Manual instrumentation and measurement

INSTRUMENTATION & MEASUREMENT Khem Gyanwali1
TRIBHUVAN UNIVERSITY
INSTITUTE OF ENGINEERING
COURSE MANUAL
ON
INSTRUMENTATION AND MEASUREMENT
(ME 553)
PREPARED BY
KHEM GYANWALI
DEPARTMENT OF MECHANICAL ENGINEERING
THAPATHALI CAMPUS, THAPATHALI
JULY, 2013

CONTENTS
CHAPTER 1: FUNDAMENTALS OF MESUREMENT ................................................................ 3
CHAPTER 2 : TIME DEPENDENT PROPERTIES OF ANALOG SIGNALS.............................. 8
CHAPTER 3 : STATIC CHARACTERISTICS OF MEASUREMENT SYSTEM....................... 13
CHAPTER 4: DYNAMIC RESPOSE OF MEASUREMET SYSTEM......................................... 21
CHAPTER 5 : SENSORS ............................................................................................................... 51
CHAPTER 6 : STRAIN GAGES.................................................................................................... 63
CHAPTER 7 : COMMON MECHANICAL MEASUREMENT SYSTEM AND TRANSDUCERS
.......................................................................................................................................................... 72
TUTORIAL NO: 1.................................................................................................................... 105
TUTORIAL NO: 2.................................................................................................................... 108
SAMPLE PROBLEMS FOR TUTORIAL NO: 2 ................................................................. 110
TUTORIAL NO: 3.................................................................................................................... 111
SAMPLE PROBLEMS FOR TUTORIAL NO: 3 ................................................................. 112
TUTORIAL NO: 4.................................................................................................................... 113

CHAPTER 1
FUNDAMENTALS OF MESUREMENT
1.1 Fundamental Methods of Measurement
Measurement of a quantity is the act or the result of a quantitative comparison between a
predefined standard and an unknown magnitude.
Process of Comparision
(Measurement)
Measurand
(Unknown quantity)
Standard
(Known quantity)
Result
(Numeric value)
Figure 1.1: Fundamental Measuring Process
If the result is to be meaningful, the act of measurement must satisfy the following requirements:
 the standard which is used for comparison must be accurately defined and internationally
accepted,
 the standard must be of the same character as the measurand, and
 the apparatus and the experimental procedure employed for obtaining the value must be
provable.
There are two basic methods of measurement: (i) direct comparison with the primary or secondary
standards and (ii) indirect comparison with standards through the use of a calibrated system.
Direct Methods
In these methods, the unknown quantity (also called the measurand) is directly compared against a
standard. The result is expressed as a numerical number and a unit.
Direct methods are quite common for the measurement of physical quantities like length, mass and
time.
These methods in most cases are inaccurate because they involve human factors. They are also less
sensitive. Hence the direct methods are not preferred and are rarely used.
Indirect Methods
Indirect comparison makes use of some form of transducing device coupled to a chain of
connecting apparatus, called generalized measurement system.

These chains of devices convert the basic form of input into an analogous form, which it then
processes, and presents at the output as a known function of the input.
Such a conversion is often necessary in order to make the desired information intelligible.
1.2 The Generalized Measurement System
The act of measurement is accomplished with a measuring instrument, an assemblage of physical
facilities known as measurement system.
Most measuring systems fall within the framework of a general arrangement consisting of three
phases or stages:
Stage I A detector-transducing or sensor stage;
Stage II An intermediate stage or signal-conditioning stage;
Stage III A terminating or read-out stage.
Figure 1.2: Generalized Measurement System
1.2.1 Sensor-Transducer Stage
The prime function of the first stage is to detect or to sense the measurand.
At the same time, ideally, it should be insensitive to every other possible input.
Unfortunately, it is rare indeed to find a detecting device that is completely selective.
1.2.2 Signal-Conditioning Stage
The purpose of the second stage of the general system is to modify the transuded information so
that it is acceptable to the third, or terminating, stage. It may perform one or more basic operations,
such as selective filtering, integration, differentiating, or telemetering, as may be required.

1.2.3 Read-Out Stage
The third stage provides the information sought in a form of comprehensible to one of the human
senses or to a controller. If the output is intended for immediate human recognition, it is, with rare
exception, presented in one of the following forms:
 As a relative displacement, such as movement of an indicating needle, displacement of
oscilloscope trace or oscillograph light beam, etc., or,
 In digital form, as presented by a counter such as an automobile odometer or one of the
modern digital voltmeter.
1.3 Calibration Concepts
Every measurement system must be provable, that is, it must prove its ability to measure reliably.
The procedure for this is called calibration.
It consists of determining the system's scale. At some point during the preparation of the system for
measurement, known magnitudes of the basic input quantity must be fed into the detector-
transducer, and the system's behavior must be observed.
If the system has been proved linear, perhaps single-point calibration will suffice, wherein the
effect of only a single value of the input is used. If the system is not linear, or if it has not been so
proved, a number of values must be used and their results observed.
1.4 Measurement Errors
1.4.1 Errors in measurement system
An error is defined as "The difference between the measured value and the actual value."
If two persons use the same instrument for measurement for finding the same measurement, it is
not essential that they may get the same results. There may arise a difference between their
measurements. This difference is referred to as an "ERROR".
Input
(True value of quantity) Output
(Measured Value of quantity)
Error = True value – Measured value
% Error = (True value – Measured value) * 100% / True value
1.4.2 Types of Error
1.4.2.1 Gross Errors
All errors committed by the users such as calculation error, observation error, reading error,
connection error, placement error etc.
Measurement System

1.4.2.2 Systematic Errors
A systematic error can be trickier to track down and is often unknown. This error is often called a
bias in the measurement. In chemistry a teacher tells the student to read the volume of liquid in a
graduated cylinder by looking at the meniscus. A student may make an error by reading the volume
by looking at the liquid level near the edge of the glass. Thus this student will always be off by a
certain amount for every reading he makes. This is a systematic error. Instruments often have both
systematic and random errors.
Figure 1.3: Distribution with systematic error
I) Instrumental Errors:
Instrumental errors are all those errors due to shortcomings of instrument such as, the error due to
defective parts. e.g. error due to less accuracy in the scale calibration, error due to the improper
tension in the spring. The corrective measures are
 Use proper instrument for proper application
 If less accuracy in scale calibration, then recalibrate the scale by comparing with standard
measurements.
 Use correction factor
II) Environmental Errors:
All the errors due to effect of surrounding such as error due to change in temperature, pressure,
humidity and also the error to the external electrostatic and magnetic field. The corrective measures
are
 To minimize the effect of change in temperature, perform the measurement in standard
condition.
 To avoid the effect of moisture, use proper casing
 To avoid the effect of fields, use proper shielding.

III) Observational Errors:
There are many sources of observational errors. As an example, the pointer of a voltmeter rests
slightly above the surface of the scale. The corrective measure for this type of error is providing
highly accurate meters are provided with mirrored scales reducing parallax errors.
1.4.2.3 Random Errors
Random errors are ones that are easier to deal with because they cause the measurements to
fluctuate around the true value. If we are trying to measure some parameter X, greater random
errors cause a greater dispersion of values, but the mean of X still represents the true value for that
instrument.
Figure 1.4: Distribution with random error

CHAPTER 2
TIME DEPENDENT PROPERTIES OF ANALOG SIGNALS
2.1 Types of Measurement Signals
A parameter common to all of measurement is time. The time dependent measurement signals are
classified as:
I. Static
II. Dynamic
A. Steady-state periodic
B. Non-repetitive or transient
1. Single pulse or aperiodic
2. Continuing or random
2.2 Harmonic Signals
A function is said to be simple harmonic in terms of a variable when its second derivative is
proportional to the function but have opposite sign.
Any signal that follows the definition of harmonic function is called harmonic signal.
In its most elementary form, simple harmonic motion is defined by the relation
s = s0 sin t ………. (2.1)
v =
dt
ds
= s0 cos t ………. (2.2)
v0 = s0 ………. (2.3)
Also,
a =
dt
dv
= – s0 sin2
t = – s2
………. (2.4)
In addition,
a0 = – s02
………. (2.5)
Equation (2.4) satisfies the definition of simple harmonic motion. The acceleration a is
proportional to the displacement s, but is of opposite sign. The proportionality factor is 2
.
The mechanical analogy for harmonic motion is Scotch-yoke mechanics. Figure 2.1 shows the
elements of the Scotch-yoke.

Figure 2.1: The Scotch-Yoke Mechanism
2.3 Periodic Signals, Fourier Series Representation
A periodic signal is one that repeats itself cyclically over and over. If this function meets the
Dirichlet conditions, i.e., if it must have a single value, be finite, and have a finite number of
discontinuities and maxima and minima in one cycle, it may be represented by a Fourier series.
That is
f(x) =
2
0A + 



1
)sincos(
n
nn nxBnxA ………. (2.7)
where An and Bn are called Fourier coefficients.
The constants terms are given as
A0 = 



dxxf )(
1
An = 


dxnxxf cos)(
1
………. (2.8)
Bn = 


dxnxxf sin)(
1
where n = 1, 2, 3, ……

Most complex dynamic-mechanical signals, steady state or transient, whether they are time
functions of pressure, displacement, strain, or something else, may be expressed as a combination
of simple harmonic components. Each component will have its own amplitude and frequency and
will be combined in various phase relations with the other components. A general mathematical
statement of this may be rewritten by replacing x by t as:
f(t) = A0 + 



1
)sincos(
n
nn tnBtnA  ………. (2.9)
where A, An and Bn = amplitude-determining constants called harmonic coefficients, and
n = integers from 0 to , called harmonic orders.
When n is unity, the corresponding sine and cosine terms are said to be fundamental. For n = 2, 3,
4, etc., the corresponding terms are referred to as second, third, fourth harmonics, and so on.
Equation (2.9) may be written in the two equivalent forms:
f(t) = A0 +  



1
cos
n
nn tnC  ………. (2.10a)
or,
f(t) = A0 +  



1
'sin
n
nn tnC  ………. (2.10b)
where the harmonic coefficients Cn are determine by the relation
Cn =
22
nn BA  ………. (2.11)
and the phase relations n and n' are determined as follows:







n
n
n
A
B
tan And 






n
n
n
B
A
'tan ………. (2.12)
The phase angles n and n' provide necessary time wise relationships among the various harmonic
components.
The Fourier series is an infinite series, and in order to get a perfect reconstruction of f(t) we would
have to add an infinite number of terms.
Since most engineering applications do not require a perfect reconstruction of f(t), generally, f(t) is
approximated by a truncated Fourier series. Very often less than ten harmonics are adequate for
most engineering applications.
After the Fourier series of a particular f(t) is found, the steady state response of any measuring
instrument can be determined by frequency response technique or principle of superposition i.e.
response to each harmonic component or sinusoidal component is found and then algebraically
added to get the total response.

2.4 Determining Fourier Coefficients
The method of obtaining the desired terms in the Fourier series depends on the nature of f(t). If f(t)
is given as a known mathematical formula, Equations (2.9) to (2.12) may be employed. If the
required integration cannot be performed analytically because of the complexity of f(t) or because
is given by a graph or table rather that a formula, then various approximate numerical methods are
available.
2.4.1 Analytical Method
The analytical method may be outlined as follows:
1. Establish the fundamental cycle and assign the values 0 to 2 to its limits. The general form
of the desired equation is then,
f(t) =
2
A
+ (A1cos + A2cos + A3cos + ….) +
(B1sin + B2sin + B3sin + ….)
2. Divide the fundamental cycle into m equal intervals, each of width, and determine the
corresponding ordinates. Do not include the ordinates for both ends of the interval being
analyzed, because this would be duplication. Select the number of intervals such that m  n,
where m is the number of intervals required per cycle and n is the order of coefficient
desired.
3. To determine a given coefficient, multiply each of the m ordinates determined in (2) by the
corresponding numerical values of the desired trigonometric function. The average value of
the resulting column of the products will be one half the coefficients being sought. For
example: To determine A2, multiply each of the values of f () as determined in step (2) by
the corresponding values of cos2. Add all the products together and divide by m/2. This
will give the numerical value of A2. Repeat this process for each of the values of A and B
that are required.
4. Determine A, which is twice the average of the values of f().
2.4.2 FFT Method
For relatively simple wave shapes, the number required numerical manipulations may easily make
the procedure prohibited from a time-benefit standpoint. This limitation becomes especially
important when the variety of non-repetitive conditions needing analysis taxes the capacity of even
the larger computers.
This has led to a truncated procedure referred to as the fast Fourier Transform, or FFT.
If N represents the number harmonic coefficients to be determined, ordinary harmonic analysis
requires roughly N2
separate computations, whereas FFT requires approximately (N) log2N

Figure 2.2 is a schematic of a fast Fourier transform analyzing system.
Figure 2.2: Block Diagram a FFT Analyzing System

CHAPTER 3
STATIC CHARACTERISTICS OF MEASUREMENT SYSTEM
3.1 Introduction
The system characteristics are to be known, to choose an instrument that most suited to a particular
measurement application. The performance characteristics may be broadly divided into two groups,
namely “static” and “dynamic” characteristics.
 Static characteristics: The performance criteria for the measurement of quantities that
remain constant, or vary only quite slowly. Normally, static characteristics of a
measurement system are, in general, those that must be considered when the system or
instrument is used to measure a condition not varying with time.
 Dynamic characteristics: The relationship between the system input and output when the
measured quantity (measurand) is varying rapidly. This is normally done with help of
differential equations.
Instruments systems are usually built up from a serial linkage of distinguishable building blocks.
The actual physical assembly may not appear to be so but it can be broken down into a
representative diagram of connected blocks. The sensor is activated by an input physical parameter
and provides an output signal to the next block that processes the signal into a more appropriate
state. A fundamental characterization of a block is to develop a relationship between the input and
output of the block. All the signals have a time characteristic. It is essential to consider the behavior
of a block in terms of both the static and dynamic states. The behavior of the static regime alone
and the combined static and dynamic regime can be found through use of an appropriate
mathematical model of each block.
The output/input ratio of a block that includes both the static and dynamic characteristics is called
the transfer function and is given the symbol G.
Gtotal = G1 * G2 *G3
Gtotal = ∑Gi
The equation for Gi can be written as two parts multiplied together.
Gi = [Static * Dynamic]
One expresses the static behavior of the block, that is, the value it has after all transient (time
varying) effects have settled to their final state. The other part tells us how that value responds
when the block is in its dynamic state. The static part is known as the transfer characteristics and is
often all that is needed to be known for block description. The static and dynamic response of the
cascade of blocks is simply the multiplication of all individual blocks.
3.2 Static characteristics of measurement system
3.2.1 Accuracy and Precision
Accuracy
It is the closeness with which an instrument reading approaches the value of the quantity being
measured. Accuracy is the ability of an instrument to show the exact reading and is always related
to the extent of the wrong reading/non accuracy.

The accuracy may be specified in terms of inaccuracy or limits of errors and can be expressed in
the following ways:
(i) Accuracy as "Percentage of Full Scale"
Accuracy (%) =
%100

scalefull
actualresult
V
VV ………. (3.1)
(ii) Accuracy as "Percentage of Actual Value"
Accuracy (%) =
%100

actual
actualresult
V
VV ………. (3.2)
Example 1:
A pressure gauge with a range between 0-1 bars with an accuracy of ± 5% fs (full-scale) has a
maximum error of:
5/100 * 1 bar = ± 0.05 bar
Notes: It is essential to choose equipment which has a suitable operating range.
Example 2:
A pressure gauge with a range between 0 - 10 bars is found to have an error of ± 0.15 bar when
calibrated by the manufacturer.
Calculate:
a. The error percentage of the gauge.
b. The error percentage when the reading obtained is 2.0 bars.
Answer:
a. Error Percentage = ± 0.15 bar/10.0 bar * 100 = ± 1.5%
b. Error Percentage = ± 0.15 bar/2.0 bar * 100 = ± 7.5 %
The gauge is not suitable for use for low range reading.
Alternative: use gauge with a suitable range.
Example 3:
Two pressure gauges (pressure gauge A and B) have a full scale accuracy of ± 5%. Sensor
A has a range of 0-1 bar and Sensor B 0-10 bar. Which gauge is more suitable to be used if the
reading is 0.9 bars?
Answer:
Sensor A:
Equipment max error = ± 5/100 * 1 bar = ± 0.05 bar
Equipment accuracy @ 0.9 bar (in %) = ± 0.05 bar/0.9 bar * 100 = ± 5.6%
Sensor B:
Equipment max error = ± 5/100 * 10 bar = ± 0.5 bar
Equipment accuracy @ 0.9 bar (in %) = ± 0.5 bar/0.9 bar * 100 = ± 55%
Conclusion:
Sensor A is more suitable to use at a reading of 0.9 bar because the error percentage (±
5.6%) is smaller compared to the percentage error of Sensor B (± 55%).

Precision
It is a measure of the reproducibility of the measurements, i.e. given a fixed value of a quantity;
precision is a measure of agreement within a group of measurements. The term precise means
clearly or sharply defined. Equipment which is precise is not necessarily accurate.
It is defined as the capability of an instrument to show the same reading when used each time
(reproducibility of the instrument).
A precise measurement may not necessarily be accurate and vice versa.
(a) High precision with
poor accuracy with poor precision
Target Plate
Bull's eye
(c) High accuracy with
high precision poor precision
Figure 3.1: Illustration of Accuracy and Precision
Accuracy vs. Precision
This is a systematic error
Figure 3.2: High Precision, but low accuracy

Figure 3.3: Illustration of accuracy and precision
High accuracy means that the mean is close to the true value, while high precision means that the
standard deviation σ is small.
3.2.2 Tolerance
Tolerance is closely related to accuracy of equipment where the accuracy of equipment is
sometimes referred to in the form of tolerance limit. It is defined as the maximum error expected in
an instrument. This explains the maximum deviation of an output component at a certain value. For
example, one resistor chosen at random from a batch having a nominal value 1000 W and tolerance
5% might have an actual value anywhere between 950W and 1050W.
3.2.3 Range or span
It is defined as the range of reading between minimum value and maximum value for the
measurement of an instrument. It has a positive value e.g..: The range or span of an instrument
which has a reading range of –100°C to 100 °C is 200 °C.
3.2.4 Linearity
It is defined as maximum deviation from linear relation between input and output. The output of an
instrument has to be linearly proportionate to the measured quantity.
Linearity is normally shown in the form of full scale percentage (% fs).
The graph shows the output reading of an instrument when a few input readings are entered.
Linearity = maximum deviation from the reading of x and the straight line.
Figure 3.4: Linearity of Measurement System

3.2.5 Sensitivity
It is defined as the ratio of change in output towards the change in input at a steady state condition.
Sensitivity (K) = Δθο/Δθi
Δθο : change in output; Δθi : change in input
Figure 3.5: Sensitivity of Measurement Sytem
Example 1:
The resistance value of a Platinum Resistance Thermometer changes when the temperature
increases. Therefore, the unit of sensitivity for this equipment is Ohm/°C.
Example 2:
Pressure sensor A with a value of 2 bars caused a deviation of 10 degrees. Therefore, the
sensitivity of the equipment is 5 degrees/bar.
Sensitivity of the whole system is (k) = k1 * k2 * k3 *... * kn
Qi Qo
Example 3:
Consider a measuring system consisting of a transducer, amplifier and a recorder, with sensitivity
for each equipment given below:
Transducer sensitivity 0.2 mV/°C
Amplifier gain 2.0 V/mV
Recorder sensitivity 5.0 mV/V
Therefore,
Sensitivity of the whole system:
(k) = k1 x k2 x k3
k = 0.2 mV/°C x 2.0 V/mV x 5.0 mV/V
k = 2.0 mV/°C
Example 4:
The output of a platinum resistance thermometer (RTD) is as follows:Calculate the sensitivity of
the equipment.
k
1
K
2
K
3

Input(°C) Output(Ohm)
0 0
100 200
200 400
300 600
400 800
Answer:
Draw an input versus output graph. From that graph, the sensitivity is the slope of the
graph.
K = Δθ0/ Δθi graph slope = (400-200) ohm/ (200-100) °C
= 2 ohm/°C
3.2.6 Threshold
When the reading of an input is increased from zero, the input reading will reach a certain value
before change occurs in the output. The minimum limit of the input reading is ‘threshold’. As an
illustration, a car speedometer typically has threshold of about 15kh/hr. This means that, if the
vehicle starts from the rest and accelerates, no output reading is observed on the speedometer until
the speed reaches 15km/hr.
3.2.7 Resolution
The smallest change in input reading that can be traced accurately. This is given in the form ‘% of
full scale (% fs)’. It is available in digital instrumentation. Using a car speedometer as an example
again, this has subdivisions of typically 20km/hr. This means that when the needle is between the
scale markings, we cannot estimate speed more accurately than to the nearest 5km/hr. This figure
of 5 km/hr thus represents the resolution of the instrument.
3.2.8 Dead space/ Dead band
It is defined as the range of input reading when there is no change in output (unresponsive system).
Blacklash in gears is a typical cause of dead space.
Figure 3.6: Dead Space
Output
Reading
Measured
Variables
Dead space
+
-

3.2.8 Hysteresis effects
Figure below illustrates the output characteristics of an instrument that exhibits hysteresis. If the
input measured quantity to the instrument is steadily increased from a negative value, the output
reading varies in the manner shown in curve (a). If the variable is then steadily decreased, the
output varies in the manner shown in curve (b). The non-coincidence between these loading and
unloading curves is known as hysteresis. Two quantities are defined, maximum input hysteresis
and maximum output hysteresis, as shown in figure. These are normally expressed as a percentage
of the full scale input or output reading respectively. For example a thermometer exposed to an
increasing temperature input (i.e. going from 0 to 100 C) may show a slightly different profile to
that for the decreasing input (i.e. decreasing from 100 to 0 C).
Hysteresis is most commonly found in instruments that contain springs, such as the passive
pressure gauge and the Prony brake (used for measuring torque). It is also evident when friction
forces in a system have different magnitudes depending on the direction of movement, such as in
the pendulum-scale mass measuring device. Devices like the mechanical fly ball (a device for
measuring rotational velocity) suffer hysteresis from both of the above sources because they have
friction in moving parts windings formed round an iron core, due to magnetic hysteresis in the iron.
This occurs in devices like the variable inductance displacement transducer, the LVDT and the
rotary differential transformer.
Figure 3.7: Instrument characteristics with hysteresis.
3.2.9 Sensitivity to disturbance
All calibrations and specifications of an instrument are only valid under controlled conditions of
temperature, pressure etc. These standard ambient conditions are usually defined in the instrument
specification. As variations occur in the ambient temperature etc. certain static instrument
characteristics change, and the sensitivity to disturbance is a measure of the magnitude of this
change. Such environmental changes affect instruments in two main ways, known as zero drift and
sensitivity drift. Zero drift is sometimes known by the alternative term, bias.
Zero drift or bias describes the effect where the zero reading of an instrument is modified by a
change in ambient conditions. This causes a constant error that exists over the full range of

measurement of the instrument. Zero drift are commonly found in instruments like voltmeters that
are affected by ambient temperature changes.
Sensitivity drift (also known as scale factor drift) defines the amount by which an instrument’s
sensitivity of measurement varies as ambient conditions change. It is quantified by sensitivity drift
coefficients that define how much drift there is for a unit change in each environmental parameter
that the instrument characteristics are sensitive to. Many components within an instrument are
affected by environmental fluctuations, such as temperature changes: for instance, the modulus of
elasticity of a spring is temperature dependent.
Figure 3.8: Effect of disturbance a) zero drift b) sensitivity drift c) zero drift plus sensitivity
drift

CHAPTER 4
DYNAMIC RESPOSE OF MEASUREMET SYSTEM
4.1 Introduction
It may be defined as an evaluation of the system's ability to faithfully sense, transmit, and present
all the pertinent information included in the measurand and to exclude all else.
Behavior of a system is explained with the help of the following response characteristics of the
particular system:
 Amplitude response,  Frequency response,
 Phase response, and  Rise time or delay.
4.1.1 Amplitude Response
Amplitude response is governed by the system's ability to treat all input amplitudes uniformly.
100
200
300
10-4
10-3
10-2
10-1
Input, ei
Gain,e/eoi
Input, e
-3
0.30
10
o
0.03
-4
10
3.00
10
i
-2 -1
10
Figure 4.1: Amplitude Response
4.1.2 Frequency Response
Good frequency response is obtained when a system treats all the frequency components with equal
faithfulness. Changing the frequency of the input signal should not alter the system's output
magnitude so long as the input amplitude remains unchanged.
Gain
3
10
100
300
200
5
104
10
Frequency, Hz
6
10
Figure 4.2: Frequency Response Curve

4.1.3 Phase Response
Amplitude and frequency response are important for all types of input waveforms, simple or
complex. Phase response, however, is of importance primarily for the complex wave only.
For the single sine-wave input, any shift would normally be unimportant. The output produced on
the oscilloscope screen could show the true waveform, and the proper parameters could be
determined. The fact that the shape being shown was actually formed a few microseconds or a few
milliseconds after being generated is of no consequence.
i
eo
t
t
1800
Figure 4.3: Phase relationship between input and output of an amplifier
103
Frequency, Hz
104
105 6
10
0
20
40
60
80
PhaseLag,degrees
Figure 4.4: Phase lag versus Frequency for an Amplifier
Let us consider, however, the complex wave made up of numerous harmonics. Suppose that each
component is delayed by a different amount. The harmonic components would then emerge from
the system in phase relations different from when they entered. The whole waveform and its
amplitudes would be changed, as result of poor phase response.

4.1.4 Delay or Rise Time
The time delay after the step is applied, but before proper output magnitude is reached is known as
rise time. It is a measure of the system's ability to handle transients. Sometimes rise time is defined
specifically as the time, t, required for the system to pass from 10 % to 90 % of its final response.
Alternatively, transients response may be characterized by the settling time required for the system
response to remain within some small percentage of its final value.
t
oe
i
t
90%
10%
Settling time
Rise time
t
Figure 4.5: Rise Time and settling Time
4.2 Mathematical Models
As in so many other areas of engineering application, the most widely useful mathematical model
for the study of measurement-system dynamic response is the ordinary linear differential equation
with constant coefficients.
A general relation between any particular input and the corresponding output, under suitable
simplifying assumptions, can be expressed by a differential equation of the form:
00
0
11
0
1
1
0
........... qa
dt
dq
a
dt
qd
a
dt
qd
a n
n
nn
n
n  

 = m
i
m
m
dt
qd
b
+ i
i
m
i
m
m qb
dt
dq
b
dt
qd
b 011
1
1 .......... 

 ………. (4.1)
where, q0 = output quantity,
qi = input quantity,
t = time, and
a's, b's = combinations of system physical parameters, assumed constant.
If we define the differential operator, D =
dt
d
, then Equation (4.1) can be written as,
(anDn
+ an-1Dn-1
+ ………. + a1D + a0)q0
= (bmDm
+ bm-1Dm-1
+ ………. + b1D + b0)qi ………. (4.2)

The complete solution of q0 is obtained in two separate parts as
q0 = q0 cf + q0 pi ………. (4.3)
where q0 cf is a complementary function part of the solution, which has n arbitrary constants; and q0
pi is the particular integral part of the solution which has not constant terms. The complementary
function part is the solution when no input is applied. This gives the natural behavior of the system.
The particular integral part is the solution due to imposed input.
4.2.1 Zero Order System
The simplest possible special case of Equation (4.1) occurs when all the a's and b's other than a0
and b0 are assumed to be zero. The differential equation then degenerates into the simple algebraic
equation
a0q0 = b0qi ………. (4.4)
Any instrument or system that closely obeys Equation (4.4) is defined to be a zero-order
instrument. Actually, two constants a0 and b0 are not necessary, and so we define the static
sensitivity (or steady-state gain) as follows:
q0 = iq
a
b
0
0
= k qi ………. (4.5)
where k =
0
0
a
b
= static sensitivity
qi q0
K
Figure 4.6: Block Diagram Representation of a Zero Order System
Since the equation q0 = kqi is a simple algebraic, it is clear that, no matter how qi might vary with
time, the instrument output (reading) follows it perfectly with no distortion or time lag of any sort
[Figure 4.7]. Thus, the zero-order instrument represents ideal or perfect dynamic performance.
qo
i
t
t
qis
Kqis
i
t
qo
t
Figure 4.7: Response of Zero Order System
Some examples of zero order system include mechanical level, light-pointer, linear electrical
potentiometer, amplifier, etc.

4.2.2 First Order System
If in Equation (3.1) all the a's and b's other than a1, a0 and b0 are taken as zero, we get
a1
dt
dq0
+ a0q0 = b0qi ………. (4.6)
Any instrument that follows this equation is, by definition, a first order instrument.
While Equation (4.6) has three parameters a1, a0 and b0, only two are really essential since the
whole equation could always be divided through a1, a0 or b0, thus making the coefficient of one of
the terms numerically equal to 1. The conventional procedure is to divide through by a0, which
gives
dt
dq
a
a 0
0
1
+ q0 =
0
0
a
b
qi ………. (4.7)
which becomes
dt
dq0
 + q0 = Kqi ………. (4.8)
or,   01 qD  = Kqi ………. (4.9)
where k =
0
0
a
b
= static sensitivity, and
 =
0
1
a
a
= time constant.
The operational transfer function of any first order instrument is
 D
q
q
i
0
=
1D
K

………. (4.10)
The block diagram representation is shown in Figure 3.8.
qiq K 0
D + 1
Figure 4.8: Block Diagram Representation of First Order Instruments
Some examples of first order system are: temperature measurement by mercury-in-glass
thermometer, thermocouples, and thermistors; build-up air pressure in bellows, RC network, RL
network, etc.

4.2.3 Second Order System
If in Equation (4.1) all the a's and b's other than a2, a1, a0 and b0 are taken as zero, the general
differential equation reduces to
a2 2
0
2
dt
qd
+ a1
dt
dq0
+ a0q0 = b0qi ………. (4.11)
Dividing Equation (3.12) by a0, we get
0
0
0
1
2
0
2
0
2
q
dt
dq
a
a
dt
qd
a
a
 = iq
a
b
0
0
………. (4.12)
The essential parameters in Equation (4.12) can be reduced to three:
K =
0
0
a
b
= static sensitivity
n =
2
0
a
a
= undamped natural frequency, (rad/s), and
 =
20
1
2 aa
a
= damping ratio, dimensionless
which gives,
02
2
1
2
q
DD
nn 











= Kqi ………. (4.13)
An instrument whose performance is governed by this equation is termed as a second order system.
The operational transfer function of a second order system is given by
 D
q
q
i
0
=
1/2/
22
 nn DD
K

………. (4.14)
D / + 2D/+ 1
qi
K q0
2
n n
2
Figure 4.9: Block Diagram Representation of Second Order Systems
The block diagram representation of second order system is shown in Figure 4.9. Some examples
of second order system are force measuring spring scale, spring mass under impressed force, L-C-R
circuits, piezoelectric pick-ups, recorders, etc.

4.3 Characteristic of First Order System
The standard test signals used for determining system performance are
 Step Input
 Ramp Input
 Impulse Input
(a) Step Input
A step input represents an application of a sudden change. Mathematically, a unit step function u(t)
is defined as:
qi(t) = 1; t  0
= 0; t < 0 ………. (4.15)
A unit step input is designated by u(t). Therefore, a unit step input is written as,
qi(t) = u(t) ………. (4.16)
Thus a unit step function represents a signal which changes its level from 0 to 1 in zero time and
the unit magnitude is maintained thereafter. The function is plotted in Figure 4.10 (a).
t
iq
u(t) Au(t)
qi
t
(a) (b)
Figure 4.10: (a) Unit Step Function(b) Step Function of magnitude A
Figure 4.10 (b) shows a step input of magnitude A wherein the magnitude changes from 0 to A in
zero time. This input is mathematically represented as
qi(t) = A; t  0
= 0; t < 0 ………. (4.17)
or, simply
qi(t) = A u(t) ………. (4.18)
(b) Ramp Input
It represents an input signal which changes at a constant rate with respect to time. A unit ramp
input signal starts at a value of zero and increases with constant slope of unity with respect to time.
A unit ramp input signal is shown in Figure 4.11(a). It can be expressed mathematically as:
qi(t) = t; t > 0
= 0; t < 0 ………. (4.19)
or, simply
qi(t) = t u(t) ………. (4.20)

(a)
i
t
(b)t
t At
t t
i
Figure 4.11: (a) Unit Ramp Function (b) Ramp Function of Magnitude A
A ramp input signal of amplitude A is shown in Figure 4.11(b). Mathematically this input can be
expressed as:
qi(t) = At; t > 0
= 0; t < 0 ………. (4.21)
or, simply
qi(t) = At u(t) ………. (4.22)
(c) Impulse Input
A unit impulse is defined as a signal which has zero value everywhere except at t = 0 where the
magnitude is finite. The function is generally called a  (delta) function. The  function has the
following properties,
(t) = 0; t  0
and 


 dtt)( = 1, where 0 ………. (4.23)
(a)
i
t
1/
A/
 t
i
(b)
Figure 4.12: (a) Unit Impulse Function (b) Impulse Function of Strength A
Since a perfect impulse function is not practically realizable, it is approximated by a pulse of small
width having a unit area as shown in Figure 4.12 (a). The diagram shows a pulse with an amplitude
of 1/ and duration . Figure 4.12 (b) shows an impulse function of strength A.
4.3.1 Step Response of First Order Instruments
To apply a step input to a system, we assume that initially it is in equilibrium, with qi = q0 0, when
at time t = 0 the input quantity increases instantly by an amount qis (Figure 4.13) i.e.,
qi = 0 for t < 0
= qis for t  0 ………. (4.24)

Hence, for t > 0, Equation (4.10) becomes
  01 qD  = Kqis ………. (4.25)
It can be shown generally (by mathematically reasoning) or in any physical problem, such as the
thermometers (by physical reasoning), that the initial condition for this situation is q0 for t = 0+
.
The complementary function solution is determined as
  01 qD  = 0
 q0 cf = C 
t
e

(Transient component) ………. (4.26)
while the particular solution is
q0 pi = Kqis (Steady state component) .……. (4.27)
Hence the complete solution is
q0 = C 
t
e

+ Kqis ………. (4.28)
Applying the initial condition (q0 = 0 when t = 0), we get
 C = – K qis ………. (4.29)
which gives finally
q0 = Kqis (1 – e-t/
) ………. (4.30)
Examination of Equation (4.31) shows that the speed of response depends on only the value of 
and is faster if  is smaller (Figure 4.13). Thus in first order systems we strive to minimize  for
faithful dynamic measurements.
These results may be non dimensionalized as
isKq
q0
= 1 – 
t
e

………. (4.31)
is
Kq
t
q o
i
qis
t
Large 
Small 
Figure 4.13: Step Response of First Order Instrument

q /Kq0 is
1.0
0.8
0.6
0.4
0.2
0
t/ q /Kq0 is
0
1
2
3
4

0
0.632
0.865
0.950
0.982
1.000
Figure 4.14: Non-dimensionalised Step Response of First Order Instrument
The dynamic error (em) at any time is the difference between the ideal (no time lag) value and the
actual measures value of the quantity is given by
em = qis –
K
q0
………. (4.33)
Substituting value of q0 from Equation (4.31) in Equation (4.33),
em = qis – qis (1 – e-t/
) = qis e-t/
………. (4.34)
which also may be nondimensionalized as
is
m
q
e
= e-t/
………. (4.35)
The normalized dynamic error em/qis versus t/ is shown in Figure 4.15.
0.4
0
0
0.2
1
0.6
0.8
1.0
m is
32 4
1.0000
t/5
3 0.050
0.018
0
4
0.368
0.1352
1
t/ e /qm is
Figure 4.15: Non-dimensionalized Dynamic Error for First Order Instrument

Time Constant
If we substitute the magnitude of one time constant for t in Equation (4.32),
q0 = Kqis (1 – e-1
)= Kqis (1 – 0.368) = 0.632 qmax
from which we see that 63.2 % of the dynamic portion of the process will have been completed.
Two time constants yield 86.5 %, three time constants yield 95.0 %; four time constants yields
988.2 %, and so on. These percentages of completed processes are important because they will
always be the same regardless of the process, provided that the process is governed by the
conditions of the step-excited first order system. It is often assumed that a process is completed
during a period of five time constants.
For a rising exponential function the time constant  is defined as the time to reach 63.2 % of its
final value. Similarly for a decaying exponential function the time constant is defined as the time
taken to fall to 36.8 % of its initial value.
4.3.2 Ramp Response of First Order Instruments
To apply a ramp input to a system, we assume that initially the system is in equilibrium, with qi =
q0 = 0, when at t = 0 the input suddenly starts to change at a constant rate of isq
.
. We thus have
qi = 0; t  0
= isq
.
t t  0 ………. (4.36)
and therefore
  01 qD  = isq
.
t ………. (4.37)
The necessary initial condition again can be shown to be q0 = 0 for t = 0+
. Solution of Equation
(4.37) gives
q0 cf = C e-t/
q0 pi = K isq
.
(t – )
q0 = C e-t/
+ K isq
.
(t – )
and applying initial condition (q0 = 0, t = 0) gives
C = – K isq
.

which gives finally
q0 = K isq
.
( e-t/
+ t – ) ………. (4.38)

We again define measurement error em by
em = qi –
K
q0
= – isq
.
 e-t/
+ isq
.
 ………. (4.39)
q i
t
(a)
q
0/K
Steady state time
lag = 
e = q m,ss is
(b)
t/
1.0
0 1 2 3 4 5
/em m,ss
Figure 4.16: Ramp Response of First Order Instrument
4.3.3 Frequency Response of First Order Instruments
The frequency response of a system consists of curves of amplitude ration and phase shift as a
function of frequency. The frequency of any linear system may be obtained by getting the
particular solution of its differential equation with
qi = 0 for t < 0
= Ai sint for t  0
Much quicker and easier methods are available. These methods depend on the concept of the
sinusoidal transfer function. The sinusoidal transfer function of a system is obtained substituting i
for D in the operational transfer function;
Sinusoidal transfer function =  i
q
q
i
0
=
     
      01
1
1
01
1
1
....
....
aiaiaia
bibibib
n
n
n
n
m
m
m
m








………. (4.40)
where i = 1 and  = angular frequency (rad/s). For any given frequency , Equation (3.40)
shows that q0/qi (i) is a complex number, which can always be put in the polar form M < . The
magnitude M of the complex number is the amplitude ration A0/Ai, while the angle  is the phase
angle by which the output q0 leads the input qi. If the output lags the input  is negative.

Equation (4.40) may be applied to the problem of finding the response of first order systems to
sinusoidal inputs. We have
 i
q
q
i
0
=
1i
K
=  



tan
122
K
………. (4.41)
Thus the amplitude ratio is
iA
A0
=  i
q
q
i
0
=
122

K
………. (4.42)
and the phase angle is
 =  i
q
q
i
0
 = tan-1
(-) ………. (4.43)
A nondimensionalized representation of the frequency response of any first order system may be
obtained by writing Equation (3.41) as
 i
q
Kq
i
/0
=  



1
22
tan
1
1
………. (4.44)
and plotting as in Figure 4.17.
0
(a)
t
1.0
0.8
0.6
0.4
0.2
0
1 2 3 4 5 6 7 8 9 10
q /K
q
0
i
-20
-80
0
-60
-40
(b)
21 3 54 6 87 9 10
0 t
Degrees
-90
Figure 3.17: Frequency Response of First Order System

4.3.4 Impulse Response of First Order Instruments
The impulse function of strength (area) A is defines by the limiting process
Impulse function of strength A = )(
0
lim
tp
T 
………. (4.46)
We now find the response of a first order system to an impulse input. We do this by finding the
response to the pulse p(t) and then applying the limiting process to the result. For 0 < t < T we have
  01 qD  = Kqi =
T
KA
………. (4.47)
Since, up until time T, this no different from a step input of size A/T, our initial condition is q0 = 0
at t = 0+
, and the complete solution is
q0 = )1( /t
e
T
KA 
 ………. (4.48)
However, this solution is valid only up to time T. At this time we have
q0|t = T = )1( /T
e
T
KA 
 ………. (4.49)
Now for t > T, differential equation becomes,
  01 qD  = Kqi = 0
which gives
q0 = C e-t/
………. (4.50)
The constant C can be found by imposing initial condition [Equation (4.49)]
)1( /T
e
T
KA 
 = C e-T/
………. (4.51)
 C = 

/
/
)1(
T
T
Te
eKA



………. (4.52)
giving finally
q0 = 

/
//
)1(
T
tT
Te
eeKA



………. (4.53)
Figure 4.18(b) shows a typical response, and Figure 4.18 (c) shows the effect of cutting T in half.
As T is made shorter and shorter, the first part (t < T) of the response becomes of negligible

consequence, so that we can get an expression for q0 by taking the limit of Equation (4.53) as T 
0.
0
lim
T



/
/
/
)1( t
T
T
e
Te
eKA 







 
= KA e-t/
0
lim
T 

/
/
1
T
T
Te
e



= KA e-t/
0
lim
T T
e T /
1 

=
0
0
which takes an indeterminate form.
Applying L'Hospital's Rule, we get
0
lim
T T
e T /
1 

=
0
lim
T 1
)/1( /
 T
e
=

1
………. (4.54)
Thus we have finally for the impulse response of a first order system
q0 =


/t
e
KA 
………. (4.55)
which is plotted in Figure 4.18(d).
(a)T
t
A/T
p(t)
A/T
T (b)
t
q
q0
i
t
T/2 (c) (d)
2A/T
q0
t
qi
q0
KA/
Figure 3.18: Impulse Response of First Order Instrument
4.4 Characteristic of Second Order System
Dynamic behavior of a second order system is given by








 1
2
2
2
nn
DD



q0 = Kqi ………. (4.14)

The transient response (complementary function) is obtained from the auxiliary equation which is
set by replacing the transfer operation D by an algebraic variable m and setting the input qi to zero.
Thus the auxiliary equation is








 1
2
2
2
nn
mm



= 0 ………. (4.56)
The response of the system depends upon the root of the characteristic Equation (3.56). The two
roots of the characteristic equation are determined as
m1, m2 = 2
222
/2
/4/4/2
n
nnn

 
m1, m2 = 12
  nn ………. (4.57)
Hence the complementary function becomes,
q0, cf =
tmtm
BeAe 21
 ………. (4.58)
where A and B are arbitrary constants to be determine from initial conditions, while m1 and m2 are
the roots of the auxiliary equation. The roots are three different types depending upon the value of
. Hence there are three types of responses and consequently three types of systems. The three
types of systems are:
Overdamped Systems
When  > 1, the characteristic equation will have two real and unequal, and the system is called
overdamped system.
Critically Damped Systems
When  = 1, the roots will be real and equal, then system is called critically damped system.
Underdamped Systems
When  < 1, the equation will have complex conjugate pairs of roots, then the system is called
underdamped system.
4.4.1 Step Response of Second Order Systems
To study the step response of a second order system, consider the differential equation,








 1
2
2
2
nn
DD



q0 = Kqis ………. (4.59)
with a set of initial conditions
q0 = 0 at t = 0+
and,
dt
dq0
= 0 at t = 0+
………. (4.60)
The particular solution of Equation (4.59) is clearly qo pi = Kqis. The complementary solution takes
one of the three possible forms, depending on whether the roots of the characteristic equation are

real and unequal (overdamped), real and equal (critically damped system), or complex
(underdamped system).
Overdamped System
The complete solution of the Equation (4.59) is given by
q0 = q0 cf + q0 pi
=
tmtm
BeAe 21
 + Kqis
=
tn
Ae
 



  12
+
tn
Be
 



  12
+ Kqis ………. (4.61)
Differentiating Equation (3.61), we get
dt
dq0
=   t
n
n
eA






 

1
2
2
1 +   t
n
n
eB






 

1
2
2
1 .. (4.62)
Now applying initial condition (q0 = 0; t = 0+
) in Equation (4.61), we get
A =
12
1
2
2





isKq and B =
12
1
2
2




isKq
The step response for an overdamped system is obtained by substituting A and B into Equation
(4.61),
q0 =
12
1
2
2





isKq
tn
e
 



  12
+
12
1
2
2




isKq
tn
e
 



  12
+ Kqis
… (4.65)
Equation (4.65) can also be expressed in nondimensional form as,
isKq
q0
=
12
1
2
2




 tn
e
 



  12
+
12
1
2
2



 tn
e
 



  12
+ 1 .… (4.66)
Critically Damped System
q0 = q0 cf + q0 pi
=   tn
eBtA 
 + Kqis ………. (4.67)

dt
dq0
=   tBA nn   1 tn
e 
………. (4.68)
) in Equation (4.67),
A = – Kqis ………. (4.69)
Again applying initial condition (dq0/dt = 0; t = 0+
B = An = – Kqisn ………. (4.70)
The step response for a critically damped system is obtained by substituting A and B into Equation
(4.67),
q0 = (– Kqis – Kqisnt)
tn
e 
+ Kqis ………. (4.71)
isKq
q0
= – (1 – nt)
tn
e 
+ 1 ………. (4.72)
Underdamped System
q0 =  22
1sin1cos 

tBtAe nn
tn
+ Kqis ……. (4.73)
dt
dq0
=
    2222
1sin11cos1   tBAtAB nnnnnn
tn
e 
… (4.74)
A = – Kqis ………. (4.75)
Again applying initial condition (dq0/dt = 0; t = 0+
B = A
2
1 


= – Kqis
2
1 


………. (4.76)
The step response for an underdamped system is obtained by substituting A and B into Equation
(4.73),

q0 =
t
nisnis
n
etKqtKq 



 










 2
2
2
1sin
1
1cos + Kqis
……. (4.77)
isKq
q0
= –
t
nn
n
ett 



 










 2
2
2
1sin
1
1cos + 1 .... (4.78)
Equation (4.78) can be expressed in an alternative form as
isKq
q0
=   t
n
n
et 





 2
2
1sin
1
1
+ 1 ………. (4.79)
where,
21
1sin   
.
Figure 4.18: Non-dimensional Step-function Response of Second Order Systems
Comparison of Step Response of Overdamped, Critically Damped and Underdamped System
 An overdamped system responds to any time varying input in a sluggish manner without any
oscillation about the final steady-state position. There is no overshoot in step response. Due to
their sluggish nature, the over damped systems are usually unsuitable for many control
applications. However, this type of system may be used in measurement, where time is not of
prime importance.
 The response of critically damped system is rapid and the system reaches its final steady-state
condition smoothly without oscillations. There is no overshot in the step response. This system
is used when fast measurement is required.

 The underdamped system follows the input with oscillation about its final steady position. The
underdamped system has very fast initial response which is useful for various measurement
applications, such as impulse measurement.
4.4.2 Ramp Response of Second Order Systems
The differential equation here is








 1
2
2
2
nn
DD



q0 = K isq
.
t ………. (4.80)
q0 = 0 at t = 0+
and,
dt
dq0
= 0 at t = 0+
………. (4.81)
The solutions are found to be
Overdamped System
K
q0
= isq
.
t –


















 



  tt
n
is nn
ee
q 





 1
2
22
1
2
22
.
22
14
1212
14
1212
1
2
.. (4.82)
K
q0
= isq
.
t – 











 
2
11
2
.
t
e
q nt
n
is n



………. (4.83)
Underdamped System
K
q0
= isq
.
t –  













 
t
eq
n
t
n
is
n
2
2
.
1sin
12
1
2
………. (4.84)
where
12
12
tan 2
2






Figure 4.19 shows the general character of the response. There is a steady state error 2 isq
.
/n.
Since the value of isq
.
is set by measured quantity, the steady –state error can reduced only by
reducing  and increasing n. For a given n, reduction of  results in larger oscillations. There is
also a steady –state time lag 2/n. Figure 4.20 gives a set of non-dimensionalized curves that
summarize system behavior.

Steady state time
lag = 2
ism,sse = 2q / 
iq = q t
/K0q
t
n
n
is
Figure 4.19: Ramp Response of Second Order System
Figure 4.20: Nondimensionalized Ramp Response
4.4.3 Frequency Response of Second Order Systems
When a sinusoidal input of the form qi = A sin t is applied to the system, a steady state solution of
the differential equation represents a sinusoidal output with a different amplitude and phase shift
but having the same frequency. The sinusoidal transfer function of a second order system is
expressed as
 j
q
q
i
0
=
12
2












nn
jj
K





………. (4.85)
which can be put in the form
 j
q
Kq
i
/0
=
    2222
/4/1
1
nn  
<  ………. (4.86)

 =


//
2
tan 1
nn 

………. (4.87)
Let u be the normalized frequency = /n, then Equations (3.86) and (3.87) can be rewritten as
iq
Kq /0
=
   222
21
1
uu 
………. (4.88)
and  = 







2
1
1
2
tan
u
u
………. (4.89)
Figure 4.21: Frequency Response of Second Order System
Figure 4.21 gives the non-dimensionalized frequency response curves. Clearly, increasing n will
increase the range of frequencies for which the amplitude–ratio curve is relatively flat; thus a high
n is needed to measure accurately high-frequency qi's. An optimum range of value for  is
indicated by both amplitude–ratio and phase–angle curves. The widest flat amplitude ratio exists
for  of about 0.6 to 0.7. While zero phase angle would be ideal, it is rarely possible to realize this
even approximately. Actually, if the main interest is in q0 reproducing the correct shape of qi and if
a time delay is acceptable, we show shortly that  need not be zero; rather, it should vary linearly
with frequency . Examining the phase curves of Figure 4.21, we note that the curves for  = 0.6 to
0.7 are nearly straight for the widest frequency range. These considerations lead to the widely
accepted choice of  = 0.6 to 0.7 as the optimum value of damping for second order instruments.

Resonant Frequency
The frequency at which the magnitude ratio q0/Kqi has the maximum value is known as the
resonant frequency. Let r be the resonant frequency and ur (= r/n) be the normalized resonant
frequency.
The normalized resonant frequency can be determined by differentiating Equation (4.88) with
respect to u and putting the differential equal to zero for u = ur,
       rrrrr uuuuu 222
3
222
81421
2
1
 

= 0
The first term of the above product expression cannot be zero, hence
  rrr uuu 22
814  = 0
or, 1 – ur
2
= 2 2
or, ur
2
= 1 – 22
 ur = 2
21  ………. (4.90)
Hence the resonant frequency is given by
r = n
2
21  ………. (4.91)
Resonant Peak
The maximum value of magnitude ratio occurs at  = r. This value is known as resonant peak.
The value of resonant peak can be determined by substituting u = ur = 2
21  in Equation (4.88),
max
0 /
iq
Kq
=
   2222
214211
1
 
=
424
844
1
 
=
 22
214
1
 

max
0 /
iq
Kq
=
2
212
1
 
………. (4.92)
It is clear from the Equation (3.88) that for   1/2 (= 0.707), there is no resonant frequency and
hence there is no resonant peak.
Bandwidth
If we examine Figure 3.21, it is found that for frequency above   r, the amplitude ratio
decreases. The frequency at which the magnitude ratio has a value of 0.707 is called cut off
frequency C. Above the cut-off frequency C, the magnitude ratio decreases below 0.707.

Frequency C represents half power point. The band of frequencies from zero to cut-off frequency
C is called the bandwidth of the system.
Measurement systems are low pass filters as the value of amplitude ratio is unity at  = 0. But as
the frequency of the input signals increases, the output gets attenuated. Bandwidth is, therefore,
indicative of the satisfactory reproduction of the input signal.
Let ub (= b/n) be normalized bandwidth where b = bandwidth in rad/s. Then from Equation
(3.88),
iq
Kq /0
=
   222
21
1
bb uu 
= 0.707
or,    222
21 bb uu  = 2
or, ub
4
+ (42
– 2) ub
2
– 1 = 0
Solving the above quadratic equation, we get
ub =  422
44221   ………. (4.93)
The denormalized bandwidth is then given by
b = n  422
44221   ………. (4.94)
4.4.4 Impulse Response of Second Order Systems
For the first order system, we showed that the impulse response is equivalent to the free response if
the initial (t = 0+
) conditions produced by the impulse are taken into account. To find the initial
conditions produced by applying an impulse of area A to a second order system, redraw the block
diagram of Figure 3.22(a) as in Figure 3.22(b). In Figure 3.22(c) the impulse is applied at qi, and
the propagation of this input signal is traced through the rest of the diagram. The analysis shows
that at t = 0+
we have q0 = 0 and 0
.
q = KAn
2
. The differential equation to be solved is then








 1
2
2
2
nn
DD



q0 = 0 ………. (4.95)
q0 = 0 at t = 0+
and,
dt
dq0
= KAn
2
at t = 0+
………. (4.96)
The solutions are found to be

Overdamped System
nKA
q

0
= 











 



  tt nn
ee


11
2
22
12
1
………. (4.97)
nKA
q

0
= t
n
n
te 
 
………. (4.98)
Underdamped System
nKA
q

0
=  te n
tn


 2
2
1sin
1
1



………. (4.99)
The results are plotted in Figure 4.23.
Figure 4.23: Nondimensionalized Impulse Response of Second Order System
4.5 Physical Examples of Zero, First and Second Order Systems
4.5.1 Zero Order System
A practical example of a zero order system is the displacement measuring potentiometer. In such
instrument a strip of resistance material is excited with a voltage and provided with a sliding
contact, as shown in Figure 3.24(a). If the resistance is distributed linearly along length L, we may
write
e0 = b
i
E
L
x
= Kxi ………. (4.100)
where K (static sensitivity) = Eb/L V/m.
From Equation (4.100), it is clear that a liner relationship exists between the input displacement
and the output voltage as shown in Figure 4.24(b).

qo
Kqis
t
iq
isq
t
LEb
x i
+
-
e 0
(a) (b)
e 0
L
Eb
x i
(c) Step Response (d) Frequency Response
qi

K
oq

Figure 4.24: Response of Resistance Potentiometer
If this device is examined more critically, it would be found that it is not exactly a zero order
instrument. This is simply a manifestation of the universal rule that no mathematical model can
exactly represent any physical system. In this present example, the output terminals must be
attached to some voltage-measuring devices (such as an oscilloscope). Such a device will always
draw some current (however small) from the potentiometer. Thus, when xi changes, the
potentiometer windings current will also change. This is itself would cause no dynamic distortion
or lag if the potentiometer were a pure resistance. However, the idea of a pure resistance is a
mathematical model, not a real system; thus the potentiometer will have some (however small)
inductance and capacitance. If xi varied relatively slow, theses parasitic inductance and capacitance
effects will not be apparent. However, for sufficiently fast variation of xi, these effects are no
longer negligible and cause dynamic errors between xi and e0. The reasons why potentiometer is
normally called a zero-order system are as follows:
 The parasitic inductance and capacitance can be made very small by design.
 The speeds (frequency) of motion to be measured are not high enough to make the
inductive or capacitive effect noticeable.
Another aspect of non-ideal behavior in a real potentiometer comes to light when we realize that
the sliding contact must be attached to the body whose motion is to be measured. Thus, there is a
mechanical loading effect, due to the inertia of the sliding contact and its friction, which will cause
the measured motion xi to be different from that which would occur if the potentiometer were not
present. Thus the effect is different in kind from the inductive and capacitive phenomena
mentioned earlier, since they affected the relation [Equation (4.100)] between e0 and xi whereas the
mechanical loading has no effect on this relation but rather, makes xi different from the undisturbed
case.

4.5.2 First Order System
Most of the temperature measurement devices behave as first order system. As an example, let us
consider the liquid-in-glass thermometer of Figure 4.25. The input (measured) quantity here is the
temperature Ti(t) of the fluid surrounding the bulb of the thermometer, and the output is the
displacement x0 of the thermometer fluid in the capillary tube. We assume the temperature Ti(t) is
uniform throughout the fluid at any given time, but may vary with time in an arbitrary fashion. The
principle of operation of such a thermometer is the thermal expansion of the filling fluid which
drives the liquid column up or down in response to temperature changes. Since this liquid column
has inertia, mechanical lags will be involved in moving the fluid from one level to another.
However, we assume that this lag is negligible compared with the thermal lag involved in
transferring heat from the surrounding fluid through the bulb wall and into the thermometer fluid.
This assumption rests on experience, judgment, order-of-magnitude calculations, and ultimately
experimental verification of the results predicted by the analysis. Assumption of negligible
mechanical lag allows us to relate the temperature of the fluid in the bulb to the reading x0 by the
instantaneous algebraic equation
x0 = tf
C
bex
T
A
VK
………. (4.101)
where x0 = displacement from reference mark, m
Ttf = temperature of fluid in bulb, Ttf = 0 when x0 = 0, 0
C
Kex = differential expansion coefficient of thermometer fluid and bulb glass, m3
/m3
.0
C
Vb = volume of bulb, m3
AC = cross-sectional area of capillary tube, m2
Figure 4.25: Liquid-in-glass Thermometer
To get a differential equation relating input and output in this thermometer, we consider
conservation of energy over an infinitesimal time dt for the thermometer bulb:
Heat in – heat out = energy stored
Assuming no heat loss,
UAb (Ti – Ttf) dt – 0 = VbCdTtf

or, UAb (Ti – Ttf) dt = VbCdTtf ………. (4.102)
where, U = overall heat transfer coefficient across bulb wall, W/m2
.0
C
Ab = heat-transfer area of bulb wall, m2
 = mass density of thermometer fluid, kg/m3
C = specific heat of thermometer fluid, J/kg.0
C.
Equation (4.102) involves many assumptions:
 The bulb wall and fluid films on each side are pure resistance to heat transfer with no heat-
storage capacity. This will be a good assumption if the heat-storage capacity (mass 
specific heat) of the bulb wall and fluid films is small compared with VbC for the bulb.
 The overall coefficient U is constant. Actually, films coefficients and bulb-wall
conductivity all change with temperature, but these changes are quite small as long as the
temperature does not vary over wide ranges.
 The heat transfer area Ab is constant. Actually, expansion and contraction would cause this
to vary, but this effect should be quite small.
 No heat is lost from the thermometer bulb by conduction up the stem. Heat loss will be
small if the stem is of small diameter, made of a poor conductor, and immersed in the fluid
over a great length and if the exposed end is subjected to an air temperature not much
different from Ti and Ttf.
 The mass of fluid in the bulb is constant. Actually mass must enter or leave the bulb
whenever the level in the capillary tube changes. For a fine capillary and a large bulb, this
effect should be small.
 The specific heat C is constant. Again, this fluid property varies with temperature, but the
variation is slight except for large temperature changes.
The above list of assumptions is not complete, but should give some appreciation of the
discrepancies between a mathematical model and the real system it represents. Many of these
assumptions could be relaxed to get a more accurate model, but we would pay a heavy price in
increased mathematical complexity. The choice of assumptions that are just good enough for the
needs of the job at hand is one of the most difficult and important tasks of the engineer.
Returning to Equation (4.102), we may write it as
VbCdTtf = UAbTidt – UAbTtfdt
or, VbCdTtf + UAbTtfdt = UAbTidt
 VbC
dt
dTtf
+ UAbTtf = UAbTi ………. (4.103)
Using Equation (3.101), we get
dt
dx
K
CA
ex
C 0
+ 0x
VK
AUA
bex
Cb
= UAbTi ………. (4.104)

which we recognize to be the form of Equation (4.9), and so we immediately define
K =
C
bex
A
VK
m/0
C ………. (4.105)
 =
b
b
UA
CV
s ………. (4.106)
As we know the fast response requires a small value of , we can examine this first-order
instrument to analyze physical changes would be needed to reduce . Equation (4.106) shows that 
may be reduced by
 Reducing , C, and Vb.
 Increasing U and Ab.
Since  and C are properties of the fluid filling the thermometer, they cannot be varied
independently of each other, and so for small  we search for fluid with a small C product. The
bulb volume Vb may be reduced, but this will also reduce Ab unless some extended surface heat-
transfer augmentation (such as fins on the bulb) is introduced. Even more significant is the effect of
reduced Vb on the static sensitivity K, as given by Equation (4.105). We see that attempts to reduce
 by decreasing Vb will result in reductions in K. Thus increased speed of response is traded off
lower sensitivity. This trade off is not unusual and will be observed in many other instruments.
The fact that  depends on U means that we cannot state that a certain thermometer has a certain
time constant and but only that a specific thermometer used in a certain fluid under certain heat
transfer conditions (say, free or forced convection) has a certain time constant. This is because U
depends partly on the value of the film coefficient of heat transfer at the outside of the bulb, which
varies greatly with changes in fluid (liquid or gas), flow velocity, etc. For example, a thermometer
in stirred oil might have a time constant of 5 s while the same thermometer in stagnant air would
have a  of perhaps 100 s. Thus we must always be careful in giving (or using) performance data to
be sure that the conditions of use correspond to those in force during calibration or that proper
corrections are applied.
4.5.3 Second Order System
A good example of second order system is the force measuring scale of Figure 4.26. We assume
the applied force fi has a frequency components only well below the natural frequency of the spring
itself. Then the main dynamic effect of the spring may be taken into account by adding one third of
the spring's mass to the main moving mass. The total mass we call M. The spring is assumed linear
with spring constant KS (in N/m). Although in real scale there might be considerable dry friction,
we assume perfect film lubrication and therefore a viscous damping effect with constant B (in
N/ms).

Figure 4.26: Force Measuring Spring Scale
The scale can be adjusted so that x0 = 0 when fi = 0 (gravity force will then drop out of the
equation), which yields
 forces = (mass)(acceleration)
fi – B
dt
dx0
2 – Ksx0 = M 2
0
2
dt
xd
………. (4.107)
which can be expressed in the form
(MD2
+ BD + KS) x0 = fi ………. (4.108)
Noting this to fit the second order model, we immediately define
K =
SK
1
m/N ………. (4.109)
 =
M
KS
rad/s ……….. (4.110)
 =
MK
B
S2
………. (4.111)

CHAPTER 5
SENSORS
5.1 Introduction
A measuring device passes through two stages while measuring a signal. First, the measurand is
sensed and then, the measured signal is transuded (or converted) into a form that is particularly
suitable for transmitting, signal conditioning, processing, or driving a controller or actuator. For
this reason, output of the transducer stage is often an electrical signal.
Measurand
(Typically
Analog Signal)
Signal
Sensor
Transducer
Transmittable Variable
(Typically Electrical)
Figure 5.1: Schematic Representation of Measuring Device
A sensor (also called detector) is a converter that measures a physical quantity and converts it into
a signal which can be read by an observer or by an (today mostly electronic) instrument. For
example, a mercury-in-glass thermometer converts the measured temperature into expansion and
contraction of a liquid which can be read on a calibrated glass tube. A thermocouple converts
temperature to an output voltage which can be read by a voltmeter. For accuracy, most sensors
are calibrated against known standards.
A transducer is a device that converts a signal in one form of energy to another form of energy.
A complex measuring device can have more than one sensing stage. Sensor and transducer stages
are functional stages, and sometimes it is not easy or even feasible to identify physical elements
associated with them. Following the common practice, the terms sensor and transducer will be used
interchangeably to denote measuring instruments.
Various transducers and their operation are listed in Table 5.1.
Table 5.1: Some Primary Detector-Transducer Elements and Operations they Perform
Element Operation
1. Mechanical
A. Contacting spindle, pin, or finger Displacement to displacement
B. Elastic member
1. Load cells
a. Tension/compression Force to linear displacement
b. Bending Force to linear displacement
c. Torsion Force to angular displacement
2. Proving ring Force to linear displacement
3. Bourdon tube Pressure to displacement
4. Bellows Pressure to displacement

5. Diaphragm Pressure to displacement
6. Helical spring Force to linear displacement
C. Mass
1. Seismic mass Force function to relative displacement
2. Pendulum Gravitational acceleration to frequency or period
3. Pendulum Force to Displacement
4. Liquid column Pressure to displacement
D. Thermal
1. Thermocouple Temperature to electric potential
2. Biomaterials (includes mercury in
glass)
Temperature to displacement
3. Thermistor Temperature to resistance change
4. Chemical phase Temperature to phase change
5. Pressure thermometer Temperature to pressure
E. Hydropneumatic
1. Static
a. Float Fluid level to displacement
b. Hydrometer Specific gravity to relative displacement
2. Dynamic
a. Orifice Fluid velocity to pressure change
b. Venturi Fluid velocity to pressure change
c. Pitot tube Fluid velocity to pressure change
d. Vanes Velocity to force
e. Turbines Linear to angular velocity
2. Electrical
Element Operation
A. Resistive
1. Contacting Displacement to resistance change
2. Variable-length conductor Displacement to resistance change
3. Variable-area conductor Displacement to resistance change
4. Variable dimensions of conductor Strain to resistance change
5. Variable receptivity of conductor Temperature to resistance change
B. Inductive
1. Variable coil dimensions Displacement to change in inductance
2. Variable air gap Displacement to change in inductance
3. Changing core material Displacement to change in inductance
4. Changing core positions Displacement to change in inductance
5. Changing coil positions Displacement to change in inductance
6. Moving coil Velocity to change in induced voltage
7. Moving permanent magnet Velocity to change in induced voltage

8. Moving core Velocity to change in induced voltage
C. Capacitive
1. Changing air gap Displacement to change in capacitance
2. Changing plate areas Displacement to change in capacitance
3. Changing dielectric constant Displacement to change in capacitance
D. Piezoelectric Displacement to voltage and/or voltage to
displacement
E. Semiconductor junction
1. Junction threshold voltage Temperature to voltage change
2. Photodiode current Light intensity to current
F. Photoelectric
1. Photovoltaic Light intensity to voltage
2. Photoconductive Light intensity to resistance change
3. Photoemissive Light intensity to current
G. Hall effect Displacement to voltage
Close scrutiny of Table 5.1 reveals that, whereas many of the mechanical sensors transduce the
input to displacement, many of the electrical sensors change displacement to an electrical output.
This is quite fortunate, for it yields practical combinations in which the mechanical sensor serves as
the primary transducers and the electrical transducers as the secondary. The two most commonly
used electrical means are variable resistance and variable inductance, although others, such as
photoelectric and piezoelectric effects, are also of considerable importance.
In addition to the inherent compatibility of the mechano-electric transducer combination, electrical
elements have several important relative advantages:
1. Amplification or attenuation can be easily obtained.
2. Mass-inertia effects are minimized.
3. The effects of friction are minimized.
4. An output power of almost any magnitude can be provided.
5. Remote indication or recording is feasible.
6. The transducer can often be miniaturized.
5.2 Classification of Transducers
All electrical transducers are broadly classified under two categories, viz., active and passive
transducers.
 Active transducers are self generating devices, operating under energy conversion principles.
They generate an equivalent electrical output signal without any external energizing source.

 Passive transducers operate under energy controlling principles. They depend upon the change
in the electrical parameter (resistance, capacitance, or inductance) whose excitation or operation
requires secondary electrical energy form an external source.
The various transduction principles under which the electrical transducers operate are given in
Table 5.2.
Table 5.2: Classification of Electrical Transducers
Active
Transducers
Passive Transducers
Thermoelectric Resistive
Piezoelectric Inductive
Photovoltaic Capacitive
Photoconductive
Piezoresistive
Magnetostrictive Magnetoresistive
Electrokinetic Thermoresistive
Electrodynamic Elastoresistive
Electromagnetic Hall effect
Pyroelectric Synchro
Galvanic Gyro
Radio-active
absorption
Ionic conduction
5.3 Force Deflection Transducers
Elastic transducers (Force-Deflection) provide an indication of the magnitude of force through
displacement measurements. Elastic elements may be subjected to one or combination of three
actions compression, tension and torsion. Elastic elements are frequently used for the measurement
of force because of their large range, continuous monitoring, ease of operation and ruggedness.
They are used for both dynamic and static force measurements. The commonly used elastic
transducers are Bourdan tube, Bellows, Diaphragm, spring, proving ring and torsion bar.
Figure 5.2: a) Bourdon tube and b) bellows

5.4 Variable Resistance and Sliding Contact Devices
Methods which involve the measurement of change in resistance are preferred to those employing
other principles because both AC as well as DC excitations are suitable for measurements. The
resistance of an electrical conductor is expressed by a simple equation:
A
L
R


Any method of varying one of the quantities involved in the above relationship can be the design
basis of an electrical resistive transducer. There are a number of ways in which resistance can be
changed by a physical phenomenon.
1. Mechanically Varied Resistance (Length of Resistor)
2. Resistivity Change By Thermal Conditions Change
3. Resistance Change Due to Strain
Resistive Potentiometer
Sliding contact resistive transducers convert a mechanical displacement input into an electrical
output, either voltage or current. Basically, a resistive potentiometer consists of a resistance
element provided with a movable contact. The contact motion may be translation, rotation or a
combination of the two (Helical motion). The resistance element is excited with either DC or AC
voltage, and the output voltage is (ideally) a linear function of the input displacement. Resistance
elements in common use may be classified as wire-wound, conductive plastic, hybrid or cermet.
Figure 5.3: Circuit diagram for potentiometer
The potentiometer can be used as a voltage divider to obtain a manually adjustable output voltage
at the slider (wiper) from a fixed input voltage applied across the two ends of the potentiometer.
This is their most common use.
The voltage across RL can be calculated by:
If RL is large compared to the other resistances (like the input to an operational amplifier), the
output voltage can be approximated by the simpler equation:
Input output relation for the realistic circuit is

    titiS
L
xxRRRxxV
V


1)(/1
1
212
which becomes ideal (R2/(R1+R2) = 0 for an open circuit) conditions
t
i
S
L
x
x
V
V

Thus for NO LOADING the input output curve is a straight line.
To achieve good linearity for a given meter of a given resistance RM, a potentiometer of sufficiently
low resistance relative to RM should be chosen.
The resolution of potentiometers is strongly influenced by the construction of the resistance
element. Other factors that should be considered during the selection of potentiometers include:
temperatures, shock and vibration, humidity, and altitude. These may act as modifying or/and
interfering inputs so as seriously to degrade instrument performance.
5.5 Resistance Strain Gauge
As described previously, the resistance of an electrical conductor with a uniform cross-section area
A and length L and resistivity ρ is given by
A
L
R


When the conductor is stretched or compressed, its resistance changes. This change in resistance is
taken as output signal. In practice, the resistance element is cemented to the surface of the member
to be strained. When the stretching or compressing force is applied, the length and area of the
resistance element will change i.e. its resistance will change. This type of sensor is used for
measuring force and/or pressure. Resistance strain gage is described in detail in the following
chapter.
5.6 Thermistors and Thermocouples
These sensors are used for measuring temperature.
Thermistors are made of semiconductor materials which include oxides of cobalt, manganese,
nickel etc. These sensors exhibit very large changes in resistance with temperature and therefore,
they can be fabricated in the form of very small bends.
The resistance-temperature relation for thermistors is given by
Where, R = resistance at temperature T,
RRef = resistance at reference temperature
T and TRef = absolute temperature, K
β = constant, characteristics of material, K

A thermocouple is a temperature measuring device whose operation depends on Seebeck effect.
The Seebeck effect states that when two dissimilar materials are brought into contact an emf exists
in the circuit which is a function of temperature at two junctions.
Because of small size, reliability and wide range of usefulness, thermocouples are widely used for
temperature measurement.
Figure 5.4: A thermocouple measuring circuit
The output voltage from the thermocouple is given by:
E0 = K (T1-T2)
Where, K = sensitivity of material combination μV/0
C
T1 and T2 = temperature at junctions 1 and 2, 0
C
5.7 Variable Inductance Transducers
Inductance is a measure that relates electrical flux to current. Inductance reactance is a measure of
the inductive effect and can be expressed as:
X = 2pfL
Where, X is the inductive reactance in ohms,
f is the frequency of the applied voltage in Hz and
L is the inductance in henries.
The inductance of a circuit is influenced by a number of factors, including:
1) The number of turns in a coil
2) The coil size
3) The permeability of the flux path
As a result of a mechanical displacement the permeability of the flux path is altered and a resulting
change in the inductance of the system occurs. Inductance is monitored through the resonant
frequency of the inductance coils to an applied voltage. As inductance changes, the resonant
frequency of the coils also changes. Electronic circuits that convert frequency to voltage are used to
gain a voltage output to inductive transducers.
Inductive transducers can be classified as follows:
1) Variable self inductance: single coil, two coil
2) Variable mutual inductance: simple two coil, three coil
3) Variable reluctance: moving iron, moving coil, moving magnet

Single coil self-inductance
In this device, the change in armature position changes the inductance of coil which is measured by
suitable circuitry indication of the value of input.
Figure 5.6: Single coil self-inductance device
Two coil self inductance
This is actually a device with one coil with a center tap. The movable magnetic core (iron core)
provides the mechanical input. The two coils usually form the two arms of a bridge which is
excited with alternating current. As the core moves, the impedance (or inductance) of coil changes
and the bridge voltage is unbalanced which is proportional to input.
Figure 5.7: Two coil self-inductance device
Mutual Inductance
In this device, the change in armature position changes inductance in the coils and causes changes
of output voltage.
Figure 5.8: Mutual inductance transducer
Differential Transformer
The linear variable differential transformer is a mechanical displacement transducer. It gives an a.c.
voltage output proportional to the distance of the transformer core to the windings. The LVDT is a
mutual-inductance device with three coils and a core. (Figure 10) An external a.c. power source
energizes the central coil and the two phase opposite end coils are used as pickup coils. The output

amplitude and phase are dependent on the relative positions between the two pickup coils and the
power coil. Theoretically there is a null or zero position between the two end coils, although in
practice this is difficult to obtain perfectly.
A typical representation of a core displacement to output voltage is shown in figure below. The
output voltage on either side of the null position is approximately proportional to the core
displacement. The phase shift that occurs on passing through the null position can be sensed by a
phase sensitive demodulator and used to detect the side that the output voltage is from.
The typical range for LVDT sensitivity is 0.4 - 2.0 mV/V. 10-3cm. LVDT are typically used in
force, displacement and pressure measurement. They offer the advantages of being relatively
insensitive to temperature changes, and providing high outputs without intermediate amplification.
The appreciable mass of the core is a disadvantage in the area of dynamic measurements.
Figure 5.9: Differential transformer schematic arrangement and its characteristics
5.8 Variable Reluctance Transducers
In transducer practice, some form of inductance device incorporating a permanent magnet is called
variable reluctance transducer. These devices are used for dynamic applications only where the flux
lines supplied by the magnet are cut by turns of coil. Some means for providing relative motion is
incorporated into the device. In its simplest form, the variable reluctance device consists simply a
coil wound on a permanent magnet core. The variation of air gap between the magnet and moving
member cause change in magnetic flux which develop voltage in the coil.

Figure 5.10: A simple variable reluctance pickup
5.9 Capacitive Transducers
The principle of operation of capacitive transducers is based upon the familiar equation for
capacitance of a parallel plate capacitor.
Capacitance (C) = εA/x = εrεoA/x
Where, A = overlapping area of plates, m2
x = distance between two plates, m
ε = εrεo = permittivity of medium, F/m
The capacitance transducer work on the principle of change of capacitance which may be caused by
1. Change in overlapping area, A
2. Change in the distance, x
3. Change in dielectric constant, ε
Figure 5.11: Variation in distance and overlapping area in capacitor
The most commonly employed method of changing capacitance is changing gap thickness and the
less frequently used method is changing overlapping area.
For capacitive transducer with variable air gap, ε =1, and the sensitivity of capacitance in plate
separation may be determined as,
dC/dx = -kA/x2
or, dC = -C/x
Therefore, dC/C = -dx/x
When the capacitor plates are stationary with a separation x0, no current flows and eo = 0. If there is
a relative displacement x1 from the x0 position, a voltage e0 is produced which is related to x1 by,
 D
x
x
1
0
=
1D
DK



5.10 Piezoelectric Transducers
A piezoelectric transducers is one in which an electric potential appears across certain surfaces if
the dimensions of the crystal of the crystal are changed by the application of a mechanical force.
The potential is produced by the displacement of charges. The effect is reversible and known as
piezoelectric effect. Piezoelectric materials such as single crystal quartz or polycrystalline barium
titanate, contain molecules with asymmetrical charge distributions. When pressure is applied, the
crystal deforms and there is a relative displacement of positive and negative charges within the
crystal.
Figure 5.12: Piezoelectric Transducer
This displacement of internal charges produces external charges of opposite sign on two surfaces of
the crystal which is determined as,
q = E0C where, C = capacitance of piezoelectric crystal and
E0 = output voltage
The surface charge q is related to the applied pressure P by equation,
q = SqAP where, Sq = charge sensitivity of the piezoelectric crystal
A = area of the electrode
5.11 Photoelectric Transducers
These types of transducers are used in certain applications when contact cannot be made with the
test specimen. Photoelectric sensors are used to monitor changes in light intensity which can be
related to the quantity being measured. Three different types of photoelectric detectors are used to
convert a radiation input to a voltage output. These include photo emissive cells, photo conductive
cells and photovoltaic cells.
The photo emissive cell contains a cathode C and an anode A mounted in a vacuum tube. The
radiation impinging on the cathode material frees electrons that flow to anode to produce an electric
current I which is proportional to illumination ψ imposed on the cathode.
I= Sψ where S= sensitivity of photoelectric cell.
Figure 5.13: Schematic of Photo emissive cell

The photo conductive cells are fabricated from semiconductor materials such as cadmium sulfide
(CdS) or cadmium selenide (CdSe) which exhibit a strong photoconductive response. The electrical
resistivity of these materials decreases when they are exposed to light.
Figure 5.14: Schematic of Photo conductive cell
The photovoltaic cells commonly used are P-N type diffused-silicon guard- ring photodiodes.
When the active area of a photodiode is illuminated and a connection is made between P and N
regions, current flows during the period of illumination. This phenomenon is known as
photovoltaic effect.
Figure 5.15: Working of Photovoltaic cell

CHAPTER 6
STRAIN GAGES
6.1 Introduction to Strain Measurement
Strain gage is one of the most popular types of transducer. It has got a wide range of applications. It
can be used for measurement of force, torque, pressure, acceleration and many other parameters.
The basic principle of operation of a strain gage is simple: when strain is applied to a thin metallic
wire, its dimension changes, thus changing the resistance of the wire.
Gage Factor
Let us consider a long straight metallic wire of length l circular cross section with diameter d (fig.).
When this wire is subjected to a force applied at the two ends, a strain will be generated and as a
result, the dimension will change (l changing to l+Δ l, d changing to d+Δd and A changing to
A+ΔA). For the time being, we are considering that all the changes are in positive direction. Now
the resistance of the wire:
R=
ρ
, where ρ is the resistivity.
From the above expression, the change in resistance due to strain:
ΔR= ) Δl + ( ) ΔA + ( Δ
= Δl - 2 ΔA + Δ
= R - R + R
Or, = - + Figure 6.1: elongation of wire
Now, for a circular cross section, A= ; from which ΔA = Δd.
Alternatively,
= 2
Hence,
= -2 +
Now, the Poisson’s Ratio is defined as:
υ = - = -
The Poisson’s Ratio is the property of the material, and does not depend on the dimension. So, it
can be rewritten as:
= (1 + 2 υ) +
Hence,
= 1 + 2 υ +

The last term in the right hand side of the above expression, represents the change in resistivity of
the material due to applied strain that occurs due to the piezo-resistance property of the material. In
fact, all the elements in the right hand side of the above equation are independent of the geometry
of the wire, subjected to strain, but rather depend on the material property of the wire.
Due to this reason, a term Gage Factor is used to characterize the performance of a strain gage.
The Gage Factor is defined as:
G= = 1 + 2 υ +
For normal metals the Poisson’s ratio υ varies in the range:
0.3 ≤ υ ≤ 0.6 ,
while the piezo-resistance coefficient varies in the range:
0.2 ≤ ≤ 0.6
Thus, the Gage Factor of metallic strain gages varies in the range 1.8 to 2.6. However, the
semiconductor type strain gages have a very large Gage Factor, in the range of 100-150. This is
attained due to dominant piezo-resistance property of semiconductors. The commercially available
strain gages have certain fixed resistance values, such as, 120Ω, 350 Ω, 1000 Ω, etc. The
manufacturer also specifies the Gage Factor and the maximum gage current to avoid self-heating
(normally in the range 15 mA to 100 mA).
The choice of material for a metallic strain gage should depend on several factors. The material
should have low temperature coefficient of resistance. It should also have low coefficient for
thermal expansion. Judging from all these factors, only few alloys qualify for a commercial
metallic strain gage. They are:
Advance (55% Cu, 45% Ni): Gage Factor between 2.0 to 2.2
Nichrome (80% Ni, 20% Co): Gage Factor between 2.2 to 2.5
Apart from these two, Isoelastic -another trademarked alloy with Gage Factor around 3.5 is also in
use. Semiconductor type strain gages, though having large Gage Factor, find limited use, because
of their high sensitivity and nonlinear characteristics.
6.2 Electrical type strain gauge: Unbonded and Bonded
6.2.1 Unbonded Strain Gauges
These strain gauges are not directly bonded (that is, pasted) onto the surface of the structure under
study. Hence they are termed as unbounded strain gauges. The unbounded strain gauge is normally
used for measuring strain (or displacement) between a fixed and a moving structure. Two frames P
and Q carrying rigidly fixed insulated pins as shown in diagram. These two frames can move
relative with respect to each other and they are held together by a spring loaded mechanism. A fine
wire resistance strain gauge is stretched around the insulated pins. The strain gauge is connected to
a wheat stone bridge.

When a force is applied on the structure under study
(frames P & Q), frames P moves relative to frame
Q, and due to this strain gauge will change in length
and cross section. That is, the strain gauge is
strained. This strain changes the resistance of
the strain gauge and this change in resistance of the
strain gauge is measured using a wheat stone
bridge. This change in resistance when calibrated
becomes a measure of the applied force and change
in dimensions of the structure under study.
It is used in force, pressure and acceleration
measurement. The range of this gauge is +/- 0.15%
strain & has very high accuracy but occupies more
space. Figure 6.2: Unbonded Strain Gauge
6.2.2 Bonded strain gauge
In the bonded strain gage, the element is fixed on a backing material, which is permanently fixed
over a structure, whose strain has to be measured, with adhesive. Most commonly used bonded
strain gages are metal foil type. The metal foil type strain gage is manufactured by photo-etching
technique. Here the thin strips of the foil are the active elements of the strain gage, while the thick
ones are for providing electrical connections. Because of large area of the thick portion, their
resistance is small and they do not contribute to any change in resistance due to strain, but increase
the heat dissipation area. Also it is easier to connect the lead wires with the strain gage.
The backing materials normally used is impregnated paper, fiber glass, etc. The bonding material
used for fixing the strain gage permanently to the structure should also be non-hygroscopic. Epoxy
and Cellulose are the bonding materials normally used.
Figure 6.3: Bonded Strain Gauge
Required characteristics for strain measuring device
An electrical strain gage meets all the required characteristics for strain measuring device which are
as follows

Manual instrumentation and measurement

Manual instrumentation and measurement

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Manual instrumentation and measurement

Similar to Manual instrumentation and measurement (20)

Recently uploaded

Recently uploaded (20)

Manual instrumentation and measurement