potential measurement in ect system

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Journal of Electrostatics 67 (2009) 27–36

Contents lists avai

Journal of Electrostatics

journal homepage: www.elsevier .com/locate/e lstat

Potential measurement in ECT system

Bin Zhou a, Jianyong Zhang b,*

a Southeast University, Chinab School of Science and Technology, University of Teesside, Middlesbrough TS1 3BA, UK

a r t i c l e i n f o

Article history:Received 12 March 2007Received in revised form3 October 2008Accepted 15 October 2008Available online 12 November 2008

Keywords:Miniature sensorElectrical capacitance tomography (ECT)Potential measuring methodGA-ECT

* Corresponding author.E-mail address: [email protected] (J. Zhang).

0304-3886/$ – see front matter Crown Copyright � 2doi:10.1016/j.elstat.2008.10.006

a b s t r a c t

Potential measurement can be applied to electrical capacitance tomography (ECT) systems. However, therelationship between potential and the capacitance is not well understood. This paper reveals thata potential is dependent on the capacitance ratio for a given excitation voltage, independent of capaci-tance themselves. The conditions for using this method are discussed. Finally, image reconstructionbased on the Generic Algorithm for Electrical Capacitance Tomography (GA-ECT) is implemented satis-factorily for square and circular sensors respectively.

Crown Copyright � 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Electrical capacitance tomography (ECT) has great potential inindustrial applications due to its non-intrusive, non-invasive,simple construction and low cost nature [1]. ECT infers a permit-tivity distribution 3(x, y) of a process inside a pipe or vessel using anarray of electrodes attached around the circumference of the pipeor vessel. Over past two decades, the research has been focused onmacro-scale industrial processes [2,3]. With the development ofnano-technology, interest has been extended to micro-scale [4].

In such ECT systems, the mutual capacitances are small due tothe size of a sensor, in particular, for very short electrodes. Thelength of electrodes is a key factor for flow profile capture. Longelectrodes would blur the image of variations in flow profilebecause of the spatial filtering effect.

According to two-dimensional (2D) supposition, the voltages onfloating electrodes are independent of the sensor’s axial length,given ideal conditions. This paper suggests that this concept can beapplied to ECT sensors.

2. Purpose of normalization

In 2D systems, normalization of capacitance data is often per-formed prior to image reconstruction using Eqs. (1) and (2) forparallel and series models respectively.

008 Published by Elsevier Ltd. All

lij ¼cm

ij � clij

chij � cl

ij

(1)

l0ij ¼

1

clij

� 1cm

ij

!, 1

clij

� 1

chij

!(2)

where cijh, cij

l , cijm are the mutual capacitances between electrodes i, j

with the measuring region completely filled with a given dielectricmaterial, completely empty, and partially filled with the givendielectric material respectively.

The normalized value lij or l0ij depends both on the sensorconfiguration (e.g. electrode angle q, the ratio Rin/Rout between theinner and the outer radius of pipe, the ratio (Rfill� Rin)/Rin betweendepth of the filling layer and inner pipe radius) and on the distri-bution of dielectric permittivity inside the sensor, but is indepen-dent of the absolute size of the pipe as shown in Fig. 1. Therefore lij

or l0ij of different pipe sizes will remain the same when theparameters mentioned above are kept unchanged. In 2D supposi-tion, one of the purposes of normalization is to eliminate the effectof actual axial length of electrodes on simulation.

The capacitances are measured before normalization usinga charge method, which is also known as ‘displacement currentmethod’. The axial length of electrodes is determined by the scaleand configuration of sensors. Because the capacitances are verysmall, the axial length of electrodes has to be large. An ECT systemwith long electrodes will have higher sensitivity, however accurate

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¦ Θ

Electrode

Pipewall

Earthed screen

Rfill

Rin

Rout

Fig. 1. Cross-section of ECT sensor.

B. Zhou, J. Zhang / Journal of Electrostatics 67 (2009) 27–3628

cross-sectional flow profile recognition will be affected. For a highprofile resolution along a pipe axis, it requires short electrodes,often resulting in small signals.

There are two possible solutions to the above problem: (1) toimprove the sensitivity of the measuring circuit without loweringthe signal-to-noise ratio (SNR), and (2) to obtain direct measure-ments characterized with capacitances’ normalization. As can beseen in Eqs. (1) and (2), the normalized capacitances are indepen-dent of the length of electrodes. It would offer improvements toboth measurement sensitivity and profile resolution along the pipeaxis provided that such a measurement method is viable.

3. Comparison of two measuring methods

The charge measurement method is traditional, which is basedon the measurement of mutual capacitance between the differentelectrode pairs surrounding a pipe/vessel. For an n-electrode ECTsensor, the number of independent capacitances is n� (n� 1)/2,whilst with the potential measurement method the voltage on eachfloating electrode with regard to the earthed screen is measured.The potential measurement discussed here is a modified version ofthe system introduced in Ref. [5], where the voltages between theelectrodes are taken as measurements. The boundary conditions ofthe mathematical models derived from the above two methods aredifferent, but the number of independent measurements is thesame, i.e. n� (n� 1)/2.

Assuming zero free charge in the pipe and the even mediumdistribution along the pipe axis, and ignoring wave propagationeffects, an ideal 8-electrode ECT model for a capacitance

Fig. 2. Comparison of two methods. (a) Low-Z

measurement system based on the charge measurement method isgoverned by Laplace’s equation and boundary conditions asfollows.

8>><>>:

V$½3ðx; yÞV4ðx; yÞ� ¼ 0;4ðx; yÞjðx;yÞ˛Gi

¼ U;4ðx; yÞjðx;yÞ˛Gj

¼ 0;4ðx; yÞjðx;yÞ˛Gs

¼ 0

8<:ðx; yÞ˛Ui˛f1;2;.;7gj˛f1;2;.;8g=i

(3)

The mathematical model and the boundary conditions fora potential measuring system can be expressed as

8>><>>:

V$½3ðx; yÞV4ðx; yÞ� ¼ 0;4ðx; yÞjðx;yÞ˛Gi

¼ U;Qj ¼ 0;4ðx; yÞjðx;yÞ˛Gs

¼ 0

8<:ðx; yÞ˛Ui˛f1;2;.;8gj˛f1;2;.;8g=i

(4)

where 3(x, y) is the dielectric permittivity distribution. The region U

is the area inside the sensor excluding the electrodes. 4(x, y) standsfor the potential distribution on U. Gi is the excitation electrodewith voltage U, Gj is the sensing electrodes with zero potential andGs represents the earthed screen.

The term 4ðx; yÞjðx;yÞ˛Gj¼ 0 in Eq. (3) is substituted by Qj¼ 0 in

Eq. (4), where Qj is the charge on the jth electrode. Assuming the sameexcitation voltage, the states of the detection electrodes for the twomeasuring methods are different. For the charge (displacementcurrent) measurement method, the charges on the virtually groun-ded electrodes are not zero, but the potentials are approximatelyequal to zero. For the potential measurement method, the charges onthe floating electrodes are zero according to the electrostatic theory,but their potentials are not zero.

Fig. 2 shows the typical circuits for these two methods, wherethe circuit in Fig. 2(a) is designed to detect mutual capacitances bymeasuring the charge (Low-Z measurement). This circuit is alsoknown as a Charge amplifier [6]. The circuit shown in Fig. 2(b) isa voltage amplifier which measures the potentials on the electrodes(High-Z measurement) [7].

It can be seen that in Fig. 2(a) and (b), C1 s C1s1, C2 s C2s1,because in a potential measurement system, all the other elec-trodes are floated when an electrode is connected to the circuit,whereas in a charge measurement system, all the other electrodesare grounded.

Fig. 3 shows that the 2D potential distributions on a cross-section of a pipe are different for ECT systems employing the abovetwo different measuring methods under the same excitationvoltage (5 V) and the same dielectric permittivity distribution(empty).

Comparing (a) and (b) in Fig. 3, it can be seen that the differencein potential distribution with the two methods is obvious. Thepotential on the detection electrode (positioned at the circle of

measurement. (b) High-Z measurement.

Fig. 3. Comparison of potential distribution. (a) Potential distribution for charge measurement. (b) Potential distribution for potential measurement.

B. Zhou, J. Zhang / Journal of Electrostatics 67 (2009) 27–36 29

10 mm radius) is zero in (a) for the charge measurement method.For potential measurement, the potential on the floating electrodein (b) is not equal to zero because the electrode is connected to thevoltage amplifier via a resistor, and the electrode is not directlyconnected to the virtual earth.

c12

c13c14

c15

c16

c17

c18

c10

8

7

6

5 4

3

2

1

Fig. 4. Equivalent circuits of 8-electrode ECT sensor.

4. Relationship between potential and capacitance

An n-electrode ECT sensor can be considered as an isolatedelectrostatic system consisting of (nþ 1) charged electrodes (nelectrodes plus the screen). According to Maxwell’s electrostaticfield theory [8], all the electric field lines emerging from theexcitation electrode sink to the detection electrodes and thescreen. In the following equations, the screen and electrodes arenumbered from 0 to n. For the charge measurement method,only one electrode is excited, and the induced charges on elec-trodes are q0, q1 ,., qn respectively. The total charge in anequilibrium state satisfies

q0 þ q1 þ/þ qn ¼ 0 (5)

According to the superposition theorem of linear systems,8>>>><>>>>:

U10 ¼ a11q1 þ a12q2 þ/þ a1kqk þ/þ a1nqn/Uk0 ¼ ak1q1 þ ak2q2 þ/þ akkqk þ/þ aknqn/Un0 ¼ an1q1 þ an2q2 þ/þ ankqk þ/þ annqn

(6)

where Uk0 is the voltage between the kth electrode and the screen.aii with identical subscripts are self-potential coefficients. aij (i s j)are mutual-potential coefficients. q0 does not appear in Eq. (6)because there are only n independent charges in Eq. (5).

The properties of the potential coefficients are as follows:

(1) all the potential coefficients are positive;(2) the self-potential coefficient ajj is larger than the mutual-

potential coefficient ajk in the same row;(3) the potential coefficient is the function of the geometry and the

dielectric medium distribution inside the pipe.

aij can be obtained by measuring the voltage and the inducedcharge. In fact, if only electrode 1 is excited (the charge q1 s 0), andall other electrodes are kept floated, we have

qi ¼ 0; i ¼ 2;.;n (7)

Applying Eq. (7) to Eq. (6) yields

ak1 ¼Uk0

q1

��q2 ¼ q3 ¼ / ¼ qk ¼ / ¼ qn ¼ 0 ðk ¼ 1;.;nÞ

(8)

The problem for this equation is that q1 on the excitation electrodehas to be detected.

The voltages (U10 ,., Uk0 ,., Un0) on floating electrodes can beobtained using an AC excitation strategy. Under the same conditionspecified in Eq. (7), using the first equation of equation set (6) todivide both sides of other equations, we have

U10

Uk0¼ a11

ak1;.;

U10

Un0¼ a11

an0(9)

ƒ'{ratios of Cij} {ratios of βij}

ƒ

ƒ*

{U}

Fig. 5. Relationship of U, the ratios of bij and the ratios of Cij.

Table 1aResults of three sets of characteristic flow profiles.

C35 C47 C35/C47

Stratified flow 4.6797e�012 1.7621e�012 2.6558Core flow 3.7238e�012 1.3058e�012 2.8518Circular flow 4.5577e�012 2.1930e�012 2.0783


This implies that the values of a11/a21 ,., a11/ak1 ,., a11/an1 can beobtained by measuring voltage on each of the electrodes.

Any electrode except for the screen can be chosen as an exci-tation electrode. If electrode k is excited and the others keep floa-ted, we have q1 ,., qk�1, qkþ1 ,., qn¼ 0.

Despite qk is unknown, the values of a1k/a2k ,., a1k/akk ,., a1k/ank can be obtained with known values of voltages (U10 ,., Uk0 ,.,Un0)k. Therefore, Eq. (6) can be rewritten as where (Uk0/U10)k is the

8>>>><>>>>:

U10 ¼ a11q1 þ a12q2 þ/þ a1kqk þ/þ a1nqn/Uk0 ¼ a11

�Uk0U10

�1q1 þ a12

�Uk0U10

�2q2 þ/þ a1k

�Uk0U10

�kqk þ/þ a1n

�Uk0U10

�nqn

/Un0 ¼ a11

�Un0U10

�1q1 þ a12

�Un0U10

�2q2 þ/þ a1k

�Un0U10

�kqk þ/þ a1n

�Un0U10

�nqn

(10)

value of Uk0/U10 when electrode k is the excitation electrode.According to the reciprocity theorem (akj¼ ajk), (ak1¼ a1k), and

ak1 can be derived from a11 as mentioned above. Therefore, aij forany i and j are known parameters for a given a11. Replacing allpotential coefficients with a11 and the known ratios of voltages, Eq.(10) becomes Eq. (11). For simplicity, Eq. (11) is expressed as matrixformat.

8>>>><>>>>:

U10 ¼ a11q1 þ a11

�U20U10

�1q2 þ/þ a11

�Uk0U10

�1qk þ/þ a11

�Un0U10

�1qn

/Uk0 ¼ a11

�Uk0U10

�1q1 þ a11

�U20U10

�1

�Uk0U10

�2q2 þ/þ a11

�Uk0U10

�1

�Uk0U10

�kqk þ/þ a11

�Un0U10

�1

�Uk0U10

�nqn

/Un0 ¼ a11

�Un0U10

�1q1 þ a11

�U20U10

�1

�Un0U10

�2q2 þ/þ a11

�Uk0U10

�1

�Un0U10

�kqk þ/þ a11

�Un0U10

�1

�Un0U10

�nqn

(11)

*U ¼ a11$T$

*q (12)

T¼

26666664

1�

U20U10

�1

/�

Uk0U10

�1

/�

Un0U10

�1/ / / / / /�

Uk0U10

�1

�U20U10

�1

�Uk0U10

�2

/�

Uk0U10

�1

�Uk0U10

�k

/�

Un0U10

�1

�Uk0U10

�n/ / / / / /�

Un0U10

�1

�U20U10

�1

�Un0U10

�2

/�

Uk0U10

�1

�Un0U10

�k

/�

Un0U10

�1

�Un0U10

�n

37777775

(13)

where T is an n� n matrix, and each entry is a known potential ratioof floating electrodes. a11 is an unknown coefficient,

*U and

*q are

potentials and charge vectors on electrodes.

{Ufloating}{ratios of βij}+{Uexcit}g

Fig. 6. Relations of Ufloating, Uexcit and the ratios of bij.

Comparing Eqs. (6) and (12), it can be seen that

a11T ¼

266664

a11 a12 / a1k / a1n/ / / / / /ak1 ak2 / akk / akn/ / / / / /an1 an2 / ank / ann

377775 (14)

Re-arranging Eq. (6), we have

8>>>><>>>>:

q1 ¼ b11U10 þ b12U20 þ/þ b1kUk0 þ/þ b1nUn0/qk ¼ bk1U10 þ/þ bkkUk0 þ/þ bknUn0/qn ¼ bn1U10 þ/þ bnkUk0 þ/þ bnnUn0

(15)

The voltage coefficients in Eq. (15) can be divided into two cate-gories: Self-induction coefficients bkk with identical subscripts, andmutual-induction coefficients bkn with different subscripts. Theyhave the following properties:

(1) the self-induction coefficients are always positive;(2) the mutual-induction coefficients are always negative;(3) the self-induction coefficient value bjj is greater than the

absolute value of mutual-induction coefficient bij.

Using the Gram algorithm, we have

bkk ¼ Akk=D; bkn ¼ Akn=D (16)

Table 1bResults of three sets of characteristic flow profiles.

jT35j jT47j C35/C47¼�jT35j/jT47j

Stratified flow �0.0056 0.0021 2.6558Core flow �0.0047 0.0016 2.8518Circular flow �0.0056 0.0027 2.0783

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6x 10-18

Relative permittivity

capa

cita

nce

vari

ance

(f)

Fig. 7. Variation of self-capacitance with permittivity.


where D¼ ja11Tj. Akk and Akn are the remaining sub-determinants ofakk and akn respectively. The induction coefficients depend only onthe system geometry and the dielectric material distribution. bkk

and bkn are unknown because a11 is unknown.For the purpose of calculation of the ratio Akk/Akn, a11 can be

eliminated.

bkk

bkn¼ Akk

Akn¼ ð � 1Þkþk

ð � 1Þkþn

jTkkjjTknj

(17)

where Tkk and Tkn are the remaining sub-determinants of tkk and tkn,and where tij is the entry (ith row, jth column) in matrix T.

The ratios of bij can be calculated from the potential measure-ments of the floating electrodes. Manipulating Eq. (15) by addingand then subtracting the same variable Uk0 to each term in the kthequation yields

qk ¼ bk1U10 þ/þ bkkUk0 þ/þ bknUn0

¼ �bk1ðUk0 � U10Þ � bk2ðUk0 � U20Þ �/� bkkðUk0 � Uk0Þ�/� bknðUk0 � Un0Þ þ ðbk1 þ bk2 þ/þ bkk þ/

þ bknÞUk0

¼ �bk1Uk1 � bk2Uk2 �/þ ðbk1 þ bk2 þ/þ bkk þ/

þ bknÞUk0 �/� bknUkn

¼ Ck1Uk1 þ Ck2Uk2 þ/þ Ck0Uk0 þ/þ CknUkn

(18)

Fig. 8. Five sensors with d

where Ck1¼�bk1, Ck2¼�bk2 ,., Ckn¼�bkn, Ck0¼ (bk1þ bk2þ/þ bkkþ/þ bkn).

Thus Eq. (15) can be expressed as a set of linear equations (19),describing the relation between the electrode charges qi and thevoltages [9].8>>>><>>>>:

q1 ¼ C10U10 þ C12U12 þ/þ C1kU1k þ/þ C1nU1n/qk ¼ Ck1Uk1 þ Ck2Uk2 þ/þ Ck0Uk0 þ/þ CknUkn/qn ¼ Cn1Un1 þ Cn2Un2 þ/þ CnkUnk þ/þ Cn0Un0

(19)

where Cij is the mutual capacitance between electrodes’ pair i and j(i,j s 0), C10, C20 ,., Ck0 ,., Cn0 are self-capacitances between theelectrodes and the screen. Ckn¼ Cnk, according to the reciprocitytheorem.

Fig. 4 shows the equivalent circuit of 8-electrode ECT sensor,where the mutual capacitances between electrode 1 and otherelectrodes are depicted. In an isolated electrostatic system with(nþ 1) electrodes, there are (nþ 1)/2 component capacitances.Based on the above analysis, the capacitance Cij can be expressed byinduction coefficients bij, which in turn can be determined bypotentials U on the floating electrodes. The relations between U, theratios of Cij and the ratios of bij are illustrated in Fig. 5. Where f0 and fare the maps from U to the ratios of bij and from the ratios of Cij tothe ratios of bij respectively, and f* is the reversible map of f.

For a given ratio of bij and excitation voltage, e.g. U10, Eq. (15)used in the potential measuring method can be written as

q1

b11¼ U10 þ

b12

b11U20 þ/þ b1k

b11Uk0 þ/þ b1n

b11Un0

/

0 ¼ bk1

b11U10 þ/þ bkk

b11Uk0 þ/þ bkn

b11Un0

/

0 ¼ bn1

b11U10 þ/þ bnk

b11Uk0 þ/þ bnn

b11Un0

8>>>>>>>><>>>>>>>>:

(20)

Taking q1/b11 as a whole, U20 ,., Uk0 ,., Un0 can be found from thelinear equations. The relationship between floating potentialsUfloating, the ratios of bij and excitations’ voltage Uexcit are illustratedin Fig. 6, where g is the map from the ratio of bij and excitations’voltage Uexcit to floating potentials Ufloating. Ufloating is independentof electrode axial length because of Uexcit and the ratios of bij do notdepend on the electrode axial length. This verifies the prediction.

The above theoretical analysis is validated by numerical simu-lation. The ratio C35/C47 is provided under three characteristic flowpatterns. The permittivity of pipe wall and filling layer is set to 5and 2.5 respectively.

The capacitances between electrodes (3, 5), (4, 7) and their ratioare given in Table 1a. The simulation capacitance ratio C35/C47¼ b35/b47¼ A35/A47¼ (�1)3þ5jTj35/(�1)4þ7jTj47 is calculatedfrom the combination of Eqs. (17) and (13) for the given flowprofiles in Table 1b. It is evident that the capacitance ratio can becalculated from potential ratio, which verifies conclusion (2).

ifferent axial lengths.

C (1,2)

0.00E+00

1.00E-12

2.00E-12

3.00E-12

4.00E-12

5.00E-12

6.00E-12

100mm 150mm 200mm 250mm 300mm

f C (1,3)

0.00E+00

5.00E-14

1.00E-13

1.50E-13

2.00E-13

2.50E-13

3.00E-13

3.50E-13

100mm 150mm 200mm 250mm 300mm

fa b

Fig. 9. Capacitances between electrodes’ pair (1, 2), (1, 3). (a) C(1, 2). (b) C(1, 3).


Fig. 7 shows that self-capacitance Ck0 between electrode k andscreen is an indication of the permittivity of the medium inside thepipe, but the variation with permittivity is very small. From Fig. 6, itcan be seen that the potential is a function of ratios of bij and Uexcit,and also dependent on the ratio of Cij and Uexcit. Therefore, thepotential contains the information of self-capacitance Ck0. Theratios between n(nþ 1)/2 capacitances (including mutual and self-capacitances) can be obtained from potential measurements, whilecharge measurement method measures n(n� 1)/2 absolute valuesof mutual capacitance. The information contained for these twomethods is slightly different.

The simulation results presented below are to verify the theo-retical analysis. It is time-consuming to make real sensors ofdifferent size. Instead, 3D simulations of five sensors with differentaxial lengths (100–300 mm, increment 50 mm) are given in Fig. 8.

Choosing electrode 1 for excitation, the capacitances between (1,2), (1, 3) are calculated with different lengths of electrodes.Comparison between capacitances of the same electrodes’ pairs indifferent sensors is given in Fig. 9. It can be seen that the capaci-tances are approximately proportional to the lengths of the elec-trodes. For other capacitances, the results are similar. The potentialmeasurements on the floating electrodes (2–8) are almost the sameas depicted in Fig. 10. Five different colour bars correspond to thesensors with different axial lengths in Fig. 8. (For interpretation ofcolours in this figure, the reader is referred to the web version of thearticle.)

An empty pipe and a pipe with an object are depicted in Fig. 11.Small sensors of 1/1000 sizes of the large sensor are also modelled.The drawing of small sensors is omitted due to the proportionalconfiguration of sensor. The difference of the capacitances betweenFig. 11(b) and (a) of the large sensor is given in Fig. 12(a). Similarresults for a small sensor can be found in Fig. 12(b). The capacitancebetween the same electrode pair for the small sensor is 1000 timessmaller than that of the large sensor. If the potential measurementsare taken from the empty pipe, there is hardly any difference on thepotentials between the large and the small sensors as depicted inFig. 12(c). Similar conclusions can be drawn for a pipe with an

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

2 3 4 5 6 7 8

v

Fig. 10. Potential measurements on floating electrodes (2–8).

object in the sensing zone as shown in Fig. 12(d). The variations inpotential are almost the same as in Fig. 12(e).

5. Conditions for practical application

The ideal situation for potential measurement is that the ‘‘load’’to the sensor is zero, i.e., there is no current flowing from thesensing system. In reality, the electrodes have to be connected toa measuring circuit, usually an amplifier.

In an ideal condition, the potential on the floating detectionelectrode, as depicted in Fig. 13, is

Vi ¼ VsC0x

C2s1 þ C0x(21)

where C0x is the equivalent capacitance between an exciting elec-trode and a detecting (floating) electrode. C2s1 is the equivalentcapacitance between the detecting electrode and the screen. Vi isthe potential in the above analysis in Eqs. (1)–(20).

When the detection electrode is connected to a voltage amplifieras shown in Fig. 2(b), the input voltage of the circuit is given by

V 0i ¼Vs juC0x

1=Ri þ ju�C2s1 þ C0x

� (22)

If jjuRiðC2s1 þ C0xÞj[1 is satisfied,

V 0i zVi (23)

To achieve this, there are two ways:

(a) To increase frequency of excitation.(b) To increase the resistance of Ri.

It usually requires careful design for high frequency circuits,while high resistance may cause difficult for DC balancing due tothe bias current, offset and noise. Therefore, it will be a compromisebetween frequency, resistance and the smallest allowable

Fig. 11. Simulation model. (a) Empty pipe. (b) A pipe with an object inside.

-5.00E-14-4.00E-14-3.00E-14-2.00E-14-1.00E-140.00E+001.00E-142.00E-143.00E-144.00E-145.00E-14

c12 c13 c14 c15 c16 c17 c18

v

-4.00E-17-3.00E-17-2.00E-17-1.00E-170.00E+001.00E-172.00E-173.00E-174.00E-175.00E-17

c12 c13 c14 c15 c16 c17 c18

f

0.00E+002.00E-014.00E-016.00E-018.00E-011.00E+001.20E+001.40E+001.60E+001.80E+002.00E+00

2 3 4 5 6 7 8

v

0.00E+002.00E-014.00E-016.00E-018.00E-011.00E+001.20E+001.40E+001.60E+001.80E+002.00E+00

1 2 3 4 5 6 7

f

-6.00E-02

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1 2 3 4 5 6 7

v

a b

c d

e

Fig. 12. Simulation results. (a) Capacitance variation in large sensor. (b) Capacitance variation in small sensor. (c) Comparison of potential measurements for empty pipe. (d)Comparison of potential measurements for pipe with an object inside. (e) Comparison of potential variation.


capacitance ðC2s1 þ C0xÞ. This shows the limitation of the potentialmeasurement method. There are alternatives, magnetic couplingwith an inductor in parallel with C2s1 to form an LC oscillator, andthis inductor as the primary winding of a transformer. Howeversuch a circuit requires more complicated design including isolatedpower supplies.

6. Image reconstruction using Generic Algorithm (GA)inversion technique

As an application example, image reconstruction usingnonlinear GA-ECT [10] is adopted and incorporated with thepotential measurements method. When using GA-ECT, thenonlinear multi-variable optimization problem (24) is constructed

Fig. 13. Potential on the floating detection electrode.

for the mathematical models in Eq. (4) based on potentialmeasuring method.8<:

Min Fð3Þ ¼ kU � Umk22

s:t: Akð3Þ4 ¼ Bk; k ¼ 1;2;.;83 ¼ f 31 / / 3n g; 3i � 1; i ¼ 1;.;n

(24)

where U is the potential obtained from the forward calculation andUm is a measurement value. Ak(3) is the discrete representation ofthe operator V$(3V). N-vector Bk is determined by the boundaryvalues. n is the number of pixels.

In practical conditions, medium components in the pipe areknown, so that the priori information l� 3� u can be added, wherel, u are the lower and upper boundaries of 3 respectively. GeneticAlgorithm (GA) can be used to solve this nonlinear multi-variableoptimization problem [11,12]. In this example, GA is chosen to solvethis problem based on its following characteristics:

(1) GA searches the solution directly based on the object function.No evaluation of the derivatives is needed. For the objectfunction F(3)¼kU�Umk2

2, it is difficult to find the derivativeexpression of U(3), the obstacle can be avoided using GA.

(2) GA searches the solution from many points while the tradi-tional algorithms start the iteration process from only onepoint with less information, which is easily trapped in the localoptimization.


To simulate the real conditions, random noise is added to thenoise-free measured data I0, using the following equation:

I ¼ I0ð1þ k� RNDÞ (25)

where I0 is the potential vector, which is calculated using FDM(finite difference method) or FEM (finite element method) given

Data noise

Reconstructedimage withoutregularization

27.86 36.56

Poten. residual 0.0079 0.0036

Corr .coeff 0.7138 0.7016

Reconstructed image withregularization

33.22 33.88Poten. residual 0.0018 0.0023Corr .coeff 0.7495 0.7435

c

Data noise

Reconstructed image without regularization

23.78 24.60


Corr .coeff 0.6999 0.6844

Reconstructed image withregularization

23.06 23.62


Corr .coeff 0.7184 0.7080

d

a

Fig. 14. Image reconstruction results for a square pipe. (a) An annular object. (b) Two rreconstruction results for the object in (b).

the test distribution. I is the potential vector contaminated by noise.RND is a random scale value drawn from a normal distribution withmean value as 0 and standard deviation as 1. By such definition, thenoise level k is equal to kI� I0k/kI0k.

To quantitatively evaluate the quality of reconstructed images,three criteria [13] were used:

38.12 35.04

0.0091 0.0125

0.6783 0.7186

36.58 33.03 0.0099 0.0124 0.7062 0.7488

28.02 30.24

0.0146 0.0194

0.6463 0.5946

26.96 30.10

0.0102 0.0120

0.6619 0.6144

b

ectangular objects. (c) Image reconstruction results for the object in (a). (d) Image

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Fig. 15. Image reconstruction results for circular pipe (2% noise). (a) Core flow. (b) Inversion result for (a). (c) Two objects. (d) Inversion result for (c). (e) Stratified flow. (f) Inversionresult for (e). (g) Three objects. (h) Inversion result for (g).


Table 2Evaluation of inversion results in Fig. 15.

Flow pattern Core Two object Stratified Three object

Image error (%) 27.06 29.62 17.96 33.13Potential residual 0.0016 0.0049 0.0062 0.0131Correlation coefficient 0.7972 0.7034 0.8419 0.6644


Relative image error :

��3* � 3��

k3k (26)

Relative capacitance or potential residual :

��Ið3Þ � I�3*��

kIð3Þk(27)

Correlation coefficient :

�3� 3; 3* � 3*

k3� 3k$

��3* � 3*�� (28)

where 3 is the true permittivity distribution of the test permittivitydistribution, 3* is the reconstructed permittivity distribution, 3 and3* are the mean value of 3 and 3* respectively.

The primary inversion results based on the potentialmeasurement method on a square pipe and circular pipes aregiven in Figs. 14 and 15. To test the GA algorithm, FDM and theFEM are used for region discretization of a square or circularcross-section. The colour bar on the right of each image indicatesthe change in permittivity from 1 to 3. (For interpretation ofcolours in this figure, the reader is referred to the web version ofthe article.)

Fig. 14 shows the image reconstruction results for an annularpattern and two adjacent square objects. In each case, thenoise-free potential data and three sets of noisy data withrelative noise level at 2%, 4% and 6% were used respectively, andthe function of regularization term is also considered. In thiscase the regularizing penalty term ak3k2 is added to the objectfunction to minimize the image energy. Other forms of regu-larizing penalty term can be used, such as smooth generalizedTikhonov regularization for the smoothly varying case or totalvariation (TV) regularization for jump changes to find anoptimal value for a.

For a more visible depiction, 3D image reconstruction on thecross-section of circular pipe is given in Fig. 15 (with relative noiselevel at 2%): (a), (c), (e) and (g) are the flow profiles set before, (b),(d), (f) and (h) are the inversion results respectively. The evaluationof inversion results is given in Table 2.

7. Conclusions

The potential measurement method has been studied in thispaper. The theoretical analysis and 3D simulation results show thatpotential measurement contains capacitances ratios, and is inde-pendent of mutual capacitance. GA is used for image reconstructionto verify the method as an alternative to the charge (displacementcurrent) method, which is very popular in ECT. The conditions foreffective applications and limitations of this method have also beendiscussed in this paper, concluding that the balance or compro-mises have to be made between excitation frequency, input resis-tance of measurement circuit and low limit of capacitance.

Acknowledgements

The research work discussed in this paper was part of researchproject RFC-CR 03005 funded by European Commission: Researchfunding for Coal and Steel. This work was also supported by theNational Natural Science Foundation of China with its key fundingprogramme project, No. 50836003.

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