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ELECTRICAL MEASUREMENTS AND
INSTRUMENTATION
EE204
LECTURER
ENG. G. KAPUNGU
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INTRODUCTION
Formal definition:
Measurement (also called metrology) is the
science of determining values of physical
variables.
Instrumentation is the technology of
measurement.
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Physical Variables
Temperature
Pressure
Light intensity
Displacement
Speed
LevelFlow-rate etc
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Typical Applications of Measurements
1. Monitoring of a process or operation to
indicate its state or condition.
Examples:
i. Monitoring environmental conditions
ii. Water and electricity meters monitor
quantity used.iii. Patient monitoring in hospitals (blood
pressure, heart beat, temperature) etc.
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Applications (contd)
iv. In vehicles various instruments areincorporated to indicate speed, fuel left intank, engine temperature etc.
2. Process control:automatic control systems incorporatemeasuring instruments at various stages of
the process.3. Data recordingthis is the recording of data
for storage and later use.
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Contd
examples of recording media are magnetic
tape, paper chart cd etc.
Block diagram of an instrumentation system
measurement target
object object
Dataacquisition
Data
processing Datadistribution
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Instrumentation system
1. Data acquisition
this is acquiring information about the
measurement object using suitable sensors
and conversion into electrical data
(transduction).
more than one variable can be measured orone variable can be measured at different
points simultaneously.
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Contd
2. Data processing
This is manipulation of measurement data in
order to achieve some desired result.
3. Data distribution
This is the supplying of data to the target
object(s) This could be a monitor, controller or a
recorder.
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Standards
1. International standards:
Defined by international agreements
2. Primary standards: Maintained at institutions around the world
Main function is checking accuracy of
secondary standards
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Contd
3. Secondary standards
Employed in industry as references for
calibration and for verifying working
standards.
4. Working standards
Used as measurement references on a day
to day basis in all electronics labs.
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Measurement errors
Measurement can thus be redefined as theprocess of comparing an unknown quantity withan expected (standard ) quantity
Usually, measurement gives a value that is notthe expected value.
The difference is measurement error.
Error is the measure of the degree that themeasured value conforms to the expected value.
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Contd
That is where and are the
expected and the measured values
respectively.
Can be
Fractional error =
And % error is fractional error is fractional
error x100%
nn XYE nY nX
n
nn
Y
XY
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Contd
Absolute fractional error = and is
always positiven
nn
Y
XY
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Types of errors (3 types)
1. Gross errors or Human errors
these result from carelessness e.g.
misreading an instrument or incorrectly
recording a reading.
2. Systematic errors -
Instrumental errors due to friction and zero
positioning.
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Contd
Environmental errorsdue to ambient
conditions i.e.
o Temperature
o Humidity
o Pressure
o Presence of electric and magnetic fields
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Measurement error combinations
Used when quantities are calculated from
measurements made from two (or more)
instruments.
it is assumed that the errors combine in a
worst possible way.
i. Sum of quantities
which give
)()(2211
VVVVV
)()( 2121 VVVVV
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Contd
ii. Difference of quantities
the error of the difference of two
measurements are again additive:
and
)()( 2211 VVVVV
)()( 2121 VVVVV
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Contd
IV. Quotient of quantities:
% error in E/I = (%error in E) + (%error in I)
v. Quantity raised to a power:%error in = B(%error in A)
BA
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Contd
Some more definitions :
i. Accuracythe degree of exactness of a
measurement compared with the expected
value.
ii. Relative accuracy, A = 1(absolute error)
iii. % accuracy = A x 100%
iv. Precisionis the consistency or repeatability
of a measurement.
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Contd
v. If i readings are taken and is the average ofthe readings, and is the reading, wedefine precision as:
vi. Instrument is a device used to indicate the presentvalue of a variable. (it can be analogue or digital)
iXiX
thi
i
ii
XXXP 1
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Characteristics (response) of
instruments
These are divided into two types depending
on:
i. Type of input to the instrument
ii. The point in time the instrument is observed
iii. The type of output
Definition: - Plot of an instruments outputagainst time is a response curve.
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Contd
a) Static response:
This describes the behavior of the instrument
when it attains steady state
( i.e. has been allowed enough time to settle
down to a steady reading)
Can also be response to a steady input.
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Contd
b) Dynamic response:
This relates instrument behavior to a varying
input
Or instrument behavior after a sudden
change in input value e.g. application of a
step input or an impulse input).
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Type of inputs
i. Ramp input
Signal amplitude changes linearly with time
time
amplituderamp f(t) = At; t > 0
= 0; t < 0
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Inputs contd
ii. Step/steady input
A
amplitude
time
f(t) = Au(t)
u(t) = 1; t > 0
= 0; t < 0
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Contd
iii. Impulse -
amplitude
time
0;0)( tt
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Contd
iv. Step/ramp -
time
amplitude
f (t) = At; t > 0
= A ;t = 0
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Contd
v. The sinusoid
Af(t) = Asint
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Contd
vi. Random _
f(t) = ?
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Static(steady state) characteristics
Listed on instrument data sheets
Apply only when instrument is used under
standard calibration conditions.
a) Accuracy and precision
b) Range or span
This is the maximum and minimum values
that an instrument is designed to measure.
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Contd
c) Bias
This is a constant error that exists over the
full range of measurement.
i.e. a reading always appears before an input
is applied.
Can be removed by calibration.
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Contd
e) Sensitivity
Change of in instrument output per unit
change in input.
i.e. sensitivity = scale deflection / value of
measurand causing the deflection.
(slope of output / input characteristic or
transfer function)
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Contd
f) Sensitivity to disturbance
Instrument must be minimally sensitive to all
other conditions except the measurand.
ambient conditions are a source of this
disturbance and the effects are in three
forms:
i. Zero driftzero reading is modified by
ambient conditions.
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Contd
E.g. in a voltmeter, zero drift coefficient
related to temperature changes is measured
in volts per degree.
An instrument can have several zero drift
coefficient related to other environmental
parameters.
Recalibration removes zero drift.
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Contd
g) Sensitivity drift
This defines the amount by which is modified
by changes in ambient conditions.
h) Zone drift
Occurs only over a certain zone.
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Illustration of drift
o/p
i/P
Nominal response
Zero drift
Sensitivity drift
both
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Static characteristics (contd)
i) Hysteresis -
This is the non-coincidence of the loading
and the unloading calibration curves.
Associated with capability of memory since
reduction of input to zero results in a
remanance of the output quantity
This property is made use of in magnetic
recording.
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Contd
j) Dead space
This is the range of input values where there
is no change in output
Also occurs in instruments which do not
exhibit hysteresis.
k) Threshold
The minimum value reached by the input
before instrument gives an output reading.
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illustration
l) Resolutionsmallest change in input thatproduces an observable change in instrument
o/p
Input variable
threshold
Dead
band
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Dynamic response
Lineartimeinvariant systems which are
asymptotically stable:
i. Linearobeys the principle of superposition
and the principle of frequency preservation.
ii. Time invariantsystem properties do not
vary with time.
iii. Asymptotically stablesystem stabilizes with
time.
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Contd
For such a system, there exists an empirical
relationship between input and output which
is:
ii
m
im
mm
im
m
oo
n
on
nn
on
n
qbdt
dqbdt
qdbdt
qdb
qadt
dqa
dt
qda
dt
qda
011
1
1
011
1
1
.......
.......
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Contd
The right hand side assumes use of known
inputs.
For a step input qi= x0for t > 0
= xrfor ramp input
= xssint
The qsare functions of time and the asandbsare constants.
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Contd
In practical instruments, certain assumptions
can be made and certain restrictions can be
imposed.
1stassume only step input. The 1stand higher
order terms on the right hand side reduce to
zero.
i.e.:
iooo
n
on
n qbqadt
dqadt
qda 01.........
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Zeroorder instrument
2ndassumption
Then
The expression describes a zeroorder
instrument and k is the static sensitivity.There is no dynamic error and output strictly
follows input.
arezerosaan 1,........,
iio
o
o kqqa
b
q
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Contd
A linear displacement potentiometer is a
typical example of a zero order instrument.
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Firstorder instrument
If all second and higher order terms are
assumed to be zero
and initial conditions are such that at t
= 0 (no bias)
Then:
Let D = d/dt
0o
q
ioooo qbqa
dt
dqa 1
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Contd
The first part on the right hand side is the
particular integral and the second the
complementary function
The c.f. reduces to zero as t becomes large
resulting in the output assuming the value of
the p.i.
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illustration
An instrument satisfying the above conditions
is a first - order instrument.
Final reading
o/p
time
90%
63%
90% lag
Dynamic error
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Contd
A typical firstorder instrument (or part of an
instrument system) is a thermocouple.
The time it takes the output to reach 63% of
final value is the time constant.
The instruments are associated with lag
quoted as % of response.
Figures used are usually the time required for
output to rise to 90%, 95% and 98%.
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Dynamic characteristics of importance
These are fidelity and speed of response:
a) Fidelity -
Quality of indication by the instrument for a
time-varying input.
Degree of closeness with which the output
reproduces the time-varying input.
Difference results in dynamic error.
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Contd
b) Speed of response
Rapidity with which the instrument response
to changes in input.
The delay is called lag.
Both dynamic error and lag must be known
for each input in order to make correct
estimation of data.
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Second - order instrument
Assumptions:
i. all third order coefficients and above are
zero.
ii. Initial conditions are such that output is zero
at t = 0.
iii. The original equation becomes:
ioo qbqaDaqDa 0012
2
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Contd
Let static sensitivity
Let un-damped natural frequency
And damping ratio
0
0
abk
2
0
aa
2
1
2aa
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Contd
then:
Any instrument which obeys the above
equation is a second-order instrument.
Dynamic behavior depends on the value of
the damping ratio
1/2/2
DD
k
q
q
i
o
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Contd
The transducer must fulfill two major
functions:
i. To sense the presence, magnitude, change in
and frequency of some measurand.
ii. To provide an electrical output which when
processed and supplied to a readout device,
gives an accurate representation of theoriginal measurand.
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Classification of transducers
These are classified as:
i. Genuine energy converters ( called active
transducers)
ii. Energy controllers (passive transducers)
Important considerations in transducer
selection:
i. Long term stability of input / output
relationships (transfer characteristic)
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Contd
ii. Size, shape and weight of the device
iii. Response to rapid changes in measurand
iv. Electrical output impedance
v. Reliability, availability and cost.
vi. Response to interfering and modifying inputs
vii. Effects of ambient conditions(temperature,humidity, vibration and supply frequency in
the case of ac powered devices.
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Interfering and modifying inputs
Transducer
Interfering i/p, Ii
Modifying i/p, Mi
Desired i/p, Di
o/p due to MI ii &
o/p
o/p due to MD ii &
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Contd
Desired input refers to the quantity the
transducer is specifically intended to respond
to.
Interfering input represents those quantitiesthe transducer is unintentionally sensitive to.
Modifying input represents the quantity
whose effect is to modify the desired andinterfering inputs.
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Contd
Common sources of modifying inputs are
ambient conditions and battery voltage.
Finally, transducer behavior can be affected by
self heating, vibration and supply frequency.
Note:
A device that convert the modified electrical
signal into a non-electrical signal is an output
transducer e.g. a radio speaker.
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Temperature transducers
These employ pure metal wire such as pureplatinum, copper, nickel, etc.
Provide definite resistance value at each
temperature.Platinum is the preferred metal because:
i. It is stable under different environmental
conditionsii. Resistance / temperature characteristic is
linear over a wide temperature range.
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Resistance temperature detectors
It is least sensitive to contamination
RTDs:
The resistance of metals increases with
temperature according to:
Platinum RTDs are either thick film or wirewound type.
The thick film type has a faster speed ofresponse.
)](1[ 1212 TTRRT
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Contd
If the two are immersed in hot water whose
temperature is T at time t = 0, response curves
are shown below:
time
T
Thickfilm
Wire wound
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Contd
Note the superior response of the thick film
type.
This transducer is a firstorder type and is an
example of a passive transducer.
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Contd
Operating temperature -60C to 150C
The resistance / temperature characteristic is
given by:
Where is a material constant which ranges
from 3000K to 5000K.
The major advantages are:
)11(
00TTeRR
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Contd
1. high sensitivity and
2. Small size.
3. Small mass (implying shorter time constant)
The small size makes them ideal for
measurement in confined places.
The major disadvantages are limited range (-
60C to 150C)
and non-linearity.
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Contd
Linearizing networks are now available
When connected to microprocessors software
does the linearising.
They also find applications in temperature
control systems where linearity is not as
important as sensitivity.
f
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Some applications of thermistors
a. Because of their negative temperaturecoefficients, they are used to compensate for
effects of temperature on circuit components.
It is mounted on or near the circuit element sothat it experiences the same temperature as the
circuit element.
It can be arranged that the equivalent resistancethat results is constant over a wide temperature
range.
d
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Contd
b. Measurement of thermal conductivity:
Two thermistors are placed in two cavities
They are connected in a bridge arrangement
so that with air in both cavities, bridge isbalanced.
Air in one cavity is replaced by, say, carbon
dioxide which has lower conductivityThe amount of bridge unbalance represents
amount of gas.
d
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Contd
c. Measurement of gas flow-rate:
One thermistor is sealed in brass and other is
placed in a hollow pipe and the two are
placed in a bridge circuit.
With no gas flow, the bridge is balance.
When gas flows, the thermistor is cooled and
its temperature is reduced and resistance
increased.
C d
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Contd
The out of balance voltage is proportional to
the flow-rate.
The thermistor is a passive transducer and is a
firstorder device.
Typical flow-rates of the order of 0.001 cubic
cm per minute have been measured by this
method.
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C d
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Contd
This effect results from the diffusion ofelectrons across the interface between the
two metals.
The material giving the electrons becomemore positive and the one receiving more
negative.
This is an active transducer which is of the firstorder type.
F b i i
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Fabrication
V
Referencejunction
Metal 2
Metal 1Sensing
junction
TTsV
C td
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Contd
Another configuration:
DVM 1J
2J
Ice
Cu
Cuconstantan
JisTV
C td
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Contd
The thermal contact (junction) can be madeby twisting, welding, soldering, pressing or
brazing.
A general two junction thermocouple:
Material 1
Material 2
C td
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Contd
An alternative expression for the transfercharacteristic is:
Where the Cs are thermoelectric constants.
These depend on the two materials used.
)()( 2
212
2211 TTCTTCV
P i i l f th l b h i
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Principles of thermocouple behavior
a) Must contain two dissimilar metals.
b) Output voltage depends only on the
difference between the two junction
temperatures
c) If a third metal is inserted into metal 1 or
metal 2, output voltage is not affected
provided the new junctions are at sametemperature.
t i l
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materials
Thermocouples are made from:
i. Copper and iron
ii. Base metal alloys of alumel, chromel,
constantan, etc.
iii. The noble metals platinum and tungsten
iv. Noble metal alloysplatinum/rhodium,
tungsten/rhenium
Th th il
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The thermopile
This consists of several thermocouplesconnected together in series.
All the reference junctions are at the same
temperature and all the hot junctions areexposed to the temperature being measured
The effect of connecting together n
thermocouples is to increase the sensitivity bya factor, n.
ill t ti
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illustration
Reference junction
V
Force transducer
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Force transducer
The strain gauge:
This is a resistive input transducer whose
resistance change is related to changes in
length.
Increase in length from L to L +
Results in increase in resistance from R to R +
Sensitivity of the gauge (or gauge factor) is
given by:
L
R
Contd
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Contd
RR
LLRRG ///
Where: R = resistance
L = length
= mechanical strain
Hookes Law: S = E
Where: E = Youngs modulusS = mechanical stress
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Contd
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Cont d
Strain gauges are used in the measurement offorce, pressure, acceleration and dc bridges
are used as conditioners.
Main source of error is resistance change dueto temperature.
Bridge methods which make use of dummy
gauges compensate for these errors.
Force measurement (Contd)
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Force measurement (Cont d)
The Piezoelectric transducer:
These are also used in the measurement of
force, acceleration and pressure
They are made from crystalline materials like
quartz, Rochelle salt and ceramics like barium
titanate
These materials generate a voltage whendeformed.
Contd
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Cont d
The crystals contain molecules withasymmetrical charge distribution
When pressure is applied, crystal deforms and
there is relative displacement of +ve andvecharges within the crystal.
This produces external charges of opposite
sign.
Contd
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Cont d
The associated output voltage is given by:
Where C is the capacitance of the crystal
The surface charge is related to the applied
pressure by:
Recall : for a parallel plate cap.
substituting for C:
Cvq o
ApSq q
d
AC ro
ro
qo
dpSv
Contd
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Cont d
Letting the voltage sensitivity =
The output voltage becomes:
Reverse Piezoelectric effect:
When an ac voltage is applied across the
piezoelectric crystal, it vibrates at a frequency
determined by its geometry and size.
This reverse effect finds applications in sound
ro
qS
dpSv vo
Contd
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Cont d
Generators found in toys, watches, electroniccalculators, electronic games, etc.
The piezo crystal can be cut to dimensions
which give it a desired natural frequency.
The frequency is very stable and is used to
stabilize oscillators in radio transmitters ,
clocks, computers, etc.
Step response of the piezoelectric
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crystal
63%
time
o/p
Applied step
Contd
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Cont d
This is a firstorder linear device.
Equivalent circuit:
pC
pR
Charge amplifier
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Charge amplifier
Output of the piezo crystal is very small andhas to be amplified.
A charge amplifier is used for this purpose.
ovpC
fC
R
Op amp
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Contd
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Cont d
And
i.e. the output is independent of the crystal
and lead capacitances.
Example:
A piezoelectric transducer has a sensitivity of
28pC/N. it is connected to a charge amplifier
with a feedback capacitor of 22nF.
f
oC
qv
Example contd
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Example cont d
Calculate the output of the at the instant astep input of 5kN is applied.
Solution:
A force of 5kN produces a charge ofq = 28 x 5000 coulomb = 140nC
= 6.36V6
9
1022
10140
x
x
C
qv
f
o
Example 2
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Example 2
A system of piezo and amplifier has a timeconstant of 90 sec. how long will it take to lose
the first 5 sec of the step input?
Solution:
The output voltage is an exponential delay of
the form:
And
5% of step is 0.95V.
/t
o Vev
sec90
Contd
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Cont d
90/95.0 tVeV
= 4.6 sec.
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Contd
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Cont d
For static or low frequency applications,
For frequencies of the order of 100kHz
Piezoresistive transducer:
This is change in crystal resistance with
pressure.
They are fabricated from semiconductor
materials e.g. silicon with boron as trace
impurity for p-type.
cZ
kZc 10
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Displacement transducers using
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resistive elements
The simplest type of displacement transduceris the a potentiometer
It consists of a resistive element with a moving
contactMotion of the contact can be translational,
rotational or a combination of the two (i.e.
helical).This is a passive device of the 0order type.
Displacement transducers (contd)
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Displacement transducers (cont d)
The resistance element is driven by a dc or anac voltage and output voltage is a linear
function of input displacement
Materials usedi. wire wound on an insulating cylindrical core,
ii. carbon film or
iii. conduct plastics.
iv. Widely used are wirewound elements
Contd
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Cont d
Major limitation is resolution (typically 40micro-meter for translational devices and 0.1for rotational pot of 5cm diameter).
Resolution is improved by use of carbon filmor conductive plastic.
Another limitation is linearity which dependson the uniformity of resistance along the
resistive element.Other major problems :
Contd
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Cont d
Spurious output voltages associated with:
i. slider contact bounce
ii. Dirt
iii. Contact wear
iv. Friction and
v. Inertia of the moving parts
Displacement transducers using
l
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capacitive elements
By definition, capacitance of a parallel platecapacitor is given by:
Any variation in any of the parameters in theexpression causes a variation in C.
Examples:
a. Variation in plate separation -
Faradsd
AC r
0
Contd.
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Cont d.
fixed plate movable plate
displacement
Use of a capacitive transducer to
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measure pressure
This uses a diaphragm and a static ( fixed)plate.
diaphragm
pressureStatic plate
Variation in area
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Variation in area
This is made of a fixed semi-circular plate arotatable semi-circular plate.
As the plate rotates, the area between the
two changes thus changing the capacitance.Illustration:
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Contd
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Cont d
It can be shown that:
ss v
d
x
CC
C
CC
Cvv
][
21
1
21
2
Contd
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Cont d
In this arrangement, P and Q are fixed andplate M moves between them.
assume M moves a displacement x towards P
we have:.
xd
andC
xd
C rr
0
20
1
Inductive transducers
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By definition, self inductance of a coil is givenby:
Where N is the number of turns and S is thereluctance of the magnetic circuit and is given
by:
Where are
S
NL
2
A
lS
r0
r &0
Contd
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The permeability of free space and therelative permeability of the material inside the
coil.
Any variation of the above parameters (usuallydue displacement ) alters the inductance.
Examples of inductance type transducers:
a. The simplest case is the change in thenumber of turns.
Contd
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A sliding contact is used to alter the numberof turns.
b. Change in permeability (or reluctance) where
a soft iron plunger is used to alterpermeability.
Linear Variable DifferentialTransformer
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This consists of a primary coil and twosecondary coils wound in opposition.
A ferromagnetic core (plunger) moves along
the axis of the three coils.since the two secondary coils are wound in
opposition,
When the core is at its central position,
In practice this condition is not met due tomismatches in the secondary coils.
21 vvvo
0ov
Contd
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When the core is moved away from thecentre, the output voltage rises.
ov
x-x0
0180
Contd
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This is a passive 0-order device whose outputdepends on both magnitude and direction of
displacement
Sensitivity ranges from 0.1V/cm to 50 mV/mThe complete displacement measurement
system based on the LVDT:
LVDT
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oscillator LVDT
Phase
Sensitivedetector
Low-pass
filter ov
Contd
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The phase sensitive detector produces anunsmoothed full-wave rectified signal which is
either +ve or -ve
This smoothed by the low-pass filter to giveThe oscillator supplies the primary coil a
reference to the phase sensitive detector.
ov
Photo-electric transducers
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These can be divided into three types:i. Photo-conductive
ii. Photo-emissive and
iii. Photo-voltaic
Photo-conductive transducer (cell):
These are elements or compounds whose
conductivity increases with intensity of
electromagnetic radiation.
Contd
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The radiation is usually in the visible or nearvisible part of the spectrum.
Examples of materials are:
Cadmium sulfide Cadmium selenide
Lead sulfide
Germanium and silicon
Contd
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The material in deposited on a ceramicsubstrate in a zig-zag fashion.
Applications:
Automatic daylight switches
Automatic street lights control
Product counting on a production line.
Sorting of objects by size etc.
Contd
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Advantages include: Inexpensive
Rugged
Will withstand shock and vibration
Operates at low voltage with a high enough
output to drive a relay.
Disadvantagescan be destroyed by strong
light and heat. Response is rather slow.
Examples of photo-conductive
transducer circuits
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transducer circuits
Simple circuits:
Photo cell
relay
Photo-emissive transducers
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Photo-emissive effect is the emission ofelectrons in a vacuum from metal orsemiconductor surfaces
This is as a result of absorption ofelectromagnetic energy by these materials.
The e.m. energy is in the visible or near visiblepart of the spectrum.
Electrons absorb enough energy from incidentphotons to escape from these surfaces.
Illustration
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e
Incident photon
Evacuated glass tube
Semi-transparent
photocathode
e
anode
+-
Contd
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Typical circuit used to measure photo-electriccurrent:
V
cellLR
pi
Lpo Riv
Contd
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Photo-emissive materials are usuallycompounds of alkali metals.
Used in conjunction with a photomultiplier,
this transducer finds applications in atomicabsorption spectro-photometry.
for chemical identification of elements.
Photo-multiplier
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Lightwindow
Photo
cathode
photon
Dynode 1 Dynode 3
Dynode 2
final
+700V
+100V +300V
+200V +600V
Contd
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The released electrons are effectivelymultiplied by the process of secondary
emission at various stages of dynodes.
The gain of the tube is given by:Where: ksare constants for a given tube.
n is number of stages
vs is voltage between stages
nk
svkG 21
Semiconductor photo-diode
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Depending on the external circuitry, thephoto-diode can be made to operate as aphoto-conductor or a photo-voltaic device (i.e.generates electrical energy.
These are fabricated from silicon and theyincorporate a p-n junction.
When reverse biased it operates as a
photoconductor and with no bias it is selfgenerating.
Contd
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-V
fR
ov
Photovoltaic
configuration
Photoconductive
configuration
Signal conditioning
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Transducers produce energy which is verysmall and often masked by noise signals.
The purpose of signal conditioning is to bring
the transducer output to a level and form thatis suitable for:
Signal conversion
Signal processing Indicating or recording
Contd
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In passive transducers excitation andamplification are necessary
In active transducers, amplification is
required. Filtering may also be required in both cases.
Excitation may be ac or dc
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Ac system
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Ac carrier system:
transdu
cer
Ac
bridge
Calibration
& zeroing
Ac
amp
PhaseSensitive
detector
Carrier
oscillator
Power
supply
LowPass
filter
Signal conditioning
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Note in each configuration, the presence of abridge after the transducer.
A bridge is used with passive transducers to
produce a change in voltage that isproportional to a parameter change (e.g. R,
C, or L)
The null indication principle is utilized.Bridges achieve a very high degree of accuracy
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The Wheatstone Bridge
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1R2R
3R 4R
1I 2I
A
B
C
D
Bridge analysis
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In the bridge circuit, constitute theratio arms
is the standard arm and is the unknown
To measure the unknown resistor, theresistance in one or both ratio arms is
adjusted until balance is achieved.
At balance,
It can then be shown that
21&RR
3R 4R
2143 & RRRR VVVV
1
324
R
RRR
Example
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Application of the Wheatstone bridge in strainmeasurement:
Assume ( strain gauge
resistance with no strain) = RAssume also that with no strain the bridge is
balanced.
Problem - assume gauge suffers a strain whichresults in a resistance change, R
4321 RRRR
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Contd
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Assume that 2R
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i.e. strain is directly proportional to thebridges out of balance voltage and the meter
can be calibrated to read strain.
This configuration is referred to as a quarterbridge
Temperature compensation:
Change in strain gauge resistance alsodepends on changes in temperature
Temperature compensation
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It becomes difficult to say whether R hasresulted exclusively from L or a contribution
from T.
In practice a second identical gauge is placedin an adjacent arm of the bridge.
Conditions:
Must be kept unstrained Must be identical to the active gauge
Contd
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Must be kept under the same temperatureconditions as the active gauge.
can be the dummy gauge
Let R be resistance change due to T then:
Therefore the out of balance voltage due to
T is zero.
32orRR
222
)(&
2
V
RR
VRRV
VV CA
Contd
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This is a temperaturecompensated quarterbridge.
The Half bridge:
To increase sensitivity, a half bridge is used.This consists of two identical active gauges in
adjacent arms.
These are mounted on opposite sides of thebeam.
The half bridge
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1R active
3R active
1I 2I
A
B
C
D
Contd
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Analysis of the half bridge:
Therefore the half bridge has twice the
sensitivity of the quarter bridge.
RRRR
VRRV
VV CA
)(
&2
R
RV
R
RV
R
VRVVAC2222
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Contd
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It can be shown that:Which is four times the sensitivity of the
quarter bridge.
Practical balancing arrangement
R
RVVAC
Summarydc bridges
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The bridge can be used in a method called thenull method
Here, one or two resistors are manually
adjusted to achieve balance. An unknown resistor can be calculated.
It can also be used in the deflection mode
where change in any resistor value by Rresults in a proportional voltage ACV
Limitations of the Wheatstone bridge
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It can measure resistances from a few ohms toseveral mega ohms.
The upper limit is set by reduction insensitivity
The lower limit is set by resistance ofconnecting leads and contact resistance.
To overcome the lower limitation, a modified
Wheatstone bridge is used. This is the Kelvinbridge
Ac Bridges
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These are similar to dc brides butconsist of:
four impedance arms,
an ac source and
an ac detector
The impedances can be combinations of
resistors, capacitors and inductors.
Contd
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They are not restricted to measuringimpedances only.
They are also very useful in:
Shifting phases Providing feedback networks for oscillators
and amplifiers
Filtering out undesirable signals
And measuring frequencies of audio signals
Analysis
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In the diagram for the dc bridge, replace allresistors by impedance equivalents so that atbalance;
This is a general bridge equation
Is applicable to all bridges at balanceregardless of the actual components in thearms.
In general the impedances are complex anddepend on the frequency of the ac signal.
3241 ZZZZ
Contd
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For balance, two conditions must be satisfiedsimultaneously:
a. Products of opposite arms of the bridge must
be equal.b. The sums of angles of opposite arms must be
equal.
c. i.e. 32413241 & ZZZZ
11 Z
Some practical ac bridges
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1. Maxwell bridge: For this bridge,
Arm AB consists of
Arm BC consists of
Arm AD consists of
Arm CD consists of
Question: find -
11parlCR
2R
3R
xxseriesLR
xx LR &
solution
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The impedances of the arms are written as:
xxx LjRZ
RZ
RZ
CRj
RZ
33
22
11
11
1
Contd
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At balance:
Equating real and imaginary parts on the left
hand and right sides:
321 ZZZZ x
32
11
1 )(1
RRLRRCj
Rxx
321
1
32 RRCjRRRLjR xx
132
1
32 & CRRLR
RRR xx
Contd
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Note that solutions to the Maxwell bridge arefrequency independent.
In general, ac bridge solutions are frequency
dependent.2. Similar angle bridge:
This bridge consists of
Arm AB =
Arm BC =
1R
2R
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Bridges with frequency dependent
solutions
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3. Opposite angle bridge This is used in the measurement of inductors
with high Q (Q >10)
Maxwell bridge is suitable for coils with Q