functional elements of measurement system,

7/27/2019 Functional elements of measurement system,

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INTRODUCTION

Measurement and Instrumentation:

Measurement is a process by which one can convert physical parameter to meaningful

numbers. The numerical measure is meaningless unless followed by a unit used, sinceit(unit) identifies the characteristic or property measured. Measurement involves the use

of instruments as a physical means of determining the value of unknown quantity or

variable.

In simple cases, an instrument consists of a single unit which gives an output reading or

signal according to the unknown variable applied to it. In more complex, measurementsituations, however, a measuring instrument m ay consists of several separate elements.

The elements may consist of transducing element which convert the measurand to an

analogous form. The analogous signal is then processed by some intermediate means an dthen fed to the end devices to present the results to measurement for the purpose of

display and or control.

Application and Need

The field of instrumentation encompasses almost all the areas of science and technology.

Even in our day to day life, instrumentation is indispensable. For ex, the ordinarywatch, an instrument for measurement of time is used by everybody, likewise an

automobile driver needs an instrument panel to facilitate him in driving the vehicle

properly. Certain common motivating factors for carrying out the measurements are asfollows.

Measurement of system parameters informations

One of the important functions of the instruments is to determine the various

parameters / informations of the system or a process. In fact, condition based system

of operation is being used very widely these days in a number of situations like medicalcare of patients or maintenance of machines.

Control of certain process or control

Another important application of measuring instruments is in the field of

automatic control system. The very concept of any control in a system requires the

measured discrepancy between the actual and the desired performance. It may be notedthat for an accurate control of any physical variable in a process or an operation, it is

important to have an accurate measurement system.

Simulation of system conditioning.

Sometimes, it may be necessary to simulate experimentally the actual conditionsof complex situations for relieving the true behavior of the system under different


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governing conditions. A scale model may be employed for this purpose and analytical

tools like dimensional analysis may also be employed to translate the experimental results

on the model to the prototype. The information thus obtained is used in the design anddevelopment of the prototype.

Experimental design studies.

The design and development of a new product generally involves trial and error

procedures which generally involve the use of empirical relations, handbook data, thestandard practices mentioned in design. For ex a design team of experienced aircraft

designers put ina number of years of effort to produce a prototype aircraft. The prototype

is flown by a test pilot to determine the various performance / operating parameters. The

prototype test data is then used to improve further the design calculations and a modifiedprototype is produced. This is carried on till eh desired design performance is achieved.

To perform various manipulations

In a number of cases, the instruments are employed to perform operations like

signal addition, subtraction, multiplication, division, differentiation, integration, signalliberalization, signal sampling, signal averaging, multi point co relations, ratio controls,

etc. In certain cases, instruments are also used to determine the solution of complex

differential equations and other mathematical manipulations. A simple pocket calculatoris an example of mathematical processing instrument of some extent. Further the modern

large memory computers are instruments that are capable of doing varied types of

mathematical manipulations.

Testing of materials, Maintenance of standards and specifications

Most countries have standard organizations that specify material standards andproduct specifications based on extensive test and measurements. These organizations

are meant to protect the interests of consumers. They ensure that the materials / products

meet the specified requirements so that they function properly and enhance the reliabilityof the system.

Verifications of Physical phenomena / scientific theories

Quite often experimental data is generated to verify certain physical phenomenon.

Whenever, a scientist oar an engineer proposes any hypothesis predicting the systems

behaviour, it needs to be checked experimentally to put the same on a sound footing. Inaddition, experimental studies play an important role in formulating certain empirical

relations where adequate theory does not exist.

Quality control in industry

It is quite common these days to have continuous quality control tests of mass

produced industrial products. This enables to discover defective components that are


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outright rejected at early stages of production. Consequently, the final assemble of the

machine / system is free from defects. This improves the reliability of product

completely.

Medical field

Instruments such as magnetometers, radiation detectors and X Ray fluoroscopes

enable sensing and detection of physical quantities, which cannot be sensed or detected

by human beings.In general, Instruments possessing additional capabilities such as storage of

measured data, correlation, computation and control function generation developed for

application in situations in which human skills and proficiency were limited made the

instruments to play a significant role in all the activities of man.

Instrument and Instrumentation

An instrument may be defined as a device or a system which is designed tomaintain a functional relationship between prescribed properties of physical variables and

must include ways and means of communications to human observer.

Instrumentation refers to the art and science of collection of several instruments

and auxiliary equipment and their utilization for conducting successfully a test or anexperiment on a system, process or plant (for measurement of a large number of variables

embracing the disciplines of physical sciences such as physics and chemistry and

engineering disciplines like electrical, mechanical, electronics, communication dn

computer engineering). Since all branches of science and technology utilize instrumentsfor measurement of quantities pertaining to their discipline, instrumentations system are

classified as belonging to chemical, aeronautical, medical, meteorological or optical. The

basic difference of one from the other are due to the nature and range of the measurandand the transducer system (used to develop output signals which are electrical by nature).

FUNCTIONAL ELEMENTS OF MEASUREMENT SYSTEMS

It is possible to describe the operation of measuring instrument and associated

equipment in terms of the functional elements of instrumentation systems, and the

performance is defined in terms of static and dynamic performance characteristics. Ageneralized measurement system consists of the following

(i) Basic functional elements(ii) Auxiliary elements

Basic functional elements are those that are integral part of all instruments. They are

a) Primary sensing element or transducer elements

b) Signal conditioning or intermediate modifying element

c) Data presentation element


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accepts a small voltage signal as input and producers an output signal that is also a

voltage signal but is some constant times the input. An element that performs such a

function is called as variable manipulation element. A variable manipulation elementdoes not necessarily follow a variable conversion element, but may precede it, appear

elsewhere in the chain, or not appear at all.

More briefly, many times it becomes necessary to perform certainoperations on the signal before it is transmitted further. These processes may be linear

like amplification, attenuation integration, differentiation, addition and subtraction.

Some non linear process like modulation detection, sampling, filtering, choppingclipping etc. are also performed on the signal to bring it to the desired form to be

accepted by the next stage of measurement system. This process of conversion is called

signal conditioning. The term signal conditioning refers to many other functions in

addition to variable conversion and variable manipulation. In general signal conditioningelement is for manipulation / processing he output of the transducer in a suitable form.

Data transmission element

When the functional elements of an instrument are actually physically separated,it becomes necessary to transmit the data from one to another. An element performing

this function is called data transmission system. For ex. A shaft and bearing assemblyor a telemetry signal. The signal conditioning and transmission stage is commonly

known as intermediate system.

3) Data presentation element

If the information about the measured quantity is to be communicated to a

human being for monitoring, control or analysis purposes, it must be put into a formrecognizable by one of the human senses [ visual sense, hearing and touch ]. An element

that perform this translation function is called a data presentation element. This

function includes the simple indication of a pointer over a scale and recording of penmoving over a chart.

Example:-Consider the DArsonval Galvanometer used for voltage measurement. A

unknown voltage to be measured is applied to the ends of eth two wires which transmit

the voltage to a coil made up of a number of turns wound on a rigid frame. The coil is

suspended in the field of permanent magnet. The resistance of the coil converts theapplied voltage to a proportional current. The interaction between the current and the

magnetic field produces a torque which is converted to an angular deflection by the

torsion spring. Hence the spindle rotates and pointer shows deflection over the graduatedscale.

In this instrument, the coil and magnet assemble probably would be considered as

primary sensing element and the leas wires serve for data transmission purpose. The coiland tortional spring acts a variable conversion element whereas the pointer and scale

arrangement is the data representation element.

The block diagram representation of the functional elements fo the measurement

system is shown


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Analog output

Digital Output

Graphical Display

Printed output

Storage device

Calibrationsignal generator

Quantity to

be measuredForce

Pressure

VoltageCurrent

PowerEtc.

Primary

Sensing

SecondarySensing

Detector Transducer stageSignal Conditioning stage

Amplifier

Clipper

Filter

A/ D converter

12344e0kkjfkl9

CD / DVD

Data Representation stage

Data Transmission

element

External Power Supply

300 V, A.C

300 V, D.C

Calibration Element

PrimarySensing

Element

Variableconversion

element

Variablemanipulation

element

Datapresentation

element

InputOutputto user


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Generalized performance characteristics of instruments

To make intelligent decisions, there must be some quantitative basis forcomparing one instrument with the possible alternatives. The treatment of instrument

performance characteristics generally has been broken down into the sub-areas of static

characteristics and dynamic characteristics.Some applications involve the measurement of quantities that are constant or vary

quite slowly. Under these conditions, it is possible to define a set of performance criteria

that give a meaningful description of the quality of measurement without becomingconcerned with dynamic descriptions. These criteria are called the static characteristics.

Many other measurement problems involve rapidly varying quantities. Here the dynamic

relations between the instrument input and output must be examined generally by the use

of differential equations. Performance criteria based on these dynamic relationsconstitute the Dynamic characteristics.

Actually, static characteristics also influence the quality of measurement under

dynamic conditions, but the static characteristics generally show up non linear or

statistical effects in the otherwise linear differential equations giving the dynamiccharacteristics. These effects would make the differential equations unmanageable, and

so the conventional approach is to treat the two aspects of the problem separately. Thesephenomena are more conveniently studied as static characteristics, and the overall

performance of an instrument is then judged by a semi quantitative superposition of the

static and dynamic characteristics. This approach is , of course, approximate but anecessary expedient.

STATIC CHARACTERISTICS.

All the static performance characteristics are obtained by one form or another of a

process called static calibration.

Static Calibration

In general, static calibration refers to a situation in which all inputs (desire,interfering, modifying) except one are kept at some constant values. Then the one input

under study is varied over some range of constant values, which causes the outputs to

vary over some range of constant values. The input output relations developed in this

way comprise a static calibration valid under the stated constant conditions of all theother inputs. This procedure may be repeated, by varying in turn each input considered

to be of interest and thus developing a family of input output relations. The statement

all other inputs are held constant refers to an ideal situation which can be onlyapproached, but never reached in practice. Measurement method describes the ideal

situation while measurement process describes the (imperfect) physical realization of the

measurement method.In performing a calibration, the following steps are necessary

(i) Examine the construction of the instrument, and identify and list all the

possible inputs


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(ii) Decide, as best you can, which of the inputs will be significant in the

application for which the instrument is to be calibrated

(iii) Procure apparatus that will allow you to vary all significant inputs over theranges considered necessary.

(iv) By holding some inputs constant, varying others, and recording the output

develop the desired static input output relations.The characteristics may be identified or classified as either general or special.

General static characteristics are of interest in only a particular instrument. We

concentrate mainly on general characteristics.The definitions are brief descriptions of the various static performance parameters

of the instruments are as follows

Accuracy

Accuracy of a measuring system is defined as the closeness of the instrument

output to the true value of the measured quantity (as per standard). By True value we

mean the average value of an infinite number of measured values when the average

deviation tends to zero. Accuracy of the instrument mainly depends on the inherentlimitations of the instruments as well as on the shortcomings in the measurement process.

The accuracy of the instruments can be specified in either of the following forms

1. Percentage of true value = Measured value True value x 100

True value

2. Percentage of full scale deflection = Measured value True value x 100

Maximum scale value

However specification of the % of full scale deflection is less accurate that the % of true

value. Accuracy depends upon various systematic errors involved in the measurement

process.

Precision

It is defined as the ability of the instrument to reproduce a certain set of readingswithin a given accuracy ie a measure of reproducibility of the measurements.

Reproducibility is the degree of closeness with which a given value may be repeatedly

measured. Thus, a highly precise instrument is the one that gives the same output

information, for a given input information when the reading is repeated a large number oftimes. The precision of the instrument depends on the factors that cause random or

accidental errors. It is instructive to note that a precise measurement may not necessarily

be accurate and vice versa. To illustrate this statement, we take the example of a persondoing shooting practice.


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Precision is always expressed in terms of the deviation in measurement

S.no Accuracy Precision1. Accuracy refers to the conformity to

true value of quantity under

measurement

Precision refers to amount of

agreement between various readings

taken of some physical quantity underreference conditions

2. Accuracy gives the maximum errors ie

maximum departure of the final result

from its true value.

Precision of a measuring system gives

its capability to reproduce a certain

reading with a given accuracy

3. Accuracy depends on the various

systematic errors involved inmeasurement process

Precision of the instrument depends on

the factors that cause random oraccidental errors

4. Accuracy is determined by proper

calibration of the instrument

Precision is determined by statistical

analysis

Resolution or Discrimination

Resolution means, the smallest change in the input signal that can be detected by

the instrument. When input to instrument is increased from some non zero arbitraryvalue, the change in output is not detected at all until a certain input increment is

exceeded, this increment is called resolution. Meters with high resolution react readily

for changes and are therefore sensitive. Resolution is expressed as a fraction or % of fullscale.

Threshold

It is particular case of resolution. It is defined as the minimum value of inputbelow which no output can be detected, ie the minimum value of the input is known as

Threshold. This phenomenon is due to input hysteresis. However threshold also dependson the output sensing device and the observer.


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Hysteresis.

Hysteresis is a phenomenon which depicts different output effects when loadingand unloading in any system. This is defined as the algebric difference between the

average errors at corresponding points of measurement when approached from opposite

directions. It arises due to fact that all the energy put into the stressed parts when loadingis not recoverable upon loading. This effect can be minimized by taking reading

corresponding to ascending and descending values of eth input and then taking their

arithmetic average.

Linearity

Linearity is a measure of the departure of the various points on the calibrationcurve from the straight line fitted into those points by the method of least squares. It is

defined as the maximum deviation of any calibration point from reference straight line.

Manufacturers of instruments always attempt to design their instruments so that the

output is a linear function of the input. In most commercial instruments, linearity isgenerally implied. In such cases, linearity specifications are equivalent to accuracy

specifications.

Drift.

Drift means that with given input the measured value do not vary with time. An

instrument is said to have no drift if it reproduces same readings at different times for

same variation in measured value (i.e) perfect reproducibility means that the instrument

has not drift. Drift may be classified as(i) Zero Drift: If entire calibration shifts due to slippage, permanent set or due to

unwarming up, zero drift occurs

(ii) Span Drift: If there is proportional change in indication all along the upwardscale, the drift is called sensitivity or span drift

(iii) Zonal Drift: If the drift occurs only over a portion of span of instrument, it is

called zonal drift.

unloading

loading

Idealised

straight

line

Actualcalibration

curve

Normal

charac.

With zerodrift With

span drift

Normalcharac.

X axis = InputY axis = output


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Dead zone

It is defined as the largest change of input quantity for which there is no output ofthe instrument. The factors which produce dead zone are friction, backlash, hysteresis.

Dead TimeIt is defined as the time required by the measurement system to begin to respond

to the change in the measurand. Dead time, infact is the time before the instrument

begins to respond after the measured quantity has been changed.

Static Sensitivity

Static sensitivity (also termed as scale factor or gain) of the instrument is

determined from the results of static calibration. This static characteristic is defined as

the ratio of the magnitude of response (output signal) to the magnitude of quantity beingmeasured (input signal).

Static Sensitivity K = Change in output signal = qo

Change in input signal qi

In other words static sensitivity is represented by the slope of the input output curve ifthe ordinates are represented in actual units. It may be noted that in certain applications

the reciprocal of the sensitivity is commonly used. This is termed as inverse sensitivityor deflection factor

Time

Measured quantity

OutputInput

Dead time

Dead zone

Input

Output

qo

qi

Sensitivity Linear caseDead zone and Dead time


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DYNAMIC CHARACTERISTICS

Invariably measurement systems, especially in industrial aerospace and biologicalapplications are subjected to inputs which are not static but are dynamic in nature (ie) the

inputs that vary with time. When dynamic or time varying are to be measured, it is

necessary to find the dynamic response characteristics of the instrument being used formeasurement. The dynamic inputs to an instrument may be of the following types.

i) Periodic inputs : Varying cyclically with time or repeating itself after a constantinterval. The input may be harmonic or non harmonic type

ii) Transient input : Varying non cyclically with time. The signal is of a definite

duration and becomes zero after a certain period of time.

iii) Random input : Varying randomly with time, with no definite period andamplitude. This may be continuous but not cyclic.

All measurement systems include one or more energy storage elements. When an inputis applied to a system, the energy storage elements do not allow an immediate flow of

energy and therefore the measurement system does not respond to the input immediately.The measurement system goes through a transient state before it finally settles to itssteady state position. The transient response if defined as the part of response which goes

to zero as time becomes large. Some measurements are made under conditions that

sufficient time is available for the measurement system to settle to its final steady stateconditions. Under such conditions, the study of behavior of the system under transient

state is not of much important. Only steady state response of the system is considered.

The steady state response of the system is its response when time tends to infinity. But insome cases for example, suppose a body is subjected to a sudden severe mechanical

impact lasting for a few milliseconds. The body is accelerated and the transient response

is of utmost importance.

The measurement systems, when subjected to periodically varying inputs exhibitin their response a magnitude and phase relationship which is different from that of input

signal because of the energy storage elements. The output is not faith-full representation

of the input. The dynamic characteristics of the measurement lag are1) Speed of response

2) Fidelity

3) Measuring lag4) Dynamic error

Periodic signal Random signalTransient signal


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1) Speed of response

It is defined as the rapidity with which a measurement system responds to

changes in measured quantity.

2) Measuring lag

It refers to the retardation or delay in the response of a measurement system to

changes in the measured quantity. The measuring lag are of two types

(a) Retardation typeIn this type, the response of an instrument begins immediately after

a change in measured quantity has occurred

(b) Time delay typeIn this type, the response of an instrument begins after a dead time,

after the application of the input. Measuring lag of this type are verysmall and in order of fraction of seconds only, so it can be ignored.

3) Fidelity

It is defined as the degree of closeness with which the system indicates or

records the changes in the measured quantity without any dynamic error. In other

words, fidelity is the ability of the system to reproduce the output in the sameform as input.

4) Dynamic error

It is defined as the difference between the true value of the measuring

quantity changing with time and the value indicated by the measuring system ifno static error is assumed i.e static error is zero. It is also known as measurement

error.

Dynamic Analysis of instruments

The dynamic characteristics of an instrument refers to the performance of the

instrument when it is subjected to time varying input. For studying the dynamiccharacteristics of an instrument or the combination of instrument, it is necessary to

represent each instrument by its mathematical model, from which the governing relation

between its input and output is obtained. Then the dynamic characteristics can bedetermined experimentally with a known dynamic input signal. In many other areas of

engineering application, the most widely useful mathematical model for the study of

measurement system is the ordinary linear differential equation with constant co


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efficient. Thus, the following steps are essential for understanding the dynamic

behaviour of an instrument.

a) To formulate its governing equations, relating dynamic input and output signalsb) To obtain the dynamic output response, for the given input, by solution of the

governing equations

c) In case the output response is not satisfactory, it may be possible to improve thesame by what is known as compensation.

We assume that the relation between any particular input (desired, interfering &

modifying) and the output can, by application of suitable simple assumptions, be put inthe form

io

i

m

i

m

mm

i

m

moo

o

n

o

n

nn

o

n

n qbdt

dqb

dt

qdb

dt

qdbqa

dt

dqa

dt

qda

dt

qda ++++=++++

............ 1

1

11

1

1

The order of the instrument is the highest derivative of the above differential equation

which describes the dynamic behaviour of the instrument for a specified input.Normally, the instruments are subjected to inputs which are random in nature. As it is

not possible to predict the random input, the following test inputs are used to determinethe dynamic behaviour of the instruments: step input, ramp input, impulse input andsinusoidal inputs.

ZERO ORDER INSTRUMENTS

The simplest possible special case of the above equation occurs when all the as

and bs except ao and bo are assumed to be zero. Then the equation becomes

iooo qbqa = io

o

o qa

bq = or io kqq = where k = static sensitivity

Any instrument or system that closely obeys this equation over its intended range ofoperating conditions is defined to be a zero order instrument. Since the equation qo= kqiis algebric, it is clear that, no matter how q i might vary with time, the instrument output

(reading) follows it perfectly with no distortion or time lag of any sort. Thus, the zero

order instrument represents ideal or perfect dynamic performance.

A practical example of a zero order instrument is the displacement measuring

potentiometer. Here a strip of resistance material is excited with a voltage and providedwith a sliding contact. If the resistance is distributed linearly along length L, we may

write

i

i

o eL

xe = or io kxe =

whereL

ek i= volts / inch


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Ofcourse, the winding will have some inductance and capacitance (very low). But yet the

potentiometer is called as zero order instrument because1) the inductance and capacitance can be made very very small by proper design.

2) The speed (frequencies) of motion to be measured are not high enough to make

the inductive and capacitive effects noticeable.

FIRST ORDER INSTRUMENTS

In the reference equation choosen, if all as and bs other than a 1 , ao and bo are

taken as zero, we get

iooo

o qbqadt

dqa =+1

Any instrument that follows this equation is, by definition a first order system. Dividingthe equation by ao we get

ii

o

oo

o

o

kqqa

bq

dt

dq

a

a==+1 where K =

o

o

a

b= static sensitivity

Taking laplace transform

( ) ( ) ( )sKQsQsSQa

aioo

o=+

1

( ) ( ) ( )sKQsQs io =+1 where0

1

a

a= = time constant

Or ( )( )

( )sQs

KsQ io

+=

1

+

eo

xi

ei

L

kqi

Time

Time

qo


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Now by applying various inputs ie by substituting for ( )sQi , the corresponding ( )sQ0 isevaluated

Step response of the First order instrument

A unit step input is defined by a signal which has a value equal to unity for t>0.At t = 0, the value of signal is zero

i.e qi (t) = 1 t > 0

qi (t) = 0 t = 0

Taking Laplace transform we get ( )s

sQi1

= . Substituting this in general first order eqn.

( )( ) s

B

s

A

ss

KsQo

++=

+=

1

1

1. Solving this we get A = 1 and B = . Therefore

( )

+

+=s

ssQo

11

or ( ) t

o etq

=1

It is observed that the instrument output reaches 63.2% of its final steady state value after

a time and the instrument takes theoretically infinite time to reach steady state value.

A dynamic characteristic useful in characterizing speed of response of any system is the

settling time and is defined as the time for the system to reach and stay within a

tolerance band. A small settling time indicates fast response. In other words a large timeconstant corresponds to slow system response and vive versa.

The dynamic or measurement error is defined as

( ) ( ) ( )tqtqte oim=

( )

=

t

mete 11

( ) t

m ete

=

= transient error which dies out as t . The steady state error is

( ) 0===

tLim

tmLim

tss etee. Thus the first order system tracks the unit step input

with zero static error.

Ramp response of a first order system

A unit ramp input is defined by a signal which changes at a constant rate withrespect to time like constant velocity

i.e( )

( ) 00

0

=

ttq

tttq

i

i

Taking Laplace Transform

time

1

qit

1

0.63

T

Time

time

qit


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Impulse Response of first order system

A unit impulse is defined as a signal which is zero value everywhere except at t =0 where the magnitude is finite

i.e

( )

( ) 01

0

=

=

asdttq

tttq

i

i

Taking laplace transform we get

Qi(S)=1

Substituting this in the general equation of first order system

( )( )

+

=

+=

S

SSQo

1

1

1

1

Therefore( )

t

o etq

=

Example

As an example of a first order instrument, let us consider a liquid(mercury) in glass

thermometer. The input quantity here is the temperature Ti(t) of the fluid surrounding the

bulb of the thermometer and the output is displacement o of the thermometer fluid in thecapillary tube. The principle of operation of such a thermometer is the thermal expansion

of the filling fluid which drives the liquid column up and down in response to

temperature changes since this liquid column has inertia, mechanical lag will be involved

in moving the fluid from one level to another level. However, this is negligible whencompared to the thermal lag involved in transferring heat from the surrounding fluid

through the bulb wall into the thermometer fluid. Hence we can say the first order tracks

the input with a time lag.

time

qit

1/

time

1/

qit

qot


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SECOND ORDER INSTRUMENTS

A second order instrument is one that follows the equation

iooo

ooqbqa

dt

dqa

dt

qda =++ 12

2

2

Dividing whole equation by ao and taking laplace transform we get

( ) ( ) ( ) ( )sKQsQsSQa

asQS

a

aioo

o

o

o

=++ 122

( )

( )11

22 ++=

Sa

aS

a

a

K

sQ

sQ

oo

i

o

( )

( )

122 0

2

20

1

2

2

+

+

=

Sa

a

aa

a

a

a

S

K

sQ

sQ

o

i

o

( )

( )1

22

2

++=

SS

K

sQ

sQ

nn

i

o

Or( )

( ) 22

2

2 nn

n

i

o

SSsQ

sQ

++=

Where

ioDampingrataa

b

requencynaturalundampedb

a

itivityStaticsensa

bK

o

o

on

o

o

2

2

2=

=

=

The characteristic equation is given by22 2 nn SS ++

Step response of second order instrument

Substituting ( )S

sQ i1

= we get ( ) 22

2

2

1

nn

n

o

SSS

sQ

++

=

The response is of three types depending upon the location of poles (ie) roots of

characteristic equation. The three different cases are

(i) Over damped system > 1


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If the > 1, the roots of the characteristic equation are real and unequal.

An over damped system responds to any time varying input in a slugging manner

with out any oscillation about the final steady state position as shown.

(ii) Critically damped system = 1

If = 1, the roots are real and equal. The response of the system in this

case is rapid and the system reaches its final steady state condition smoothly

without oscillations as shown.

(iii) Under damped system < 1

If < 1, then the roots of characteristic equation are complex conjugatepair. The under damped system follows the input with oscillations about its final

steady state position.

The speed of response is determined by the rise time which is the time taken torise from o to 90% of its final value. For a overdamped system, the rise time is very

large, for critically damped it is optimum where as for underdamped system, though rise

time is very low, the oscillations should be controlled in order that the response isfollowing the input. An increase in value of damping ratio reduces the oscillations

but slows the response. n is an indication of the speed of response since doubling its

value will reduce the time t to half its value for achieving a given output response.Further the peak value of qo over and above the value of q i called peak overshoot should

compromise has to be made while choosing value of during design state in order to

achieve a reasonably fast response and small peak overshoot.

The second order system tracks step input without any error.

Ramp response of second order system.

Time Time Time

qi

t,qot

qi

t,qot

qi

t,qot

qit q

it q

itq

0t

q0t

q0t

Over damped system Critically damped system Under damped system


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The response looks similar to that of first order instrument except the transient

portion. After transient error has vanished, the output lags the input by a time periodgiven by 2 / nThe second order system tracks ramp input with steady state error = 2 T

ExampleA good example of a second order instrument is the force measuring spring

scale. We assume the applied force qi has frequency components only well below the

natural frequency of the spring itself. Then the main dynamic effect of the spring may betaken into account by adding one third of the springs mass to the main moving mass.

This total mass we call M. The spring is assumed linear with spring constant K

Newtons / meter. Assuming perfect film lubrication, a viscous damping effect isaccounted as constant B ( Newton / meter / second ). The scale can be adjusted so that qo

= 0 when qi = 0 (i.e force input = 0 ) force = (mass)(acceleration)

ERRORS IN MEASUREMENT SYSTEMS

No measurement can be made with perfect accuracy and there is always somedifference between actual value or true value and the observed vale. This difference is

called as error. It is necessary to reduce this error. Therefore, it is instructive to know

the various types of errors and the uncertainties that are in general, associated withmeasurement system. Further, it is also important to know how those errors are

propagated.

This is because if an error is detected, then it can be eliminated or its effects can beaccounted for in the form of suitable correction. On the other hand if an error goes

unrecognized then it would make experimental data ureliable.

Types of errors

Time

qit

qot

2


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These may be effects of temperature, pressure, humidity, dust, windforces,

magnetic or electrostatic fields.

Elimination

1) Conditions should be kept constant as nearly as possible

2) Using equipment which is immune to these effects3) Employing techniques which eliminate the effects of these disturbances /

applying computed corrections

4) Proper shields may be provided

c) Observational errors

There are many sources of observational errors. An error on account of

Parallax will be incurred unless the line of vision of the observer is exactlyabove the pointer. There are observational errors in measurement involving

timing of an event especially when sound and light measurements are involved

since no two observers possess the same physical response.

Elimination

1) Parallax error is eliminated by having the pointer and the scale in the sameplane

2) Digital display of output eliminates the errors on account of observational or

sensing powers

Accidental or Random errors

These errors are caused due to random variations in the parameter or the systemof measurement. Such errors vary in magnitude and may be either positive or negative.

The main contributing factors to random error are

Inconsistencies associated with accurate measurement of small

quantities

Presence of certain system defects such as large dimensional

tolerance / friction

Effect of randomly variable parameters

Some happenings or disturbances which we are unaware are lumped

together

This error cannot be eliminated as such

Miscellaneous type of Gross Errors

There are certain errors that cannot be strictly classified as either systematic or

random as they are partly systematic and partly random. Therefore, such errors aretermed miscellaneous type of gross errors. This class of errors is mainly callused by the

following

Personal or human errors ie due to oversight / transpose the reading

Errors due to faulty components / adjustments like misalignment of

moving parts, electrical leakage etc.


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Improper application of the instrument like extreme vibration,

mechanical shock, electrical noise.

Elimination

1) Great care should be taken in reading and recording data

2) Three or more readings should be taken for quantity under measurement

Statistical analysis of Measured data and errors

As seen from the above discussion, the systematic errors and the gross errors are removed

where as there remain random errors in the final result. There is no information available

on any of the disturbing factors. The outcome of certain measurement (With random

errors) may be predicted by statistical analysis. For this, a large number of measurementsare usually used. The collection of measured data is called the sample data. This

experimental data is obtained in two form of tests:

(i) Multi-sample test : - In this test, repeated measurement of a given quantity are done

using different conditions such as different instruments, different ways of measurementand by employing different observers.

(ii) Single-sample test :- A single measurement (or successive measurement ) doneunder identical conditions excepting for time.

Many of data may repeat a number of time.The number of repetition of a datum is

called its frequency. The sample data may

be represented by a graph known as

Histogram or Frequency distribution curvewith more and more data taken at smaller

and smaller increments the histogram wouldfinally change into a smooth curve, asindicated by the dashed line. The smooth

curve is symmetrical with respect to the

central value. For statistical analysis wecalculate some numbers known as statistical

descriptors. The simplest of the descriptors

is the arithmetic mean of the data

Arithmetic Mean

The arithmetic mean of a number of readings gives the most probable value of the

measured variable. The result will be closed to the actual value if the number of readingsis very large. Ideally an infinite number of reading would give the true value. However

in practice, only a finite but a large number of measurements can be taken. If x1,x2,x3xnare n readings, n being large, the arithmetic mean x is given by

n

xxxxx n

++++=

......321or

n

x

x

n

i

i== 1

Quantity under measurement

No.ofobservedreading


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Mostly sample data are given in the form of the frequency is number of time same

reading is measured say value 1x is measured 1f times, 2x is measured 2f times as so

on, for this frequency table of reading, the arithmetic mean is given as follows

n

nn

fff

fxfxfxfxx

+++

++++=

.......

......

21

332211or

=

=n

i i

ii

f

fxx

1

DeviationThe deviation of a reading from the mean value is a measure of error in the

reading (i.e) Deviation is defined as the departure of the observed reading from the

arithmetic mean of the group of the reading. Let the deviation of reading 1x be d1 and

that of reading 2x be d2 then

xxd =11

xxd =22 and so on

Algebric sum of deviation = d1 +d2 +..dn

= ( xx 1 )+( xx 2 )+..( xxn )

= (x1 +x2 +..xn)-n x

= 0

Average Deviation

The average deviation is an indication of the accuracy and precision of the

instrument. A precise instrument will yield a low average deviation. It is defined as the

average of the absolute values of the deviations of the readings.

==

++++=

n

i

i

n

n

dd

n

ddddd

1

321 .......

Standard deviationThe root mean square (r.m.s) deviation, in the statistical analysis is known as

standard deviation

n

d

n

ddd

n

i

i

n

==

+++=

1

2

22

2

2

1.....

Where n is very large and definitely greater than 20

Variance

The variance(V) is defined as the mean square deviation and is the square ofstandard deviation


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n

d

n

dddV

n

i

i

n

==

+++==

1

2

22

2

2

12 .....

Probable error

Consider two points r and +r so located that the area bounded by the curve, the x axis

and the ordinates erected at x = r and x = +r is equal to half of the total area under thecurve. That is half of deviations lie between x = r

A convenient measure of precision is the quantity r. It is called Probable error. In

terms of it is represented asr = 0.6745

Specifying measurement dataAfter carrying out statistical analysis of multi sample data, the results of

measurements must be specified. The results are expressed as deviations about a meanvalue. The deviations are expressed as(i) Standard deviation

The result is expressed as X . The error limit in this case is the standard

deviation. This means that 0.6828(about 68%) of the readings are within limits = 1

approximately.(ii) Probable error

The results is expressed as 6745.0X i.e rX . This means that 50% of thereadings lie within limits.(iii) 2 limit.

The result is expressed as 2X . In this case the probability range is increased

i.e about 95% of readings fall within limit.(iv) 3 limit.

The result is expressed as 3X . In this case the probability range is increased

more i.e about 99% of readings fall within limit.

Limiting errors.

Manufacturers specify the deviations from the nominal value of a particular quantity. The

limits of these deviations from the specified value are defined as limiting errors. Relativelimiting error is defined as the ratio of the error to the specified magnitude of a quantity.

1. A moving coil voltmeter has a uniform scale with 100 divisions and gives full scalereading of 200V. The instrument can read upto 1/5th of a scale division with fair degree

of certainity. Determine the resolution of instrument in volt

Major division = 200 / 100 = 2VEach major division has 5 small division

Therefore resolution = 2/5 = 0.4V


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2. The dead space in a certain pyrometer is 0.12% of span. The calibration is 500C to

1250 C. Determine the temperature change that might occur before it is detected.

Span = 1250 500 = 750

Dead space = 0.12 * 750 = 0.9 C

3. Given the following set of voltage measurements taken from the voltmeter, find their

(i) average value, (ii) average deviation, (iii) standard deviation, (iv) probable error, (v)probable error of mean

Quantity deviation deviation2

153 4.6 21.16

162 -4.4 19.36

157 0.6 0.36

161 -3.4 11.56155 2.6 6.76

Av 157.6 3.12 59.2

Therefore Arithmetic mean = 157.6Average deviation = 3.12

Standard deviation = sqrt (59.2 / 5 1 ) = 3.847

Probable error r = 0.6745 S.D = 2.595Probable error of mean= r / sqrt(5 1) = 1.297

4. During a test run, measurement of weight were made 100 times with variation in

apparatus and procedure. After applying corrections for known systematic errors, the

following data were obtained

Weight (kg) 397 398 399 400 401 402 403 404 405

Frequency 1 3 12 23 37 16 4 2 2

Calculate: (i) Arithmetic mean, (ii) Mean deviation (iii) Standard deviation (iv)

probable error of one reading (v) probable error of mean

T f T * f d [d] d * f d2 d2*f

397 1 397 -3.78 3.78 3.78

14.28

84

14.28

84

398 3 1194 -2.78 2.78 8.347.728

423.18

52

399 12 4788 -1.78 1.78 21.363.168

438.02

08

400 23 9200 -0.78 0.78 17.940.608

413.99

32401 37 14837 0.22 0.22 8.14 0.048 1.790


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4 8

402 16 6432 1.22 1.22 19.521.488

423.81

44

403 4 1612 2.22 2.22 8.884.928

419.71

36

404 2 808 3.22 3.22 6.4410.36

8420.73

68

405 2 810 4.22 4.22 8.4417.80

8435.61

68

40078102.8

4191.1

6Average =

400.78

1.0284

Mean deviation= 1.0284Standarddeviation =

1.382606

Probable error of onereading= 0.932568

Probable error of mean=0.0932

57

functional elements of measurement system,

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