8 Sensing Elements
8.2 Capacitive sensing elements
The simplest capacitor or condenser consists of two parallel metal plates separated by a dielectric or insulating material (Figure 8.9). The capacitance of this parallel plate capacitor is given by:
C= [8.17]
where ε0is the permittivity of free space (vacuum) of magnitude 8.85 pF m−1, ε is the relative permittivity or dielectric constant of the insulating material, A m2is the area of overlap of the plates, and d m is their separation. From [8.17] we see that C
ε0ε A d Figure 8.8 Organic
polymer resistive gas sensors:
(a) Characteristics of NO2sensor
(b) Typical construction of single sensor (c) Array of six sensors (after Trident Exhibitions and Dr A. Cranny[6]).
8.2 CAPACITIVE SENSING ELEMENTS 161
can be changed by changing either d, A or ε; Figure 8.9 shows capacitive displace-ment sensors using each of these methods. If the displacedisplace-ment x causes the plate separation to increase to d+ x the capacitance of the sensor is:
Variable separation
displacement sensor [8.18]
i.e. there is a non-linear relation between C and x. In the variable area type, the dis-placement x causes the overlap area to decrease by ∆A = wx, where w is the width of the plates, giving:
Variable area
displacement sensor [8.19]
In the variable dielectric type, the displacement x changes the amount of dielectric material ε2(ε2>ε1) inserted between the plates. The total capacitance of the sensor
C = ε εd0 ( A−wx)
C A
d x
= + ε ε0
Figure 8.9 Capacitive sensing elements.
is the sum of two capacitances, one with area A1and dielectric constant ε1, and one
A commonly used capacitive pressure sensor is shown in Figure 8.9. Here one plate is a fixed metal disc, the other is a flexible flat circular diaphragm, clamped around its circumference; the dielectric material is air (ε ≈ 1). The diaphragm is an elastic sensing element (Section 8.6) which is bent into a curve by the applied pressure P.
The deflection y at any radius r is given by:
y= (a2− r2)2P [8.21]
where
a= radius of diaphragm t= thickness of diaphragm E= Young’s modulus ν = Poisson’s ratio.
The deformation of the diaphragm means that the average separation of the plates is reduced. The resulting increase in capacitance ∆C is given by[7]
Capacitive pressure
sensor [8.22]
where d is the initial separation of the plates and C=ε0πa2/d the capacitance at zero pressure.
The variable separation displacement sensor has the disadvantage of being non-linear (eqn [8.18]). This problem is overcome by using the three-plate differential or push-pull displacement sensor shown in Figure 8.9. This consists of a plate M moving between two fixed plates F1and F2; if x is the displacement of M from the centre line AB, then the capacitances C1and C2formed by MF1and MF2respectively are:
Differential capacitive
displacement sensor [8.23]
The relations between C1, C2and x are still non-linear, but when C1and C2are incor-porated into the a.c. deflection bridge described in Section 9.1.3, the overall relationship between bridge output voltage and x is linear (eqn [9.23]).
8.2 CAPACITIVE SENSING ELEMENTS 163
The next sensing element shown in Figure 8.9 is a level sensor consisting of two concentric metal cylinders. The space between the cylinders contains liquid to the height h of the liquid in the vessel. If the liquid is non-conducting (electrical con-ductivity less than 0.1µmho cm−3), it forms a suitable dielectric and the total capa-citance of the sensor is the sum of liquid and air capacapa-citances. The capacapa-citance/unit length of two coaxial cylinders, radii b and a (b > a), separated by a dielectric ε is 2πε0ε /loge(b/a). Assuming the dielectric constant of air is unity, the capacitance of the level sensor is given by:
Capacitive level sensor
[8.24]
The sensor can be incorporated into the a.c. deflection bridge of Figure 9.5(a).
The final sensor shown in Figure 8.9 is a thin-film capacitive humidity sensor.[8]
The dielectric is a polymer which has the ability to absorb water molecules; the resulting change in dielectric constant and therefore capacitance is proportional to the percentage relative humidity of the surrounding atmosphere. One capacitor plate consists of a layer of tantalum deposited on a glass substrate; the layer of polymer dielectric is then added, followed by the second plate, which is a thin layer of chromium. The chromium layer is under high tensile stress so that it cracks into a fine mosaic which allows water molecules to pass into the dieletric. The stress in the chromium also causes the polymer to crack into a mosaic structure. A sensor of this type has a input range of 0 to 100% RH, a capacitance of 375 pF at 0% RH and a linear sensitivity of 1.7 pF/% RH. The capacitance–humidity relation is therefore the linear equation:
C= 375 + 1.7 RH pF
The maximum departure from this line is 2% due to non-linearity and 1% due to hysteresis.
The basic capacitive pressure sensor shown in Figure 8.9 has some limitations. It consists of two circular metal plates or diaphragms, one flexible, one rigid, with air as dielectric. Because the dielectric constant ε for air is only 1, changes in capa-citance ∆C caused by the applied pressure P will be very small. To increase ∆C, the thickness t of the flexible plate/diaphragm can be reduced; this gives greater deformation y for a given pressure P but reduces mechanical strength. Also many metals have limited resistance to corrosion. These problems can be largely overcome by using liquid-filled ceramic membrane sensors. Dielectric liquids such as silicon oil have a much higher ε than air, so that significant capacitance changes ∆C can be obtained with small membrane displacements of the order of 10 µm. These small displacements, together with lower Young’s modulus for ceramics than metals, mean that ceramic membranes can be made much thicker than metal diaphragms – between 0.25 and 1.5 mm is typical. This gives greater mechanical strength and ease
of maintenance; furthermore, ceramics such as aluminium oxide have excellent resistance to corrosion.
Figure 8.10 shows a differential pressure sensor using this technology. Process fluid at pressure p1 in the left-hand chamber and at pressure p2in the right-hand chamber is in contact with the ceramic membrane. The membrane is supported on a ceramic substrate. Two capacitances C1and C2are formed using gold plates and silicon oil as dielectric. An increase in differential pressure p1− p2causes the left-hand membrane to move to the right, causing C1to increase by ∆C. The pressure change is transmitted through the fluid to the right, causing the right-hand membrane to move to the right and C2to decrease by ∆C. C1and C2are then incorporated in an a.c. bridge circuit (Sections 9.1.3 and 9.4.2).
Capacitive sensing elements are incorporated in either a.c. deflection bridge cir-cuits (Section 9.1) or oscillator circir-cuits (Section 9.5.1). Capacitive sensors are not pure capacitances but have an associated resistance R in parallel, to represent losses in the dielectric; this has an important influence on the design of circuits, particularly oscillator circuits. For example, a typical capacitive humidity sensor would have an approximate parallel dielectric loss resistance R of 100 kΩ at 100 kHz. The quality of a dielectric is often expressed in terms of its ‘loss tangent’ tan δ where:
tan δ =
In this example, if C = 500 pF then tan δ ≈ 0.03. Figure 9.25(b) shows the capa-citive sensor incorporated with a pure inductance into an oscillator circuit. The
‘quality factor’ Q of the circuit is given by
and thus depends crucially on R. If the natural frequency fnof the circuit with the above humidity sensor is fn= 105Hz, then the circuit Q factor is approximately 30.
Q R C
L nCR = =ω
1 ωCR Figure 8.10 Ceramic
liquid-filled differential pressure sensor (after Endress and Hauser Ltd.).