Capacitance transducers – Basic principles
By
Erling A Hammer
Introduction
The principle of a capacitance transducer is very simple, but in practice there are many factors
that influence the measurements in such a way that the results can easily be miss-interpreted if
the basic principles are not thoroughly understood.
Capacitance sensors can be used to predict the concentration ratios in two-phase or twocomponent flow if the two components or the two phases have different electrical permittivity
(dielectric constant).
The principle is based on the fact that the difference in permittivity of the different
components or phases flowing between two capacitance plates (electrodes) makes the
capacitance between these two plates dependent on the ratio of concentration of the
components or the phases in the flow.
The connection, however, between the concentration ratio and the permittivity of the mixture,
and hence, the sensor capacitance, is not simple and depends on the distribution of the
components or the phases in the mixture (the flow regime). Even if the capacitance technique
is flow regime dependent it can still be used for concentration measurements if the
components are homogeneously mixed .
The sensor
The simplest capacitance sensor consists of two parallel metal plates separated a distance d
from each other (figure 1).
Sensing
electrode
Cs
0 r
Guard
electrodes
d
Exitation electrode
Figure 1. Basic capacitance sensor
If the guard electrodes are kept at the same potential as the sensing electrode but electrically
insulated from it, so that the influence of the fringe field or edge effect of the electrodes is
eliminated, the measured capacitance will be:
Cs 
 r 0 A
d
(1)
where A is the area of the sensing electrode (m2)
d is the distance between the electrodes (m)
r is relative permittivity of the material between the electrode plates
 is permittivity of free space (8.85410-12 F/m)
If now the material between the electrodes consists of components 1 and 2 and these
components have different permittivity the volumes of these two components can be found if
we know how the permittivity depends on the component fraction.
1
A more practical sensor is shown in Figure 2.
detector electronics
detector electronics
Guard
ectrodes
Bare
inner
electrodes
Surface plate
electrodes
D
D
D
Electrical
insulating
material
Steel pipe
(Screen)
Electrical
insulating
material
Liner
a
Steel pipe
(Screen)
Figure 2. Surface plate electrodes
Liner
b
Figure 2b. Innside naked electrodes
The electric field between the electrode plates will, for this type of sensors, not be
homogeneous and it will be more sensitive to concentration variations in the surroundings of
the gaps between the plates. However, the guard arrangement shown in Figure 2a will
eliminate the influence of the strongest inhomogeneous field at the edges of the electrodes and
thus make the sensor less sensitive for inhomogeneities in the flowing medium. In Figure 2b
the electrode plates are placed inside the liner. They are exposed direct to the mixture and
have therefore a very high electrode capacitance Ce that will increase the sensitivity.
Another used capacitance sensor is the coaxial electrode sensor. This sensor consists of an
inner electrode supported in the centre of the pipe as shown in Figure 3.
Inner electrode
Outher electrode
Pipe
Inner electrode
d
r
D
Figure 3. The principle of the coaxial electrode sensor
The electric field from this sensor is strongly inhomogeneous but symmetric around the pipe
axis. The sensitivity to variation in concentration of the liquid will be highest close to the
inner electrode. The capacitance pr unit length of this sensor is
2m 0
Cs 
(2)
ln Dd
where m is the relative permittivity of the mixture flowing through the sensor head.
2
Guarding and screening of the capacitance electrodes
In order to perform as accurate measurements as possible it is necessary to both guard and
screen the capacitance sensor. In real sensor the electrodes are arranged as shown in Figure 2.
To make it easier to display the different spread capacitances a parallel capacitance sensor is
sketched as shown in Figure 4. From this figure we see that the equivalent diagram of a
capacitance sensor will be as shown in Figure 5.
A
Guard
Screen
CA
CG
CAG
CX
CBG
CB
B
Figure 4. Capacitance sensor with guard electrode and screen. Electrode A is the detector
electrode and electrode B is the exitation electrode
Cs
CX
A
CA
B
CB
CAG
CBG
S
CG
G
Figure 5. Equivalent diagram of capacitance sensor head equipped with guard electrode and
screen
For all practical sizes of sensors the parasite capacitances CA and CB are considerable, thus
resulting in an unwanted shunting of CX. It is also a disadvantage that both CA and CB will
change when the permittivity of the mixture in the pipe changes due to displacement of the
field lines between the electrodes and the screen. However, the influence of all the parasite
capacitances shown in Figure 4 and 5 can be diminished if a proper guard-driver detector is
used.
The basic capacitance detector system
- There is in principle only one method used to eliminate the influence of the various
parasite capacitances in the sensor head. This principle is based on the virtual
ground circuit.
3
Figure 6. shows the basic principle of this circuit. Neither CA, CB nor CAG, CBG nor CG will
influence on the output voltage Uout, if the operational amplifier has a high open loop gain
AOL, so that the differential input voltage e  0 .
ZF
CX
B
A
e
CA
CB
Uosc
AOL
CAG
CBG
Uout
S
CG
G
Figure 6. Capacitance transducer with grounded guard and screen and guarded electrode (A)
connected to a virtual ground circuit
It can easily be seen that if AOL is large, the differential input voltage e  0 and the output
signal of the circuit given in Figure 6 is:
(3)
uout  uosc zF jcX
and if the feedback impedance ZF is a capacitor C0 (ZF = 1/jC0) equation (3) can be written:
uout  uosc
Cx
C0
(4)
Other used detector circuits are the oscillator circuit and the charge/discharge circuit. They are
both based on the virtual ground principle.
Interpreting the capacitance signals
To utilise the information gained by using capacitance transducers it is of great importance to
know the basic theory that underlies this technique. This is not as simple as many believe.
It is not difficult to foresee that the capacitance sensors will be dependent on the distribution
of the different components in a mixture. It is therefore quite obvious that reliable
measurements can be made only if the flow regime is constant and the most stable regime is
the homogeneous flow where the two components are well mixed.
Maxwell (1873) [1], Bruggeman (1935) and many others have developed formulae for the
permittivity and conductivity of homogeneous mixtures of two different materials.
On the basis of a model developed by van Beek (1967), Ramu and Rao (1973) have derived
formulae, which are also valid, if one of the components in the mixture has a high
conductivity.
If  1   2 and 1<<2 Ramu and Raos equations for the expression of the mixture
permittivity and conductivity [4] can be written as:
4
 m  1
1  2
1 
 m  1
(5)
1  2
1 
(6)
which is valid for component 1 as the continuous phase and
2
(7)
 m'   2
3 
2
(8)
 m'   2
3 
for component 2 as the continuous phase in the mixture.
Here
m is the relative permittivity of the mixture when component 1 makes the continuous phase
'm is the relative permittivity of the mixture when component 2 makes the continuous phase
m is the conductivity of the mixture when component 1 makes the continuous phase
m is the conductivity of the mixture when component 2 makes the continuous phase
1 is the relative permittivity of component 1
2 is the relative permittivity of component 2
1 is the conductivity of component 1
2 is the conductivity of component 2
is the volume fraction of the component 2
Let us assume that the conductivity of component 1 is zero and makes the continuous phase
when   0.5 and that component 2 is conductive (2 0) and is the continuous phase when
  0.5 . The relative mixture permittivity will then be as shown in Figure 7.
m
m
2
Mixture conductivity
Mixture permittivity
2
m
m
0.42
0.42
m
41
m

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

Figure 7 Mixture permittivity and conductivity versus the volume fraction of component 2 of a
homogeneous mixture of component 1 and 2. Only component 2 is conductive
To understand how a capacitance sensor reacts to changes in the permittivity of the mixture it
is useful to work out an equivalent diagram for the sensor. Such a diagram is shown in
Figure 8.
5
Ce1
Rm
Cm
Z
Ce2
Figure 8. Equivalent diagram of a capacitance sensor when all the spread capacitances are
eliminated
In Figure 8, C m and Rm are the capacitance and the resistance between imaginary electrodes,
placed at the mixture/liner interface, and with the same area as the sensor electrodes.
C e 1 and C e 2 are resultant capacitances between the sensor electrodes and the mixture with the
electrode insulating material as dielectric. If component 1 is non-conducting and makes the
continuous phase in the flow, Rm   , and the measured capacitance will be:
CeCm
Cs 
(9)
Ce  Cm
where Ce = 1/2C1=1/2C2
If component 2 is conducting and makes the continuous phase then the current through Cm
will be by-passed by Rm. It can be shown that the current through Cm is equal to the current
through Rm if:
2
fe 
(10)
20 2
An example of this can be the measured capacitance for a capacitance sensor used for
measuring the water content in a mixture of crude oil/saline water (w). This is shown in
Figure 9. The transition point is here chosen to be c= 0.7
Cs
Ce
Sensor capacitance
f<<w/20w
fw
Coil

0
Water cut
0.7
1
Figure 9. Measured capacitance of a capacitance sensor as function of the water fraction of a
homogeneous mixture of oil and water. Excitation frequency f   water / 20 water  The
dotted line indicates the characteristic if f   water / 20 water
6
Conclusions
As we have seen, the water content in oil can only be determined accurately, when the
distribution of the water is exactly known. In practice this means that usually only
homogeneous mixtures of oil and water can reliably be determined.
The continuous component in a process oil/water mixture changes from oil to water at around
20 to 40% water concentration. The transition point in a mixture of crude oil and water will
occur somewhere between 60% and 80% water fraction, depending on the type of crude,
temperature and content of emulsion breaker etc.
Whether the water concentration is detectable or not above the transition point is dependent
on the water conductivity and sensor excitation frequency. The sensor capacitance will
increase with increasing water fraction, even above the transition point (>C), if the sensor
excitation frequency
1 w
f 
 fe
(11)
2  0 w
In North Sea oil the conductivity of the water component in the crude will approximately be 5
S/m and the relative permittivity approximately 70 giving a critical excitation frequency
of f e  1.3  10 9 Hz.
However, higher excitation frequencies than 1.3.109 Hz cannot be used because the
capacitance detector will not eliminate the influence of the parasitic capacitances at
frequencies higher than approximately 1 MHz.
The water content of North Sea crude can therefore not be measured with these types of
capacitance sensors if the mixture is water continuous (>C).
Some capacitance sensors on the marked to day are bare (naked) electrodes inside an
insulating liner as shown in Figure 10. These sensors have large interface capacitances
Ce1 and Ce2 (Approximately 1-5 F/cm2 ) resulting in high sensitivity for Cm and Rm . Still the
sensor cannot be used for capacitance measurements in water continuous phases but the
electrodes can be used for conductivity measurements and thus the water content can be
determined in water continuous phases.
Figure 10. Bare (naked) inside capacitance electrodes (Roxar Flow Measurement AS)
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References
[1]
Hammer, E.A., Chapter 14.1.1 Capacitance transducers – Basic Principles.
Multiphase Flow Handbook, 2003. Crowe, C (editor), Multiphase Flow Handbook,
(Taylor and Francis Group) ISBN 0-8493-1280-9.
[2]
Maxwell, J.C., A Treatise on Electricity & Magnetism,
The Clarendon Press, Oxford, Vol. 1, 1st edition 1873.
[3]
Bruggeman, D.A.G., Berechnung verschiedener physikalischer Konstanten von
heterogenen Substanzen, Annalen der Physik, 5. Folge, Band 24, 1935
[4]
Van Beek, L., Dielectric Behaviour of Heterogeneous Systems. Progress in
Dielectrics, Vol 7, 1967
[5]
Ramu, T.S. and Narayana Rao, Y., On the Evaluation of Conductivity of Mixtures of
Liquid Dielectrics, IEEE Transactions on Electrical Insulation. Vol. E1-8, No. 2, June
1973.
[6]
Hammer, E.A., Three-Component Flow Measurement in Oil/Gas/Water Mixtures
using Capacitance Transducers, Ph.D. thesis, University of Manchester,. CMI-No.
831251-2, 1983
[7]
Hammer, E.A., Multi modality tomography systems - State of the art and possible
future applications. 5th International Symposium on Process Tomography in Poland,
Zakopane, 25–26.08.2008
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