20.1.5 The resolver
The resolver, also known as a
synchro-resolver, is an electromechanical device that gives an analogue output
by transformer action. Physically, resolvers resemble a small a.c. motor and
have a diameter ranging from 10 mm to 100 mm. They are friction[1]less and reliable
in operation because they have no contacting moving surfaces, and consequently
they have a long life. The best devices give measurement resolutions of 0.1%.
Resolvers have two stator windings,
which are mounted at right angles to one another, and a rotor, which can have
either one or two windings. As the angular position of the rotor changes, the
output voltage changes. The simpler configuration of a resolver with only one
winding on the rotor is illustrated in Figure 20.8. This exists in two separate
forms that are distinguished according to whether the output voltage changes in
amplitude or changes in phase as the rotor rotates relative to the stator
winding.
Varying amplitude output resolver
The stator of this type of resolver
is excited with a single-phase sinusoidal voltage of frequency ω, where the
amplitudes in the two windings are given by:
Varying phase output resolver
This is a less common form of
resolver but it is used in a few applications. The stator windings are excited
with a two-phase sinusoidal voltage of frequency ω, and the instantaneous
voltage amplitudes in the two windings are given by:
V1 = Vs
sin(ωt); V2 = Vs sin(ωt + π/2) = Vs cos(ωt)
The net output voltage in the rotor
winding is the sum of the voltages induced due to each stator winding. This is
given by:
This represents a linear relationship
between shaft angle and the phase shift of the rotor output relative to the
stator excitation voltage. The accuracy of shaft rotation measurement depends
on the accuracy with which the phase shift can be measured. This can be improved
by increasing the excitation frequency, ω, and it is possible to reduce
inaccuracy to ±0.1%. However, increasing the excitation frequency also
increases magnetizing losses. Consequently, a compromise excitation frequency
of about 400 Hz is used.
20.1.6 The synchro
Like the resolver, the synchro is a
motor-like, electromechanical device with an analogue output. Apart from having
three stator windings instead of two, the instrument is similar in appearance
and operation to the resolver and has the same range of physical dimensions.
The rotor usually has a dumb-bell shape and, like the resolver, can have either
one or two windings.
Synchros have been in use for many
years for the measurement of angular posi[1]tions,
especially in military applications, and achieve similar levels of accuracy and
measurement resolution to digital encoders. One common application is axis
measurement in machine tools, where the translational motion of the tool is
translated into a rotational displacement by suitable gearing. Synchros are
tolerant to high temperatures, high humidity, shock and vibration and are
therefore suitable for operation in such harsh environmental conditions. Some
maintenance problems are associated with the slip ring and brush system used to
supply power to the rotor. However, the only major source of error in the
instrument is asymmetry in the windings, and reduction of measurement
inaccuracy down to ±0.5% is easily achievable.
Figure 20.9 shows the simpler form of
synchro with a single rotor winding. If an a.c. excitation voltage is applied
to the rotor via slip rings and brushes, this sets up a certain pattern of
fluxes and induced voltages in the stator windings by transformer action. For a
rotor excitation voltage, Vr, given by:
Vr = V sin(ωt)
the voltages induced in the three
stator windings are:
If the rotor is turned at constant
velocity through one full revolution, the voltage waveform induced in each
stator winding is as shown in Figure 20.10. This has the form of a
carrier-modulated waveform, in which the carrier frequency corresponds to the
excitation frequency, ω. It follows that if the rotor is stopped at any
particular angle, β’ , the peak-to-peak amplitude of the stator
voltage is a function of β’. If therefore the stator winding voltage
is measured, generally as its root-mean-squared (r.m.s.) value, this indicates
the magnitude of the rotor rotation away from the null position. The direction
of rotation is determined by the phase difference between the stator voltages,
which is indicated by their relative instantaneous magnitudes.
Although a single synchro thus
provides a means of measuring angular displacements, it is much more common to
find a pair of them used for this purpose. When used in pairs, one member of
the pair is known as the synchro transmitter and the other as the synchro
transformer, and the two sets of stator windings are connected together, as
shown in Figure 20.11. Each synchro is of the form shown in Figure 20.9, but
the rotor of the transformer is fixed for displacement-measuring applications.
A sinusoidal excitation voltage is applied to the rotor of the transmitter,
setting up a pattern of fluxes and induced voltages in the transmitter stator
windings. These voltages are transmitted to the transformer stator windings
where a similar flux pattern is established. This in turn causes a sinusoidal
voltage to be induced in the fixed trans[1]former
rotor winding. For an excitation voltage, V sin(ωt), applied to the transmitter
rotor, the voltage measured in the transformer rotor is given by:
V0
= V sin(ωt) sin(θ)
Where θ is the relative angle between
the two rotor windings.
Apart from their use as a
displacement transducer, such synchro pairs are commonly used to transmit
angular displacement information over some distance, for instance to transmit
gyro compass measurements in an aircraft to remote meters. They are also used
for load positioning, allowing a load connected to the transformer rotor shaft
to be controlled remotely by turning the transmitter rotor. For these
applications, the transformer rotor is free to rotate and is also damped to
prevent oscillatory motions. In the simplest arrangement, a common sinusoidal
excitation voltage is applied to both rotors. If the transmitter rotor is
turned, this causes an imbalance in the magnetic flux patterns and results in a
torque on the transformer rotor that tends to bring it into line with the
transmitter rotor. This torque is typically small for small displacements, and
so this technique is only useful if the load torque on the transformer shaft is
very small. In other circumstances, it is necessary to incorporate the synchro
pair within a servomechanism, where the output voltage induced in the
transformer rotor winding is amplified and applied to a servomotor that drives
the transformer rotor shaft until it is aligned with the transmitter shaft.
20.1.7 The induction potentiometer
These instruments belong to the same
class as resolvers and synchros but have only one rotor winding and one stator
winding. They are of a similar size and appearance to other devices in the
class. A single-phase sinusoidal excitation is applied to the rotor winding and
this causes an output voltage in the stator winding through the mutual
inductance linking the two windings. The magnitude of this induced stator
voltage varies with the rotation of the rotor. The variation of the output with
rotation is naturally sinusoidal if the coils are wound such that their field
is concentrated at one point, and only small excursions can be made away from
the null position if the output relationship is to remain approximately linear.
However, if the rotor and stator windings are distributed around the
circumference in a special way, an approximately linear relationship for
angular displacements of up to ±90° can be obtained.
20.1.8 The rotary inductosyn
This instrument is similar in
operation to the linear inductosyn, except that it measures rotary
displacements and has tracks that are arranged radially on two circular discs,
as shown in Figure 20.12. Typical diameters of the instrument vary between 75
mm and 300 mm. The larger versions give a measurement resolution of up to 0.05
seconds of arc. Like its linear equivalent, however, the rotary inductosyn has
a very small measurement range, and a lower-resolution, rotary displacement
transducer with a larger measurement range must be used in conjunction with it.
20.1.9 Gyroscopes
Gyroscopes measure both absolute
angular displacement and absolute angular velocity. The predominance of
mechanical, spinning-wheel gyroscopes in the market place is now being
challenged by recently introduced optical gyroscopes.
Mechanical gyroscopes
Mechanical gyroscopes consist
essentially of a large, motor driven wheel whose angular momentum is such that
the axis of rotation tends to remain fixed in space, thus acting as a reference
point. The gyro frame is attached to the body whose motion is to be measured.
The output is measured in terms of the angle between the frame and the axis of
the spinning wheel. Two different forms of mechanical gyroscope are used for
measuring angular displacement, the free gyro and the rate-integrating gyro. A
third type of mechanical gyroscope, the rate gyro, measures angular velocity
and is described in section 20.2.
Free gyroscope
The free gyroscope is illustrated in
Figure 20.13. This measures the absolute angular rotation of the body to which
its frame is attached about two perpendicular axes. Two alternative methods of
driving the wheel are used in different versions of the instrument. One of
these is to enclose the wheel in stator-like coils that are excited with a
sinusoidal voltage. A voltage is applied to the wheel via slip rings at both
ends of the spindle carrying the wheel. The wheel behaves as a rotor and motion
is produced by motor action. The other, less common, method is to fix vanes on
the wheel that is then driven by directing a jet of air onto the vanes.
The instrument can measure angular
displacements of up to 10° with a high accuracy. For greater angular
displacements, interaction between the measurements on the two perpendicular
axes starts to cause a serious loss of accuracy. The physical size of the coils
in the motor-action driven system also limits the measurement range to 10°. For
these reasons, this type of gyroscope is only suitable for measuring rotational
displacements of up to 10°. A further operational problem of free gyroscopes is
the presence of angular drift (precession) due to bearing friction torque. This
has a typical magnitude of 0.5° per minute and means that the instrument can
only be used over short time intervals of say, 5 minutes. This time duration
can be extended if the angular momentum of the spinning wheel is increased.
A major application of gyroscopes is
in inertial navigation systems. Only two free gyros mounted along orthogonal
axes are needed to monitor motions in three dimensions, because each instrument
measures displacement about two axes. The limited
angular range of measurement is not
usually a problem in such applications, as control action prevents the error in
the direction of motion about any axis ever exceeding one or two degrees.
Precession is a much greater problem, however, and for this reason, the
rate-integrating gyro is used much more commonly.
Rate integrating gyroscope
The rate-integrating gyroscope, or
integrating gyro as it is commonly known, is illustrated in Figure 20.14. It
measures angular displacements about a single axis only, and therefore three
instruments are required in a typical inertial navigation system. The major
advantage of the instrument over the free gyro is the almost total absence of
precession, with typical specifications quoting drifts of only 0.01°/hour. The
instrument has a first order type of response given by:
where K = H/β, t = M/β, θi is the
input angle, θo is the output angle, D is the D-operator, H is the angular
momentum, M is the moment of inertia of the system about the measurement axis
and β is the damping coefficient.
Inspection of equation (20.1) shows
that to obtain a high value of measurement sensitivity, K, a high value of H
and low value of β are required. A large H is normally obtained by driving the
wheel with a hysteresis-type motor revolving at high speeds of up to 24 000
rpm. The damping coefficient β can only be reduced so far, however, because a
small value of β results in a large value for the system time constant, t, and
an unacceptably low speed of system response. Therefore, the value of β has to be
chosen as a compromise between these constraints.
Besides their use as a fixed
reference in inertial guidance systems, integrating gyros are also commonly
used within aircraft autopilot systems and in military applications such as
stabilizing weapon systems in tanks.
Optical gyroscopes
Optical gyroscopes have been
developed only recently and come in two forms, the ring laser gyroscope and the
fibre-optic gyroscope.
The ring laser gyroscope consists of
a glass ceramic chamber containing a helium–neon gas mixture in which two laser
beams are generated by a single anode/twin cathode system, as shown in Figure
20.15. Three mirrors, supported by the
ceramic block and mounted in a
triangular arrangement, direct the pair of laser beams around the cavity in opposite
directions. Any rotation of the ring affects the coherence of the two beams,
raising one in frequency and lowering the other. The clockwise and
anticlockwise beams are directed into a photodetector that measures the beat
frequency according to the frequency difference, which is proportional to the
angle of rotation. A more detailed description of the mode of operation can be
found elsewhere (Nuttall, 1987). The advantages of the ring laser gyroscope are
considerable. The measurement accuracy obtained is substantially better than
that afforded by mechanical gyros in a similar price range. The device is also
considerably smaller physically, which is of considerable benefit in many
applications.
The fibre-optic gyroscope measures
angular velocity and is described in section 20.2.
20.1.10 Choice between rotational
displacement transducers
Choice between the various rotational
displacement transducers that might be used in any particular measurement
situation depends first of all upon whether absolute measurement of angular
position is required or whether the measurement of rotation relative to some
arbitrary starting point is acceptable. Other factors affecting the choice
between instruments are the required measurement range, the resolution of the
transducer and the measurement accuracy afforded.
Where only measurement of relative
angular position is required, the incremental encoder is a very suitable
instrument. The best commercial instruments of this type can measure rotations
to a resolution of 1 part in 20 000 of a full revolution, and the measurement
range is an infinite number of revolutions. Instruments with such a high
measurement resolution are very expensive, but much cheaper versions are
available according to what lower level of measurement resolution is
acceptable.
All the other instruments presented
in this chapter provide an absolute measurement of angular position. The
required measurement range is a dominant factor in the choice between these. If
this exceeds one full revolution, then the only instrument available is the
helical potentiometer. Such devices can measure rotations of up to 60 full
turns, but they are expensive because the procedure involved in manufacturing a
helical resistance element to a reasonable standard of accuracy is difficult.
For measurements of less than one
full revolution, the range of available instruments widens. The cheapest one
available is the circular potentiometer, but much better measurement accuracy
and resolution is obtained from coded-disc encoders. The cheapest of these is
the optical form, but certain operating environments necessitate the use of the
alternative contacting (electrical) and magnetic versions. All types of
coded-disc encoder are very reliable and are particularly attractive in computer
control schemes, as the output is in digital form. A varying phase output
resolver is yet another instrument that can measure angular displacements up to
one full revolution in magnitude. Unfortunately, this instrument is expensive
because of the complicated electronics incorporated to measure the phase variation
and convert it to a varying-amplitude output signal, and hence use is not
common.
An even greater range of instruments
becomes available as the required measurement range is reduced further. These
include the synchro (±90°), the varying amplitude output resolver (±90°), the
induction potentiometer (±90°) and the differential transformer (±40°). All
these instruments have a high reliability and a long service life.
Finally, two further instruments are
available for satisfying special measurement requirements, the rotary
inductosyn and the gyroscope. The rotary inductosyn is used in applications
where very high measurement resolution is required, although the measurement
range afforded is extremely small and a coarser-resolution instrument must be
used in parallel to extend the measurement range. Gyroscopes, in both
mechanical and optical forms, are used to measure small angular displacements
up to ±10° in magnitude in inertial navigation systems and similar
applications.
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