Bef 23901 Chapter 1 Measurement and Errors

CHAPTER 1

MEASUREMENT AND ERRORS

Principles of Measurement

Measurement is the process of comparing an unknown quantity with an accepted standard quantity.

It involves connecting a measuring instrument into the system under consideration and observing the resulting response on the instrument.

The term measurement can also be used to refer to a specific result obtained from the measurement process.

Electrical Measurement (BEF 23903) 2

Measurand (Unknown quantity to be measured)

Process of comparison (measurement)

Standard (known quantity)

Result

(Read out)

Measurand

• The physical quantity or the characteristic condition which is the object of measurement in an instrumentation system.

• Also called i. Measurement variable

ii. Instrumentation variable

iii. Process variable

• The measurand may be:

- Fundamental quantity, e.g. length, mass, and time;

- Derived quantity, e.g. speed, velocity, acceleration.


Before measurement process we have to ensure:

Methods/procedures of measurement

Characteristics of the parameter

Quality: time and cost, instrument capabilities, knowledge of measurement, acceptable result

What instrument to use

During the measurements we have to ensure:

Quality- best instrument chosen, suitable position when taking the data, etc..

Safety- electric shock, overloaded, instrument limits, read instruction manual

Sampling – observe parameter changing, taking enough sample

After measurement Analyse the data mathematically/statistically Full result must be reported completely and accurately


What is an instrument?

1. Device that communicates, denotes, detects, indicates, measures, observes, records, or signals a quantity or phenomenon, or controlsor manipulates another device.

2. A tool or device used for a particular purpose; especially : a tool or device designed to do careful and exact work.


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Instrumentation

• The technology of using instruments to measure and control the physical and chemical properties of materials is called instrumentation.

• When the instruments are used for the measurement and control of industrial manufacturing, conversion or treatment process, the term “process instrumentation” is used.

• When the measurement and control instruments are combined so that measurements provide impulse for remote automatic action, the result is called a Control system.


The basic requirements for getting meaningful result of measurement are:

1. The standard employed for comparison purpose may be accurately defined and should be commonly acceptable.

2. The standard must be of the same character as the measurand and usually but not always, is prescribed and defined by a legal or recognised organisation, e.g. the International Organisation of Standards (ISO).

3. The apparatus used and method adopted for the comparison purposes must be provable.


Why do we need measurements?

The significance of measurement is discussed below:

1. Measurement plays a very significant role in every branch of scientific research and engineering processes which include the following:

- Control systems;

- Process instrumentation

- Data reduction.

2. The whole area of automation or automatic control is based on measurements. The very concept of control is based on the comparison of the actual condition and the desired performance. The exactness of error depends on the precision and accuracy of measurement made.


Why do we need measurements?

3. The measurements confirm the validity of a hypothesis and also add to its understanding. This eventually leads to new discoveries that require new and sophisticated measuring techniques.

4. Through measurements a product can be designed or a process be operated with maximum efficiency, minimum cost, and with desired degree of reliability and maintainability.


SI Systems

S.I Unit

Base units Derived unitsSupplementary

units


International system of units (S.I) are divided into three classes:

S.I Base Units


Base Quantity Base Unit

Name Symbol

Length Meter m

Mass Kilogram Kg

Time Second s

Electric Current Ampere A

Thermodynamic Temperature Kelvin K

Amount of substance Mole mol

Luminous Intensity Candela cd

Derived Unit

• Derived Quantities are formed by combining two or more of the fundamental quantities.

Examples:

Area = length x width

Volume = length x width x height

Speed = distance/time

Density = mass/volume

Most of the units in the International System are derived units, that is units defined in terms of base units and supplementary units. Derived units can be divided into two groups - those that have a special name and symbol, and those that do not.



Without Names and Symbols

Measure of Derivation

acceleration m/s2

angular acceleration rad/s2

angular velocity rad/s

density kg/m3

electric field strength V/m

magnetic field strength A/m

velocity m/s


With Names and Symbols

Unit Measure of Symbol Derivation

coulomb electric charge C A·s

farad electric capacitance F A·s/V

henry inductance H V·s/A

hertz frequency Hz cycles/s

joule quantity of energy J N·m

ohm electric resistance Ω V/A

tesla magnetic flux density T Wb/m2

volt voltage V W/A

watt power W J/s

weber magnetic flux Wb V·s

Supplementary Units

Third class of S.I units

Supplementary units may be regarded either as base units or as derived units

Example of S.I derived units formed by using supplementary units

- Angular velocity ( )

- Angular acceleration ( )


Quantity S.I Units

Name Symbol

Plane angle radian rad

Solid angle steradian sr

2/ srad

srad /

Instrument’s Performance Characteristic

Performance Characteristics - characteristics that show the performance of an instrument.

Allows users to select the most suitable instrument for a specific measuring jobs.

Two basic characteristics :

1. Static

2. Dynamic


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Static Characteristic

I. Accuracy – the degree of exactness (closeness) of measurement

compared to the expected (desired) value.

II. Resolution – the smallest change in a measurement variable to which an

instrument will respond.

III. Precision – a measure of consistency or repeatability of measurement,

i.e successive reading do not differ.

IV. Sensitivity – ratio of change in the output (response) of instrument to a

change of input or measured variable.

V. Error – the deviation of the true value from the desired value


Errors in Measurement

Various types of errors in measurement:

i. Absolute error

ii. Gross Errors

iii. Systematic Error

iv. Random Error

v. Limiting Error

Static error of a measuring instrument – The numerical difference between the true value of a quantity and its value as obtained by measurement (i.e. repeated measurement of the same quantity gives different indications).


i) Absolute error

Error - The difference between the expected value of the variable and the measured value of the variable:

e = Yn – Xn

where:

e = absolute error

Yn = expected value

Xn = measured value

Percentage Error:


100

nYnXnY

% error =

Relative accuracy (A):

% Accuracy:

a = 100% - % error

=

Precision:


n

nn

Y

XYA

1

n

nn

X

XXP

1

100A

Wherevalue of the nth measurement

average set of measurement

nX

nX

Example 1.1

Given expected voltage value across a resistor is 80V. The measurement is 79V. Calculate,

1. The absolute error

2. The % of error

3. The relative accuracy

4. The % of accuracy


Solution 1.1


Example 1.2

From the value in table below, calculate the precision of 6th

measurement?


No Xn

1 98

2 101

3 102

4 97

5 101

6 100

7 103

8 98

9 106

10 99

Solution 1.2


ii) Gross Error

Due to human mistakes in reading or in using instruments or error in recording observations.

Example: incorrect reading, incorrect recording, improper use of instruments, etc.

To minimize:

Take at least 3 separate reading.

Take proper care in reading & recording.


iii) Systematic Error

Due to shortcoming of the instruments or environmental effects or observational errors.

example

defective or worn parts

ageing

parallax error

wrong estimation reading scale


Types of Systematic Errorsa) Instrumental errors : Due to friction in the bearings of the meter movement, incorrect spring

tension, improper calibration, or faulty instruments. Can be reduced by proper maintenance, use, and handling of instruments.

b) Environmental errors : Due to external condition of the measuring device. Example: effects of change in temperature, humidity, barometric pressure,

electrostatic fields etc. Can be avoided by: air conditioning, hermetically sealing certain

components in the instrument and using magnetic shields.

c) Observational errors : Errors that introduced by the observer. The two most common observational errors are probably the parallax

error introduced in reading a meter scale and the error of estimation when obtaining a reading from a meter scale.


Example 1.3 (a)

A voltmeter having a sensitivity of 1 k/V is connected across an unknown resistance in series with a milliammeter reading 80 V on 150 V scale. When the milliammeter reads 10 mA, calculate the

i. apparent resistance of the unknown resistance,

ii. actual resistance of the unknown resistance, and

iii. error due to the loading effect of the voltmeter.


Solution 1.3(a)


Example 1.3 (b)

Referring to example 1.3 (a), if the milliammeter reads 600 mA and the voltmeter reads 30 V on a 150 V scale, calculate the following:

i. Apparent, resistance of the unknown resistance,

ii. Actual resistance of the unknown resistance,

iii. Error due to loading effect of the voltmeter.

Comment on the loading effect due to the voltmeter for both Examples 1.3 (a) and (b). (Voltmeter sensitivity given 1000 /V.)


Solution 1.3(b)


iv) Random Errors

Errors that remain after gross and systematic errors have been substantially reduced.

Are generally the accumulation of a large number of small effects.

May be of real concern only in measurements requiring a high degree of accuracy.

Such errors can only be analyzed statistically.

These errors are due to unknown causes, not determinable in the ordinary process of making measurements.


iv) Limiting Errors

Most manufacturers of instruments state that an instrument is accurate within a certain percentage of a full-scale reading.

Eg: a voltmeter is accurate within ±2% at full-scale deflection (limiting errors).

However, with reading less than full-scale, the limiting error will increase.

therefore, it is important to obtain measurements as close as possible to full scale


Example 1.4

A 300-V voltmeter is specified to be accurate within ±2% at full scale. Calculate the limiting error when the instrument is used to measure a 120-V source?


Solution 1.4


Example 1.5

A voltmeter reading 70 V on its 100 V range and an ammeter reading 80 mA on its 150 mA range are used to determine the power dissipated in a resistor. Both these instruments are guaranteed to be accurate within ±1.5% at full scale deflection. Determine the limiting error of the power.


Solution 1.5


Dynamic Characteristic

Dynamic – measuring a varying process condition.

Instruments rarely respond instantaneously to changes in the measured variables due to such things as mass, thermal capacitance, fluid capacitance or electrical capacitance.

The dynamic characteristics of an instrument are:

I. Speed of response

II. Dynamic errorThe difference between the true and measured value with no static error.

III. Lag – response delay

IV. Fidelity – the degree to which an instrument indicates the changes in the measured variable without dynamic error (faithful reproduction).


Statistical Analysis of Measurement Data

Important because it allows an analytical determination of the uncertainty of the final result.

A large number of measurements is usually required.

Can be divided into 4:


Arithmetic mean / Average

Deviation

Average deviation

Standard deviation

i

ii

iii

iv

i) Arithmetic mean/average:

The most probable value of measured variable.

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


n

1i

n321 x x x x

n

x

nx i

n = total number of readingxn = nth reading takenxi = set of number

ii) Deviation:

The difference between each piece of data and arithmetic mean

Algebraic sum of deviation,


xxd nn

1 1d x x

2 2d x x

0 21 ntotal dddd

iii) Average deviation (D):

Indication of the precision of a measuring instrument used in measuremnt

high D low precision

low D high precision


n

dddD n

21

iv) Standard Deviation

Also known as root mean square deviation.

The most important factor in statistical analysis.

Reduction means improvement in measurement.


2 2 2 2

1 2 3 ....

1

nd d d d

n

2

1

nd

n

Example 1.4


For the following data

Compute:

(a) The arithmetic mean.

(b) The deviation of each value.

(c) The algebraic sum of the deviation.

(d) The average deviation.

(e) The standard deviation.

No. 𝑥𝑛

1 49.7

2 50.1

3 50.2

4 49.6

5 49.7

Solution 1.4


END
