A High-sensitivity Flexible-excitation Electrical Capacitance Tomography System - c Gamio Phd Thesis

7/23/2019 A High-sensitivity Flexible-excitation Electrical Capacitance Tomography System - c Gamio Phd Thesis

1/221

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


2/221

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

i


3/221

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

ii


4/221

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

iii


5/221

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

iv


6/221

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

v


7/221

APPENDIX C: Papers produced as a result of this

work ...................................... 210

vi


8/221

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

vii


9/221

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.

viii


10/221

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.

ix


11/221

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.

x


12/221

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

1


13/221

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

2


14/221

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

3


15/221

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,

4


16/221

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.

5


17/221

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

6


18/221

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.

7


19/221

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

8


20/221

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)

9


21/221

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)

10


22/221

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),

11


23/221

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

12


24/221

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.

13


25/221

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.

14


26/221

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.

15


27/221

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.

16


28/221

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)

17


29/221

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)

18


30/221

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,

19


31/221

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.

20


32/221

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

21


33/221

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,

22


34/221

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.

23


35/221

Fig. 2.4 General flow diagram of iterative reconstructionFig. 2.4Fig. 2.4

algorithms

24


36/221

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

25


37/221

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.

26


38/221

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).

27


39/221

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

28


40/221

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

29


41/221

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.

30


42/221

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

31


43/221

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

32


44/221

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

33


45/221

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.

34


46/221

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,

35


47/221

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

36


48/221

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.

37


49/221

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

38


50/221

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

39


51/221

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.

40


52/221

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

41


53/221

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

42


54/221

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.

43


55/221

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

44


56/221

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

45


57/221

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)

46


58/221

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.

47


59/221

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

48


60/221

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.

49


61/221

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.

Documents

A High-sensitivity Flexible-excitation Electrical Capacitance Tomography System - c Gamio Phd Thesis