Types of Error

Measurement And

Instrumentations

DR. Eng. Samir Elshamy

DR. Eng. Samir Elshamy1

Definition of TermsReadability of an instrument:This term indicates the closeness with which the scale of the instrument

may be read; an instrument with a 12-cm scale would have a higher readability than an instrument with a 6-cm scale and the same range.

The least count:is the smallest difference between two indications that can be detected on the instrument scale.


The sensitivity of an instrument:is the ratio of the linear movement of the pointer on an analog instrument to the change in the measured variable causing this motion.For example, a 1-mV recorder might have a 25-cm scale length. Its sensitivity would be 25 cm/mV


Accuracy and Precision


The accuracy of an instrument

indicates the deviation of the reading from a known input. Accuracy is

frequently expressed as a percentage of full-scale reading, so that a 100-kPa

pressure gage having an accuracy of 1 percent would be accurate within ±1

kPa over the entire range of the gage.


The precision of an instrument

indicates its ability to reproduce a certain reading with a given accuracy. As an

example of the distinction between precision and accuracy, consider the

measurement of a known voltage of 100 volts (V) with a certain meter. Four

readings are taken, and the indicated values are 104, 103, 105, and 105 V. From

these values it is seen that the instrument could not be depended on for an

accuracy of better than 5 percent (5 V), while a precision of ±1 percent is

indicated since the maximum deviation from the mean reading of 104 V is only

1 V. DR. Eng. Samir Elshamy6


Calibration

The calibration of all instruments is important, for it affords the opportunity to check the instrument against a known standard and subsequently to reduce errors in accuracy. Calibration procedures involve a comparison of the particular instrument with either(1)a primary standard, (2) a secondary standard with a higher accuracy than the instrument to be calibrated(3) a known input source.


For example, a flow meter might be calibrated by

(1) comparing it with a standard flow-measurement facility

of the National Institute for Standards and Technology (NIST),

(2) comparing it with another flow meter of known accuracy,

(3) directly calibrating with a primary measurement such as

weighing a certain amount of water in a tank and recording the

time elapsed for this quantity to flow through the meter.


The Generalized Measurement System منظام القياس المعم

Most measurement systems may be divided into three parts:1. A detector-transducer stage, which detects the physical variable

and performs either a mechanical or an electrical transformation to convert the signal into a more usable form.

2. Some intermediate stage, (Signal conditioning)which modifies the direct signal by amplification, filtering, or other means so that a desirable output is available.

3. A final or terminating stage, which acts to indicate, record, or control the variable being measured. The output may also be digital or analog. DR. Eng. Samir Elshamy10

Bourdon-tube pressure gage

1) The bourdon tube is the detector-transducer stage because it converts the pressure signal into a mechanical displacement of the tube.

2) The intermediate stage consists of the gearing arrangement, which amplifies the displacement of the end of the tube so that a relatively small displacement at that point produces as much as three-quarters of a revolution of the center gear.

3) The final indicator stage consists of the pointer and the dial arrangement, which, when calibrated with known pressure inputs, gives an indication of the pressure signal impressed on the bourdon tube.




Analysis of Experimental DataIn this section we present a discussion of some of the types of

errors that may be present in experimental data and begin to

indicate the way these data may be handled.

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Types of Errors

Random

Systematic

Gross error

Environmental

Observational

Instrumental

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An experimental error is an experimental error. If the experimenter knew what

the error was, he would correct it and it would no longer be an error. In other

words, the real errors in experimental data are those factors that are always

vague to some extent and carry some amount of uncertaintyعدم اليقين.

Our task is to determine just how uncertain a particular observation may be

and to devise a consistent way of specifying the uncertainty in analytical form.

A reasonable definition of experimental uncertainty may be taken as the

possible value the error may have. This uncertainty may vary a great deal

depending on the circumstances of the experiment. Perhaps it is better to

speak of experimental uncertainty instead of experimental error because the

magnitude of an error is always uncertain. Both terms are used in practice,

however, so the reader should be familiar with the meaning attached to the

terms and the ways that they relate to each other.

Error Analysis on a Commonsense Basis تحليل الخطأ علي أساس المنطق

15


Consider the calculation of electric power from

P = EIwhere E and I are measured asE = 100 V ± 2 VI = 10 A ± 0.2 AThe nominal value of the power is 100×10 = 1000W. By taking the worst possible variations in voltage and current, we could calculatePmax = (100 + 2)(10 + 0.2) = 1040.4 WPmin = (100 − 2)(10 − 0.2) = 960.4 WThus, using this method of calculationthe uncertainty in the power is= +4.04 % & −3.96 %.


Static error (A) :

is defined as the difference between the best measured value

and the true value A=A1-A

Where A1=measured value & A =true value

Relative static error(uncertainty)= (A/A)


The result R is a given function of the independent variables x1, x2, x3, . . . , xn. Thus, R = R(x1, x2, x3, . . . , xn)Let ωR be the uncertainty in the result and ω1, ω 2, . . . , ωn be the uncertainties in the independent variables.

If this relation is applied to the electric-power relation of the previous section, the expected uncertainty is 2.83 percent instead of 4.04 percent.


Example 3.1 :UNCERTAINTY OF RESISTANCE OF A COPPER WIRE.

The resistance of a certain size of copper wire is given as

R = R0[1 + α(T − 20)]

where R0 = 6 ± 0.3 percent is the resistance at 20◦C,

α = 0.004◦C−1 ± 1 percent is the temperature coefficient of

resistance, and the temperature of the wire is T = 30±1◦C.

Calculate the resistance of the wire and its uncertainty.


Thanks for your kind attention


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Types of Error