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c h a p t e r
2
Resistance
No pain, no palm; no thorns, no throne; no gall, no glory; no cross,
no crown.
—William Penn
Historical Profiles
Georg Simon Ohm (1787–1854), a German physicist, in 1826
experimentally determined the most basic law relating voltage and current for a resistor. Ohm’s work was initially denied by critics.
Born of humble beginnings in Erlangen, Bavaria, Ohm threw himself into electrical research. Ohm’s major interest was current electricity, which had recently been advanced by Alessandro Volta’s invention
of the battery. Using the results of his experiments, Ohm was able to
define the fundamental relationship among voltage, current, and resistance. This resulted in his famous law—Ohm’s law—which will be covered in this chapter. He was awarded the Copley Medal in 1841 by the
Royal Society of London. He was also given the Professor of Physics
chair in 1849 by the University of Munich. To honor him, the unit of
resistance is named the ohm.
Ernst Werner von Siemens (1816–1892) was a German electrical
engineer and industrialist who played an important role in the development of the telegraph.
Siemens was born at Lenthe in Hanover, Germany, the oldest of
four brothers—all of whom were distinguished engineers and industrialists. After attending grammar school at Lübeck, Siemens joined the
Prussian artillery at age 17 for the training in engineering that his father
could not afford. Looking at an early model of an electric telegraph,
invented by Charles Wheatstone in 1837, Siemens realized its possibilities for making improvements and for international communication.
He invented a telegraph that used a needle to point to the right letter,
instead of using Morse code. He laid the first telegraph line in Germany
with his brothers, William Siemens and Carl von Siemens. The unit of
conductance is named in his honor.
Georg Simon Ohm
© SSPL via Getty Images
Ernst Werner von Siemens
© Hulton Archive/Getty
23
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Chapter 2
24
2.1
Resistance
Introduction
In the last chapter, we introduced some basic concepts such as current,
voltage, and power in an electric circuit. To actually determine the values of these variables in a given circuit requires that we understand some
fundamental laws that govern electric circuits. These laws—known as
Ohm’s law and Kirchhoff’s laws—form the foundation upon which
electric circuit analysis is built. Ohm’s law will be covered in this chapter, while Kirchhoff’s laws will be covered in Chapters 4 and 5.
We begin the chapter by first discussing resistance—its nature and
characteristics. We then cover Ohm’s law, conductance, and circular
wires. We present color coding for physically small resistors. We will
finally apply the concepts covered in this chapter to dc measurements.
2.2
Resistance
Materials in general have a characteristic behavior of opposing the flow
of electric charge. This opposition is due to the collisions between electrons that make up the materials. This physical property, or ability to
resist current, is known as resistance and is represented by the symbol
R. Resistance is expressed in ohms (after Georg Simon Ohm), which
is symbolized by the capital Greek letter omega (). The schematic
symbol for resistance or resistor is shown in Fig. 2.1, where R stands
for the resistance of the resistor.
The resistance R of an element denotes its ability to resist the flow
of electric current; it is measured in ohms ().
R
The resistance of any material is dictated by four factors:
Figure 2.1
Circuit symbol for resistance.
l
Material with
resistivity ␳
Cross-sectional
area A
1. Material property—each material will oppose the flow of current
differently.
2. Length—the longer the length , the more is the probability of collisions and, hence, the larger the resistance.
3. Cross-sectional area—the larger the area A, the easier it becomes
for electrons to flow and, hence, the lower the resistance.
4. Temperature—typically, for metals, as temperature increases, the
resistance increases.
Thus, the resistance R of any material with a uniform cross-sectional area
A and length (as shown in Fig. 2.2) is directly proportional to the length
and inversely proportional to its cross-sectional area. In mathematical form,
Rr
/
A
(2.1)
Figure 2.2
A conductor with uniform cross section.
where the Greek letter rho r is known as the resistivity of the material. Resistivity is a physical property of the material and is measured
in ohm-meters (-m).
The cross section of an element can be circular, square, rectangular, and so on. Because most conductors are circular in cross-section,
the cross-sectional area may be determined in terms of the radius r or
diameter d of the conductor as
d 2 pd2
A pr2 pa b (2.2)
2
4
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2.2
Resistance
TABLE 2.1
Resistivities of common materials.
Material
Silver
Copper
Aluminum
Gold
Iron
Lead
Germanium
Silicon
Paper
Mica
Glass
Teflon
Resistivity (-m)
8
1.64 10
1.72 108
2.8 108
2.45 108
1.23 107
2.2 107
4.7 101
6.4 102
1010
5 1011
1012
3 1012
Usage
Conductor
Conductor
Conductor
Conductor
Conductor
Conductor
Semiconductor
Semiconductor
Insulator
Insulator
Insulator
Insulator
The resisitivity r varies with temperature and is often specified for
room temperature.
Table 2.1 presents the values of r for some common materials at
room temperature (20°C). The table also shows that materials can be
classified into three groups according to their usage: conductors, insulators, and semiconductors. Good conductors, such as copper and aluminum, have low resistivities. Of those materials shown in Table 2.1,
silver is the best conductor. However, a lot of wires are made of copper because copper is almost as good and is much cheaper. In general,
the resistance of a conductor increases with a rise in temperature. Insulators, such as mica and paper, have high resistivities. They are used
in forming the insulating coating of copper wires. Semiconductors,
such as germanium and silicon, have resistivities that are neither high
nor low. They are used in making transistors and integrated circuits.
There is even a considerable range within the conductor group.
Nichrome (an alloy of nickel, chrome, and iron) has resistivity roughly
58 times greater than that of copper. For this reason, Nichrome is used
in making resistors and heating elements.
The circuit element used to model the current-resisting behavior
of a material is the resistor. For the purpose of constructing circuits,
resistors shown in Fig. 2.3 are usually made from metallic alloys and
carbon compounds. The resistor is the simplest passive element.
Figure 2.3
From top to bottom 14-W, 12 -W, and 1-W resistors.
© Sarhan M. Musa
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Chapter 2
26
Example 2.1
Resistance
Calculate the resistance of an aluminum wire that is 2 m long and of
circular cross section with a diameter of 1.5 mm.
Solution:
We first calculate the cross-sectional area:
A
p(1.5 10 3 ) 2
pd 2
1.767 106 m2
4
4
From Table 2.1, we obtain the resistivity of aluminum as r 2.8 108 -m. Thus,
R
r/
2.8 108 2
A
1.767 106
31.69 m
Practice Problem 2.1
Determine the resistance of an iron wire having a diameter of 2 mm
and a length of 30 m.
Answer: 1.174 Example 2.2
A copper bus bar is shown in Fig. 2.4. Calculate the length of the bar
that will produce a resistance of 0.5 .
Solution:
The bus bar has a uniform cross section so that Eq. (2.1) applies. But
the cross section is rectangular so that the cross-sectional area is
A Width Breadth (2 103 ) (3 103 )
6 106 m2 6 mm2
l
3 mm
2 mm
Figure 2.4
A copper bus bar; for Example 2.2.
From Table 2.1, the resistivity of copper is obtained as r 1.72 108 -m. Thus,
Rr
/
A
¡ /
/
RA
r
0.5 6 106
174.4 m
1.72 10 8
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2.3
Ohm’s Law
A conducting bar with triangular cross section is shown in Fig. 2.5. If
the bar is made of lead, determine the length of the bar that will produce a resistance of 1.25 m.
27
Practice Problem 2.2
4 mm
6 mm
Figure 2.5
For Practice Problem 2.2.
Answer: 6.82 cm
2.3
Ohm’s Law
Georg Simon Ohm (1787–1854), a German physicist, is credited with
finding the relationship between current and voltage for a resistor. This
relationship is known as Ohm’s law. That is,
V r I
(2.3)
Ohm’s law states that the voltage V across a resistor is directly proportional to the current I flowing through the resistor.
Ohm defined the constant of proportionality for a resistor to be the
resistance R. (The resistance is a material property that could change
if the internal or external conditions of the element were altered, e.g.,
if there were changes in the temperature.) Thus, Eq. (2.3) becomes
V IR
(2.4)
which is the mathematical form of Ohm’s law. In Eq. (2.4), we recall
that the voltage V is measured in volts, the current I is measured in
amperes, and the resistance R is measured in ohms. We may deduce
from Eq. (2.4) that
V
R
(2.5)
I
so that
1 1 V1 A
(2.6)
We may also deduce from Eq. (2.4) that
V
I
(2.7)
R
Thus, Ohm’s law can be stated in three different ways, as in Eqs. (2.4),
(2.5), and (2.7).
To apply Ohm’s law as stated in Eq. (2.4), for example, we must
pay careful attention to the current direction and voltage polarity. The
direction of current I and the polarity of voltage V must conform with
the convention shown in Fig. 2.6. This implies that current flows from
I
+
V
–
R
Figure 2.6
Direction of current I and polarity of voltage V across a resistor R.
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Chapter 2
28
+
I
V=0 R=0
Source
−
Resistance
a higher potential to a lower potential in order for V IR. If current
flows from a lower potential to a higher potential, then V IR.
(When the polarity of the voltage across the resistor is not specified,
always place the plus sign at the terminal where the current enters.)
Because the value of R can range from zero to infinity, it is important that we consider the two extreme possible values of R. An element
with R 0 is called a short circuit, as shown in Fig. 2.7(a). For a short
circuit,
(a)
V IR 0
+
Source
I=0
(2.8)
showing that the voltage is zero but the current could be anything. In
practice, a short circuit is usually a connecting wire assumed to be a
perfect conductor. Thus
V R=∞
A short circuit is a circuit element with resistance approaching zero.
−
Similarly, an element with R is known as an open circuit, as
shown in Fig. 2.7(b). For an open circuit,
(b)
Figure 2.7
(a) Short circuit (R 0); (b) open circuit
(R ).
I
V
V
0
R
(2.9)
indicating that the current is zero though the voltage could be anything.
Thus,
An open circuit is a circuit element with resistance approaching infinity.
Example 2.3
An electric iron draws 2 A at 120 V. Find its resistance.
Solution:
From Ohm’s law,
R
Practice Problem 2.3
V
120
60 I
2
The essential component of a toaster is an electrical element (a resistor) that converts electrical energy to heat energy. How much current
is drawn by a toaster with resistance of 12 at 110 V?
Answer: 9.17 A
Example 2.4
In the circuit shown in Fig. 2.8, calculate the current I.
+
30 V
Figure 2.8
For Example 2.4.
V
−
I
5 kΩ
Solution:
The voltage across the resistor is the same as the source voltage (30 V)
because the resistor and the voltage source are connected to the same
pair of terminals. Hence,
I
30
V
6 mA
R
5 103
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2.4
Conductance
If I 8 mA in the circuit shown in Fig. 2.9, determine the value of
resistance R.
29
Practice Problem 2.4
I
Answer: 1.5 k
R
12 V
2.4
Conductance
Figure 2.9
A useful quantity in circuit analysis is the reciprocal of resistance R,
known as conductance and denoted by G:
G
1
I
R
V
For Practice Problem 2.4.
(2.10)
The conductance is a measure of how well an element will conduct
electric current. The old unit of conductance is the mho (ohm spelled
backward) with symbol , the inverted omega. Although engineers
still use mhos, in this book we will prefer to use the SI unit of conductance, the siemens (S), in honor of Werner von Siemens:
1S1
1 A1 V
(2.11)
Thus,
Conductance is the ability of an element to conduct electric current;
it is measured in siemens (S).
[We should not confuse S for siemens with s (seconds) for time.] The
same resistance can be expressed in ohms or siemens. For example,
10 is the same as 0.1 S. From Eqs. (2.1) and (2.10), we may write
G
A
sA
r/
/
(2.12)
where the Greek letter sigma s 1r conductivity of the material
(in S/m).
Find the conductance of the following resistors: (a) 125 (b) 42 k
Example 2.5
Solution:
(a) G 1R 1 (125 ) 8 mS
(b) G 1R 1 (42 103 ) 23.8 mS
Determine the conductance of the following resistors:
(a) 120 (b) 25 M
Answers: (a) 8.33 mS (b) 40 nS
Practice Problem 2.5
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Chapter 2
Resistance
2.5
Circular Wires
Circular wires are commonly used in several applications. We use wires
to connect elements, but those wires have resistance and a maximum
allowable current. So we need to choose the right size. Wires are
arranged in standard gauge numbers, known as AWG (American Wire
Gauge). This designation of cables and wires is in the English system.
In the English system,
1,000 mils 1 in
(2.13a)
or
1 mil 1
in 0.001 in
1000
(2.13b)
A unit of cross-sectional area used for wires is the circular mil (CM),
which is the area of a circle having diameter of 1 mil. From Eq. (2.2),
A
p(1 mil) 2
pd2
p
sq mil
4
4
4
(2.14)
Thus,
1 CM p
sq mil
4
(2.15a)
4
CM
p
(2.15b)
or
1 sq mil If the diameter of a circular wire is in mils, the area in circular mils is
ACM d 2mil
(2.16)
A listing of data for standard bare copper wires is provided in
Table 2.2, where d is the diameter and R is the resistance for 1000 ft.
(Notice the wire diameter decreases as the gauge number increases.)
As you might guess, the maximum allowable currents are just a rule
of thumb. The steel industry uses a different numbering system for their
wire thickness gages (e.g., U.S. Steel Wire Gauge) so that the data in
Table 2.2 do not apply to steel wire. See Fig. 2.10 for different sizes
of wires. Typical household wiring is AWG number 12 or 14. Telephone wire is usually 22, 24, or 26 gauge. The following examples will
illustrate how to use the table.
Figure 2.10
Insulated wires of different gauges.
© Sarhan M. Musa
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2.5
Circular Wires
TABLE 2.2
American wire gauge (AWG) sizes at 20°C.
AWG #
d(mil)
0000
000
00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
460
409.6
364.8
324.9
289.3
257.8
229.4
204.3
181.9
162
144
128.5
114.4
101.9
90.74
80.81
71.96
64.08
57.07
50.82
45.26
40.30
35.89
31.96
28.46
25.3
22.6
20.1
17.9
15.94
14.2
12.6
11.26
10.03
8.928
7.95
7.08
6.305
5.6
5
4.5
3.965
3.531
3.145
Area (CM)
211,600
167,810
133,080
105,530
83,694
66,373
52,634
41,740
33,102
26,250
20,820
16,510
13,090
10,381
8,234
6,530
5,178
4,107
3,257
2,583
2,048
1,624
1,288
1,022
810.10
642.40
509.5
404.01
320.40
254.10
201.50
159.79
126.72
100.50
79.70
63.21
50.13
39.75
31.52
25
19.83
15.72
12.47
9.89
R (/1000 ft)
0.0490
0.0618
0.0780
0.0983
0.1240
0.1563
0.1970
0.2485
0.3133
0.3951
0.4982
0.6282
0.7921
0.9989
1.260
1.588
2.003
2.525
3.184
4.016
5.064
6.385
8.051
10.15
12.80
16.14
20.36
25.67
32.37
40.81
51.57
64.90
81.83
103.2
130.1
164.1
206.9
260.9
329.0
414.8
523.1
659.6
831.8
1049
Maximum
allowable
current (A)
230
200
175
150
130
115
100
85
—
65
—
50
—
30
—
20
—
15
31
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Chapter 2
32
Example 2.6
Resistance
Calculate the resistance of 840 ft of AWG #6 copper wire.
Solution:
From Table 2.2, the resistance of 1000 ft of AWG #6 is 0.3951 .
Hence, for a length of 840 ft,
R 840 ft a
Practice Problem 2.6
0.3951 b 0.3319 1000 ft
Find the resistance of 1200 ft of AWG #10 copper wire.
Answer: 199 Example 2.7
Find the cross-sectional area of a AWG #9 having a diameter of
114.4 mil.
ACM (114.4) 2 13,087 CM
Practice Problem 2.7
What is the cross-sectional area in CM of a wire with a diameter of
0.0036 in.?
Answer: 12.96 CM
2.6
Types of Resistors
Different types of resistors have been created to meet different requirements. Some resistors are shown in Fig. 2.11. The primary functions
of resistors are to limit current, divide voltage, and dissipate heat.
A resistor is either fixed or variable. Most resistors are of the fixed
type; that is, their resistance remains constant. The two common types
Figure 2.11
Different types of resistors.
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2.6
Types of Resistors
of fixed resistors (wirewound and composition) are shown in Fig. 2.12.
Wirewound resistors are used when there is a need to dissipate a large
amount of heat, while the composition resistors are used when large
resistance is needed. The circuit symbol in Fig. 2.1 is for a fixed resistor. Variable resistors have adjustable resistance. The symbol for a variable resistor is shown in Fig. 2.13. There are two main types of variable
resistors: potentiometer and rheostat. The potentiometer or pot for
short, is a three-terminal element with a sliding contact or wiper. By
sliding the wiper, the resistances between the wiper terminal and the
fixed terminals vary. The potentiometer is used to adjust the amount of
voltage provided to a circuit, as typically shown in Fig. 2.14. A potentiometer with its adjuster is shown in Fig. 2.15. The rheostat is a twoor three-terminal device that is used to control the amount of current
within a circuit, as typically shown in Fig. 2.16. As the rheostat is
adjusted for more resistance and less current flow, and the motor slows
down and vice versa. It is possible to use the same variable resistor as
a potentiometer or a rheostat, depending on how it is connected. Like
fixed resistors, variable resistors can either be of wirewound or composition type, as shown in Fig. 2.17. Although fixed resistors shown in
Fig. 2.12 are used in circuit designs, today, most circuit components
(including resistors) are either surface mounted or integrated, as typically shown in Fig. 2.18. Surface mount technology (SMT) is being
used to implement both digital and analog circuits. An SMT resistor is
shown in Fig. 2.19.
It should be pointed out that not all resistors obey Ohm’s law. A
resistor that obeys Ohm’s law is known as a linear resistor. It has a constant resistance, and thus its voltage-current characteristic is as illustrated in Fig. 2.20(a); that is, its V-I graph is a straight line passing
through the origin. A nonlinear resistor does not obey Ohm’s law. Its
resistance varies with current and its V-I characteristic is typically shown
33
(a)
(b)
Figure 2.12
Fixed resistors: (a) wirewound type;
(b) carbon film type.
Courtesy of Tech America
(a)
(b)
Figure 2.13
Circuit symbols for a variable resistor.
V
R
Figure 2.14
Variable resistor used as a potentiometer.
Figure 2.15
Potentiometers with their adjusters.
© Sarhan M. Musa
R
V
Motor
(a)
(b)
Figure 2.16
Figure 2.17
Variable resistor used as a rheostat.
Variable resistors: (a) composition type; (b) slider pot.
Courtesy of Tech America
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Chapter 2
Resistance
Figure 2.18
Resistors in an integrated circuit board.
© Eric Tomey/Alamy RF
V
Slope = R
I
(a)
V
Slope = R
I
(b)
Figure 2.20
Figure 2.19
Surface mount resistor.
© Greg Ordy
The V-I characteristics of a
(a) linear resistor;
(b) nonlinear resistor.
Figure 2.21
Diodes.
© Sarhan M. Musa
in Fig. 2.20(b). Examples of devices with nonlinear resistance are the
lightbulb and the diode1 (see Fig. 2.21). Although all practical resistors
may exhibit nonlinear behavior under certain conditions, we will assume
in this book that all objects actually designated as resistors are linear.
1
A diode is a semiconductor device that acts like a switch; it allows charge/current to
flow in only one direction.
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2.7
2.7
Resistor Color Code
35
Resistor Color Code
Some resistors are physically large enough to have their values printed
on them. Other resistors are too small to have their values printed on
them. For such small resistors, color coding provides a way of determining the value of resistance. As shown in Fig. 2.22, the color coding consists of three, four, or five bands of color around the resistor.
The bands are illustrated in Table 2.3 and explained as follows:
A B C D E
Figure 2.22
Resistor color codes.
0
1
2
3
4
5
6
7
8
9
A First significant figure of resistance value
B Second significant figure of resistance value
C Multiplier of resistance for resistance value
D Tolerance rating (in %)
E Reliability factor (in %)
*We read the bands from left to right.
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
The first three bands (A, B, and C) specify the value of the resistance. Figure 2.23
Memory aid for color codes.
Bands A and B represent the first and second digits of the resistance
value. Band C is usually given as a power of 10 as in Table 2.3. If
present, the fourth band (D) indicates the tolerance percentage. For
example, a 5 percent tolerance indicates that the actual value of the
resistance is within 5 of the color-coded value. When the fourth band
is absent, the tolerance is taken by default to be 20 percent. The fifth
band (E), if present, is used to indicate a reliability factor, which is a
statistical indication of the expected number of components that will
fail to have the indicated resistance after working for 1,000 hours. As
shown in Fig. 2.23, the statement “Big Boys Race Our Young Girls,
But Violet Generally Wins” can serve as a memory aid in remembering the color code.
TABLE 2.3
Resistor color code.
Color
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
Gold
Silver
No color
Band A
Band B
significant significant Band C
Band D
Band E
figure
figure
multiplier tolerance reliability
N/A
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
100
101
102
103
104
105
106
107
108
109
0.1
0.01
1%
0.1%
0.01%
0.001%
5%
10%
20%
Big
Boys
Race
Our
Young
Girls
But
Violet
Generally
Wins
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Chapter 2
36
Example 2.8
Figure 2.24
For Example 2.8.
Resistance
Determine the resistance value of the color-coded resistor shown in
Fig. 2.24.
Solution:
Band A is blue (6); band B is red (2); band C is orange (3); band D is
gold (5%); and band E is red (0.1%). Hence,
R 62 103 5% tolerance with a reliability of 0.1%
62 k 3.1 k with a reliability of 0.1%
This means that the actual resistance of the color-coded resistor will
fall between 58.9 k (62 3.1) k and 65.1 k (62 3.1) k. The
reliability of 0.1% indicates that 1 out of 1,000 will fail to fall within
the tolerance range after 1,000 hours of service.
Practice Problem 2.8
What is the resistance value, tolerance, and reliability of the colorcoded resistor shown in Fig. 2.25?
Answer: 3.3 M 10% with a reliability of 1%
Figure 2.25
For Practice Problem 2.8.
Example 2.9
A resistor has three bands only—in order green, black, and silver. Find
the resistance value and tolerance of the resistor.
Solution:
Band A is green (5); band B is black (0); and band C is silver (0.01).
Hence
R 50 0.01 0.5 Because the fourth band is absent, the tolerance is, by default, 20 percent.
Practice Problem 2.9
What is the resistance value and tolerance of a resistor having bands
colored in the order yellow, violet, white, and gold?
Answer: 47 G 5%
Example 2.10
A company manufactures resistors of 5.4 k with a tolerance of
10 percent. Determine the color code of the resistor.
Solution:
R 5.4 103 54 102
From Table 2.3, green represents 5; yellow stands for 4; while red
stands for102. The tolerance of 10 percent corresponds to silver. Hence,
the color code of the resistor is:
Green, yellow, red, silver
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2.8
Standard Resistor Values
If the company in Example 2.10 also produces resistors of 7.2 M
with a tolerance of 5 percent and reliability of 1 percent, what will be
the color codes on the resistor?
Answer: Violet, red, green, gold, brown
2.8
Standard Resistor Values
One would expect resistor values are commercially available in all values. For practical reasons, this would not make sense. Only a limited
number of resistor values are commercially available at reasonable cost.
The list of standard values of commercially available resistors is presented in Table 2.4. These are the standard values that have been agreed
to for carbon composition resistors. Notice that the values range from
0.1 to 22 M. While 10 percent tolerance resistors are available only
for those values in bold type at reasonable cost, 5 percent tolerance
resistors are available in all values. For example, a 330- resistor could
be available either as a 5 or 10 percent tolerance component, while a
110-k resistor is available only as 5 percent tolerance component.
When designing a circuit, the calculated values are seldom standard. One may select the nearest standard values or combine the standard values. In most cases, selecting the nearest standard value may
TABLE 2.4
Standard values of commercially available resistors.
Ohms
()
0.10
0.11
0.12
0.13
0.15
0.16
0.18
0.20
0.22
0.24
0.27
0.30
0.33
0.36
0.39
0.43
0.47
0.51
0.56
0.62
0.68
0.75
0.82
0.91
1.0
1.1
1.2
1.3
1.5
1.6
1.8
2.0
2.2
2.4
2.7
3.0
3.3
3.6
3.9
4.3
4.7
5.1
5.6
6.2
6.8
7.5
8.2
9.1
10
11
12
13
15
16
18
20
22
24
27
30
33
36
39
43
47
51
56
62
68
75
82
92
Kilohms
(k)
100
110
120
130
150
160
180
200
220
240
270
300
330
360
390
430
470
510
560
620
680
750
820
910
1000
1100
1200
1300
1500
1600
1800
2000
2200
2400
2700
3000
3300
3600
3900
4300
4700
5100
5600
6200
6800
7500
8200
9100
10
11
12
13
15
16
18
20
22
24
27
30
33
36
39
43
47
51
56
62
68
75
82
91
100
110
120
130
150
160
180
200
220
240
270
300
330
360
390
430
470
510
560
620
680
750
820
910
Megohms
(M)
1.0
1.1
1.2
1.3
1.5
1.6
1.8
2.0
2.2
2.4
2.7
3.0
3.3
3.6
3.9
4.3
4.7
5.1
5.6
6.2
6.8
7.5
8.2
9.1
10.0
11.0
12.0
13.0
15.0
16.0
18.0
20.0
22.0
37
Practice Problem 2.10
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Chapter 2
38
Resistance
provide adequate performance. To ease calculations, most of the resistor values used in this book are nonstandard.
2.9
Applications: Measurements
Resistors are often used to model devices that convert electrical energy
into heat or other forms of energy. Such devices include conducting
wires, lightbulbs, electric heaters, stoves, ovens, and loudspeakers.
Also, by their nature, resistors are used to control the flow of current.
We take advantage of this property in several applications such as in
potentiometers and meters. In this section, we will consider meters—
the ammeter, voltmeter, and ohmmeter, which measure current, voltage, and resistance, respectively. Being able to measure current I,
voltage V, and resistance R is very important.
The voltmeter is the instrument used to measure voltage; the ammeter
is the instrument used to measure current; and the ohmmeter is the
instrument used to measure resistance.
It is common these days to have the three instruments combined into
one instrument known as a multimeter, which may be analog or digital.
An analog meter is one that uses a needle and calibrated meter to display
the measured value; that is, the measured value is indicated by the pointer
of the meter. A digital meter is one in which the measured valued is shown
in form of a digital display. The digital meters are more commonly used
today. Because both analog and digital meters are used in the industry,
one should be familiar with both. Figure 2.26 illustrates a typical analog
multimeter (combining voltmeter, ammeter, and ohmmeter) and a typical
digital multimeter. The digital multimeter (DMM) is the most widely used
instrument. Its analog counterpart is the volt-ohm-milliammeter (VOM).
To measure voltage, we connect the voltmeter/multimeter across
the element for which the voltage is desired, as shown in Fig. 2.27.
The voltmeter measures the voltage across the load and is therefore
connected in parallel2 with the element.
(a)
(b)
Figure 2.26
(a) Analog multimeter; (b) digital multimeter.
(a) © iStock; (b) © Oleksy Maksymenko/Alamy RF
2
Two elements are in parallel if they are connected to the same two points.
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2.9
V
+
−
+
V
+
Applications: Measurements
R
−
39
+ Voltmeter
−
−
Figure 2.27
Measuring voltage.
To measure current, we connect the ammeter/multimeter in series3
with the element under test, as shown in Fig. 2.28. The meter must be
connected such that the current enters through the positive terminal to
get a positive reading. The circuit must be “broken”; that is, the current path must be interrupted so that the current must flow through the
ammeter. (The ampclamp is another device for measuring ac current.)
I
+
+−
mA
−
−
R
+
V
+
Ammeter
−
R
+
−
Figure 2.29
Figure 2.28
Measuring resistance.
Measuring current.
To measure resistance of an element, connect the ohmmeter/
multimeter across it, as shown in Fig. 2.29. If the element is connected
to a circuit, one end of the element must first be disconnected from the
circuit before we measure its resistance. Because the resistance of a
wire with no breaks is zero, the ohmmeter can be used to test for continuity. If the wire has a break, the ohmmeter connected across it will
read infinity. Thus, the ohmmeter can be used to detect a short circuit
(low resistance) and an open circuit (high resistance).
When working with any of the meters mentioned in this section,
it is good practice to observe the following:
1. If possible, turn the circuit power off before connecting the meter.
2. To avoid damaging the instrument, it is best to always set the meter
on the highest range and then move down to the appropriate range.
(Most DMMs are auto-ranging.)
3. When measuring dc current or voltage, observe proper polarity.
4. When using a multimeter, make sure you set the meter in the correct mode (ac, dc, V, A, ), including moving the test idea to the
appropriate jacks.
5. When the measurement is completed, turn off the meter to avoid
draining the internal battery of the meter.
This leads to the issue of safety in electrical measurement.
3
Two elements are in series if they are cascaded or connected sequentially.
+ Ohmmeter
−
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Chapter 2
Resistance
2.10
Electrical Safety Precautions
Now that we have learned how to measure current, voltage, and resistance, we need to be careful how we handle the instruments so as to
avoid electric shock or harm. Because electricity can kill, being able
to make safe and accurate measurements is an integral part of the
knowledge that you must acquire.
2.10.1 Electric Shock
When working on electric circuits, there is the possibility of receiving
an electric shock. The shock is due to the passage of current through
your body. An electric shock can startle you and cause you to fall down
or be thrown down. It may cause severe, rigid contractions of the muscles, which in turn may result in fractures, dislocations, and loss of
consciousness. The respiratory system may be paralyzed and the
heart may beat irregularly or even stop beating altogether. Electrical
burns may be present on the skin and extend into deeper tissue. High
current may cause death of tissues between the entry and exit point of
the current. Massive swelling of the tissues may follow as the blood in
the veins coagulates and the muscles swell. Thus, electric shock can
cause muscle spasms, weakness, shallow breathing, rapid pulse, severe
burns, unconsciousness, or death.
Electric shock is an injury caused by an electrical current passing
through the body.
The human body has resistance that depends on several factors
such as body mass, skin moisture, and points of contact of the body
with the electric appliance. The effects of various amounts of current
in milliamperes (mA) is shown in Table 2.5.
2.10.2 Precautions
Working with electricity can be dangerous unless you adhere strictly
to certain rules. The following safety rules should be followed whenever you are working with electricity:
• Always make sure that the circuit is actually dead before you begin
working on it.
• Always unplug any appliance or lamp before repairing it.
• Always tape over the main switch, empty fuse socket, or circuit
breaker when you’re working. Leave a note there so no one will
accidentally turn on the electricity. Keep any fuses you’ve removed
in your pocket.
TABLE 2.5
Electric shock
Electric Current
Less than 1mA
1 mA
5–20 mA
20–100 mA
Physiological effect
No sensation or feeling
Tingling sensation
Involuntary muscle contraction
Loss of breathing, fatal if continued
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2.11
Summary
• Handle tools properly and make sure that the insulation on metal
tools is in good condition.
• If measuring V or I, turn on the power and record reading. If measuring R, do not turn on power.
• Refrain from wearing loose clothing. Loose clothes can get caught
in an operating appliance.
• Always wear long-legged and long-sleeved clothes and shoes and
keep them dry.
• Do not stand on a metal or wet floor. (Electricity and water do not
mix.)
• Make sure there is adequate illumination around the work area.
• Do not work while wearing rings, watches, bracelets, or other
jewelry.
• Do not work by yourself.
• Discharge any capacitor that may retain high voltage.
• Work with only one hand a time in areas where voltage may be high.
Protecting yourself from injury and harm is absolutely imperative. If
we follow these safety rules, we can avoid shock and related accidents.
Thus, our rule should always be “safety first.”
2.11
Summary
1. A resistor is an element in which the voltage, V, across it is directly
proportional to the current, I, through it. That is, a resistor is an
element that obeys Ohm’s law.
V IR
where R is the resistance of the resistor.
2. The resistance R of an object with uniform cross-sectional area A
is evaluated as resistivity r times length divided by the crosssection area A, that is,
R
r/
A
3. A short circuit is a resistor (a perfectly conducting wire) with zero
resistance (R 0). An open circuit is a resistor with infinite resistance (R ) .
4. The conductance G of a resistor is the reciprocal of its resistance R:
G
1
R
5. For a circular wire, the cross-sectional area is measured in circular mils (CM). The diameter in mils is related to the area in CM as
ACM d2mil
6. American Wire Gauge is a standard system for designating the
diameter of wires.
7. There are different types of resistors: fixed or variable, linear or
nonlinear. Potentiometer and rheostat are variable resistors that are
used to adjust voltage and current, respectively. Common types of
41
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Chapter 2
42
8.
9.
10.
11.
Resistance
resistors include carbon or composition resistors, wirewound resistors, chip resistors, film resistors, and power resistors.
A resistor is usually color coded when it is not physically large
enough to print the numerical value of the resistor on it. The statement “Big Boys Race Our Young Girls, But Violet Generally
Wins” is a memory aid for the color code: black, brown, red,
orange, yellow, green, blue, violet, gray, and white.
For carbon composition resistors, standard values are commercially available in the range of 0.1 to 22 M.
Voltage, current, and resistance are measured using a voltmeter,
ammeter, and ohmmeter, respectively. The three are measured
using a multimeter such as a digital multimeter (DMM) or a
volt-ohm-milliammeter (VOM).
Safety is all about preventing accidents. If we follow some safety
precautions, we should have no problems working on electric
circuits.
Review Questions
2.1
2.2
Which of the following materials is not a conductor?
(a) Copper
(b) Silver
(d) Gold
(e) Lead
2.6
(c) Mica
The main purpose of a resistor in a circuit is to:
2.7
(c ) 10 S
(d) 100 S
Potentiometers are types of:
(a) fixed resistors
(b) variable resistors
(c) meters
(d) voltage regulators
2.8
An element draws 10 A from a 120-V line. The
resistance of the element is:
(a) 1200 (b) 120 (c) 12 (d) 1.2 2.9
What is the area in circular mils of a wire that has a
diameter of 0.03 in.?
(a) 0.0009
(b) 9
(c ) 90
(d) 900
All resistors are color coded.
(a) True
(b) False
The reciprocal of resistance is:
(a) voltage
(b) current
(c) conductance
(d) power
2.10 Digital multimeters (DMM) are the most widely
used type of electronic measuring instrument.
(a) True
(b) False
Which of these is not the unit of conductance?
(a) ohm
(b) Siemen
(c) mho
(d)
2.5
(b) 0.1 S
(b) produce heat
(d) limit current
2.4
(a) 0.1 mS
(a) resist change in current
(c) increase current
2.3
The conductance of a 10-m resistor is:
Answers: 2.1c, 2.2d, 2.3c, 2.4c, 2.5a, 2.6d, 2.7b, 2.8d,
2.9b, 2.10a
Problems
Section 2.2 Resistance
2.1
A 250-m-long copper wire has a diameter of 2.2 mm.
Calculate the resistance of wire.
2.2
Find the length of a copper wire that has a resistance
of 0.5 and a diameter of 2 mm.
2.3
A 2-in. 2-in. square bar of copper is 4 ft long. Find
its resistance.
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Problems
2.4
If an electrical hotplate has a power rating of 1200 W
and draws a current of 6 A, determine the resistance
of the hotplate.
2.5
A Nichrome (r 100 108 m) wire is used
to construct heating elements. What length of a
2-mm-diameter wire will produce a resistance
of 1.2 ?
2.6
An aluminum wire of radius 3 mm has a resistance
of 6 . How long is the wire?
2.7
A graphite cylinder with a diameter of 0.4 mm and a
length of 4 cm has resistance of 2.1 . Determine
the resistivity of the cylinder.
2.8
A certain circular wire of length 50 m and diameter
0.5 m has a resistance of 410 at room temperature.
Determine the material the wire is made of.
2.9
If we shorten the length of a conductor, why does the
conductor decrease in resistance?
2.10 Two wires are made of the same material. The first
wire has a resistance of 0.2 . The second wire is
twice as long as the first wire and has a radius that is
half of the first wire. Determine the resistance of the
second wire.
2.11 Two wires have the same resistance and length. The
first wire is made of copper, while the second wire is
made of aluminum. Find the ratio of the crosssectional area of the copper wire to that of the
aluminum wire.
2.12 High-voltage power lines are used in transmitting
large amounts of power over long distances.
Aluminum cable is preferred over copper cable due
to low cost. Assume that the aluminum wire used for
high-voltage power lines has a cross-sectional area
of 4.7 104 m2. Find the resistance of 20 km of
this wire.
V
I
(a)
V
I
(b)
V
I
(c)
Figure 2.30
For Problem 2.13.
2.19 If a current of 30 mA flows through a 5.4-M
resistor, what is the voltage?
2.20 A current of 2 mA flows through a 25- resistor.
Find the voltage drop across it.
Section 2.3 Ohm’s Law
2.13 Which of the graphs in Fig. 2.30 represent Ohm’s law?
2.14 When the voltage across a resistor is 60 V, the
current through it is 50 mA. Determine its
resistance.
2.15 The voltage across a 5-k resistor is 16 V. Find the
current through the resistor.
2.16 A resistor is connected to a 12-V battery. Calculate
the current if the resistor is:
(a) 2 k
43
(b) 6.2 k
2.17 An air-conditioning compressor has resistance 6 .
When the compressor is connected to a 240-V
source, determine the current through the circuit.
2.18 A source of 12 V is connected to a purely resistive
lamp and draws 3 A. What is the resistance of the
lamp?
2.21 An element allows 28 mA of current to flow through
it when a 12-V battery is connected to its terminals.
Calculate the resistance of the element.
2.22 Find the voltage of a source which produces a
current of 10 mA in a 50- resistor.
2.23 A nonlinear resistor has I 4 102 V2. Find I for
V 10, 20, and 50 V.
2.24 Determine the magnitude and direction of the current
associated with the resistor in each of the circuits in
Fig. 2.31.
2.25 Determine the magnitude and polarity of the voltage
across the resistor in each of the circuits in Fig. 2.32.
2.26 A flashlight uses two 3-V batteries in series to
provide a current of 0.7 A in the filament. (a) Find
the potential difference across the flashlight bulb.
(b) Calculate the resistance of the filament.
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Chapter 2
44
15 V +
−
10 Ω
Resistance
9V
+
−
12/5/11
(a)
10 Ω
30 V
(b)
+
−
sad28078_ch02_023-046.qxd
6Ω
(c)
Figure 2.31
For Problem 2.24.
10 Ω
4A
10 Ω
20 mA
(a)
2Ω
6 mA
(b)
(c)
Figure 2.32
For Problem 2.25.
Section 2.4 Conductance
2.34 Find the diameter in inches for wires having the
following cross-sectional areas:
2.27 Determine the conductance of each of the following
resistances:
(a) 2.5 (b) 40 k
(c) 12 M
2.28 Find the resistance for each of the following
conductances:
(a) 10 mS
(b) 0.25 S
(c) 50 S
2.29 When the voltage across a resistor is 120 V, the
current through it is 2.5 mA. Calculate its
conductance.
2.30 A copper rod has a length of 4 cm and a conductance
of 500 mS. Find its diameter.
2.31 Determine the battery voltage V in the circuit shown
in Fig. 2.33.
(a) 420 CM
(b) 980 CM
2.35 Calculate the area in circular mils of the following
conductors:
(a) circular wire with diameter 0.012 in.
(b) rectangular bus bar with dimensions
0.2 in. 0.5 in.
2.36 How much current will flow in a #16 copper wire
1 mi long connected to a 1.5-V battery?
Section 2.7 Resistor Color Code
2.37 Find the resistance value having the following color
codes:
(a) blue, red, violet, silver
I = 4 mA
(b) green, black, orange, gold
+
V
−
5 mS
Figure 2.33
For Problem 2.31.
Section 2.5 Circular Wires
2.32 Using Table 2.2, determine the resistance of 600 ft of
#10 and #16 AWG copper.
2.33 The resistance of a copper transmission line cannot
exceed 0.001 , and the maximum current drawn by
the load is 120 A. What gauge wire is appropriate?
Assume a length of 10 ft.
2.38 Determine the range (in ohms) in which a resistor
having the following bands must exist.
(a)
(b)
(c)
Band A
Band B
Band C
Band D
Brown
Red
White
Violet
Black
Red
Green
Orange
Gray
Silver
Gold
—
2.39 Determine the color codes of the following resistors
with 5 percent tolerance.
(a) 52 (b) 320 (c) 6.8 k
(d) 3.2 M
2.40 Find the color codes of the following resistors:
(a) 240 (b) 45 k
(c) 5.6 M
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Problems
2.41 For each of the resistors in Problem 2.37, find the
minimum and maximum resistance within the
tolerance limits.
45
20
(a) 10 , 10 percent tolerance
5
0
AC C
D
2
1
0
0
AC S 0
P
AM
dB
m
(b) 7.4 k, 5 percent tolerance
0
15
6
3
10
4
2
4
10
2
0
2
6
4
1
25
0
50
10
20
8
4
2
10 5
5
8
10
0 AC
30 0 DC
6 2
1
AC PS
AM
0
OH
MS
2
200
40
8
0
1k
150
30
6
100
20
4
50
10
2
20
0
2.42 Give the color coding for the following resistors:
3
10
20
50
4
5
MS
OH
3
12 0
11 dB 6
m
(c) 12 M, 20 percent tolerance
Analog Multimeter
Section 2.9 Applications: Measurements
0
1
dB
m
0
0
AC S 0
P
AM
20
AC C
D
10
4
2
0
2
4
6
20
8
4
25
10
5
8
10
DC Vo
lts
mA
DC
15
6
3
10
4
2
2
Figure 2.35
DC Vo
lts
mA
DC
–
300
60
12
3
x1
x10
x100
x1K
x100K
Ohms
Adj
V
+
Vs
−
Ohm
s
120
For Problem 2.44.
600
s
Volt
0.06
1.2
12
+
AC
600
300
60
12
3
0.3
x1
x10
x100
x1K
x100K
Ohms
Adj
–
3
12 0
11 dB 6
m
Analog Multimeter
OFF
120
300
60
12
3
Ohm
s
5
0
0.06
1.2
12
1
25
0
50
10
0 AC
30 0 DC
6 2
1
AC PS
AM
0
OH
MS
2
200
40
8
0
50
10
2
20
150
30
6
100
20
4
600
s
Volt
0
1k
3
10
20
50
4
5
MS
OH
OFF
AC
600
300
60
12
3
0.3
2.43 How much voltage is the multimeter in Fig. 2.34
reading?
Lamp
A
+
Figure 2.36
For Problem 2.46.
Figure 2.34
For Problem 2.43.
2.44 Determine the voltage reading for the multimeter in
Fig. 2.35.
2.45 You are supposed to check a lightbulb to see whether
is burned out or not. Using an ohmmeter, how would
you do this?
2.46 What is wrong with the measuring scheme in
Fig. 2.36?
2.47 Show how you would place a voltmeter to measure
the voltage across resistor R1 in Fig. 2.37.
2.48 Show how you would place an ammeter to measure
the current through resistor R2 in Fig. 2.37.
2.49 Explain how you would connect an ohmmeter to
measure the resistance R2 in Fig. 2.37.
2.50 How would you use an ohmmeter to determine the
on and off states of a switch?
R1
+
V1
−
R2
Figure 2.37
For Problems 2.47, 2.48, and 2.49.
Section 2.10 Electrical Safety Precautions
2.51 What causes electric shock?
2.52 Mention at least four safety precautions you would
take when taking measurements.
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