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2007 Pearson Education, Inc. All rights reserved.

Measurement andInstrumentation

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Course Books

1)Experimental Methods For Engineers-Holman

2) lme Teknii-Osman Fevzi Genceli

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1.1. Simple Definition and Brief Story

Measurement is an evaluation of an unknownquantity comparing with a known quantity.

Measurement techniques have been of greatimportance since the start of civilization, speciallysince the industrial revolution.

During this period in 19th century, new

measurement techniques and their instruments weredeveloped rapidly.

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1.1. Brief Story of Measurement

Since the industrial revolution, newmeasurement techniques and their instruments

were developed rapidly. The many developmentswere also realized in electronics, productiontechniques and computer science.

These new conditions brought with some new

requirements in measurement techiques such astighter accuracy and lower production costs.

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1.1. Brief Story of Measurement

Therefore, all instrument manufacturers focuson these problems and they improved new

solutions using digital computing techniques.

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1.2. Basic Measurement System Design

A measurement system exists to provide informationabout the physical value of some variable being

measured.A measurement system usually consists of severalseparate components, although only one componentmight be involved for some very simple measurementtasks.

These components might be contained within oneor more boxes, and the boxes holding individualmeasurement elements.

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The first element in any measurement system is thesensor: this gives an output of the measurand. Someexamples of sensors are a liquid-in glass thermometer,and a strain gauge.

Variable conversion elements convert the inconvenientform of a measured variable to a more convenient form.

For instance, the measuring strain gauge has an outputin the form of a varying resistance. Because the resistancechange cannot be measured easily, it is converted to achange in voltage by a variable conversion element.

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In many cases, the primary sensor and variableconversion element are combined, this combination isknown as a transducer.

Signal processing elements improve the quality of theoutput of a measurement system in some way. A verycommon type of signal processing element is theelectronic amplifier, which amplifies the output of thetransducer.

In addition to these components, some measurementsystems have one or two other components, first to

transmit the signal to some remote point and second todisplay or record the signal.

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Suppose we want to measure the profile of a surface at ananometer scale using an atomic force microscobe. If thiscantilever is translated over the surface, the cantilever will deflect,

indicating the height of the surface. The cantilever beam is asensor and the movement of the cantilever will deflect the laser. Atransducer converts the sensed information into a detectablesignal.

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Consider the simple bourdon-tube pressure gage. This gage offers amechanical example of the generalized measurement system. In thiscase the bourdon tube is the detector-transducer stage because it

converts the pressure signal into a mechanical displacement of thetube. The intermediate stage consists of the gearing arrangement,which amplifies the displacement of the end of the tube. The finalindicator stage consists of the pointer and the dial arrangement.

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1.3. Measurement System Applications

Current applications of measurement instruments can beclassified into three major areas.

The first of these is their use in trade, applyinginstruments that measure physical quantities such as

length, volume, and mass in terms of standard units.The second application area of measuring instruments isin monitoring functions. These provide information thatenables people to take some precautions or decisions about

their actions accordingly.Use as part of automatic feedback control systems formsthe third application area of measurement systems.

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1.3. Measurement System Applications

Figure 1.4 shows a functional block diagram of a simpletemperature control system in which temperature Taof aroom is maintained at reference value Td. The value of thecontrolled variable, Ta, as determined by a temperature-

measuring device, is compared with the reference value, Tdand the difference, e, is applied as an error signal to theheater.

The heater then modifies the room temperature until Ta=

Td. The characteristics of the measuring instrumentsusedin any feedback control system are of fundamentalimportance to the quality of control achieved.

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1.4. Static Characteristics of the Instruments

Readability:This term indicates the wideness of the

instrument scale. Wide range scale should be preferred. Aninstrument with a 12-in scale has a higher readability than aninstrument with a 6-in scale.

Least Count: It is the smallest difference between two valuesthat can be detected on the instrument scale. It means that theleast value of the change in physical magnitude sensed byinstrument. Both readability and Least Count depend on scalelength, graduation and size of pointer.

Accuracy:The difference between the real value of measurand

and the value can be read from instrument scale.

Sensitivityis the ratio of the linear movement of theinstrument pointer on an analog instrument to the change inthe measured variable.

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307 200

314 230

321 260

328 290
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Range: The minimum and maximum values of theinstrument. These limits define operating range of the

system.Resolution: The smallest variation of the measured physicalmagnitude can be read in the instrument scale. One of themajor factors influencing the resolution of an instrument is

how finely its output scale is divided into subdivisions.Linearity is the deviation of the input, output values from alinear function. It is normally desirable that the outputreading of an instrument is linearly proportional to the

quantity being measured. Nonlinearity is defined as themaximum deviation of the output from the straight line.

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Precision is the property of an instrument is to give the samevalues for different measurements. As an example of the

difference between precision and accuracy, consider a knownvoltage of 100V for 10 different readings respectively102,104,104,103,105, 106,103,105,102,106 V.

Mean Value=104V

Accuracy= (106-100)/100=%6Precision= (106-104)/104=%2

Accuracy=(max.measured value-real value)/real value

Precision =(max.measured value-mean value)/mean value

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Toleranceis a term that is closely related to accuracy anddefines the maximum error that is to be expected in some value.

ExampleA packet of resistors bought in an electronics component shopgives the nominal resistance value as I000 and themanufacturing tolerance as 5%. If one resistor is chosen at

random from the packet, what is the minimum and maximumresistance value that this particular resistor is likely to have?

Solution

The minimum likely value is 1000 - 5% = 950 .

The maximum likely value is 1000 + 5% = 1050 .

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Hysteresis: Different values in reading depend on the approaching to themeasured quantity from above or below. It means that the instrument showsthe above or below values from the real values. Hysteresis may be result ofmechanical friction, magnetic and thermal effects, elastic deformation.

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Dead-Space: It is the space which the instrument can notdisplay the small change in pysical magnitude. Dead-Space also

may be result of mechanical friction, magnetic and thermaleffects, elastic deformation.

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Calibration is the operation to decrease the measurement errorsand to control the instruments. It is applied to the measurement

systema)According to the Basic Standarts

b)Using an other instrument has the higher accuracy

c)Respect to the known input value

Static Calibration : The most common type of calibration isknown as a static calibration. In this procedure, a known value isinput to the system under calibration and the system output isrecorded. By applying a range of known input values and by

observing the system output values, a direct calibration curvecan be developed for the measurement system. The staticcalibration curve describes the static inputoutput relationshipfor a measurement system.

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2.Dynamic Characteristics of Instruments

2.1. Measurement System Model

Consider the following model of a measurement system which consist of ageneral nth-order linear ordinary diff. equation in terms of a general outputsignal, the following general relation between output, represented by variable

y(t)and input signal, represented by the forcing function, F(t) can bewritten as:

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Example 2.1

Consider the seismic accelerometer in Figure 2.1. It is used in seismic and

vibration engineering to determine the motion of large bodies. As theaccelerometer mass reacts to motion, it places the piezoelectric crystal intocompression or tension, causing a surface charge to develop on the crystal.The charge is proportional to the motion. Let y denote the position of themass and x denote the displacement of the body. Solving Newtonssecond

law for the second-order linear, ordinary differential equation

Comparing this to the general form for a second-order equation fromEquation 2.1

we can see that a2=m, a1=b1=c, a0=b0=k.

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The stiffness of the spring, k, provides a restoring force

to move the accelerometer mass back to equilibrium

while frictional damping, c, opposes any displacement

away from equilibrium.
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where K=1/a0. K is called the static sensitivity or static gainof the system.

This property is the relation between the change in output associated with achange in static input. Any instrument that behaves according to (2.2) it is

a zero-order type.
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2.2.1. Zero-Order Systems

In ZOS, following a step change in the input (mesured value) at time

t, the instrument output moves instantly to a new value at the samet(Figure 2.2).

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2.2.1. Zero-Order Systems

They are the simplest model of the measurement systems and for realsystems, the zero-order system concept is used to model the time-independent measurement system response to static inputs. In fact,the zero-order concept appropriately models any system during astatic calibration.

Determination of K

The static sensitivity is found from the static calibration of the

measurement system. It is the slope of the calibration curve, K=dy/dx.

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Example 2.2

A pencil-type pressure gauge commonly used to measure tire pressure can bemodeled at static equilibrium by considering the force balance on the gaugesensor, a piston that slides up and down a cylinder. Considering the piston

free-body at static equilibrium in Figure 2.3b, the static force balance, F= 0,gives

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Eq. (2.3) can be written as
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When we apply a step function input to a measurementsystem, we obtain information about how quickly a system

will respond to a change in input signal.

Let us apply a step function as an input to the general first-order system.
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Setting F(t)=AU(t) in Equation 2.4 gives

The solution of the differential equation, y(t), is the responseof the system. Equation 2.5 describes the behavior of thesystem to a step change in input. This means that y(t) is infact the output indicated by the display stage of the system.

We have simply used mathematics to simulate this response.
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The general solution of (2.7) yields the measurement systemoutput signal, that is, the time response to the applied input,

y(t):

where the value for C depends on the initial conditions. Theoutput signal, y(t), of Equation (2.8) consists of a transientand a steady response. The first term on the right side is the

transient response. Transient response is important onlyduring the initial period following the application of the newinput. We already have information about the systemtransient response from the step function study.
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We focus our attention on the second term, the steadyresponse. This term persists for as long as the periodic input

is maintained. Equation 3.8 can be rewritten in a generalform:
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2.2.3 Second-Order Systems

A system modeled by a second-order differential equation is

called a second-order . Examples of second-order instrumentsinclude accelerometers and pressure transducers.
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Step Function Input
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Step Function Input
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Step Function Input
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di i l i f h f ll i i i (1)
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We discuss uncertainty analysis for the following measurement situations: (1)

design stage, where tests are planned but information is limited; (2) advanced

stage or single measurement, where additional information about process control

can be used to improve a design-stage uncertainty estimate; and (3) multiple

measurements, where all available test information is combined to assess the

uncertainty in a test result.

3.2. Design-Stage Uncertainty

Design-stage uncertainty analysis refers to an analysis performed in theformulation stage prior to a test. It provides only an estimate of the minimum

uncertainty based on the instruments and method chosen. If this uncertainty

value is too large, then alternate approaches will need to be found. So, it is useful

for selecting instruments and selecting measurement techniques.

In the design stage, distinguishing between systematic and random errors mightbe too difficult to be of concern. So for this initial discussion, consider only

sources of error and their assigned uncertainty in general.

E h i di id l i i h h ff h
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Each individual measurement error interacts with other errors to affect the

uncertainty of a measurement. This is called uncertainty propagation. Each

individual error is called an elementalerror. For example, the sensitivity error

and linearity error of a transducer are two elemental errors, and the numbers

associated with these are their uncertainties.

Consider a measurement ofxthat is subject to someKelements of error, each of

uncertainty uk, where k = 1,2, . . . , K. A realistic estimate of the uncertainty in

the measured variable, ux, due to these elemental errors can be computed using

the root-sum-squares (RSS) method to propagate the elemental uncertainties:

Th d i i f i h d i
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The design-stage uncertainty, ud, for an instrument or measurement method is an

interval found by combining the instrument uncertainty with the zero-order

uncertainty:

Final uncertanties are usually reported at a 95 % probability.
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Uncertainty Propagation

Suppose we have a set of measurement and these measurements are then

used to calculate the some results of the experiments. We wish to estimate

the uncertainty in the calculated result. The result R is a given function of

the independent variables x1,x2,x3,,xn.

Let uR be the uncertainty in the result and u1,u2,,unbe the

uncertainties in the variables. Then we obtain :
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4. Statistical Analysis of Experimental Data

When a set of readings of an instrument is taken, theindividual readings will vary from each other, and theexperimenter should have the information about mean value ofall the readings. If each reading is denoted byxiand there are n

readings, the arithmetic mean is given by

The deviation difor each reading is defined by
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The average of the absolute values of the deviations is given by

The standart deviationpresents the distribution of the datadeviation from the arithmetic mean value. The standart

deviationor root-mean-square deviation is defined by

and the square of the standard deviation 2is called thevariance.
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A Different Presentation

For any set of n measurementsx1, x2.xn, of a constant quantity, the mostlikely true value is the mean given by

The median is an other approximation to the mean that can be written down

without having to sum the measurements. The median is the middle valuein the data set. For a set of n measurementsx1, x2.xn, the median value isgiven by

Thus, for a set of nine measurementsx1, x2...x9arranged in order ofmagnitude, the median value is x5. For an even number of measurements,the median value is midway between the two center values, that is, for 10measurementsx1x10, the median value is given by (x5+ x6)/2.
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Thus, the smaller the spread of the measurements, the more confidence wehave in the mean or median value calculated.

Let us now see what happens if we increase the number of measurements byextending measurement set B to 23 measurements. We will call thismeasurement set C.

409 406 402 407 405 404 407 404 407 407 408 406 410 406 405 408 406409 406 405 409 406 407 (Measurement set C)

Now, mean = 406.5 and median = 406

We can say that the median value tends toward the mean value as thenumber of measurements increases.
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These definitions for the ariance and standard de iation of data are made

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These definitions for the variance and standard deviation of data are madewith respect to an infinite population of data values whereas, in allpractical situations, we can only have a finite set of measurements.

A better prediction of the variance of the infinite population can beobtained using the formula given below:

This leads to a similar better prediction of the standard deviation:

:
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:

:
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:

Sometimes it is appropriate to use a geometric mean when studying
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Sometimes it is appropriate to use ageometric meanwhen studyingphenomena which grow in proportion to their size. This would apply tocertain biological processes and to growth rates in financial resources. Thegeometric mean is defined by
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Example 4.1: The following readings are taken of a certain physical length.Compute the mean reading, standard deviation, variance.
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5. Probability

Probabilityis a measure or estimation of likelihood ofoccurrence of an event.

Suppose we throw a coin, the probability that we get a head is for one toss, the probability of tail will be also . Somebody

may get the same numbers of heads and tail for if the coin isthrown large number of times. In other words, thefrequency ofoccurrence is the same for both heads or tails for a very largenumber of tosses.
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Suppose we toss a horseshoe some distancex. Even though we make aneffort to toss the horseshoe the same distance each time, we would not

always meet with success. On the first toss the horseshoe might travel adistancex1, on the second toss a distance ofx2, and so forth. If one is agood player of the game, there would be more tosses which have anxdistance equal to that of the objective.

Since eachx distance will vary from otherx distances, we may calculatethe probability of a toss landing in a certain increment of x betweenx and

x + Dx.

When this calculation is made, we might get something like the situationshown in Fig. 5.1. For a good player the maximum probability is expectedto surround the distancexmdesignating the position of the target.
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A particular probability distribution is the binomial distribution. Thisdistribution gives the number of successes n forN possible independent

events when each event has a probability of successp.

It will be noted that the quantity (1p) is the probability of failure of eachindependent event.

Example 5 1 A coin is tossed three times Calculate the probability of
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Example 5.1. A coin is tossed three times. Calculate the probability ofgetting zero, one, two, or three heads in these tosses.

Solution

The binomial distribution applies in this case since the probability of eachtoss of the coin is independent of previous or successive toss. Theprobability of getting a head on each throw isp = 1/ 2 andN = 3, while ntakes on the values 0, 1, 2, and 3.
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Matlab Application

5.1.1. Histograms
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We have noted that a probability distribution like Fig. 5.1 is obtained whenwe observe frequency of occurrence over a large number of observations.When a limited number of observations is made and the raw data are

plotted, we call the plot a histogram.For example, the following distribution of throws might be observed for ahorseshoes player:

These data are plotted in Fig. 5.2 using increments of 10 cm inx. The same

d t l tt d i Fi i f
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data are plotted in Fig. 5.3 using ax of 20 cm.
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Calculation the number of intervals

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Using Matlab Codes
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5.1.2. The Gaussian or Normal Error Distribution

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Suppose we have a set of experimental results and they have been subjectedto many random errors. Thegaussian or normal error distributionisdesigned to find the probability that a result (measurement) will lie withina specific interval ( betweenx andx+dx )

In this expressionxmis the mean reading and is the standard deviation.P(x) is called astheprobability density.
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PROBABILITY FOR DEVIATION FROM MEAN VALUE

Example A: Calculate the probabilities that a measurement will fall
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Example A: Calculate the probabilities that a measurement will fallwithin one, two, and three standard deviations of the mean value andcompare them with the values in Table 5.3
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5.1.3 Rejection of Data (Chauvenets Criterion)
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There exist a number of methods to eliminate values that have lowprobability. The most common method is Chauvenet Criterion.
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Calculate mean value (xm)

Calculate di=|xi-x |
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Calculate di |xi xm|

Find di/s

5.1.3 Rejection of Data (Chauvenets Criterion)
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There exist a number of methods to eliminate values that have lowprobability. The most common method is Chauvenet Criterion.

5.1.4 Confidence Interval and Level of Significance

The confidence interval expresses the probability that the mean value will
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The confidence interval expresses the probability that the mean value willlie within a certain number of values and is given by the symbol z.

where D is the standart deviation of the mean values, s is the standartdeviation of the measurement group ,z is the confidence linterval and nis the

numbers of measurements.
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5.1.3 Probability Graph Paper
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5.1.5 The Chi-Square Test

This test allows us to find the distributions of the random errors
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This test allows us to find the distributions of the random errors.

Calculations have been made of the probability that the actual

measurements match the expected distribution, and these probabilities aregiven in Table 5.5. In this table F represents the number of degrees offreedom in the measurements and is given by F = n k (5.9)

where n is the number of cells and k is the number of imposed conditionson the expected distribution.

( n : gzlem eidi (grubu)

k: beklenen dalm etkileyen koullarn says)
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If 0.1 < P

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If P0.98, it doesnt match to the normal distribution

Example: A coin is tossed 20 times, resulting in 6 heads and 14 tails.Using the chi-square test, estimate the probability distribution of the coin.

5.1.6 Method of Least Squares
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a) The Systems Have One Variable

a) The Systems Have Two Variables
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a) The Systems Have Two Variables

Suppose that the two variablesx andy are measured over a range ofvalues. We wish to obtain a simple analytical expression fory as a function of

x.
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5.1.7 The Correlation Coeffient
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It is explained that how a suitable correlation graphic can be obtianedbetweenxandyin the previous chapter, using least-squares analysis or

graphical curve fitting. Now, we want to know how good this fit with theexperimental results. The parameter which conveys this information is thecorrelation coefficient r defined by

Example: Calculate the correlation coefficient for the preivous example.
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From previous example

5.1.8 Standart Deviation of the Mean
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We have taken the arithmetic mean value as the best estimate of the truevalue of a set of experimental measurements. But one very important

question has not yet been answered: Howgood (or precise) is this arithmeticmean value which is taken as the best estimate of the true value of a set ofreadings? Statistical analysis gives us an equation to respond this question.

Example:
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1 2 3 4 5 6 7 8 9 10

11.2 9.3 12.3 9.2 11.0 14.1 8.9 9.7 10.3 10.0

Standart Deviation s=1.61 xm = 10.6

Using related table for s, 2 s , 3 s
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Example:

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6.1.1. Vernier Caliper, Micrometer Caliper

Because the Rulers can be scaled to 0 5 mm at least the more
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Because the Rulers can be scaled to 0.5 mm at least, the moreaccurate instruments are needed for more accurate

measurements sucs as Calipers and Micrometers.

a) Vernier Caliper
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6.1.2. Optical Methods

An optical method for measuring dimensions very accurately
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p g y yis based on the principle of light interference. The instrument

based on this principle is called an interferometer and is usedfor the calibration of meausurement devices and otherdimensional standards.

Consider the two sets of light beams shown in Fig. 6.1. InFig. 6.1a the two beams are in phase so that the brightness at
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point P is augmented when they intersect.

In Fig. 6.1b the beams are out of phase by half a wavelength sothat a cancellation is observed, and the light waves are said tointerferewith each other. This is the essence of theinterference principle.

Now, let us apply the interference principle to dimensionalmeasurements. Consider the two parallel plates shown in Fig.
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6.2. One plate is a transparent, glass accurately polished flat.The other plate has a reflecting metal surface. The glass plateis called an optical flat. Parallel light beamsA and B areprojected on the plates from a suitable source. The separationdistance between the plates d is assumed to be quite small.

The reflected beamA intersects the incoming beam B at

point P. Since the reflected beam has traveled farther thanbeam B by a distance of 2d, it will create an interference atpoint P if this incremental distance is an odd multiple of/2.If the distance 2d is an even multiple of/2, the reflected

beam will augment beam B. Thus, for 2d =/2, 3/2, etc., thescreen Swill detect no reflected light.
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For practical purposes the interferometer, as indicated

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p p pschematically in Fig. 6.3, is employed. Monochromatic light

from the source is collimated by the lens L onto the splitterplate S2, which is a half-silvered mirror that reflects half ofthe light toward the optically flat mirrorM and allowstransmission of the other half toward the workpiece W. Bothbeams are reflected back and recombined at the splitter plate

S2 and then transmitted to the screen. Fringes may appear on the screen resulting from differences

in the optical path lengths of the two beams. If theinstrument is properly constructed, these differences will

arise from dimensional variations of the workpiece.

6.1.3. Pneumatic Displacement Gage
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Consider the system shown in Fig. 6.4. Air is supplied at a constantpressurep1. The flow through the orifice and through the outlet ofdiameter d2 is governed by the separation distancex between the outlet

and the workpiece. The change in flow withxwill be indicated by achange in the pressure downstream from the orifice P2. Thus, ameasurement of this pressure may be taken as an indication of theseparation distancex. For purposes of analysis we assume incompressibleflow.

6.1.4. The Planimeter for Area Measurement
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Let the length of the tracing arm BT be L and the distance frompoint B to the wheel be a. The diameter of the wheel is D. Thedi OB i k R N h BT i d
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distance OB is taken as R. Now, suppose the arm BT is rotatedan angle d and the arm OB through an angle d as a result ofmovement of the tracing point. The area swept out by the armsBT and OB is
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6.1.5 Graphical and Numerical Methods for Area Measurement

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A very simple method of plane-area measurement is to place

the figure on coordinate paper and to count the number ofsquares enclosed by the figure. An appropriate scale factor isthen applied to determine the area. Numerical integration iscommonly applied to determine the area under an irregularcurve. Perhaps the two most common methods are the

trapezoidal rule and Simpsons rule. Consider the area shown inFig. 6.6. The area under the curve is

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6.1.5 Surface Areas
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6.1.5 Surface Areas
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Consider the general three-dimensional surface shown in Fig.

6.7. The surface is described by the function z =f(x, y)

If the function z is known and well behaved, the integral in Eq.(1) may be evaluated directly. Let us consider the case where the

function is not given but specific values of z are known forincremental changes inx andy.

6.1.5 Surface Areas

The increments inx andy are denoted byx andy, while thel f i d d b h h b i f h
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value of z is denoted by , where the subscript n refers to thexincrements and the superscriptp refers to they increment. Wethus have the approximations
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