1E6 Electrical Engineering
DC Circuit Analysis
Lecture 2: Resistance and Resistors
2.1 Introduction
Fig. 1 again shows the overlap of the conduction and valence bands of
conductors. In pure metals, such as Copper, Iron, Silver or Gold and in metal
alloys such as Steel or Brass, there is a large degree of overlap in the bands and
a plentiful supply of free energy levels for charge to move between with ease.
Consequently, these materials conduct electric current readily and a small
electric field placed along them will result in a very high degree of conduction
and high values of current. Hence, interconnecting wires and cables are made of
such metals where they are intended to offer the least possible impediment to the
flow of current. They are used only to provide a path for current to flow in an
electric circuit and to connect points together electrically where the potential or
emf is intended to be the same. However, practical electric circuits must consist
of conducting elements other than interconnecting wires or there would be little
point to their existence. The real motivation of the electric circuit is to transfer
energy from its electric form into some other form such as thermal or
mechanical energy in order to do work. In an electric kettle for example the
electrical energy drawn from the mains supply is converted into thermal energy
to heat water, while in an electric motor it is converted into mechanical energy
to drive a physical load such as the drum of a washing machine.
Fig. 1
Energy Bands in Solid Materials
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2.2 Resistivity
In electric circuits, and in particular in electronic circuits, it is generally
required to control the voltages and currents present at various points in the
circuit to defined values for given purposes, like the biasing of the
semiconductor devices in a hi-fi amplifier, for example. In this case it is not just
a matter of using wires to create circuits for current to flow in but rather one of
limiting the currents and defining the potentials at specific values at particular
locations within a circuit. In this case materials are used to create elements
which provide specific degrees of impediment or resistance to the flow of current
through them. Conductors which are not pure metals have a much lower
number of free energy levels in their conduction bands and hence do not conduct
current as easily as pure metals. In these materials a much larger emf or
potential drop is required across them to enable the flow of current and the
magnitudes of the resulting currents are much lower than in pure metals.
Materials like Carbon, Cobalt and Ferrite compounds as well as some metal
oxides are popular for this purpose.
Conducting materials possess the property of resistivity. Resistivity is a
measure of how strongly a material opposes the flow of electric current through
it when subjected to the influence of an electric field or emf. A low value of
resistivity means the material readily allows the movement of charge through it
while a high value of resistivity means a high degree of opposition to the
movement of charge. The resistivity, also known as the specific resistance, of a
material depends on its particular atomic structure and is given the symbol ρ
and has units of Ohm-metres.
2.3 Resistance
The resistance of a piece of conducting material is essentially the
combined effect of its resistivity established for the piece of material as a whole.
It depends on the resistivity of the material as an element or compound and the
physical dimensions of the piece of the material in question as well as the way in
which the emf is applied to it.
Fig. 2 A Piece of Resistive Material which Electrical Contacts on Both Ends
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Fig. 2 shows a piece of resistive material having resistivity, ρ, uniform
cross-sectional area, A and length, l. Electrical contacts are made uniformly over
the cross-sectional area at both ends. This allows the uniform flow of charge
through the material when an electric field is applied along its length. The total
resistance, R, of this piece of material as a whole is given as:
R=ρ
l
A
Ohms (Ω )
Resistance has units of Ohms and uses the Greek letter Omega, Ω, as the symbol
of this unit, which is named after the German physicist Georg Ohm (1789 1854) who first formally described the property.
2.4 Case Study 1
A piece of material having a resistivity of 2.5 x 10-5 Ωm has a cylindrical shape
with a diameter of 6mm and a length of 12cm and has electrical connections
made across its area at both ends. Determine the total resistance of the piece of
material.
Area A = πr2 = 3.14 x (3x10-3)2 = 3.14 x 9 x 10-6 = 28.8 x 10-6 m2
Length l = 12 cm = 0.12 m
Resistivity ρ = 2.5 x 10-5 Ωm
Then:
R=ρ
l
0.12
3
= 2.5 × 10 −5 ×
=
= 0 .1 Ω
A
28.8 × 10 − 6 28.8
This value is seen to be very low in absolute terms. Interconnecting wires in an
electrical circuit will have values much lower than this but circuit components
will require values which are much higher.
The resistance can be increased by increasing the resistivity of the material used
but there is a limit to what can be achieved here. It can also be increased by
either increasing the length or reducing the conducting surface area of the piece
of material. The latter is the more practical option in today’s manufacturing
environment.
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2.5 Case Study 2
A resistor is formed of a hollow cylinder of internal diameter 2mm and
length 8mm. The Carbon material used has a resistivity of 3.5 x 10-5 Ωm.
Determine the thickness of the cylinder required to manufacture a resistor of
value 10 kΩ.
r1
r2
tρ
Fig. 3 A Resistor of Hollow Cylindrical Construction
The area of the cylinder forming the resistor is the difference in the areas of the
bases of cylinders having the radii r1 and r2 as indicated in Fig. 3. Then:
A2 = πr22 = π (r1 + t ρ )
2
A1 = πr12
A = A2 − A1
A = πr12 + 2πr1t ρ + πt ρ2 − πr12 = 2πr1t ρ + πt ρ2
If r1 >> tρ then this approximates to:
A = 2πr1t ρ
so that:
R=ρ
l
l
=ρ
A
2πr1t ρ
Rearranging gives:
tρ = ρ
l
2πr1 R
Finally substituting gives:
8 × 10 −3
28 × 10 −9
t ρ = 3.5 × 10 ×
=
= 4.45 × 10 −9 = 4.45nm
−3
4
2 × π × 10 × 10
6.28
−5
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2.6 Resistors
Resistors are electrical circuit elements specifically manufactured to
exploit their properties of resistivity. The resistors typically used as components
in electronic circuits are manufactured using amorphous Carbon as the
material. This is a form of Graphite having no consistent crystalline structure
and providing a resistivity ranging from 1.5 – 4.5 x 10-5 Ωm, with a value of 3.5 x
10-5 Ωm being popular. Resistors are usually cylindrical in shape with electrical
connections at both circular ends to wires which can be soldered into a circuit.
This familiar form of resistor is shown in Fig. 4 below where the usual colour
bands printed on the component are used as a numbering system to indicate the
value of its resistance.
Fig. 4 Typical Resistors for Electronic Circuits
The resistor shown can be fabricated in the form of cylinder or rod of
conductive carbon granules bound in a resin compound. The more modern
method is to deposit a thin film of carbon onto a cylindrical base made of an
insulating material such as glass or ceramic. In some cases the deposited layer of
Carbon is cut into a spiral shape to allow higher values of resistance to be
obtained. Metal oxides such as tin-oxide are sometimes used as an alternative to
Carbon when other properties of the resistor such as a low temperature
coefficient are important.
Resistors are manufactured in ranges or series which have different
degrees of refinement of the materials and control of the process. This produces
resistors having different ranges of accuracy or manufacturing tolerance. The
most common ranges of manufacturing tolerance today are ±5%, ±2%, ±1%,
while the most common series of values are the E12 and E24.
A colour-code printed onto the body of the resistor is used internationally
to specify the value of a resistor. Using the colour code the value of resistance
which may range over several decades can be determined for any individual
resistor. The colour code is shown printed on the body of a typical resistor in
Fig. 5. This code consists typically of four coloured bands. The three left hand
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bands indicate the value of the resistance while the fourth band on the right
hand side indicates the manufacturing tolerance.
Fig. 5 The Resistor Colour Code
Each of the left hand group of three bands represents one decimal digit in
the value of resistance. The first and second digits are the numerical significant
digits while the third is the multiplier digit which gives the numerical value of
the power of ten which the two significant digits in the value are multiplied by.
The decimal numbers associated with each colour in the resistor value are given
in Table 1 while the value of the tolerance corresponding to the colour of the
fourth band is given in Table 2.
Table 1 Resistor Value
Number
Table 2 Manufacturing Tolerance
Tolerance
Colour
Colour
0
Black
±1%
brown
1
Brown
±2%
red
2
Red
±5%
gold
3
Orange
±10%
silver
4
Yellow
5
Green
6
Blue
7
Violet
8
Grey
9
White
6
From this the value of the resistor in Fig. 5 is obtained as:
Digit 1
Yellow = 4
Digit 2
Violet = 7
Digit 3 Multiplier Red = 2
Value is 47 x 102 = 47 x 100 = 4700 Ω = 4.7 kΩ.
Digit 4
Gold
Tolerance = ±5%
The E12 series of resistors is so called because it has 12 values of resistance per
decade. This means that Digits 1 and 2 in the code have 12 values. The E24 series
has 24 values per decade. The values in each of these series are given in Table 3.
Table 3.
E12 series
E12 and E24 Series of Resistors
tol. ±5%, ±10%
10
E24 series
10
11
12
12
13
15
15
16
18
18
20
22
22
24
27
27
30
33
33
36
39
39
43
47
47
51
56
56
62
68
68
75
82
82
91
7
tol. ±5%
Fixed resistors, having specific values in particular ranges, come in a number of
different forms. The form shown in Fig. 2 is the most familiar in low-and
medium power electronic circuits such as those implemented on printed circuit
boards. However, for higher power ratings larger packages are used as shown in
Fig. 6 and the resistors are often constructed of resistance wire wound on an
insulated former.
Fig. 6
A Range of Resistors of Various Power Ratings
In modern low-power electronic circuits where long-term battery operation and
miniaturisation of size is of importance other forms of packages are available
such as pin-arrays and surface mounted resistors as shown in Fig. 7.
Fig. 7 Miniaturised Resistor Packages
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There is also a wide variety of variable resistors whose value can be
altered to suit particular circumstances or to achieve a specific purpose in
electric and electronic circuits. Small low-power trimmers are used to make incircuit adjustments to cancel out unwanted errors such as offset voltages in
semiconductor devices. Examples of these are shown in Fig. 8.
Fig. 8 Low-Power Trimmer Resistors
Larger manually variable resistors, known as potentiometers, are used as
volume controls, for example in radios of hi-fi systems. Examples of these are
shown in Fig. 9.
Fig. 9
Potentiometers Used as Volume and Tone Controls
The conventional symbols used for fixed and variable resistors in
schematic diagrams of electric circuits are shown in Fig. 10.
fixed resistor
variable resistor
or potentiometer
R
RV
Fig. 10 Schematic Symbols for Resistors
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