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8. SENSORS
8.2. POSITION AND SPEED MEASUREMENT
(1) Proximity Sensors and Switches
Magnetic, electrical capacitance, inductance, and eddy current methods are used.
Photoemitter-detector pair: interruption or reflection of a light beam is used to detect an
object in a noncontact manner as illustrated in Fig. 8.1.
Applications of proximity sensors and limit switches: counting moving objects (e.g., cans
on a convey belt), and limiting the traverse of a mechanism.
Switches are characterized by the number of poles (P) and throws (T) and whether
connections are normally open (NO) or normally closed (NC).
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(2) Potentiometer
Variable resistance device (a wiper makes contact with a resistive element, and as the
contact point moves, the resistance between the wiper and end leads of the device changes).
Through voltage division, the change in resistance can be used to create an output voltage
that is proportional to the input displacement.
(3) Linear Variable Differential Transformer (LVDT)
Operation Thoery
Transducer for measuring linear displacements.
It consists of primary and secondary windings and a movable iron core.
As with any transformer, the voltage of the induced signal in the secondary coil is linearly
related to the number of coils, i.e.,in
out
in
out
N
N
V
V= .
As the core is displaced, the number of coils in the secondary coil changes linearly.
Therefore the amplitude of the induced signal varies linearly with displacement.
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With two secondary coils connected in series-opposing configurations, the output signal
includes both the magnitude and direction of the core motion.
At the null position (midpoint in the cores position), the voltage induced in each coil will
be of the same amplitude and 180o
out of phase, producing a null output.
As the core moves from the null position, the output amplitude will increase a proportional
amount over a linear range around null.
Demodulation and Low-pass filter
The diode bridges in this circuit produce a positive or negative rectified sine wave
depending on what side of the null position the core is on.
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A low-pass filter is used to convert the rectified output into a smoothed signal. The cutoff
frequency of the low-pass filter should be higher than that of the core motion (at least 10
times the maximum expected frequency of the core motion).
Characteristics
Advantages: accuracy over the linear range, an analog output that may not require
amplification, and insensitivity to wide ranges in temperature.
Disadvantages: limited range of motion and limited frequency response.
(4) Digital Optical Encoder
Encoders have both linear and rotary configurations.
The absolute encoder where a unique digital word corresponds to each rotational position
of the shaft, and the incremental encoder, which produces digital pulses as the shaft rotates,
allowing measurement of relative position of shaft.
Rotary encoders are composed of a glass or plastic code disk and photoemitter-detector
pairs. As redial lines in each track interrupt the beam between a photoemitter-detector pair,
digital pulses are produced.
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Absolute Encoder
Producing a digital word that distinguishes Ndistrict positions of the shaft. For example, if
there are 8 tracks, the encoder is capable of producing 28=256 district positions or angular
resolution of 1.406o
(360o/256).
Gray and binary codes for numerical encoding of the absolute encoder.
[4-bit Gray code] [4-bit Binary code]
For the gray code, the uncertainty during a transition is only one count, unlike with the
binary code, where the uncertainty could be multiple counts.
The gray code provides data with the least uncertainty, but the binary code is the preferred
choice for direct interface to computers.
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A simple circuit to convert from gray to binary code (binary bit (iB ), gray bit ( iG ), and
is the exclusive OR gate):
01012123233 ,,, GBBGBBGBBGB ====
Incremental Encoder
It consists of two tracks and two sensors whose outputs are called channels A and B.
The A and B channels are used to determine the direction of rotation by assessing which
channel leads the other. The signals from the two channels are a 1/4 out of phase with
each other and are known as quadrature signals.
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The third channel, called INDEX, yields one pulse per revolution, which is useful in
counting full revolution. It also useful as a reference to define a home base or zero position.
There are two separate tracks for the A and B channels, but a more common configuration
uses a single track with the A and B sensors offset a 1/4 cycle on the track to yield the same
signal pattern.
Decoder circuits: A= with B=1 implies a CW pulse, and B= with A=1 implies a CCW
pulse, in the 1X mode. The D flip-flops decode whether the shaft is rotating CW or CCW,
and this information is used to drive an up-down counter to keep the current pulse count.
An incremental encoder can be used in conjunction with a limit switch to define absolute
position relative to some home position defined by the switch.
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8.3. STRESS AND STRAIN MEASUREMENT
- Stress, force, pressure and temperature can be determined from strain measurements, i.e.,
the electrical resistance strain gage.
(1) Electrical Resistance Strain Gage
The metal foil strain gage illustrated in Fig. 8.17 consists of a thin foil of metal (usually
constantan) and etched in a grid pattern onto a thin plastic backing material (usually
polyimide). The foil pattern is terminated at both ends with large metallic pads for allowing
leadwires to be attached with solder. The size of gage is typically 5mm to 15 mm long.
The gage is adhesively bonded directly to the surface of a mechanical component, usually
with epoxy.
Fig. 8.17 Metal foil gage.
When the component is loaded, the resistance of the metal foil changes, and if this
resistance change is measured accurately, the strain on the surface of the component can be
determined in the following ways:
The metal foil grid lines in the active portion of the gage can be approximated by a single
rectangular conductor. The total resistance is given by
A
LR
=
where is the foil metal resistivity, L is the total length of the grid lines, and A is the
cross-sectional area.
To see how the resistance changes under deformation, we need to take the differential of
the above equation.
ALR lnlnlnln += AdALdLdRdR //// += (Eq. 8.4)
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The resistance increase ( 0>dR ) with increased resistivity and increased length, and
decreases with increased cross-sectional area.
The cross-sectional area and the area differential are
whA = w
dw
h
dh
hw
dwhdhw
A
dA+=
+=
From the Poissons Ratio (see Appendix 2),
L
dL
h
dh= and
L
dL
w
dw= axial
L
dL
A
dA 22 ==
where axial is the axial strain. When the conductor is elongated ( 0>axial ), the cross-
sectional area decreases ( 0/
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(2) Measuring Resistance Changes with a Wheatstone Bridge
- It consists of a four-resistor network excited by a dc voltage, in order to measure small
changes in resistance.
Static Balanced Mode
2R and 3R are precision resistors, 4R is a precision potentiometer with an accuratescale for displaying the resistance value, and
1R is the strain gage resistance.
To balance the bridge, the variable resistor is adjusted until the voltage between nodes A
and B is zero.
In the balanced state, the voltages at A and B must be equal so
2211 RiRi = (Eq. 8.14)
With the high-input impedance voltmeter,
41
41RR
Vii ex
+== , and
32
32RR
Vii ex
+== .
Substituting these expressions into Eq. 8.14 gives
3
2
4
1
R
R
R
R= .
So, we can calculate the unknown resistance 1R as (the result is independent of exV )
3
241
R
RRR =
Dynamic Deflection Mode
The changes in the strain gage resistance 1R that occur when the mechanical component
is loaded can be determined from changes in the output voltage.
The output voltage is expressed in terms of the resistor currents as
324122110RiRiRiRiV +==
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The excitation voltage can be related to the same current as
)()(322411
RRiRRiVex
+=+=
Eliminating the currents from these equations results in
++= 322
41
10RR
R
RR
RVV ex
When the bridge is balanced,0V is zero and 1R has a known value. When 1R changes
value, the voltage change0
V can be related to the change in resistance1
R . To find
this relation, we can replace 1R by 11 RR + and 0V by 0V .
+
++
+=
32
2
411
110
RR
R
RRR
RR
V
V
ex
1
132
20
32
20
1
4
1
1
+
++
=
RR
R
V
V
RR
R
V
V
R
R
R
R
ex
ex
By measuring the change in the output voltage 0V , we can determine the gage resistance
change 1R .
Leadwire Effects
When using a strain gage located far from the bridge circuit, each of the leadwire
resistances R add to the resistance of the strain gage. The problem is that if the leadwire
temperature changes, it will cause changes in the resistance of the bridge.
As shown in Fig. 8.22b, a 3-wire connection can solve this problem. Equal resistances are
added to adjacent branches in the bridge so the effects of changes in the leadwire
resistances offset each other.
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Fig. 8.22 Leadwire effects in 1/4 bridge circuit.
Temperature Compensation
In addition to temperature effects in leadwires, temperature changes in the actual strain
gage can cause significant changes in resistance.
A method for eliminating this effect is to use a 1/2 bridge circuit where two of the four
bridge legs contain strain gages.
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(3) Measuring Different States of Stress with Strain Gages
Uniaxial Stress (1 strain gage)
When a component is loaded only in one direction in tension and compression.
By measuring the strain x , the stress is known from Hooks Law to be
xx E = . The axial stress in the bar
x is given by
A
Px=
where A is the bars cross-sectional area.
The force P can be determined from the strain gage measurement:
xAEP =
Biaxial Stress (2 strain gages)
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When a component is loaded in two orthogonal direction in tension or compression.
By measuring the strainsx and y , the stresses in the tank shell can be determined by
EE
yx
x
= and
EE
xy
y
=
Solving for the stress components gives
)(1 2
yxx
E
+
= and )(
1 2xyy
E
+
= .
For a thin-walled pressure vessel (i.e., t/r < 1/10), the stresses are given by
t
prx= and
t
pry
2= .
The pressure of vessel can be calculated by
)()1(2 yx
x
r
tE
r
tp
+== or )()1(
222 xy
y
r
tE
r
t
p
+== .
General Planar Stress (3 strain gages)
For uniaxial and biaxial loading, we already know the directions of principal stresses.
However, when the loading is more complex or when the geometry is more complex, we
have to use three gages in three different directions.
An assembly of strain gages is referred to as astrain gage rosette.
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(4) Force Measurement with Load Cells
A load cell is a sensor used to measure a force. It contains an internal flexural element,
usually with several gages mounted to its surface. The flexural elements shape is designed
so that the strain gage outputs can be related to the applied force.
A typical connection is that
Strain gage (or load cell) Amplifier Low-pass filter A/D Converter.
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8.4. TEMPERATURE MEASUREMENT
(1) Liquid-in-Glass Thermometer
(2) Bimetallic Strip
It is composed of two or more metal layers having different coefficients of thermal
expansion. So, the deflection is related to the temperature of the strip.
It is used in household thermostats where the mechanical motion of the strip makes or
breaks an electrical contact to turn a heating or cooling system on or off.
(3) Electrical resistance Thermometer
Resistance Temperature Device (RTD)
It is constructed of metallic wire wound around a ceramic or glass core and hermetically
sealed.
The resistance-temperature relationship is approximated by
)](1[ 00 TTRR += .
The most common metal used in RTDs is platinum, and the operating range for a typical
platinum RTD is 220oC to 750
oC.
Thermistor
It is a semiconductor device whose resistance changes exponentially with temperature
given by
=)
11(
00TTeRR
The accuracy of thermistors is better than that of RTDs, but thermistors have much
narrower operating ranges than RTDs.
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(4) Thermocouple
Two dissimilar metals in contact form a thermoelectric junction that produces a voltage
proportional to the temperature of the junction known as the Seebeck effect.
The wires of metals A and B form junctions at different temperatures1
T and2
T , resulting
in a potential V that can be measured. The thermocouple voltage V depends on the
metal properties of A and B and the difference between the junction temperatures1T and
2T given by
)(21 TTV = where is called the Seebeck coefficient.
A standard configuration is shown in Fig. 8.40. The reference junction is used to establish a
temperature reference.
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8.5. VIBRATION AND ACCELERATION MEASUREMENT
Accelerometer
It detects acceleration along one axis and is insensitive in orthogonal directions.
As a position transducer, strain gages or piezoelectric elements converting vibration into a
voltage signal are used.
Through the frequency response analysis, we can relate the displacement transducer output
to acceleration of the object.
The relative displacementrx between the seismic mass ( 0x ) and the object ( ix ) is
defined asirxxx = 0 .
The equation of the motion for the seismic mass is
0xmFF bk &&= 0xmxbkx rr &&& = )( irrr xxmxbkx &&&&& +=
irrrxmkxxbxm &&&&& =++
whererik
kxxxkF == )(0
,ribxbxxbF &&& == )(
0and
irxxx &&&&&& +=
0.
Referring to the analysis of a second order system in Chapter 4, the amplitude ratio and
phase angle can be obtained as
2/12
2
22
2
41
)/(
+
=
nn
n
i
r
X
X
and
=
2
1
1
2
tan
n
n
where the input and output displacements are )sin()( tXtxii
= and
)sin()( += tXtx rr .
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To relate the output displacement signal rx to the input acceleration ix&& , differentiate the
displacements:
)sin()( 2 tXtx ii =&& and )sin()(2 += tXtx rr&&
Rearranging the amplitude ratio with the amplitudes of the input and output acceleration
gives
2/12
2
22
2
2
41
1)(
+
==
nn
i
nra
X
XH
.
If we design the accelerometer so that )(aH is nearly 1 over a large frequency range,
then the input acceleration amplitude is given directly in terms of the relative displacement
amplitude scaled by a constant factor 2n :
rni XX )(amplitude,onAccelerati22 =
The largest frequency range resulting in a unity amplitude ratio occurs when the damping
ratio is 0.707 and when the natural frequency n is as small as possible. We can
make the natural frequency, mkn /= , large by choosing a small seismic mass and a
large spring constant.
Vibrometer
Instead of measuring acceleration, this device measures displacement.
The displacement ratio can be defined asi
rd
X
XH =)( . The input displacement amplitude
iX is related to the measured relative displacement amplitude rX as
r
d
ri X
H
XX =
)(
where )(dH is nearly 1 over a large frequency range.
As seen in Fig. 8.47, the largest frequency range resulting in a unity amplitude ratio occurs
when the damping ratio is 0.707 and when the natural frequency n is as small as
possible.
We can make the natural frequency small by choosing a large seismic mass and a small
spring constant. This explains the large size of seismographs used to measure the earths
displacement during an earthquake.
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Piezoelectric Accelerometers
As a displacement sensor, high quality accelerometers use a piezoelectric crystal, a
material whose deformation results in charge polarization across the crystal. In a reciprocal
manner, application of an electric filed to a piezoelectric material results in deformation.
An equivalent circuit for a piezoelectric crystal is shown in the following figure. The
crystal is effectively a capacitor and a charge source Q generating charge across the
capacitor plates proportional to the deformation of the crystal. By a Thevinin equivalent
circuit, the open circuit voltage V is equal to the charge (typically in pico-coulomb)
divided by the capacitance (typically in the pico-farad range):
pC
QV =
The sensitivity of the accelerometer is the ratio of the charge output to the acceleration of
the housing expressed in pC/g, (rms pC)/(rms g), or (peak pC)/(peak g), where g is the
gravity acceleration.
(read Ch 8.6 and 8.7)