CHAPTER 1
UNITS, DIMENSIONS, AND
MEASUREMENT ERRORS
Learning Outcomes
 At the end of the chapter, students should
be able to:
 Discuss the fundamental mechanical and
Electrical units in the SI system and derived
units.
 Define the dimensions of various quantities.
 Define and explain the types of
measurement errors.
 Explain and apply measurement terms.
Introduction
 Before standard systems of measurement were invented,
many approximate units were used.
 With the development of science and engineering, more
accurate units had to be devised.
 It is necessary to establish a single system of units of
measurement that would be acceptable internationally
because of the increase of world trade and exchange of
scientific information.
Units
 Units of measurement define the definite magnitude of
physical quantity which adopt convention and law.
 e.g. Unit for physical quantity length is metre
 The International System of units (SI unit) is a form of
metric system and divided in 3 classes:
 Base units.
 Derived units.
 Supplementary units.
SYSTEMS OF UNITS

In the past, the systems of units most commonly used were the
English and metric.

Note that while the English system is based on a single
standard, the metric is subdivided into two interrelated
standards: the MKS and the CGS.

The MKS and CGS systems draw their names from the units of
measurement used with each system; the MKS system uses
Meters, Kilograms, and Seconds, while the CGS system uses
Centimeters, Grams, and Seconds.
English
SI System
 SI stands for Systéme International d’Unités, i.e. the
International System of Units. SI is the abbreviation used in all
languages to indicate the system.
 The SI is constructed from seven base units, which are defined
in physical terms.
 By combining these units in accordance with simple geometrical
and physical laws, we can arrive at the derived units.
 In principle, the SI covers all application areas, although certain
units outside SI are so useful that they are accepted for general
use together with the SI (e.g degree, hour, day, minute)
Base Units (seven base units)
 Fundamental unit refers to quantity
NAME
SYMBOL
QUANTITY
Kilogram
kg
Mass
Second
s
Time
Meter
m
Length
Ampere
A
Electrical current
Kelvin
K
Temperature
Mole
mol
Amount of substance
Candela
cd
Luminous intensity
By combining these units in accordance with simple
geometrical and physical laws, we can arrive at the derived
units.
Derived Units
 Derivation/further ext./combination . unit of base unit
Derived quantity
Derived unit
Symbol
Area
Square meter
m2
Volume
Cubic meter
m3
Speed, velocity
Meter per second
m∙s-1
Acceleration
Meter per second square
m∙s-2
Angular velocity
Radian per second
rad∙s-1
Angular acceleration
Radian per second square
rad∙s-2
Density
Kilogram per cubic meter
Kg∙m-3
Magnetic field intensity,
(Linear current density)
Ampere per meter
A∙m-1
Derived quantity
SI derived unit
name
Symbol
Frequency
Hertz
Hz
s-1
Force
Newton
N
m∙kg∙s-2
Pressure, stress
Pascal
Pa
N∙m-2
m-1 ∙kg ∙s-2
Energy, work, heat quantity
Joule
J
N∙m
m2 ∙kg ∙s-2
Power, radiant flux
Watt
W
J/s
m2 ∙kg ∙s-3
Electric charge
Coulomb
C
Electric potential difference
Volt
V
W/A
m2 ∙kg ∙s-3 ∙ A-1
Electric capacitance
farad
F
C/V
m-2 ∙kg-1∙s4 ∙ A2
Electric resistance
ohm
V/A
m2 ∙kg ∙s-3 ∙ A-2
Electric conductance
Siemens
A/V
m-2 ∙kg-1 ∙s3 ∙ A2
S
In SI
units
In SI base units
s∙A
Supplementary Units
 Unit outside of SI but accepted
Quantity
Unit
Symbol
Value in SI units
Time
Minute, hour,
day
Min, h, d
1 min = 60 s
1 h = 60 min = 3600s
1 day = 24 h = 1440 min =
86400 s
Plane angle
Degree, minute,
second, grad
̊ ’ ” gon
1 ̊ = (π/180) rad
1’ = (1/60)’ = (π/10 800) rad
1” = (1/60)” (π/648 000) rad
1 gon = (π/200) rad ;
400 gon = 360 ̊
Volume
litre
l, L
1 l = 1 dm3 = 10-3 m3
Mass
Metric tonne
t
1 t = 103 kg
Pressure in air, fluid
bar
bar
1 bar = 105 Pa
TABLE: Comparison of the English and metric
systems of units.
SI Mechanical Units
 Unit of Force
 Force which will give a mass of 1 kg an acceleration
of 1 meter per second per second.
F  ma

Work
 The product of the force and the distance
W  F.d
 Energy
 The capacity for doing work.
 Energy is measured in the same units as work.

Power
 The time rate of work done
W
P
t
SI Electrical Units
 Units of Current and Charge
 Current is the quantity of electricity that passes a
given point in a conductor during a time of 1 s.
Q
coulombs
I
 ampere 
t
seconds

Voltage
 The potential difference between two points on a
conductor carrying a constant current of 1 ampere
when the power dissipated is 1 watt.
 Resistance and Conductance

Conductance is the reciprocal of resistance.
1
condutance 
resistance

Unit : Siemens (S)
Prefixes
10 n
Prefix
Sym
1024
yotta
1021
Short scale Long scale
Decimal equivalent
Y
septillion
Quadrillion
1 000 000 000 000 000 000 000 000
zetta
Z
sextillion
Trilliard
1 000 000 000 000 000 000 000
1018
exa
E
Quintillion Trillion
1 000 000 000 000 000 000
1015
peta
P
Quadrillio
n
Billiard
1 000 000 000 000 000
1012
tera
T
Trillion
Billion
1 000 000 000 000
109
giga
G
Billion
Milliard
1 000 000 000
106
mega
M
Million
1 000 000
103
kilo
k
Thousand
1 000
102
hecto
h
Hundred
100
101
deca
da
Ten
10
Prefixes
10 n
Prefix
Sym
100
none
none
one
1
10-1
Deci
d
tenth
0.1
10-2
centi
c
hundredth
0.01
10-3
milli
m
thousandth
0.001
10-6
micro
µ
millionth
0.000 001
10-9
nano
n
Billionth
Milliardth
0.000 000 001
10-12
pico
p
Trillionth
Billionth
0.000 000 000 001
10-15
temto
f
Quadrilliont
Billiardth
0.000 000 000 000 001
10-18
atto
a
Quintillionth
Trillionth
0.000 000 000 000 000 001
10-21
zapto
z
Sextillionth
trilliardth
0.000 000 000 000 000 000 001
10-24
yocto
y
septillionth
quadrillionth
0.000 000 000 000 000 000 000
0001
Short scale
Long scale
Decimal equivalent
Dimensions
 Parameter or measurement


used
to
describe
some
relevant characteristic of an
object.
Dimensions is describing the
size or spatial characteristic of
an object: length, width, and
height .
Also
for
other
physical
parameters such as the mass
and electric charge of an
object.
3-Dimension of gear system
 Uses symbol M (mass), L (length), T (time) – known as mech.
unit, Q (e’ charge), I or A (current)
 A derived unit of physical quantity
 Example 1;
- Dimension of physical quantity SPEED is L/T (or in units
m/s, km/h, mph)
QUANTITY
UNIT
DIMENSION
SPEED
m/s
L/T
 Dimension of a physical quantity is the total of all units
attached to it.
 For example, speed is given as distance / time;
metres/second (m/s) MKS and centimetres/second (cm/s)
in CGS system.
 Dimension of measurement of speed ,
[v] = [L]/[T]
Example
 A bar magnet with a 1 inch square cross sectional area is

said to have a total magnetic flux of 500 Maxwell.
Determine the flux density in tesla.
Solution:
Total flux,
Area,
Flux density,
Example 2
 Determine the dimensions of velocity, acceleration and

force.
Solution:
Velocity = length/time
[v] = [L]/[T] = [LT-1]
Acceleration = velocity/time
[a] = [v]/[T] = [LT-1]/[T1] = [LT-2]
Force = mass × acceleration
[F] = [M] • [LT-2] = [MLT-2]
Static Characteristics
Instrument: A device or mechanism used to determine the present
value of the quantity under measurement.
Measurement: The process of determining the amount, degree, or
capacity by comparison (direct or indirect) with the
accepted standards of the system units being used.
Accuracy: The degree of exactness (closeness) of a measurement
compared to the expected (desired) value.
Precision: A measure of consistency or repeatability of
measurement, i.e. successive reading do not differ.
Expected value: The design value, i.e. the most probable
value that calculations indicate one should
expect to measurement.
Error:
The deviation of the true value from the desired value.
Sensitivity:
the ratio of the change in output (response) of the
instrument to a change of input or measured variable.
Error in measurement
Any measurement is affected by many variables, therefore the
results rarely reflect the expected value.
The degree to which a measurements nears the expected value is
expressed in terms of the error of the measurement.
Error may be expressed either as absolute or as percentage of error.
Absolute error:
may be defined as the difference between the expected value of
the variable and the measured value of the variable.
Where e= absolute error
Yn= expected value
Xn=measured values
e= Yn-Xn
Percentage error:
% e= (absolute value*100)/Expected value = e*100/Yn
It is more frequency expressed as accuracy rather than error
Per-unit Accuracy:
Accuracy A= 1-(Yn-Xn)/Yn
Percentage Accuracy:
a=% A= (100%)- %e
The precision of a measurement
is numerical indication of the closeness with
which a repeated set of measurement of the same variable agree
with the average set of measurement.
Where
The value of the n th measurement
The average set of measurement
Example
An analog voltmeter is used to measure voltage of 50V
across a resistor. The reading value is 49 V. Find
a) Absolute Error
b) Relative Error
c) Accuracy
d) Percent Accuracy
Solution
a) e  X t  X m  50V  49V  1V
b) % E rror 
Xt X m
50V  49V
 100% 
 100%  2%
Xt
50V
c) A  1  % E rror  1  2%  0.98
d) % A cc  100%  2%  98%
TYPE OF STATIC ERROR
The static error of a measuring instrument is 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 give
difference indications.
1. Gross Error
• These errors are mainly due to human mistakes in reading or in using
instruments or errors in recording observations.
• if the accuracy of an instrument has not been calibrated.
• With analog instruments if the pointer has not been adjusted before us.
• Gross errors can be avoided with care.
2. Systematic errors:
• These errors due to shortcomings of the instrument, such as defective
or worn parts, or ageing or effects of the environment on the instrument.
• Occur because the measurement system affects the measured quantity.
• Errors that are the result of instrument inaccuracy.
I. Instrumental Errors
 Friction of bearings.
 irregular spring tensions.
 Zero positioning.
II. Environment Errors
 Temperature.
 Humidity.
 Pressure.
3. Observational Error
These errors are introduced by the observer. The most common
error is:
 the parallax error introduced in reading a meter scale.
 the error of estimation when obtaining a reading from a meter scale
4. Random Errors
These errors are due to unknown causes, not determinable in the
ordinary process of making measurements.
Random errors can thus be treated mathematically.
Accuracy and Precision
 Accuracy – the closeness agreement
between a measurement result & the
actual measured quantity.
 Precision – the closeness with a
repeated set of measurements of the
same variable agrees with the average
of the set of measurements.
Resolution
• Resolution – The smallest change in a measured variable
to which an instrument will respond.
Arithmetic Mean Value
 When a number of measurements of a quantity are made
and the measurements are not exactly equal, the best
approximation to the actual value is found by calculating
the average value of the results.
is the arithmetic mean
is n the reading taken
n is the total number of reading
Deviation
 Deviation: The difference between any one measured value
and the arithmetic mean of a series of measurements.
The deviation from the mean can be expressed as:
 The average deviation: may be calculated as the average of
the absolute values of deviations, neglecting plus and minus
sign.
Standard Deviation
 The mean-squared value of the deviations can also be

calculated by first squaring each deviation value before
determining the average, which gives a quantity known as
variance.
Variance: the mean-squared value of the deviations
2
2
2
d
+
d
+
...+
d
2
n
s2 = 1
n
Standard deviation: Taking the square root of the
variance produces the root mean square (rms) value.
Probable Error
 Probable error: error in any one measurement for the
case of a large number of measurements in which only
random errors are present.
For the case of a large number of measurements in
which only random errors are present, it can be shown
that the probable error in any one measurement is
0.6745 times the standard deviation
Example The accuracy of five digital voltmeters are checked by using each of them
to measure a standard 1.0000V from a calibration instrument. The voltmeter
readings are as follows: V1 = 1.001 V, V2 = 1.002, V3 = 0.999, V4 = 0.998, and
V5 = 1.000. Calculate the average measured voltage, the average deviation, standard
deviation and probable error
Solution
V 1 V 2 V 3 V 4 V 5
5
1 .0 0 1  1 .0 0 2  0 .9 9 9  0 .9 9 8  1 .0 0 0

 1 .0 0 0V
5
V av 
d 1  V 1 V av  1.001  1.000  0.001V
&
d 2  V 2 V av  1.002  1.000  0.002V
d 3  V 3 V av  0.999  1.000   0.001V
&
d 4  V 4 V av  0.998  1.000   0.002V
d 5  V 5 V av  1.000  1.000  0V
D 
d 1  d 2  ...  d 5 0.001  0.002  0.001  0.002  0

 0.0012V
5
5
 
d 1  d 2  ...  d 5

5
2
2
2
2
2
2
2
 0.001   0.002    0.001   0.002   0  0.0014V
probable error  0.6745    0.6745  0.0014V
 0.94 m V
5
Limiting Error
Most manufacturers of measuring instruments specify accuracy
within a certain % of a full scale reading.
For example, the manufacturer of a certain voltmeter may specify
the instrument to be accurate within 2% with full scale deflection.
This specification is called the limiting error.
Measurement Error Combinations
Errors in measurement systems often arise from two or more
different sources, and these must be aggregated in the correct
way in order to obtain a prediction of the total likely error in
output readings from the measurement system.
Two different forms of aggregation are required.
Firstly, a single measurement component may have both
systematic and random errors and, secondly, a measurement
system may consist of several measurement components that
each have separate errors.
Example
A circuit requirement for a resistance for 550 ohm is satisfied by
connected together two resistors of nominal values 220 ohm and
330 ohm in series. If each has a tolerance of 2%.
Calculate the absolute error, the percentage error, likely maximum
error and percent of this error.
[11Ω, 2%, 7.93 ohm, 1.4%]
Estimate the percentage error, the likely maximum error and the
percentage error of the likely maximum error in this case .
Example
A fluid flow rated in calculated from the difference in pressure
measured on both sides of orifice plate. If the pressure measurements
are 10.0 bar and 9.5 bar and the error in the pressure measuring
instruments is specified as 0.1%. Find the absolute error, the
percentage error, the likely maximum error and its percentage error.
[±0.0195bar ±3.9, 0.0138bar and 2.76%]
Difference of Quantities

The error of the difference of two measurements are again additive
Estimate the absolute error, percentage error, the likely
maximum error and its percentage error in this case.
The maximum error in the protect is: (a+b).The better estimate
of the likely maximum error e in the product P, provided that the
Measurements are uncorrelated, is given by Topping(1962).
Product of Quantities
• When a calculated quantity is the product of two or more quantities, the
percentage error is the sum of the percentage errors in each quantity
P  EI
 E  ΔE I  ΔI 
 EI  E Δ I  I Δ E  ΔEΔI
since ΔEΔI is very small ,
P  EI
 E Δ I  I Δ E 
E  I  I E
 100 %
EI
 I   E 


  100 %
I
E

 

percentage error 
% error in P  % error in I   % error in E 
Estimate the absolute error, percentage error, the likely
maximum error and its percentage error in this case
48
Thus the maximum error in the quotients is (a+b). However, using the
same argument as made above for the product of measurements,
A statistically better estimate (Topping 1962) of the likely maximum
Error in the quotient Q, provided measurements are uncorrelated.
Quotient of Quantities
E
% error in
 % error in E   % error in I 
I
Estimate the absolute error, percentage error, the likely
maximum error and its percentage error in this case
Quantity Raised to a Power
% error in A
B
 B % error in A 
The final case to be covered in where the final measurement is
calculated from several measurements that are combined in a way
that involves more than one type of the arithmetic operation.
The error involved in each stage of arithmetic are cumulative, and
error can be calculated by adding together all error values.
Example