Table of Contents
The measurement of displacement, position, or location is an important topic in the process industries. Examples of industrial requirements to measure these variables are many and varied, and the required sensors are also of greatly varied designs. To give a few examples of measurement needs: (1) location and position of objects on a conveyor system, (2) orientation of steel plates in a rolling mill., (3) liquid/solid level measurements, (4) location and position of work piece in automatic milling operations, and (5) conversion of pressure to a physical displacement that is measured to indicate pressure. In the following sections, the basic principles of several common types of displacement, position, and location sensors are given.
The simplest type of displacement sensor involves the action of displacement in moving the wiper of a potentiometer. This device then converts linear or angular motion into a changing resistance that may be converted directly to voltage and/ or current signals. Such potentiometric devices often suffer from the obvious problems of mechanical wear, friction in the wiper action, limited resolution in wire-wound units, and high electronic noise. (See Figure 5.1.)
A potentiometric displacement sensor is to be used to measure work-piece motion from 0 to 10 cm. The resistance changes linearly over this range from 0 to 1 kW. Develop signal conditioning to provide a linear, 0- to 10-volt output.
The key thing here is not to lose the linearity of the resistance versus displacement. We cannot use a voltage divider because the voltage versus resistance variation is not linear for that circuit. We can use an op amp circuit, however, because the gain and therefore the output voltage is linearly dependent on the feedback resistor.
Figure 5.1 Potentiometric displacement sensor.
Figure 5.2 Circuit for Example 5.1.
Therefore, the circuit of Figure 5.2 will satisfy this problem. The output voltage is
Vout = -(RD)/1kW)(-10V)
Vout = 0.01RD
As RD varies from 0 to 1 kW, the output will change linearly from 0 to 10 volts.
Capacitive and Inductive
A second class of sensors for displacement measurement involves changes in capacity or inductance.
Capacitive The basic operation of a capacitive sensor can be seen from the familiar equation for a parallel-plate capacitor
where K = the dielectric constant
Eo = permittivity = 8.85 pF/m
A = plate common area
d = plate separation
There are three ways to change the capacity: variation of the distance between the plates (d), variation of the shared area of the plates (A), and variation of the dielectric constant (K). The former two methods are shown in Figure 5.3. The last method is illustrated later in this chapter. An ac bridge circuit or other active electronic circuit is employed to convert the capacity change to a current or voltage signal.
Figure 5.3 Capacity varies with the distance between the plates and the common area. Both ef-fects are used in sensors.
Figure 5.4 shows a capacitive-displacement sensor designed to monitor small changes in work-piece position. The two metal cylinders are separated by a plastic sheath/bearing of thickness 1 mm and dielectric constant at 1 kHz of 2.5. If the radius is 2.5 cm, find the sensitivity in pF/m as the upper cylinder slides in and out of the lower cylinder. What is the range of capacity if h varies from 1.0 to 2.0 cm?
The capacity is given by Equation (5.1). The net area is the area of the shared cylindrical area, which has a radius r and height h. Thus, A = 2nrh, so the capacity can be expressed as
The sensitivity with respect to the height, h, is defined by how C changes with h; that is, it is given by the derivative,
Substituting for the given values, we get
Since the function is linear with respect to h, we find the capacity range as Cmin = (3475 pF/m) (10-2 m) = 34.75 pF to Cmax = (3475 pF/m) (2 X 10-2 m) = 69.50 pF.
Inductive If a permeable core is inserted into an inductor as shown in Figure 5.5, the net inductance is increased. Every new position of the core produces a different inductance. In this fashion, the inductor and movable core assembly may be used as a displacement sensor. An ac bridge or other active electronic circuit sensitive to inductance then may be employed for signal conditioning.
Figure 5.4 Capacitive-displacement sensor for Example 5.2.
Figure 5.5 This variable-reluctance displacement sensor changes the inductance in a coil in response to core motion.
LVDT The LVDT is an important and common sensor for displacement in the industrial environment. Figure 5.6 shows that an LVDT consists of three coils of wire wound on a hollow form. A core of permeable material can slide freely through the center of the form. The inner coil is the primary, which is excited by some ac source as shown. Flux formed by the primary is linked to the two secondary coils, inducing an ac voltage in each coil.
When the core is centrally located in the assembly, the voltage induced in each primary is equal. If the core moves to one side or the other, a larger ac voltage will be induced in one coil and a smaller ac voltage in the other because of changes in the flux linkage associated with the core.
Figure 5.6 The LVDT has a movable core with the three coils as shown.
If the two secondary coils are wired in series opposition, as shown in Figure 5.6, then the two voltages will subtract; that is, the differential voltage is formed. When the core is centrally located, the net voltage is zero. When the core is moved to one side, the net voltage amplitude will increase. In addition, there is a change in phase with respect to the source when the core is moved to one side or the other.
A remarkable result, shown in Figure 5.7, is that the differential amplitude is found to increase linearly as the core is moved to one side or the other. In addition, as noted, there is a phase change as the core moves through the central location. Thus, by measurement of the voltage amplitude and phase, one can determine the direction and extent of the core motion, that is, the displacement.
Figure 5.7 LVDT net secondary voltage amplitude for a series opposition connection varies linearly with displacement.
It turns out that a carefully manufactured LVDT can provide an output linear within ±0.25% over a range of core motion and with a very fine resolution, limited primarily by the ability to measure voltage changes.
The signal conditioning for LVDTs consists primarily of circuits that perform a phase-sensitive detection of the differential secondary voltage. The output is thus a dc voltage whose amplitude relates the extent of the displacement, and the polarity indicates the direction of the displacement. Figure 5.8 shows a simple circuit for providing such an output. An important limitation of this circuit is that the differential secondary voltage must be at least as large as the forward voltage drop of the diodes. The use of op amp detectors can alleviate this problem.
Figure 5.8 This simple circuit produces a bipolar dc voltage that varies with core displacement.
Figure 5.9 shows a more practical detection scheme, typically provided as a single integrated circuit (1C) manufactured specifically for LVDTs. The system contains a signal generator for the primary, a phase-sensitive detector (PSD) and amplifier/filter circuitry.
Figure 5.9 A more sophisticated LVDT signal-conditioning circuit uses phase-sensitive detection to produce a bipolar dc voltage output.
A broad range of LVDTs is available with linear ranges at least from ±25 cm down to ±1 mm. The time response is dependent on the equipment to which the core is connected. The static transfer function is typically given in millivolts per millimeter (mV/mm) for a given primary amplitude. Also specified are the range of linearity and the extent of linearity.
An LVDT has a maximum core motion of ±1.5 cm with a linearity of ±0.3% over that range. The transfer function is 23.8 mV/mm. If used to track work-piece motion from —1.2 to +1.4 cm, what is the expected output voltage? What is the error in position determination due to nonlinearity?
Using the known transfer function, the output voltages can easily be found, V(-1.2 cm) = (23.8 mV/mm) (-12 mm) = -285.6 mV and V(1.4 cm) = (23.8 mV/mm) (14 mm) = 333 mV. The linearity deviation shows up in deviations of the transfer function. Thus, the transfer function has an uncertainty of (±0.003) (23.8 mV/mm) = ±0.0714 mV/mm. This means that a measured voltage, Vm (in mV), could be interpreted as a displacement that ranges from Vm/23.73 to Vm/23.87 mm, which is approximately ±0.3%, as expected.
The measurement of solid or liquid level calls for a special class of displacement sensors. The level measured is most commonly associated with material in a tank or hopper. A great variety of measurement techniques exist, as the following representative examples show.
Mechanical One of the most common techniques for level measurement, particularly for liquids, is a float that is allowed to ride up and down with level changes. This float, as shown in Figure 5.10a, is connected by linkages to a secondary displacement measuring system such as a potentiometric device or an LVDT core.
Figure 5.10 There are many level-measurement techniques.
Electrical There are several purely electrical methods of measuring level. For example, one may use the inherent conductivity of a liquid or solid to vary the resistance seen by probes inserted into the material. Another common technique is illustrated in Figure 5.10b. In this case, two concentric cylinders are contained in a liquid tank. The level of the liquid partially occupies the space between the cylinders, with air in the remaining part. This device acts like two capacitors in parallel, one with the dielectric constant of air (=1) and the other with that of the liquid. Thus, variation of liquid level causes variation of the electrical capacity measured between the cylinders.
Figure 5.11 Ultrasonic level measurement needs no physical contact with the material, just a transmitter T and receiver R.
Ultrasonic The use of ultrasonic reflection to measure level is favored because it is a "noninvasive" technique; that is, it does not involve placing anything in the material. Figure 5.11 shows the external and internal techniques. Obviously, the external technique is better suited to solid-material level measurement. In both cases the measurement depends on the length of time taken for reflections of an ultrasonic pulse from the surface of the material. Ultrasonic techniques based on reflection time also have become popular for ranging measurements.
Pressure For liquid measurement, it is also possible to make a noncontact measurement of level if the density of the liquid is known. This method is based on the well-known relationship between pressure at the bottom of a tank, and the height and density of the liquid. This is addressed further in Section 5.5.1.
The level of ethyl alcohol is to be measured from 0 to 5 m using a capacitive system such as that shown in Figure 5.10b. The following specifications define the system:
For ethyl alcohol: K = 26 (for air, K = 1)
Cylinder separation: d = 0.5 cm
Plate area: A = pRL
R = 5.75 cm = average radius
L = distance along cylinder axis
Find the range of capacity variation as the alcohol level varies from 0 to 5 m.
We saw earlier that the capacity is given by C = KEo(A/d). Therefore, all we need to do is find the capacity for the entire cylinder with no alcohol, and then multiply that by 26.
A = 2pRL = 2p(0.0575 m)(5 m) = 1.806 m2
Thus, for air
C = (1)(8.85 pF/M)(1.806 m2/0.005 m)
C = 3196 pF = 0.0032 mF
With the ethyl alcohol, the capacity becomes
C= 26(0.0032 mF)
C = 0.0832 mF
The range is 0.0032 to 0.0832 mF.
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Reader Comments | Submit a comment »
For the solution of EXAMPLE 5.4 I think the A = pRL is right instead of A = 2pRL. Am i correct? Thank you.
- email@example.com - Nov 2, 2009
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