Chapter 1Measurement Practice and Error

Introduction

• Measurement systems • Systems which physical quantities are measured to

obtain data and transmitted to recording and displayunit.

• Physical quantities: e.g. light, sound, heat, motion, etc. All physical quantities must be changed into electrical signals (quantities) such as voltage, current, resistance, capacitance, inductance, frequency, phase and etc.

Definition of Measurement

1. Accuracy • Closeness with which an instrument reading approaches

the TRUE VALUE of the variable being measured (how close the measured value to the true value).

• Given V = 10 V, R = 5 Ω. By calculation, I = 2.0 A. I. Calculated (theoretical) value: Actual (TRUE) value:

2.0AII. Measured (practical) value: Using analog Ammeter,

I =2.15 AIII. Measured (practical) value: Using digital Ammeter,

I = 2.00789 A. More accurate: close to actual value

Definition of Measurement

2. Precision i. a measure of the reproducibility of the measurements

(ability to give the same reading if it is measured for many times); i.e. given a fixed value of a variable, precision is a measure of the degree to which successive measurements differ from one another.

ii. Measurement precision depends on the smallest change that can be observed in the measured quantity.

Definition of Measurement

• Conclusion: Ammeter1 is more precise than Ammeter2

Readings Ammeter1 (A) Ammeter2 (A)1 2.15 2.152 2.15 2.143 2.15 2.134 2.15 2.135 2.15 2.10

Definition of Measurement

3. Resolution i. The resolution means the smallest change in a

measured value that the instrument can detect. ii. It may be expressed as an actual value or as a fraction

or percentage of the full scale value. iii. Resolution is also known as sensitivity.

Definition of Measurement

4. Significant Figure i. An indication of the precision of the measurement is

obtained from the number of significant figures in which result is expressed.

ii. Significant figures convey actual information regarding the magnitude and the measurement precision of a quantity.

iii. More the significant figures, greater the precision of measurement.

Definition of Measurement

Example 1 :The significant figure of 0.0005 is 4 and the significant figure for 0.05 is 2.So 0.0005 is more precise than 0.05 since it has more significant figures.

Definition of Measurement• Error

i. deviation of theoretical (true) value from the practical or measured variable.

ii. Error = practical value - theoretical value

%

100

iii. Generally, if error greater than 20% it is rejected/failed.iv. In certain cases error is not allowed (0% error) or very

small percentage error is allowed depends on the requirements/applications.

Type of Error

1. Gross error• The errors occur because of mistakes in observed

readings, or using instruments and in recording and calculating measurement results.

• These errors usually occur because of human mistakes and these may be of any magnitude and cannot be subjected to mathematical treatment.

• One common gross error is frequently committed during improper use of the measuring instrument.

Type of Error

Error will occur in the measurement

result if proper zero setting is not

being done

Gross error will occur in the

measurement result if the pointer deflects between two scaling points

Parallax error will occur in the

measurement result if the pointer is not set in the vertical

position

Type of Error

2. Systematic error• errors that remain constant or change according to a

definite law on repeated measurement of the given quantity.

• There are two types of systematic errors:i. Instrumental errorii. Environmental error

Type of Errori. Instrumental error• inherent in the measuring instruments because of their

mechanical structure and calibration or operation of the apparatus used.

• may be avoided by:i. Selecting a proper measuring device for the particular

applicationii. Calibrating the measuring device or instrument against

a standardiii. Applying correction factors after determining the

magnitude of instrumental errors

Type of Errorii. Environmental error• These errors are introduced due to using an instrument in

different conditions than in which it was assembled and calibrated.

• These errors can be eliminated or reduced by:i. Use the measuring instrument in the same atmospheric

conditions in which it was assembled and calibrated.ii. The deviation in local conditions must be determined

and suitable compensations are applied in the instrumental reading.

iii. Automatic compensation, employing sophisticated devices for such deviations.

Type of Error

3. Random error• Originated from unexplained source (parameters or

systems). • The presence of random errors become evident when

different results are obtained on repeated measurements of one and the same quantity.

• The effect of random errors is minimised by measuring the given quantity many times under the same conditions and calculating the arithmetical mean of the results obtained.

Error in Measurements1. Absolute Error• The absolute error is the difference between the

measured value and the actual value. • Stated in the form of value and unit of measured quantity

(e.g. ± 10Ω, ± 3 V, ± 0.5 A, etc). • Absolute error is simply the amount of physical error in a

measurement.

• For example, if you know a length is 3.535 m + 0.004 m, then 0.004m is an absolute error.

Error in MeasurementsExample 2:The measured value of a resistance is 10.25, where as its value is 10.22. Determine the absolute error of measurement.

Solution: Measured value, = 10.25True value = 10.22Absolute error = Measured value – True value

= 10.25 – 10.22 = 0.03

Error in Measurements2. Relative Error• Relative error is the ratio of the absolute error of the

measurement to the accepted measurement. • stated in percentage (e.g. ±5%, ±10%, ±20%, etc) or

fractional ( e.g. ±1/10, ±¼, ±½, ±3/4, etc).

Error in Measurements2. Relative Error• Relative accuracy is how close a measured value is to the

standard value on relative terms.

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Error in MeasurementsExample 3:The measured value of capacitor is 205.3F, where as its true value is 201.4F. Determine the relative error.Solution: Measured value, = 205.3FTrue value = 201.4F

205.3 201.4201.4 0.0194

Limiting Errors

• In most indicating instruments, the accuracy is guaranteed to a certain percentage of full-scale reading.

• Circuit components (such as resistors, capacitors, etc.) are guaranteed within a certain percentage of their rated value.

• The limits of these deviations from the specified values are known as limiting errors or guarantee errors.

Limiting Errors

• For example, if the resistance of a resistor is given as 500 Ω ± 10%, the manufacturer guarantees that the resistance falls between the limits 450 Ω and 550 Ω. The maker is not specifying a standard deviation or a probable error, but promises that the error is no greater than the limits set.

Limiting ErrorsExample 4:A 0 – 150V voltmeter has a guaranteed accuracy of 1% full-scale reading. The voltage measured by this instrument is 83 V. Calculate the limiting error in percentage.Solution: The magnitude of limiting error = 1% or 0.01 x 150 V = 1.5VThe percentage error at a voltmeter indication of 83 V is( 1.5V / 83 V ) x 100 = 1.81 %• a voltmeter is guaranteed to have an accuracy of better

than 1% of the full-scale reading, but when the meter reads 83 V, the limiting error increases to 1.81 %.

Limiting Errors

• Correspondingly, when a smaller voltage is measured, the limiting error will increase further. If the meter reads 30V, the limiting error is

( 1.5 V / 30 V ) x 100 = 5 %

• The increase in percentage limiting error, as smaller voltages are measured, occurs because the magnitude of the limiting error is a fixed quantity based on the full-scale reading of the meter.

Probability and Error Control

• The statistical analysis of measurement data is important because it allows an analytical determination of the uncertainty of the final test result.

• To make statistical analysis meaningful, a large number of measurements is usually required.

• Systematic errors should be small compared to random errors, because statistical analysis of data cannot remove a fixed bias contained in all measurements.

Probability and Error Control1. Arithmetic Mean

• The most probable value of a measured variable is the arithmetic mean of the number of readings taken.

• The best approximation is possible when the number of readings of the same quantity is very large.

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Probability and Error Control2. Deviation from the Mean

• The difference between an observed value of a variable and its mean.

• If the deviation of the first reading, x1 is called d1, and that of the second reading x2 is called d2, and so on

• The deviations from the mean can be expressed as

• The deviation may be positive or negative. • The algebraic sum of all the deviations must be zero.

Probability and Error Control3. Average Deviations

• The average deviation is an indication of the precision of the instrument used in measurement.

• Average deviation is defined as the sum of the absolute values of the deviation divided by the number of readings.

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• Highly precise instruments yield a low average deviation between readings.

Probability and Error Control

4. Standard Deviation• The standard deviation of an infinite number of data is

the square root of the sum of all the individual deviations squared, divided by the number of readings.

• Reduction in this quantity effectively means improvement in measurement. For small readings (n < 30), the denominator is frequently expressed as (n - 1) to obtain a more accurate value for the standard deviation.

Probability and Error Control5. Precision of measurement.

• Precision is a measure of how well a result can be determined (without reference to a theoretical or true value). It is the degree of consistency and agreement among independent measurements of the same quantity; also the reliability or reproducibility of the result.

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Probability and Error ControlExample 5:Givenx1 = 49.7; x2 = 50.1; x3 = 50.2; x4 = 49.6; x5 = 49.7 Determine:(i) Arithmetic mean(ii) Deviation of each value(iii) Algebraic sum of the deviations(iv)Average deviation (v) Standard deviation(vi)Precision of 3rd measurement

Solutions(i) Arithmetic mean

Solutions(ii) Deviation of each value

Solutions(iii) Algebraic sum of the deviations

Solutions(iv)Average deviation

Solutions(v) Standard deviation

Solutions(vi)Precision of 3rd measurement

System of Unit

• The International System of Units (SI, abbreviated from the Système international) is the modern form of the metric system.

• The SI system also provides twenty prefixes to the unit names and unit symbols that may be used when specifying power-of-ten (i.e. decimal) multiples and sub-multiples of SI units.

Standards of Measurement1. International Standards • defined by international agreement and they represent

certain units of measurement to the closest possible accuracy that production and measurement technology allow.

• periodically evaluated and checked by absolute measurements in terms of the fundamental units.

• Such standards are maintained at the International Bureau of Weights and Measures and are not available to the ordinary users of measuring instruments for the purpose of comparison or calibration.

Standards of Measurement2. Primary Standards • maintained by National Standard Laboratories• represent the fundamental units and some of the

derived mechanical and electrical units. • calibrated independently by absolute measurements at

each of the National Laboratories. • results are then compared against each other, leading

to a world average figure for the primary standards. • Main function of primary standards is the• verification and calibration of secondary standards

Standards of Measurement3. Secondary Standards • basic reference standards used in industrial

measurement laboratories. • maintained by the particular involved industry and are

checked locally against reference standards. • Such standards are calibrated and compared against

the primary standards in national standard laboratories on a periodic basis.

• These laboratories issue a certificate of their measured value in terms of the primary standards.

Standards of Measurement4. Working Standards • Working standards are the principal tools of a

measurement laboratory.• Are used to check and calibrate general laboratory

instruments for accuracy and performance or to perform comparison measurements in industrial applications.