (LBP) algorithm [3]. The basic idea of
this qualitative algorithm, which is an adaptation of a
method used in medical tomography, is to do a weighted
back-project or smearing of each one of the n(n-1)
normalised measurements along its sensing zone, given by the
corresponding sensitivity map.
2.2.2.1 THE SENSITIVITY MAPS
Let us consider an n-electrode sensor and an image made of m
equal-area pixels. For each pair of electrodes i (source) and
j (detector), a capacitance sensitivity map can be defined by
Ci j( k ) - C i j emp Qi j( k ) - Q i j empSi j(k)= = (2.10)
Ci j full - Ci j emp Qi j full - Qi j emp
where k is the pixel number (from 1 to m), Qi j(k) is the
charge induced on electrode j by electrode i when the region
of pixel k is full of high-permittivity material while the
rest of the sensing area is full of low-permittivity
material, Qi j full and Qi j emp are the charge induced on electrode
j by electrode i when the sensor is full of high- and
low-permittivity material, respectively, and Cij are the
corresponding mutual capacitances of electrodes i and j under
the same conditions.
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a) Location of electrodes b) S 13
c) S15 d) S17
Figure 2.3 Typical sensitivity maps for single-electrodeFigure 2.3Figure 2.3
excitation
The sensitivity maps can be determined experimentally for a
particular sensor by probing the sensing area using a test
rod, although this is a very time-consuming task. A more
practical approach is to use computer simulation techniques
to model the sensor using the finite-element method (FEM).
The author chose the latter and, in his work, used PC-OPERA,
a commercially available FEM software package for
electromagnetic analysis and simulation. A more detailed
description of the procedures used to calculate the
sensitivity maps is given in chapter 3 and in appendix B.
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Regardless of the method used, it is not necessary to
determine the sensitivity maps for all possible electrode
pair combinations, since, due to sensor symmetry, all the
sensitivity maps can be obtained by rotation from (for a
12-electrode sensor) the following basic set of 6: S12 , S13 ,
S14 , S15 , S16 and S17.
Figure 2.3 shows typical sensitivity maps for several
electrode combinations, which were calculated for imaging oil
and gas mixtures ( high = 2.1 and low = 1), with m = 313. As
might be expected, the sensitivity maps show that each
electrode pair responds mainly to the material lying between
the electrodes, albeit in a very non-uniform way.
2.2.2.2 THE NORMALISED MEASUREMENTS
Prior to back-projection, the measurements obtained with each
electrode pair are normalised according to
Ci j meas - Ci j emp Qi j meas - Qi j emp i j= = (2.11)
Ci j full - Ci j emp Qi j full - Qi j emp
where i j is the normalised measurement (charge or
capacitance) corresponding to electrodes i (source) and j
(detector), Qi j meas is the measured charge induced on electrode
j by electrode i, Qi j full and Qi j emp are the charge induced on
electrode j by electrode i when the sensor is full of high-
and low-permittivity material, respectively, and Cij are the
corresponding mutual capacitances of electrodes i and j.
Normalised values are used so that the same software will
cope with systems having different electrode lengths, which
produce different absolute measurements but the same
normalised ones.
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2.2.2.3 THE WEIGHTED BACK-PROJECTION OPERATION
Mathematically, for an n-electrode sensor and an m-pixel
image, the LBP algorithm calculates the grey level G(k) for
each pixel k as
G(k)
n 1
i 1
n
j i 1
i j Si j (k)
n 1
i 1
n
j i 1
Si j (k)
(k 1..m) (2.12)
where
i j are the normalised measurements defined by equation(2.11) and Si j are the sensitivity maps defined by equation
(2.10). The actual back-projection operation occurs in the
numerator of equation (2.12), while the quantity in the
denominator serves as a position-dependent weighting factor
used to compensate for the decrease in sensitivity towards
the centre of the sensor.
2.2.3 MULTIPLE-ELECTRODE EXCITATION METHODS
Although they have not been used in ECT, multiple-electrode
excitation techniques have been around for quite a while in
other EIT modes and are especially popular in electrical
resistance tomography (ERT) work [30,31], where they can be
found under various names like adaptive , multi-reference ,
and optimal currents methods.
If there are n electrodes in an ECT sensor, there is no
reason why we should not simultaneously apply excitation
voltages to more than one of them. In fact we can define
excitation voltage vectors of the form
V = [V1 , V2 , ..., Vn ] (2.13)
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If we apply this excitation voltage vector to an ECT sensor
and then measure all the electrode charges, we can form
another vector
Q = [Q1 , Q2 , ..., Qn ] (2.14)
Let us consider two different permittivity distributions
(p), corresponding to a known reference state such as an
empty sensor, and (p), corresponding to some unknown
material distribution, where p denotes a point inside the
sensor. When we apply a voltage vector V to these two
permittivity distributions we get two charge vectors Q and
Q . Let us now define our measurement signal as the change in
the electrode charges due to the permittivity distribution
changing from to , i.e.
Q = Q - Q (2.15)
We can then define a vector of measured signals m as
m = Q - Q = [ Q1 , Q2 , ..., Qn ] (2.16)
Following the ideas of Issacson [30], we define thedistinguishability of with respect to when exciting
with V as
Q = m 2 = Q - Q
2 = [ Q1 , Q2 , ..., Qn ] 2 (2.17)
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additionally, from equation (2.3) we have
Q = Q - Q
2
= (C - C )V 2
= DV 2
(2.18)
where C and C are the capacitance matrices corresponding to
and , respectively, and D = C - C is called the
distinguishability matrix.
Equation (2.17) shows that Q is an indicator of the
magnitude of the detection signals (defined as the change in
electrode charge Q). Obviously, the larger the detection
signals the easier it will be to detect the change in
permittivity distribution from to . On the other hand,
equation (2.18) shows that Q depends on the distingui-
shability matrix (which ultimately depends on and ),
but, more importantly, also on the excitation vector V.
In other words, for given permittivity distributions and
, not all excitation vectors will produce the same signal
level, and there will be some excitation vectors that are the
best choice to distinguish between and , in the sense
that they will produce the largest detection signals, and,
therefore, the best signal-to-noise ratio (SNR). This is the
main justification for the use of multiple-electrode
excitation, since with single-electrode excitation the choice
of excitation vectors is limited to those having only one
non-zero element.
Extending to ECT the result obtained by Issacson in [30], we
can say that the best voltage vectors to distinguish
between and
are the eigenvectors of D having the
largest eigenvalue. In general, excitation patterns having
a low spatial frequency will yield measurements which are
more sensitive to changes in permittivity near the centre,
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while high frequency patterns yield measurements sensitive
mainly to changes near the electrodes [30].
Now, although the multiple-electrode excitation method yields
measurements with the best SNR, there is a price to pay both
in terms of the system hardware and software. Firstly, the
hardware becomes more complex and expensive, since it must
now include n independent voltage sources. Secondly, unlike
with single-electrode excitation, what is measured is no
longer capacitance but the response of the sensor (in the
form of changes in electrode charge Q) to a set of
excitation vectors Vk, with k = 1, ..., L. Because of this, the
practical LBP algorithm, which is based on capacitance
measurements, cannot be used, at least not directly.
As it will soon become clear, it can be advantageous to
derive an expression for the measurements Q as a function of
the inter-electrode voltages, and to re-define the excitation
vectors in terms of the latter instead of the electrode
voltages themselves. We shall, for the sake of simplicity,
use a 4-electrode sensor (figure 2.2) to illustrate these
concepts, although the same ideas apply fully to sensors with
any number of electrodes.
From equation (2.7) we can arrive at the following system of
equations describing the sensor
Q1 =
C1V1 +
C12 (V1-V2) +
C13 (V1-V3) +
C14 (V1-V4) Q2 = C2V2 + C21 (V2-V1) + C23 (V2-V3) + C24 (V2-V4)
Q3 = C3V3 + C31 (V3-V1) + C32 (V3-V2) + C34 (V3-V4) (2.19)
Q4 = C4V4 + C41 (V4-V1) + C42 (V4-V2) + C43 (V4-V3)
Now, because of the geometry of the sensor (assuming that the
inter-electrode gaps are small) we have Ci 0, i = 1, .., n.
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In other words, the change in the capacitances to ground is
negligible because they do not depend on the permittivity of
the material inside the sensing area, but only on that of the
material in the region between the electrodes and the outer
screen. Therefore, equation (2.19) becomes
Q1 = C12 (V1-V2) + C13 (V1-V3) + C14 (V1-V4)
Q2 = C21 (V2-V1) + C23 (V2-V3) + C24 (V2-V4)
Q3 = C31 (V3-V1) + C32 (V3-V2) + C34 (V3-V4) (2.20)
Q4 = C41 (V4-V1) + C42 (V4-V2) + C43 (V4-V3)
For our 4-electrode sensor (n = 4) we can then define the
following excitation vectors, of size n2-n:
U = [ (V1-V2),(V1-V3),(V1-V4),(V2-V1),(V2-V3),(V2-V4),
(V3-V1),(V3-V2),(V3-V4),(V4-V1),(V4-V2),(V4-V3) ] (2.21)
In equation (2.20) we can see that our measurement signals,
considered as the change in charge Qi, do not depend on Vi,
the actual voltage on the measuring electrode, but only on
the voltage differences between the electrodes. However, in
practice the system cannot measure Qi directly, it has to
measure Qi and Qi separately and from them calculate
Qi = Qi - Qi
. From equation (2.7) we see that the voltage Vi
on the measuring electrode can have a considerable effect on
its charge Qi, since the capacitances to ground Ci (which in
an actual system include the capacitance of the cable
connecting the electrode to the instrument) are much larger
than the inter-electrode ones, whose effect will be obscured.
In order to avoid having to measure the electrode charge due
to Ci, the voltage Vi on the measuring electrode should be set
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to zero. The advantage of defining the measurement signals Q
and the excitation vectors U in terms of the inter-electrode
voltages is that, once the most suitable vectors have been
chosen (according to some criterion), the voltage on the
measuring electrode can always be set to zero and the rest of
the electrode voltages adjusted accordingly to satisfy the
particular excitation vector being applied.
2.2.3.1 RECONSTRUCTION WITH MULTIPLE-ELECTRODE EXCITATION
The idea of multiple-electrode excitation brings about the
important question of how to reconstruct an image from the
knowledge of the applied excitation vectors and the vector ofmeasurements. For a quantitative reconstruction directly from
the charge measurements, we have to resort to iterative
algorithms like the those used in electrical resistance
tomography (ERT) [32,33], which are normally based on some
variant of Newton s method.
The formal development of iterative image reconstruction
algorithms based on optimisation is a vast and complex task
that could itself be the subject of another PhD thesis,
involving a considerable amount of advanced mathematics, and
is not within the scope of this work. The main objective of
this work, as far as multiple-electrode excitation is
concerned, is to design and build an actual system and use it
to confirm experimentally that multiple excitation can indeed
improve distinguishability and the SNR of the measurements
(chapters 4 and 5), a purpose which can be achieved without
having to produce any images. Nevertheless, without going
into details, we shall present a general description of some
of the methods that can be used for iterative image
reconstruction.
Let us assume that we apply a set of L = n - 1 linearly
independent inter-electrode voltage excitation vectors Uk,
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k = 1, .., L. The excitation vectors are applied one by one
and, for each vector, the change in electrode charge Q is
measured for every electrode. So, for each excitation vector
Uk we can form an n-dimensional vector of measurements
mk like
the one in equation (2.16). The total number of independent
measurements is, therefore, N = nL = n (n - 1), and they can be
stacked in a long vector
M = [m1 ,m2 , ...,mN ]. (2.22)
We know that M is a linear function of the excitation
vectors, which can be stacked in a vector E = [U1, U2, ..., Uk ],
and a non-linear function of the reference and unknown
permittivity distributions and , i.e. M = (E, , ).
As shown in figure 2.4, iterative algorithms start with an
initial guess of the unknown permittivity distribution, 0 ,
which is then used to calculate (E, , 0 ) using a
finite-element model of the sensor. Then the iterative part
commences, by comparing the simulated measurements with the
actual ones, the difference between the two being calculated
as i = (E, , i ) - M 2, i = 0, 1, 2, ...; if i is smaller
than some specified tolerance then we can consider i as the
true permittivity distribution, otherwise this error is fed
to an optimiser (based on some Newton-like formula) that
produces a new estimated permittivity distribution i+1 ,
which is used to calculate the next simulated measurements.
The cycle is repeated until the error is within the accepted
tolerance. Despite requiring great computing power and being
relatively slow,iterative algorithms like this produce
quantitative reconstruction, unlike LBP. One serious problem,
however, is that the noise level of the measurements can
severely affect the algorithm s convergence and accuracy. The
better SNR achieved through multiple-electrode excitation
would be particularly useful in alleviating this problem.
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Fig. 2.4 General flow diagram of iterative reconstructionFig. 2.4Fig. 2.4
algorithms
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It really seems rather strange that iterative image
reconstruction algorithms based on optimisation theory, which
are mathematically sound and yield quantitative results, have
not been applied to the specific case of ECT, whereas in ERT
they are very popular and there has been a lot of research
into the subject, with the publication of many papers and
several PhD theses.
An alternative approach to image reconstruction for
multiple-electrode excitation ECT systems involves using
equation (2.20) to recover the inter-electrode capacitance
changes Cij from the measurements M and the inter-electrode
excitation voltages E. This can easily be done by solving n
systems of n - 1 linear equations each. Once we have the
inter-electrode capacitance changes Cij , which represent the
equivalent single-electrode excitation measurements, we can
use the standard LBP algorithm to perform a qualitative image
reconstruction.
2.2.3.2 OPTIMAL MULTIPLE-ELECTRODE EXCITATION PATTERNS
The best inter-electrode voltage vectors U are the ones
that maximise the changes in electrode charge Q (equation
(2.20)), and they can be determined from the distingui-
shability matrix D = C -C .
Let us see, first, how the D matrix can be determined. For
i j, we have Dij = cij - cij . For i=j, we have Dii = cii - cii
, but
from equation (2.8)
n
cii = Ci - cij (2.23)j=1
(i j)
therefore
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n n
Dii = Ci - cij - Ci + cij
(2.24)j=1 j=1
(i j) (i j)
Recalling that the capacitances to ground do not depend on
the permittivity distribution inside the imaging area, we
have Ci = Ci , so they cancel out and equation (2.24) becomes
n n
Dii = - cij - cij (2.25)
j=1 j=1
(i j) (i
j)
or
n
Dii = - Dij (2.26)j=1
(i j)
From the foregoing discussion we can conclude that the matrix
D is completely determined by the mutual capacitances cij and
cij (with i j) and does not depend on the self-capacitances
cii and cii . The mutual capacitances can easily be determined
using the single-electrode excitation method described in
section 2.2.1. In a practical situation, cij , the mutual
capacitances for the reference state (empty sensor), could be
measured in advance and stored, so that it would only be
necessary to measure the mutual capacitances corresponding to
the unknown permittivity distribution
in order to
determine D.
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As mentioned earlier, the optimal excitation voltage vectors
are the eigenvectors of D corresponding to the largest
eigenvalues [30]. Many methods can be used to obtain the
eigensystem of the symmetric matrix D, like the QR algorithm,
singular value decomposition (SVD) and the Jacobi method. The
SVD method is preferred because it is stable and
straightforward to obtain [31]. Using SVD we factorise D as
D = X Y T (2.27)
where X and Y are orthogonal matrices and is a diagonal
matrix whose entries are the singular values. The
eigenvectors of D (i.e. the optimal excitation voltage
vectors) are given by the columns of X, while its eigenvalues
are equal to the singular values. The best excitation vector
is the eigenvector corresponding to the largest eigenvalue.
The last eigenvalue is always equal to zero, since the rank
of D is n - 1, and the corresponding excitation vector is not
used. If necessary, the electrode voltage vectors thus
obtained can be re-scaled, in order to fully exploit the
dynamic range of the voltage sources employed.
In this way we end up with an optimal set of orthogonal
n-dimensional electrode-voltage unit vectors Vk ,
k = 1, .., (n - 1), which will produce the largest measurements
for a particular permittivity distribution , compared with
other vectors of unit length. It was shown earlier that,because of the need to always set the measuring electrode
voltage to zero, it is more convenient to work with
inter-electrode voltages. The optimal inter-electrode voltage
vectors Uk can be obtained from Vk using equations (2.13)
and (2.21).
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2.3 ECT TRANSDUCERS
The function of an ECT transducer is to measure the charge Q
on a detection electrode. On a single-electrode excitation
system, where only two electrodes are involved in each
measurement, this charge Q is also a measure of the
capacitance between the two electrodes, given by Cij = - Qi /Vj .
Among the desired characteristics of an ECT transducer are:
a) It must be stray-immune. This means that the capacitance
between the measuring electrode and ground should not have
an effect on the measurement. This is normally achieved by
ensuring that the detection electrode potential is
maintained at zero during measurement (thus making Vi = 0
in equation (2.7)).
b) Its range has to match the sensor and the characteristics
of the materials to be imaged. For example, for a typical
12-electrode sensor used to image oil ( r = 2.1) and gas
( r = 1) mixtures, the inter-electrode capacitances can lie
anywhere between 10 and 600 femtoFarads approximately.
c) It should be able to measure small inter-electrode
capacitance changes in the presence of large standing
values. Again for the same situation as in (b), we have
that, depending on the particular electrode pair, the
full-scale capacitance change can be as low as 15% of the
standing capacitance. The transducer must have some means
of balancing these standing capacitances.
d) The resolution must be high enough, and the noise level
low enough, to allow the detection of capacitance changes
of a few tenths of a femtoFarad.
e) If the system is to be used in fast real-time applications
like flow imaging, the transducer must have a fast dynamic
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response. Considering a frame rate of, say, 100 frame/s,
the time available for collecting all the data for one
frame is 10 ms.
2.3.1 THE CHARGE-DISCHARGE CAPACITANCE TRANSDUCER CIRCUIT
The single-electrode excitation ECT system developed at UMIST
uses capacitance transducers based on the charge transfer
principle [9,34]. This type of transducer is stray-immune
(i.e. they are insensitive to the capacitances to ground) and
its main attractive is its simplicity and relatively low
cost.
Fig. 2.5 The charge-discharge capacitance measuring circuitFig. 2.5Fig. 2.5
The charge transfer transducer circuit is shown in
figure 2.5. The device works by repeatedly charging and
discharging the unknown capacitance Cx through the combined
action of semiconductor switches S1
to S4. First, S
1 and S
2 are
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closed (keeping S3 and S4 open) to charge Cx to voltage Vc, and
the charging current flows into the current-to-voltage
converter formed by operational amplifier (op-amp) A1 and its
feed-back resistor Rf, causing a negative output voltage. Inthe second half of the operating cycle, S
1 and S
2 are open
while S3 and S4 are closed, thus discharging Cx to ground. The
discharge current flows out of the current-to-voltage
converter formed by op-amp A2
and its feed-back resistor Rf,
producing a positive output voltage. This charge-discharge
cycle repeats itself at a frequency f (up to 2 MHz) and the
successive charging and discharging current pulses are
averaged in the two current detectors, producing two dc
output voltages:
V1 = - f V c Rf Cx + e1 (2.28)
V2 = f Vc Rf Cx + e2 (2.29)
where e1 and e2 are the output offset voltages of the current
to voltage converters. Since the detection electrode of Cx is
always connected to a virtual earth point, any stray
capacitance to ground Cs1 is always short-circuited and has no
effect on the measurement. And, because the excitation
electrode is always being driven by a low-impedance voltage
source, its stray capacitance Cs2 has no effect either.
The voltage difference V2 - V1 is taken as the output, giving
Vo = V2 - V1 = 2 f Vc Rf Cx + e2 - e1 (2.30)
this has the advantage of doubling the sensitivity and that
the offset signals e1 and e2 tend to cancel each other.
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Capacitors C at the input of the op-amps ensure a stable
virtual earth during the fast charge and discharge of Cx.
The bandwidth of the circuit is set by Rf
and Cf
through
1B = (2.31)
2 Rf Cx
and the maximum bandwidth achievable with a particular op-amp
was calculated by Huang [35] as
ABmax = 0.1 (2.32)
T (C+Cf)Rf
where A is the open-loop gain and T the open-loop time
constant of the op-amp.
The main limitation of the charge-discharge transducer is
that it has a lower signal-to-noise ratio than AC-based
circuits. For instance, in the system developed at UMIST,
which uses a sensor with 12 10-cm-long electrodes, the peak
noise level at the system s output is equivalent to an input
capacitance 0.26 femtoFarads, with a transducer bandwidth of
about 10 kHz. This figure essentially sets the resolution of
the system, and corresponds to a change of 2% in the gas void
fraction of an oil/gas mixture occurring in the middle of the
sensor (which is the least sensitive area) [9]. Clearly,
under these conditions, the measurements will not be too
reliable or accurate, since the signal-to-noise ratio is
equal to 1. If we want to use shorter electrodes in order to
reduce the averaging effect along the axial direction, or
increase the number of electrodes to, say, 16 in order to
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increase the image spatial resolution, then the electrode
area would be reduced and more sensitive transducers would be
required to detect the smaller capacitances produced. The
same is true for imaging mixtures with low permittivity
contrast or lean flows.
Another drawback of the charge-discharge transducer is that
it is susceptible to the effects of conductance losses, i.e,
it is not phase-sensitive. This can be a problem in
applications involving water or in flame imaging, for
example.
Finally, this transducer suffers from charge injection caused
by the feed-through of gate control signals in the switching
semiconductor devices. This charge injection appears as an
offset voltage at the output, which is the main component of
e1 and e2 in equation (2.30). The effect is temperature-
dependent and, for a temperature change of 15C, it is
equivalent to an input capacitance of up to 5 femtoFarads
[9]. These offsets cause a baseline drift and, in order to
maintain accuracy, they need to be periodically monitored and
compensated.
2.3.2 AC-BASED ECT TRANSDUCERS
The charge-discharge transducer effectively uses square-wave
excitation. Although this type of excitation signal has the
advantage of being easily implemented by switching between
two DC levels, it has been recognized that by employingAC-based circuits, i.e., circuits based on sine-wave
excitation, a higher signal to noise ratio can be achieved
[9,35]. The main reason for this is that the use of
single-frequency excitation allows the use of narrow-band
filtering techniques based on phase-sensitive demodulators,
which can greatly reduce the noise bandwidth [45] and also
provide the means to discriminate between capacitive and
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conductive effects. Additionally, switch-related problems
like charge injection and the generation of glitches are
eliminated. Likewise, drift is no longer such a big concern.
Although there are numerous AC-based methods to measure small
capacitance values (for a good review of many of them see
[35]), not all of them lend themselves well to a
multi-channel application like ECT. In particular, we are
interested in transducer circuits based on the use of an
operational amplifier as a current detector [36-40], an area
relatively new compared with other methods.
In order to be consistent with the concept of multiple-
electrode excitation we shall consider the ECT transducer as
a charge rather than capacitance sensor. Figure 2.6 shows a
charge detector (or charge amplifier) [41] based on an
op-amp, which will be used as the basic component for the
design of a more sensitive ECT transducer in chapter 4.
Fig. 2.6 Basic charge detectorFig. 2.6Fig. 2.6
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Essentially, the circuit consists of an operational amplifier
(op-amp) with capacitive feedback. The resistor R provides a
path for the op-amp s DC bias current to avoid saturation of
the device due to that current charging Cf
. It has negligible
effect on the op-amp output at the frequency of operation.
Assuming that amplifier has a high gain at the frequency of
operation, in the sinusoidal steady state, the circuit is
described by Vo
(j ) = - Zf Iin(j ) = j Iin(j )/ Cf where Iin(j )V (j ) I (j ) I (j ) I (j )V (j ) I (j ) I (j ) I (j )
is the input current. But Iin(t) = - dQ(t)/dt , or in theI (t) Q(t)I (t) Q(t)
frequency domain Iin
(j ) = - j Q(j ) where Q(j ) is theI (j ) Q(j ) Q(j )I (j ) Q(j ) Q(j )
electrode charge. Therefore we have that Vo(j ) = Q(j )/Cf .V (j ) Q(j )V (j ) Q(j )
Thus, the circuit can be considered as an AC charge to
voltage (Q-V) converter. Because of the feedback action of
the op-amp, this circuit has the important advantage of
keeping the measuring electrode at virtual earth, avoiding
the appearance of the comparatively large charge due to the
stray capacitance to ground (C1 in figure 2.6). In other
words, the circuit is stray immune.
This simple circuit measures the charge induced on the
detection electrode. If only one electrode is used for
excitation with a voltage Vexc (i.e. in the single-electrode
excitation method) we can get the capacitance between the
electrode pair from Cx = - Q/Vexc , and the circuit can be used
as a capacitance transducer with its output given by
Vo = - (Vexc /Cf ) Cx . The equivalent circuit for this case is
shown in figure 2.7. The capacitance to ground of the
detection electrode, CD , has no effect on Vo , since the
voltage across it is very close to zero, whereas that of the
excitation electrode, CE , is driven by a low-impedance
voltage source and so it does not affect Vo either. However,
if more than one electrode are used simultaneously for
(multi-electrode) excitation then we cannot say precisely
which capacitance is being measured, and it is more
convenient to view the circuit as a charge detector.
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Fig. 2.7 The charge detector as a capacitance meterFig. 2.7Fig. 2.7
A full analysis of the capabilities and limitations of the
proposed charge detector circuit will be presented in
chapter 4, which shows the detailed design of an AC-based
multiple-electrode excitation ECT system.
2.4 DISCUSSION AND CONCLUSIONS
2.4.1 SINGLE- VS. MULTIPLE-ELECTRODE EXCITATION
It has been shown earlier that multiple-electrode excitation
using optimal inter-electrode voltage vectors U can produce
larger detection signals Q and, thus, increase the SNR of
measurements compared with single-electrode excitation. This,
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however, comes at the cost of more complicated and expensive
hardware, since n independent voltage sources are required.
Another point to consider is parallel measurement. In
single-electrode excitation systems all electrodes but one
are kept at zero volts, and they can all be put to measure
simultaneously (i.e. parallel measurement). This is no longer
achievable with multiple-electrode excitation, because, for
any particular application of an inter-electrode voltage
vector U, generally only a few electrodes will be at zero
potential (i.e. available for measurement) and, therefore,
each inter-electrode voltage vector will actually have to be
applied several times, until all electrodes have had a chance
to be at zero volts and be measured. This, of course, takes
time and has the effect of slowing down the system.
Considering the foregoing, it is the view of the author that,
although multiple-electrode excitation can indeed be very
useful in special applications where the absolute maximum
sensitivity is desired, for industrial process applications
where low-cost and speed of operation are important,
single-electrode excitation ECT systems, with its straight-
forward approach to both hardware design and image
reconstruction (LBP algorithm), would be a more sensible
choice.
2.4.2 CHARGE-DISCHARGE VS. AC-BASED TRANSDUCERS
The use of AC-based ECT transducers result in better noiseperformance, and hence, better resolution and SNR. Later on,
in chapter 4, a system based on this type of transducer will
be designed and the results of its experimental evaluation
presented, showing that a 10-fold improvement in SNR over the
charge-discharge system is possible in practice. Once again,
there is a price to pay, and it comes in the form of more
complex and expensive electronics, mainly due to the fact
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that one demodulator per channel is required in order to have
parallel measurement capability for fast operation (see
chapter 4).
Despite its higher cost, AC-based transducers are a good
choice in application that require higher resolution and SNR,
like when working with low-contrast mixtures or with lean
flows, or in systems where the electrodes are small, either
because they are short (say less that 10 cm) for better axial
resolution, or because there are a large number of them (more
that 12, say 16) for improved spatial resolution.
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C H A P T E R 3 :
FINITE-ELEMENT SIMULATION OF SINGLE-ELECTRODE AND
PARALLEL-FIELD EXCITATION
3.1 INTRODUCTION
Of the many possible multiple-electrode excitation
arrangements, so-called parallel-field excitation deserves
special attention. At the beginning of this work, some ECT
researchers shared the idea that the problem of low
sensitivity in the centre of the sensor and the
non-uniformity of the sensitivity maps (figure 2.3) had
something to do with the uneven distribution of electric
force lines which occurs when single-electrode excitation is
employed [21,22]. This type of excitation results in the
electric field being very strong near the excitation
electrode, rapidly weakening as we move away (as shown in
figure 3.4). It was thought that the ideal situation would
rather be to have a parallel electric field uniformly
distributed across the entire sensing area. By so trying to
mimic X-ray tomography, it was believed that increased
sensitivity in the central region would be achieved and also
that the quality of the reconstructed images could be
improved.
In this chapter, we show how parallel-field excitation can be
realised by applying specific excitation voltages to all
electrodes in the sensor. We present the results of
simulation experiments carried out to determine what effects
would the use of parallel-field excitation have, both on the
shape of the sensitivity maps and on the reconstructed
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images, compared to single-electrode excitation. Finally, we
consider the question of whether or not parallel field
excitation can provide a complete set of independent
measurements required to determine the sensor mutual
capacitances.
3.2 FINITE-ELEMENT MODELLING OF ECT SENSORS
In order to investigate the effects of parallel-field
excitation, simulation experiments were carried out using
finite-element (FE) models of the sensor. PC-OPERA, a
commercially available software package for 2-dimensional
electromagnetic field analysis based on the finite-element
method (FEM), was used to perform the simulation. Note that,
because of the two extra cylindrical guarding electrodes
(grounded) used in the actual sensor on each side of the
sensing electrodes in the axial direction (figure 3.1), 2-D
simulation can be used to model the sensor [4], albeit we are
restricted to work with 2-dimensional material distributions.
Fig. 3.1 Side view of ECT sensor showing guard electrodesFig. 3.1Fig. 3.1
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We are going to use the FEM package to solve the following
basic problem:
Given a number of conducting bodies with known applied
potentials placed in a dielectric medium with a known
permittivity distribution, find the resultant electric
field distribution and the self and mutual capacitances
characterising the system
Essentially what PC-OPERA does is to find the value of the
electric potential at a large number of points or nodes in a
mesh of contiguous triangular elements used to represent the
actual physical system. The potential data can then be used
to calculate the electric field vectors (by E = -
V) orEE
other parameters of interest.
Given a relative permittivity distribution
(x,y), PC-OPERA
finds the potential distribution V(x,y) by numerically
solving the following partial differential equation (where o
is the free-space permittivity)
[ o (x,y) V(x,y)] = 0 (3.1)
subject to the corresponding Dirichlet conditions (known
potentials on the boundary).
Working with PC-OPERA involves the following three steps:
1) First, in the preprocessing stage, a model of the problem
is generated using an interactive pre-and-postprocessor
program. In this phase two files are generated, one with
extension "MES" which contains the geometry of the mesh,
and another with extension "OP2" which contains the
boundary conditions, material properties, etc.
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2) Second, the static analysis program is used to generate
the solution (i.e. the potential distribution). The input
to this program are the two files created in the previous
step, while its output is a solution file with extension
"ST".
3) In the third and final step the solution is viewed and the
parameters of interest (i.e. electrode charge or
capacitance) are calculated, using once again the pre-and-
postprocessor program.
The main parameter of interest, in our case, is the detection
electrode charge Q, which can be calculated using Gauss Law:
Q DDDdsss (3.2)
however, since we are working in two dimensions, we will not
integrate over a closed surface, but over a closed line
around the electrode. It will not be a surface integral but
a line integral, which is evaluated using one of the
pre-and-postprocessor commands. And, of course, Q then
represents the charge per unit length.
Once the charge is known, the capacitance per unit length can
be easily found (for single-electrode excitation) as
C = - Q/Vexc , where Vexc is the voltage on the excitation
electrode.
PC-OPERA can be run interactively or in an
off-line
mode.
Using this option, the three basic steps mentioned earlier
are automatically executed in sequence. First, a series of
optional pre-and-postprocessor commands (contained in a
command input file) are executed on the specified input
model, then the solution program is run, and finally another
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set of pre-and-postprocessor commands (contained in another
command input file) is executed on the solution file to
calculate the parameters of interest.
For our work, two FE models of a 12-electrode ECT sensor were
created as shown in figures 3.2 and 3.3. In this type of
2-dimensional problems the absolute dimensions are
irrelevant, so we state the geometric characteristics of the
model in terms of the inner radius R of the pipe (i.e. the
radius of the imaging area). In this way, the thickness of
the pipe wall is 0.1R, and the distance between the external
side of the pipe and the outer screen is 0.2R. The relative
permittivity of the region between the pipe and the screen is
set to 1 (air), while that of the pipe is set to 2.5
(perspex). The electrode angle is 26, with inter-electrode
gaps of 4.
a) Model geometry b) FE mesh (5881 elements)
Figure 3.2 Basic FE sensor modelFigure 3.2Figure 3.2
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a) Model geometry b) FE mesh (9912 elements)
Figure 3.3 FE sensor model with polar gridFigure 3.3Figure 3.3
One of the models (figure 3.3) includes a polar grid composed
of 313 equal-area regions and is used in the determination of
the sensitivity maps required for the LBP algorithm. The
permittivity of each one of this regions can be independently
set to any value, hence allowing the simulation of arbitrary
material distributions. A polar grid was chosen instead of
square one because of its particular symmetry, which allows
the model behaviour to be orientation-independent.
By setting the proper boundary conditions, the application ofany excitation voltage vector can be simulated.
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3.3 SINGLE-ELECTRODE EXCITATION
Figure 3.4 shows the simulation results for single-electrode
excitation. The excitation voltage is applied to electrode
one and the figure shows the equipotential lines. Clearly,
with this type of excitation the electric field distribution
is quite uneven, with the field concentrated near the
excitation electrode.
Fig. 3.4 Equipotential lines for single-electrodeFig. 3.4Fig. 3.4
excitation
In order to calculate the sensitivity maps (as defined by
equation (2.10) in chapter 2), a program was written inQuickBasic 4.5, which iteratively runs PC-OPERA off-line
using the 313-region polar-grid sensor model of figure 3.3
with the electrode potentials set for single-electrode
excitation. At the start of the ith iteration, the QuickBasic
program generates a command input file that will be used by
PC-OPERA to set the relative permittivity of in-pipe region
i equal to 2.1 ( oil ) while that of all the others will be set
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to 1. Then the program calls PC-OPERA off-line, which finds
the solution and calculates the charge on all electrodes
using another command input file previously written for this
purpose. The process is repeated 313 times (for each one of
the in-pipe regions). In this manner, all the sensitivity
maps associated with electrode 1 are obtained (i.e. S1 1 to
S1 12 ). The full program can be found in appendix B. The
sensitivity maps associated with the electrodes 2 to 12 are
obtained by rotation of those calculated for electrode 1.
Finally we end up with 144 sensitivity maps, of which only 66
are really needed for use with the LBP image reconstruction
algorithm (one for each measurement, see sections 2.2.1 and
2.2.2 in chapter 2).
Typical examples of sensitivity maps for single-electrode
excitation obtained in the way described above are shown in
figure 2.3 (chapter 2). It can be observed that the detection
areas of the sensor form clearly defined
channels
between
the detection and excitation electrodes. This is a desirable
feature in a tomography sensor, since each detector
looks
only to a specific area. However, the single-electrode
excitation ECT sensor departs from the ideal situation in
that:
a) The
channels
are not straight, and
b) Their
height
is not constant, that is to say, the
response of the detectors is lower in the middle of the
channel
and higher near the electrodes.
3.4
IDEAL
SENSITIVITY MAPS
In an ideal situation, we would like to see something similar
to the case of parallel-beam X-ray tomography, where each
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detector is sensitive over a very narrow and straight
channel
of constant
height
.
For an ECT sensor, though, the sensitivity
channels
could
not be narrowed too much without having to reduce the
electrode width to such an extent that measurement signals
would become undetectable. However, it might seem natural to
think that if multiple-electrode excitation is used so as to
create a parallel electric field inside the sensor, at least
sensitivity maps forming straight and uniform
channels
could be obtained, probably something similar to figure 3.5.
a) Location of electrodes b) Electrode 6
Figure 3.5Figure 3.5Figure 3.5
c) Electrode 5
Ideal
ECT sensor
s e n s i t i v i t y m a p s
intuitively
expected
w i t h p a r a l le l - fi e l d
excitation (field parallel
to the x axis)
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In order to verify the previous conjecture, in the next
sections we shall describe how a parallel field can be
created inside an ECT sensor, and the actual characteristics
of the resulting sensitivity maps, which were calculated by
FE simulation, will also be reported.
3.5 PARALLEL-FIELD EXCITATION
3.5.1 PARALLEL-FIELD GENERATION
A parallel field can be approximated inside an n-electrode
ECT sensor by applying electrode potentials according to
sin ( i - )
Vi = E (3.3)n
MAX [ sin( i - )]
i = 1
where, referring to figure 3.6, i indicates the electrode
number (1 to n), is the angle between the field direction
and the y axis, E is a voltage constant (determined by
hardware constraints), and i is the angular position of the
centre of the ith electrode, given by
360 i = ( i - 0.5 ) (3.4)
n
By using this voltage distribution, the potential difference
between pairs of electrodes facing each other in the
direction of the field is made proportional to the separation
between their centres.
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Fig. 3.6 Parallel electric field inside an ECT sensorFig. 3.6Fig. 3.6
For example, to produce a parallel field along the y axis
(
= 0) on a 12-electrode sensor, with E = 15 V, the voltages
shown on table 3.1 would have to be used.
Table 3.1 Electrode potentials for parallel-field
excitation
ELECTRODE
VOLTAGE
ELECTRODE
VOLTAGE
1
4.02
7
-4.02
2 10.98 8 -10.98
3
15.00
9
-15.00
4
15.00
10
-15.00
5 10.98 11 -10.98
6 4.02 12 -4.02
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The generation of a parallel field inside the sensor using
the method described above was confirmed by FE simulation as
shown on figure 3.7.
Fig. 3.7 Equipotential lines for parallel-fieldFig. 3.7Fig. 3.7
excitation using the voltages of Table 3.1
By shifting the potentials one electrode position, the field
can be rotated and different
projections
(in the sense of
conventional X-ray computed tomography) can be defined, each
one associated with a particular direction of the field. For
a 12-electrode sensor, rotating the field in this way means
that we can define six different projections (numbered 1
to 6), corresponding to
equal to 0, 30, 60, 90, 120
and 150. Note that the remaining six projections (180,
210, 240, 270, 300 and 330) do not contribute any
additional information since they are just a sign-changed
repetition of the first ones. For every projection, the
signal from each one of the twelve electrodes can be
measured, giving a total of 6 12 = 72 measurements.
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3.5.2 SENSITIVITY MAPS FOR PARALLEL-FIELD EXCITATION
The sensitivity map for parallel-field excitation is defined
as follows, for projection i and electrode j:
Qi j(k) - Q i j empSi j(k)= (3.5)
Qi j full - Qi j emp
were k = 1, .., 313 is the region (or pixel) number, Qi j(k) is
the charge of electrode j in projection i when region k isfull of high-permittivity material and the rest of the
sensing area is full of low-permittivity material, while Qi j full
and Qi j emp are the charge of electrode j in projection i when
the sensor is full of high- and low-permittivity material,
respectively.
The sensitivity maps were calculated for projection 1 using
the same QuickBasic 4.5 program described in section 3.3,
which runs of PC-OPERA iteratively, but this time with the
electrode potentials set for parallel-field according to
table 3.1, and are shown in figure 3.8. The 12 sensitivity
maps for projection 1 were then rotated to obtain those for
projections 2 to 6, giving a total of 72 sensitivity maps.
Unfortunately, it can be seen in figure 3.8 that the actual
sensitivity maps for parallel-field excitation bear no
resemblance whatsoever with the ideal ones of figure 3.5.