19.1 Displacement
Translational displacement
transducers are instruments that measure the motion of a body in a straight
line between two points. Apart from their use as a primary trans[1]ducer measuring
the motion of a body, translational displacement transducers are also widely
used as a secondary component in measurement systems, where some other physical
quantity such as pressure, force, acceleration or temperature is translated
into a translational motion by the primary measurement transducer. Many
different types of translational displacement transducer exist and these, along
with their relative merits and characteristics, are discussed in the following
sections of this chapter. The factors governing the choice of a suitable type
of instrument in any particular measurement situation are considered in the
final section at the end of the chapter.
19.1.1 The resistive potentiometer
The resistive potentiometer is
perhaps the best-known displacement-measuring device. It consists of a
resistance element with a movable contact as shown in Figure 19.1. A voltage Vs
is applied across the two ends A and B of the resistance element and an output
voltage V0 is measured between the point of contact C of the sliding element
and the end of the resistance element A. A linear relationship exists between
the output voltage V0 and the distance AC, which can be expressed by:
V0/Vs
= AC/AB (19.1)
The body whose motion is being
measured is connected to the sliding element of the potentiometer, so that
translational motion of the body causes a motion of equal magnitude of the
slider along the resistance element and a corresponding change in the output
voltage V0. Three different types of
potentiometers consist of a coil of
resistance wire wound on a non-conducting former. As the slider moves along the
potentiometer track, it makes contact with successive turns of the wire coil.
This limits the resolution of the instrument to the distance from one coil to
the next. Much better measurement resolution is obtained from potentiometers
using either a carbon film or a conducting plastic film for the resistance
element. Theoretically, the resolution of these is limited only by the grain
size of the particles in the film, suggesting that measurement resolutions up
to 10-4 ought to be attainable. In practice, the resolution is
limited by mechanical difficulties in constructing the spring system that
maintains the slider in contact with the resistance track, although these types
are still considerably better than wire-wound types.
Operational problems of
potentiometers all occur at the point of contact between the sliding element and
the resistance track. The most common problem is dirt under the slider, which
increases the resistance and thereby gives a false output voltage reading, or
in the worst case causes a total loss of output. High-speed motion of the
slider can also cause the contact to bounce, giving an intermittent output.
Friction between the slider and the track can also be a problem in some
measurement systems where the body whose motion is being measured is moved by
only a small force of a similar magnitude to these friction forces.
The life expectancy of potentiometers
is normally quoted as a number of reversals, i.e. as the number of times the
slider can be moved backwards and forwards along the track. The figures quoted
for wire-wound, carbon-film and plastic-film types are respectively 1 million,
5 million and 30 million. In terms of both life expectancy and measurement
resolution, therefore, the carbon and plastic film types are clearly superior,
although wire-wound types do have one advantage in respect of their lower
temperature coefficient. This means that wire-wound types exhibit much less
variation in their characteristics in the presence of varying ambient
temperature conditions.
A typical inaccuracy figure that is
quoted for translational motion resistive potentiometers is ±1% of full-scale
reading. Manufacturers produce potentiometers to cover a large span of
measurement ranges. At the bottom end of this span, instruments with a range of
±2 mm are available whilst at the top end, instruments with a range of ±1 m are
produced.
The resistance of the instrument
measuring the output voltage at the potentiometer slider can affect the value
of the output reading, as discussed in Chapter 3. As the slider moves along the
potentiometer track, the ratio of the measured resistance to that of the
measuring instrument varies, and thus the linear relationship between the
measured displacement and the voltage output is distorted as well. This effect
is minimized when the potentiometer resistance is small relative to that of the
measuring instrument. This is achieved firstly by using a very high-impedance
measuring instrument and secondly by keeping the potentiometer resistance as
small as possible. Unfortunately, the latter is incompatible with achieving
high measurement sensitivity since this requires a high potentiometer
resistance. A compromise between these two factors is therefore necessary. The
alternative strategy of obtaining high measurement sensitivity by keeping the
potentiometer resistance low and increasing the excitation voltage is not
possible in practice because of the power rating limitation. This restricts the
allowable power loss in the potentiometer to its heat dissipation capacity.
The process of choosing the best
potentiometer from a range of instruments that are available, taking into
account power rating and measurement linearity considerations, is illustrated
in the example below.
Example
The output voltage from a
translational motion potentiometer of stroke length 0.1 metre is to be measured
by an instrument whose resistance is 10 kΩ. The maximum measurement error,
which occurs when the slider is positioned two-thirds of the way along the
element (i.e. when AC = 2AB/3 in Figure 19.1), must not exceed 1% of the
full-scale reading. The highest possible measurement sensitivity is also
required. A family of potentiometers having a power rating of 1 watt per 0.01
metre and resistances ranging from 100 Ω to 10 kΩ in 100 Ω steps is available.
Choose the most suitable potentiometer from this range and calculate the
sensitivity of measurement that it gives.
Solution
Referring to the labelling used in
Figure 19.1, let the resistance of portion AC of the resistance element Ri
and that of the whole length AB of the element be Rt. Also, let the
resistance of the measuring instrument be Rm and the output voltage
measured by it be Vm. When the voltage-measuring instrument is
connected to the potentiometer, the net resistance across AC is the sum of two
resistances in parallel (Ri and Rm) given by:
If we express the voltage that exists
across AC in the absence of the measuring instru[1]ment
as V0, then we can express the error due to the loading effect of the measuring
instrument as Error = V0 - Vm. From equation (19.1), V0
= (RiV) /Rt. Thus,
19.1.2 Linear variable differential
transformer (LVDT)
The linear variable differential
transformer, which is commonly known by the abbreviation LVDT, consists of a
transformer with a single primary winding and two secondary windings connected
in the series opposing manner shown in Figure 19.2. The object whose
translational displacement is to be measured is physically attached to the
central iron core of the transformer, so that all motions of the body are
transferred to the core. For an excitation voltage Vs given by Vs
= Vp sin (ωt), the e.m.f.s induced in the secondary windings Va
and Vb are given by:
The parameters Ka and Kb
depend on the amount of coupling between the respective secondary and primary
windings and hence on the position of the iron core. With the core in the
central position, Ka = Kb, and we have:
Because of the series opposition mode
of connection of the secondary windings, V0 = Va - Vb,
and hence with the core in the central position, V0 = 0. Suppose now
that the core is displaced upwards (i.e. towards winding A) by a distance x. If
then Ka = K1 and Kb = K2, we have:
If, alternatively, the core were
displaced downwards from the null position (i.e. towards winding B) by a
distance x, the values of Ka and Kb would then be Ka
= K2 and Kb = K1, and we would have:
Thus for equal magnitude displacements
+x and -x of the core away from the central (null) position, the magnitude of
the output voltage V0 is the same in both cases. The only
information about the direction of movement of the core is contained in the
phase of the output voltage, which differs between the two cases by 180°. If,
therefore, measurements of core position on both sides of the null position are
required, it is necessary to measure the phase as well as the magnitude of the
output voltage. The relationship between the magnitude of the output voltage
and the core position is approximately linear over a reasonable range of
movement of the core on either side of the null position, and is expressed
using a constant of proportionality C as V0 = Cx. The only moving
part in an LVDT is the central iron core. As the core is only moving in the air
gap between the windings, there is no friction or wear during operation. For
this reason, the instrument is a very popular one for measuring linear
displacements and has a quoted life expectancy of 200 years. The typical
inaccuracy is ±0.5% of full[1]scale reading and
measurement resolution is almost infinite. Instruments are available to measure
a wide span of measurements from ±100 µm to ±100 mm. The instrument can be made
suitable for operation in corrosive environments by enclosing the windings
within a non-metallic barrier, which leaves the magnetic flux paths between the
core and windings undisturbed. An epoxy resin is commonly used to encapsulate
the coils for this purpose. One further operational advantage of the instrument
is its insensitivity to mechanical shock and vibration.
Some problems that affect the
accuracy of the LVDT are the presence of harmonics in the excitation voltage
and stray capacitances, both of which cause a non-zero output of low magnitude
when the core is in the null position. It is also impossible in practice to
produce two identical secondary windings, and the small asymmetry that
invariably exists between the secondary windings adds to this non-zero null
output. The magnitude of this is always less than 1% of the full-scale output
and in many measurement situations is of little consequence. Where necessary,
the magnitude of these effects can be measured by applying known displacements
to the instrument. Following this, appropriate compensation can be applied to
subsequent measurements.
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