-
23
ResistanceNo pain, no palm; no thorns, no throne; no gall, no
glory; no cross,no crown.
William Penn
c h a p t e r
2Historical Profiles
Georg Simon Ohm (17871854), a German physicist, in
1826experimentally determined the most basic law relating voltage
and cur-rent for a resistor. Ohms work was initially denied by
critics.
Born of humble beginnings in Erlangen, Bavaria, Ohm threw
him-self into electrical research. Ohms major interest was current
electric-ity, which had recently been advanced by Alessandro Voltas
inventionof the battery. Using the results of his experiments, Ohm
was able todefine the fundamental relationship among voltage,
current, and resist-ance. This resulted in his famous lawOhms
lawwhich will be cov-ered in this chapter. He was awarded the
Copley Medal in 1841 by theRoyal Society of London. He was also
given the Professor of Physicschair in 1849 by the University of
Munich. To honor him, the unit ofresistance is named the ohm.
Ernst Werner von Siemens (18161892) was a German
electricalengineer and industrialist who played an important role
in the devel-opment of the telegraph.
Siemens was born at Lenthe in Hanover, Germany, the oldest
offour brothersall of whom were distinguished engineers and
industri-alists. After attending grammar school at Lbeck, Siemens
joined thePrussian artillery at age 17 for the training in
engineering that his fathercould not afford. Looking at an early
model of an electric telegraph,invented by Charles Wheatstone in
1837, Siemens realized its possi-bilities 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 Germanywith his brothers,
William Siemens and Carl von Siemens. The unit ofconductance is
named in his honor.
Georg Simon Ohm SSPL via Getty Images
Ernst Werner von Siemens Hulton Archive/Getty
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IntroductionIn the last chapter, we introduced some basic
concepts such as current,voltage, and power in an electric circuit.
To actually determine the val-ues of these variables in a given
circuit requires that we understand somefundamental laws that
govern electric circuits. These lawsknown asOhms law and Kirchhoffs
lawsform the foundation upon whichelectric circuit analysis is
built. Ohms law will be covered in this chap-ter, while Kirchhoffs
laws will be covered in Chapters 4 and 5.
We begin the chapter by first discussing resistanceits nature
andcharacteristics. We then cover Ohms law, conductance, and
circularwires. We present color coding for physically small
resistors. We willfinally apply the concepts covered in this
chapter to dc measurements.
ResistanceMaterials in general have a characteristic behavior of
opposing the flowof electric charge. This opposition is due to the
collisions between elec-trons that make up the materials. This
physical property, or ability toresist current, is known as
resistance and is represented by the symbolR. Resistance is
expressed in ohms (after Georg Simon Ohm), whichis symbolized by
the capital Greek letter omega (). The schematicsymbol for
resistance or resistor is shown in Fig. 2.1, where R standsfor the
resistance of the resistor.
The resistance of any material is dictated by four factors:1.
Material propertyeach material will oppose the flow of current
differently.2. Lengththe longer the length , the more is the
probability of col-
lisions and, hence, the larger the resistance.3. Cross-sectional
areathe larger the area A, the easier it becomes
for electrons to flow and, hence, the lower the resistance.4.
Temperaturetypically, for metals, as temperature increases, the
resistance increases.Thus, the resistance R of any material with
a uniform cross-sectional areaA and length (as shown in Fig. 2.2)
is directly proportional to the lengthand inversely proportional to
its cross-sectional area. In mathematical form,
(2.1)
where the Greek letter rho r is known as the resistivity of the
mate-rial. Resistivity is a physical property of the material and
is measuredin ohm-meters (-m).
The cross section of an element can be circular, square,
rectangu-lar, and so on. Because most conductors are circular in
cross-section,the cross-sectional area may be determined in terms
of the radius r ordiameter d of the conductor as
(2.2)A pr2 pad2b2 pd2
4
R r /A
The resistance R of an element denotes its ability to resist the
flowof electric current; it is measured in ohms ().
2.2
2.1
24 Chapter 2 Resistance
R
Figure 2.1Circuit symbol for resistance.
l
Cross-sectionalarea A
Material withresistivity
Figure 2.2A conductor with uniform cross section.
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The resisitivity r varies with temperature and is often
specified forroom temperature.
Table 2.1 presents the values of r for some common materials
atroom temperature (20C). The table also shows that materials can
beclassified into three groups according to their usage:
conductors, insu-lators, and semiconductors. Good conductors, such
as copper and alu-minum, have low resistivities. Of those materials
shown in Table 2.1,silver is the best conductor. However, a lot of
wires are made of cop-per because copper is almost as good and is
much cheaper. In general,the resistance of a conductor increases
with a rise in temperature. Insu-lators, such as mica and paper,
have high resistivities. They are usedin forming the insulating
coating of copper wires. Semiconductors,such as germanium and
silicon, have resistivities that are neither highnor 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 roughly58 times greater
than that of copper. For this reason, Nichrome is usedin making
resistors and heating elements.
The circuit element used to model the current-resisting
behaviorof a material is the resistor. For the purpose of
constructing circuits,resistors shown in Fig. 2.3 are usually made
from metallic alloys andcarbon compounds. The resistor is the
simplest passive element.
2.2 Resistance 25
TABLE 2.1
Resistivities of common materials.
Material Resistivity (-m) UsageSilver 1.64 108 ConductorCopper
1.72 108 ConductorAluminum 2.8 108 ConductorGold 2.45 108
ConductorIron 1.23 107 ConductorLead 2.2 107 ConductorGermanium 4.7
101 SemiconductorSilicon 6.4 102 SemiconductorPaper 1010
InsulatorMica 5 1011 InsulatorGlass 1012 InsulatorTeflon 3 1012
Insulator
Figure 2.3From top to bottom -W, -W, and 1-W resistors. Sarhan
M. Musa
12
14
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From Table 2.1, the resistivity of copper is obtained as r .
Thus,
/ 0.5 6 106
1.72 108 174.4 m
R r/A
/ RAr
1.72 108 -m
26 Chapter 2 Resistance
Calculate the resistance of an aluminum wire that is 2 m long
and ofcircular cross section with a diameter of 1.5 mm.
Solution:We first calculate the cross-sectional area:
From Table 2.1, we obtain the resistivity of aluminum as r -m.
Thus,
31.69 m
R r/A
2.8 108 21.767 106
2.8 108
A pd2
4p(1.5 103)2
4 1.767 106 m2
Example 2.1
Determine the resistance of an iron wire having a diameter of 2
mmand a length of 30 m.
Answer: 1.174
Practice Problem 2.1
A copper bus bar is shown in Fig. 2.4. Calculate the length of
the barthat will produce a resistance of 0.5 .
Solution:The bus bar has a uniform cross section so that Eq.
(2.1) applies. Butthe cross section is rectangular so that the
cross-sectional area is
6 106 m2 6 mm2 A Width Breadth (2 103) (3 103)
Example 2.2
3 mm
2 mm
l
Figure 2.4A copper bus bar; for Example 2.2.
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2.3 Ohms Law 27
A conducting bar with triangular cross section is shown in Fig.
2.5. Ifthe bar is made of lead, determine the length of the bar
that will pro-duce a resistance of 1.25 m.
Practice Problem 2.2
6 mm
4 mm
Figure 2.5For Practice Problem 2.2.
Answer: 6.82 cm
Ohms LawGeorg Simon Ohm (17871854), a German physicist, is
credited withfinding the relationship between current and voltage
for a resistor. Thisrelationship is known as Ohms law. That is,
(2.3)
Ohm defined the constant of proportionality for a resistor to be
theresistance R. (The resistance is a material property that could
changeif the internal or external conditions of the element were
altered, e.g.,if there were changes in the temperature.) Thus, Eq.
(2.3) becomes
(2.4)
which is the mathematical form of Ohms law. In Eq. (2.4), we
recallthat the voltage V is measured in volts, the current I is
measured inamperes, and the resistance R is measured in ohms. We
may deducefrom Eq. (2.4) that
(2.5)so that
(2.6)We may also deduce from Eq. (2.4) that
(2.7)Thus, Ohms law can be stated in three different ways, as in
Eqs. (2.4),(2.5), and (2.7).
To apply Ohms law as stated in Eq. (2.4), for example, we
mustpay careful attention to the current direction and voltage
polarity. Thedirection of current I and the polarity of voltage V
must conform withthe convention shown in Fig. 2.6. This implies
that current flows from
I VR
1 1 V1 A
R VI
V IR
Ohms law states that the voltage V across a resistor is directly
pro-portional to the current I flowing through the resistor.
V r I
2.3
V R
I
+
Figure 2.6Direction of current I and polarity of volt-age V
across a resistor R.
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a higher potential to a lower potential in order for . If
currentflows from a lower potential to a higher potential, then
.(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
impor-tant that we consider the two extreme possible values of R.
An elementwith R 0 is called a short circuit, as shown in Fig.
2.7(a). For a shortcircuit,
(2.8)showing that the voltage is zero but the current could be
anything. Inpractice, a short circuit is usually a connecting wire
assumed to be aperfect conductor. Thus
Similarly, an element with is known as an open circuit, asshown
in Fig. 2.7(b). For an open circuit,
(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.
I VR
V
0
R
A short circuit is a circuit element with resistance approaching
zero.
V IR 0
V IRV IR
28 Chapter 2 Resistance
(a)
(b)
R = 0
I
R =
I = 0
V = 0Source
Source
+
V
+
Figure 2.7(a) Short circuit (R 0); (b) open circuit( ).R
An electric iron draws 2 A at 120 V. Find its resistance.
Solution:From Ohms law,
R VI
1202
60
Example 2.3
The essential component of a toaster is an electrical element (a
resis-tor) that converts electrical energy to heat energy. How much
currentis drawn by a toaster with resistance of 12 at 110 V?
Answer: 9.17 A
Practice Problem 2.3
In the circuit shown in Fig. 2.8, calculate the current I.
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 samepair of terminals. Hence,
I VR
30
5 103 6 mA
Example 2.4
5 kV+
30 V
I
Figure 2.8For Example 2.4.
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2.4 Conductance 29
R12 V
I
Figure 2.9For Practice Problem 2.4.
Answer: 1.5 k
If I 8 mA in the circuit shown in Fig. 2.9, determine the value
ofresistance R.
Practice Problem 2.4
ConductanceA useful quantity in circuit analysis is the
reciprocal of resistance R,known as conductance and denoted by
G:
(2.10)
The conductance is a measure of how well an element will
conductelectric current. The old unit of conductance is the mho
(ohm spelledbackward) with symbol , the inverted omega. Although
engineersstill use mhos, in this book we will prefer to use the SI
unit of con-ductance, the siemens (S), in honor of Werner von
Siemens:
1 S 1 1 A1 V (2.11)Thus,
[We should not confuse S for siemens with s (seconds) for time.]
Thesame 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
(2.12)
where the Greek letter sigma conductivity of the material(in
S/m).
s 1r
G Ar/
sA/
Conductance is the ability of an element to conduct electric
current;it is measured in siemens (S).
G 1R
IV
2.4
Find the conductance of the following resistors: (a) 125 (b) 42
k
Solution:(a)(b) mSG 1R 1 (42 103 ) 23.8
G 1R 1 (125 ) 8 mS
Example 2.5
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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Circular WiresCircular wires are commonly used in several
applications. We use wiresto connect elements, but those wires have
resistance and a maximumallowable current. So we need to choose the
right size. Wires arearranged in standard gauge numbers, known as
AWG (American WireGauge). This designation of cables and wires is
in the English system.In the English system,
1,000 mils 1 in (2.13a)or
(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),
(2.14)
Thus,
(2.15a)
or
(2.15b)
If the diameter of a circular wire is in mils, the area in
circular mils is
(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 ruleof thumb. The steel industry uses a different numbering
system for theirwire thickness gages (e.g., U.S. Steel Wire Gauge)
so that the data inTable 2.2 do not apply to steel wire. See Fig.
2.10 for different sizesof wires. Typical household wiring is AWG
number 12 or 14. Tele-phone wire is usually 22, 24, or 26 gauge.
The following examples willillustrate how to use the table.
ACM d2mil
1 sq mil 4p
CM
1 CM p
4 sq mil
A pd2
4p(1 mil)2
4p
4 sq mil
1 mil 1
1000 in 0.001 in
2.5
30 Chapter 2 Resistance
Figure 2.10Insulated wires of different gauges. Sarhan M.
Musa
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2.5 Circular Wires 31
TABLE 2.2
American wire gauge (AWG) sizes at 20C.
Maximum allowable
AWG # d(mil) Area (CM) R (/1000 ft) current (A)0000 460 211,600
0.0490 230
000 409.6 167,810 0.0618 20000 364.8 133,080 0.0780 175
0 324.9 105,530 0.0983 1501 289.3 83,694 0.1240 1302 257.8
66,373 0.1563 1153 229.4 52,634 0.1970 1004 204.3 41,740 0.2485 855
181.9 33,102 0.3133 6 162 26,250 0.3951 657 144 20,820 0.4982 8
128.5 16,510 0.6282 509 114.4 13,090 0.7921
10 101.9 10,381 0.9989 3011 90.74 8,234 1.260 12 80.81 6,530
1.588 2013 71.96 5,178 2.003 14 64.08 4,107 2.525 1515 57.07 3,257
3.18416 50.82 2,583 4.01617 45.26 2,048 5.06418 40.30 1,624 6.38519
35.89 1,288 8.05120 31.96 1,022 10.1521 28.46 810.10 12.8022 25.3
642.40 16.1423 22.6 509.5 20.3624 20.1 404.01 25.6725 17.9 320.40
32.3726 15.94 254.10 40.8127 14.2 201.50 51.5728 12.6 159.79
64.9029 11.26 126.72 81.8330 10.03 100.50 103.231 8.928 79.70
130.132 7.95 63.21 164.133 7.08 50.13 206.934 6.305 39.75 260.935
5.6 31.52 329.036 5 25 414.837 4.5 19.83 523.138 3.965 15.72
659.639 3.531 12.47 831.840 3.145 9.89 1049
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32 Chapter 2 Resistance
Find the resistance of 1200 ft of AWG #10 copper wire.
Answer: 199
Practice Problem 2.6
Find the cross-sectional area of a AWG #9 having a diameter
of114.4 mil.
ACM (114.4)2 13,087 CM
Example 2.7
What is the cross-sectional area in CM of a wire with a diameter
of0.0036 in.?
Answer: 12.96 CM
Practice Problem 2.7
Types of ResistorsDifferent types of resistors have been created
to meet different require-ments. Some resistors are shown in Fig.
2.11. The primary functionsof resistors are to limit current,
divide voltage, and dissipate heat.
A resistor is either fixed or variable. Most resistors are of
the fixedtype; that is, their resistance remains constant. The two
common types
2.6
Figure 2.11Different types of resistors. Sarhan M. Musa
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 a0.3951 1000 ft
b 0.3319
Example 2.6
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of fixed resistors (wirewound and composition) are shown in Fig.
2.12.Wirewound resistors are used when there is a need to dissipate
a largeamount of heat, while the composition resistors are used
when largeresistance is needed. The circuit symbol in Fig. 2.1 is
for a fixed resis-tor. Variable resistors have adjustable
resistance. The symbol for a vari-able resistor is shown in Fig.
2.13. There are two main types of variableresistors: potentiometer
and rheostat. The potentiometer or pot forshort, is a
three-terminal element with a sliding contact or wiper. Bysliding
the wiper, the resistances between the wiper terminal and thefixed
terminals vary. The potentiometer is used to adjust the amount
ofvoltage provided to a circuit, as typically shown in Fig. 2.14. A
poten-tiometer with its adjuster is shown in Fig. 2.15. The
rheostat is a two-or three-terminal device that is used to control
the amount of currentwithin a circuit, as typically shown in Fig.
2.16. As the rheostat isadjusted for more resistance and less
current flow, and the motor slowsdown and vice versa. It is
possible to use the same variable resistor asa potentiometer or a
rheostat, depending on how it is connected. Likefixed resistors,
variable resistors can either be of wirewound or com-position type,
as shown in Fig. 2.17. Although fixed resistors shown inFig. 2.12
are used in circuit designs, today, most circuit
components(including resistors) are either surface mounted or
integrated, as typi-cally shown in Fig. 2.18. Surface mount
technology (SMT) is beingused to implement both digital and analog
circuits. An SMT resistor isshown in Fig. 2.19.
It should be pointed out that not all resistors obey Ohms law.
Aresistor that obeys Ohms law is known as a linear resistor. It has
a con-stant resistance, and thus its voltage-current characteristic
is as illus-trated in Fig. 2.20(a); that is, its V-I graph is a
straight line passingthrough the origin. A nonlinear resistor does
not obey Ohms law. Itsresistance varies with current and its V-I
characteristic is typically shown
2.6 Types of Resistors 33
(a) (b)
Figure 2.13Circuit symbols for a variable resistor.
RV
Figure 2.14Variable resistor used as a potentiometer.
R
V Motor
Figure 2.16Variable resistor used as a rheostat.
(a) (b)
Figure 2.17Variable resistors: (a) composition type; (b) slider
pot.Courtesy of Tech America
Figure 2.15Potentiometers with their adjusters. Sarhan M.
Musa
(a)
(b)
Figure 2.12Fixed resistors: (a) wirewound type; (b) carbon film
type. Courtesy of Tech America
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34 Chapter 2 Resistance
Figure 2.18Resistors in an integrated circuit board. Eric
Tomey/Alamy RF
Figure 2.19Surface mount resistor. Greg Ordy
Slope = R
(a)
V
I
Slope = R
(b)
V
I
Figure 2.20The V-I characteristics of a (a) linear resistor; (b)
nonlinear resistor.
Figure 2.21Diodes. Sarhan M. Musa
in Fig. 2.20(b). Examples of devices with nonlinear resistance
are thelightbulb and the diode1 (see Fig. 2.21). Although all
practical resistorsmay exhibit nonlinear behavior under certain
conditions, we will assumein 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 toflow in
only one direction.
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Resistor Color CodeSome resistors are physically large enough to
have their values printedon them. Other resistors are too small to
have their values printed onthem. For such small resistors, color
coding provides a way of deter-mining the value of resistance. As
shown in Fig. 2.22, the color cod-ing 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 First significant figure of resistance valueB Second
significant figure of resistance valueC Multiplier of resistance
for resistance valueD Tolerance rating (in %)E Reliability factor
(in %)
*We read the bands from left to right.
The first three bands (A, B, and C) specify the value of the
resistance.Bands A and B represent the first and second digits of
the resistancevalue. Band C is usually given as a power of 10 as in
Table 2.3. Ifpresent, the fourth band (D) indicates the tolerance
percentage. Forexample, a 5 percent tolerance indicates that the
actual value of theresistance is within 5 of the color-coded value.
When the fourth bandis absent, the tolerance is taken by default to
be 20 percent. The fifthband (E), if present, is used to indicate a
reliability factor, which is astatistical indication of the
expected number of components that willfail to have the indicated
resistance after working for 1,000 hours. Asshown in Fig. 2.23, the
statement Big Boys Race Our Young Girls,But Violet Generally Wins
can serve as a memory aid in remember-ing the color code.
2.7
2.7 Resistor Color Code 35
A B C D E
Figure 2.22Resistor color codes.
0 Black Big1 Brown Boys2 Red Race3 Orange Our4 Yellow Young5
Green Girls6 Blue But7 Violet Violet8 Gray Generally9 White
Wins
Figure 2.23Memory aid for color codes.
TABLE 2.3
Resistor color code.
Band A Band Bsignificant significant Band C Band D Band E
Color figure figure multiplier tolerance reliabilityBlack N/A 0
100Brown 1 1 101 1%Red 2 2 102 0.1%Orange 3 3 103 0.01%Yellow 4 4
104 0.001%Green 5 5 105Blue 6 6 106Violet 7 7 107Gray 8 8 108White
9 9 109Gold 0.1 5%Silver 0.01 10%No color 20%
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Solution:Band A is blue (6); band B is red (2); band C is orange
(3); band D isgold (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 willfall between 58.9 k (62 3.1) k and 65.1 k (62 3.1) k.
Thereliability of 0.1% indicates that 1 out of 1,000 will fail to
fall withinthe tolerance range after 1,000 hours of service.
36 Chapter 2 Resistance
Figure 2.24For Example 2.8.
Determine the resistance value of the color-coded resistor shown
inFig. 2.24.
Example 2.8
What is the resistance value, tolerance, and reliability of the
color-coded resistor shown in Fig. 2.25?
Practice Problem 2.8
Figure 2.25For Practice Problem 2.8.
Answer: 3.3 M 10% with a reliability of 1%
A resistor has three bands onlyin order green, black, and
silver. Findthe 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.
Example 2.9
What is the resistance value and tolerance of a resistor having
bandscolored in the order yellow, violet, white, and gold?
Answer: 47 G 5%
Practice Problem 2.9
A company manufactures resistors of 5.4 k with a tolerance of10
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
redstands for102. The tolerance of 10 percent corresponds to
silver. Hence,the color code of the resistor is:
Green, yellow, red, silver
Example 2.10
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2.8 Standard Resistor Values 37
Standard Resistor ValuesOne would expect resistor values are
commercially available in all val-ues. For practical reasons, this
would not make sense. Only a limitednumber of resistor values are
commercially available at reasonable cost.The list of standard
values of commercially available resistors is pre-sented in Table
2.4. These are the standard values that have been agreedto for
carbon composition resistors. Notice that the values range from0.1
to 22 M. While 10 percent tolerance resistors are available onlyfor
those values in bold type at reasonable cost, 5 percent
toleranceresistors are available in all values. For example, a 330-
resistor couldbe available either as a 5 or 10 percent tolerance
component, while a110-k resistor is available only as 5 percent
tolerance component.
When designing a circuit, the calculated values are seldom
stan-dard. One may select the nearest standard values or combine
the stan-dard values. In most cases, selecting the nearest standard
value may
2.8
If the company in Example 2.10 also produces resistors of 7.2
Mwith a tolerance of 5 percent and reliability of 1 percent, what
will bethe color codes on the resistor?
Answer: Violet, red, green, gold, brown
Practice Problem 2.10
TABLE 2.4
Standard values of commercially available resistors.
Ohms Kilohms Megohms() (k) (M)
0.10 1.0 10 100 1000 10 100 1.0 10.00.11 1.1 11 110 1100 11 110
1.1 11.00.12 1.2 12 120 1200 12 120 1.2 12.00.13 1.3 13 130 1300 13
130 1.3 13.00.15 1.5 15 150 1500 15 150 1.5 15.00.16 1.6 16 160
1600 16 160 1.6 16.00.18 1.8 18 180 1800 18 180 1.8 18.00.20 2.0 20
200 2000 20 200 2.0 20.00.22 2.2 22 220 2200 22 220 2.2 22.00.24
2.4 24 240 2400 24 240 2.40.27 2.7 27 270 2700 27 270 2.70.30 3.0
30 300 3000 30 300 3.00.33 3.3 33 330 3300 33 330 3.30.36 3.6 36
360 3600 36 360 3.60.39 3.9 39 390 3900 39 390 3.90.43 4.3 43 430
4300 43 430 4.30.47 4.7 47 470 4700 47 470 4.70.51 5.1 51 510 5100
51 510 5.10.56 5.6 56 560 5600 56 560 5.60.62 6.2 62 620 6200 62
620 6.20.68 6.8 68 680 6800 68 680 6.80.75 7.5 75 750 7500 75 750
7.50.82 8.2 82 820 8200 82 820 8.20.91 9.1 92 910 9100 91 910
9.1
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provide adequate performance. To ease calculations, most of the
resis-tor values used in this book are nonstandard.
Applications: MeasurementsResistors are often used to model
devices that convert electrical energyinto heat or other forms of
energy. Such devices include conductingwires, 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 inpotentiometers
and meters. In this section, we will consider metersthe ammeter,
voltmeter, and ohmmeter, which measure current, volt-age, and
resistance, respectively. Being able to measure current I,voltage
V, and resistance R is very important.
It is common these days to have the three instruments combined
intoone instrument known as a multimeter, which may be analog or
digital.An analog meter is one that uses a needle and calibrated
meter to displaythe measured value; that is, the measured value is
indicated by the pointerof the meter. A digital meter is one in
which the measured valued is shownin form of a digital display. The
digital meters are more commonly usedtoday. Because both analog and
digital meters are used in the industry,one should be familiar with
both. Figure 2.26 illustrates a typical analogmultimeter (combining
voltmeter, ammeter, and ohmmeter) and a typicaldigital multimeter.
The digital multimeter (DMM) is the most widely usedinstrument. Its
analog counterpart is the volt-ohm-milliammeter (VOM).
To measure voltage, we connect the voltmeter/multimeter
acrossthe element for which the voltage is desired, as shown in
Fig. 2.27.The voltmeter measures the voltage across the load and is
thereforeconnected in parallel2 with the element.
The voltmeter is the instrument used to measure voltage; the
ammeteris the instrument used to measure current; and the ohmmeter
is theinstrument used to measure resistance.
2.9
38 Chapter 2 Resistance
2 Two elements are in parallel if they are connected to the same
two points.
Figure 2.26(a) Analog multimeter; (b) digital multimeter.(a)
iStock; (b) Oleksy Maksymenko/Alamy RF
(a) (b)
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2.9 Applications: Measurements 39
VoltmeterRV
+
+ V
+
+
Figure 2.27Measuring voltage.
+
+ mA
+
+
Ammeter
RV
I
Figure 2.28Measuring current.
To measure current, we connect the ammeter/multimeter in
series3with the element under test, as shown in Fig. 2.28. The
meter must beconnected such that the current enters through the
positive terminal toget a positive reading. The circuit must be
broken; that is, the cur-rent path must be interrupted so that the
current must flow through theammeter. (The ampclamp is another
device for measuring ac current.)
+
R Ohmmeter+
Figure 2.29Measuring resistance.
3 Two elements are in series if they are cascaded or connected
sequentially.
To measure resistance of an element, connect the
ohmmeter/multimeter across it, as shown in Fig. 2.29. If the
element is connectedto a circuit, one end of the element must first
be disconnected from thecircuit before we measure its resistance.
Because the resistance of awire with no breaks is zero, the
ohmmeter can be used to test for con-tinuity. If the wire has a
break, the ohmmeter connected across it willread 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 cor-
rect mode (ac, dc, V, A, ), including moving the test idea to
theappropriate jacks.
5. When the measurement is completed, turn off the meter to
avoiddraining the internal battery of the meter.
This leads to the issue of safety in electrical measurement.
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Electrical Safety PrecautionsNow that we have learned how to
measure current, voltage, and resist-ance, we need to be careful
how we handle the instruments so as toavoid electric shock or harm.
Because electricity can kill, being ableto make safe and accurate
measurements is an integral part of theknowledge that you must
acquire.
2.10.1 Electric Shock
When working on electric circuits, there is the possibility of
receivingan electric shock. The shock is due to the passage of
current throughyour body. An electric shock can startle you and
cause you to fall downor be thrown down. It may cause severe, rigid
contractions of the mus-cles, which in turn may result in
fractures, dislocations, and loss ofconsciousness. The respiratory
system may be paralyzed and theheart may beat irregularly or even
stop beating altogether. Electricalburns may be present on the skin
and extend into deeper tissue. Highcurrent may cause death of
tissues between the entry and exit point ofthe current. Massive
swelling of the tissues may follow as the blood inthe veins
coagulates and the muscles swell. Thus, electric shock cancause
muscle spasms, weakness, shallow breathing, rapid pulse,
severeburns, unconsciousness, or death.
The human body has resistance that depends on several
factorssuch as body mass, skin moisture, and points of contact of
the bodywith the electric appliance. The effects of various amounts
of currentin milliamperes (mA) is shown in Table 2.5.
2.10.2 Precautions
Working with electricity can be dangerous unless you adhere
strictlyto certain rules. The following safety rules should be
followed when-ever you are working with electricity:
Always make sure that the circuit is actually dead before you
beginworking on it.
Always unplug any appliance or lamp before repairing it. Always
tape over the main switch, empty fuse socket, or circuit
breaker when youre working. Leave a note there so no one
willaccidentally turn on the electricity. Keep any fuses youve
removedin your pocket.
Electric shock is an injury caused by an electrical current
passingthrough the body.
2.10
40 Chapter 2 Resistance
TABLE 2.5
Electric shock
Electric Current Physiological effectLess than 1mA No sensation
or feeling
1 mA Tingling sensation520 mA Involuntary muscle contraction
20100 mA Loss of breathing, fatal if continued
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2.11 Summary 41
Handle tools properly and make sure that the insulation on
metaltools is in good condition.
If measuring V or I, turn on the power and record reading. If
meas-uring R, do not turn on power.
Refrain from wearing loose clothing. Loose clothes can get
caughtin an operating appliance.
Always wear long-legged and long-sleeved clothes and shoes
andkeep them dry.
Do not stand on a metal or wet floor. (Electricity and water do
notmix.)
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. Ifwe follow these safety rules, we can avoid shock and
related accidents.Thus, our rule should always be safety first.
Summary
1. A resistor is an element in which the voltage, V, across it
is directlyproportional to the current, I, through it. That is, a
resistor is anelement that obeys Ohms 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
cross-section area A, that is,
3. A short circuit is a resistor (a perfectly conducting wire)
with zeroresistance (R 0). An open circuit is a resistor with
infinite resist-ance .
4. The conductance G of a resistor is the reciprocal of its
resistance R:
5. For a circular wire, the cross-sectional area is measured in
circu-lar mils (CM). The diameter in mils is related to the area in
CM as
6. American Wire Gauge is a standard system for designating
thediameter of wires.
7. There are different types of resistors: fixed or variable,
linear ornonlinear. Potentiometer and rheostat are variable
resistors that areused to adjust voltage and current, respectively.
Common types of
ACM d2mil
G 1R
(R )
R r/A
2.11
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42 Chapter 2 Resistance
resistors include carbon or composition resistors, wirewound
resis-tors, chip resistors, film resistors, and power
resistors.
8. A resistor is usually color coded when it is not physically
largeenough to print the numerical value of the resistor on it. The
state-ment Big Boys Race Our Young Girls, But Violet GenerallyWins
is a memory aid for the color code: black, brown, red,orange,
yellow, green, blue, violet, gray, and white.
9. For carbon composition resistors, standard values are
commer-cially available in the range of 0.1 to 22 M.
10. Voltage, current, and resistance are measured using a
voltmeter,ammeter, and ohmmeter, respectively. The three are
measuredusing a multimeter such as a digital multimeter (DMM) or a
volt-ohm-milliammeter (VOM).
11. Safety is all about preventing accidents. If we follow some
safetyprecautions, we should have no problems working on
electriccircuits.
2.6 The conductance of a 10-m resistor is:
(a) 0.1 mS (b) 0.1 S(c ) 10 S (d) 100 S
2.7 Potentiometers are types of:
(a) fixed resistors (b) variable resistors(c) meters (d) voltage
regulators
2.8 What is the area in circular mils of a wire that has
adiameter of 0.03 in.?
(a) 0.0009 (b) 9(c ) 90 (d) 900
2.9 All resistors are color coded.
(a) True (b) False2.10 Digital multimeters (DMM) are the most
widely
used type of electronic measuring instrument.
(a) True (b) False
Answers: 2.1c, 2.2d, 2.3c, 2.4c, 2.5a, 2.6d, 2.7b, 2.8d,2.9b,
2.10a
Review Questions
2.1 Which of the following materials is not a conductor?
(a) Copper (b) Silver (c) Mica(d) Gold (e) Lead
2.2 The main purpose of a resistor in a circuit is to:
(a) resist change in current(b) produce heat(c) increase
current(d) limit current
2.3 An element draws 10 A from a 120-V line. Theresistance of
the element is:
(a) 1200 (b) 120 (c) 12 (d) 1.2
2.4 The reciprocal of resistance is:
(a) voltage (b) current(c) conductance (d) power
2.5 Which of these is not the unit of conductance?
(a) ohm (b) Siemen(c) mho (d)
Problems
2.2 Find the length of a copper wire that has a resistanceof 0.5
and a diameter of 2 mm.
2.3 A 2-in. 2-in. square bar of copper is 4 ft long. Findits
resistance.
Section 2.2 Resistance
2.1 A 250-m-long copper wire has a diameter of 2.2 mm.Calculate
the resistance of wire.
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Problems 43
2.4 If an electrical hotplate has a power rating of 1200 Wand
draws a current of 6 A, determine the resistanceof the
hotplate.
2.5 A Nichrome (r 100 108 m) wire is usedto 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 resistanceof 6 . How
long is the wire?
2.7 A graphite cylinder with a diameter of 0.4 mm and alength of
4 cm has resistance of 2.1 . Determinethe resistivity of the
cylinder.
2.8 A certain circular wire of length 50 m and diameter0.5 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
theconductor decrease in resistance?
2.10 Two wires are made of the same material. The firstwire has
a resistance of 0.2 . The second wire istwice as long as the first
wire and has a radius that ishalf of the first wire. Determine the
resistance of thesecond wire.
2.11 Two wires have the same resistance and length. Thefirst
wire is made of copper, while the second wire ismade of aluminum.
Find the ratio of the cross-sectional area of the copper wire to
that of thealuminum wire.
2.12 High-voltage power lines are used in transmittinglarge
amounts of power over long distances.Aluminum cable is preferred
over copper cable dueto low cost. Assume that the aluminum wire
used forhigh-voltage power lines has a cross-sectional areaof 4.7
104 m2. Find the resistance of 20 km ofthis wire.
Section 2.3 Ohms Law
2.13 Which of the graphs in Fig. 2.30 represent Ohms law?
2.14 When the voltage across a resistor is 60 V, thecurrent
through it is 50 mA. Determine its resistance.
2.15 The voltage across a 5-k resistor is 16 V. Find thecurrent
through the resistor.
2.16 A resistor is connected to a 12-V battery. Calculatethe
current if the resistor is:
(a) 2 k (b) 6.2 k2.17 An air-conditioning compressor has
resistance 6 .
When the compressor is connected to a 240-Vsource, determine the
current through the circuit.
2.18 A source of 12 V is connected to a purely resistivelamp and
draws 3 A. What is the resistance of thelamp?
2.19 If a current of 30 mA flows through a 5.4-Mresistor, what
is the voltage?
2.20 A current of 2 mA flows through a 25- resistor.Find the
voltage drop across it.
2.21 An element allows 28 mA of current to flow throughit 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 acurrent of 10
mA in a 50- resistor.
2.23 A nonlinear resistor has I 4 102 V2. Find I forV 10, 20,
and 50 V.
2.24 Determine the magnitude and direction of the
currentassociated with the resistor in each of the circuits inFig.
2.31.
2.25 Determine the magnitude and polarity of the voltageacross
the resistor in each of the circuits in Fig. 2.32.
2.26 A flashlight uses two 3-V batteries in series toprovide a
current of 0.7 A in the filament. (a) Findthe potential difference
across the flashlight bulb.(b) Calculate the resistance of the
filament.
Figure 2.30For Problem 2.13.
(a)
V
I
(b)
V
I
(c)
V
I
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44 Chapter 2 Resistance
Figure 2.31For Problem 2.24.
Figure 2.32For Problem 2.25.
10
(a) (b) (c)
4 A 10 20 mA 2 6 mA
10
(a) (b) (c)
15 V 10 9 V 6 30 V+
+
+
Section 2.4 Conductance
2.27 Determine the conductance of each of the
followingresistances:
(a) 2.5 (b) 40 k (c) 12 M2.28 Find the resistance for each of
the following
conductances:
(a) 10 mS (b) 0.25 S (c) 50 S2.29 When the voltage across a
resistor is 120 V, the
current through it is 2.5 mA. Calculate itsconductance.
2.30 A copper rod has a length of 4 cm and a conductanceof 500
mS. Find its diameter.
2.31 Determine the battery voltage V in the circuit shownin Fig.
2.33.
Figure 2.33For 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 cannotexceed
0.001 , and the maximum current drawn bythe load is 120 A. What
gauge wire is appropriate?Assume a length of 10 ft.
I = 4 mA
5 mSV
+
2.34 Find the diameter in inches for wires having thefollowing
cross-sectional areas:
(a) 420 CM (b) 980 CM2.35 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 wire1 mi long
connected to a 1.5-V battery?
Section 2.7 Resistor Color Code
2.37 Find the resistance value having the following
colorcodes:
(a) blue, red, violet, silver(b) green, black, orange, gold
2.38 Determine the range (in ohms) in which a resistorhaving the
following bands must exist.
Band A Band B Band C Band D(a) Brown Violet Green Silver(b) Red
Black Orange Gold(c) White Red Gray
2.39 Determine the color codes of the following resistorswith 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 45
Section 2.10 Electrical Safety Precautions
2.51 What causes electric shock?
2.52 Mention at least four safety precautions you wouldtake when
taking measurements.
2.41 For each of the resistors in Problem 2.37, find theminimum
and maximum resistance within thetolerance limits.
2.42 Give the color coding for the following resistors:
(a) 10 , 10 percent tolerance(b) 7.4 k, 5 percent tolerance(c)
12 M, 20 percent tolerance
Section 2.9 Applications: Measurements
2.43 How much voltage is the multimeter in Fig. 2.34reading?
Figure 2.34For Problem 2.43.
2.44 Determine the voltage reading for the multimeter inFig.
2.35.
2.45 You are supposed to check a lightbulb to see whetheris
burned out or not. Using an ohmmeter, how wouldyou do this?
2.46 What is wrong with the measuring scheme inFig. 2.36?
2.47 Show how you would place a voltmeter to measurethe voltage
across resistor R1 in Fig. 2.37.
2.48 Show how you would place an ammeter to measurethe current
through resistor R2 in Fig. 2.37.
2.49 Explain how you would connect an ohmmeter tomeasure the
resistance R2 in Fig. 2.37.
2.50 How would you use an ohmmeter to determine theon and off
states of a switch?
0.3
0.061.2
12120
x1x10
x100x1K
x100K
OhmsAdj
+
33
1212
60 60
300 300600 600
OFF
Analog Multimeter
AC Volts O
hms
DC
Vol
ts
DC
mA
OHMS OHMS
DCAC
ACdBm
dBm
AC
ACDC
AMPS AMPS
1
2345
10
100204
150306
200408 25050
10
3006012
20
50
00
0
102
50
200
1k
0
410
2
1563
20
20
104 2
2 46 8
1011
0
84 2510
53012
6
5
00
0
21
Figure 2.35For Problem 2.44.
Figure 2.36For Problem 2.46.
Figure 2.37For Problems 2.47, 2.48, and 2.49.
V1
+
R1
R2
LampVs
+
A
V
0.3
0.061.2
12120
x1x10
x100x1K
x100K
OhmsAdj
+
33
1212
60 60
300 300600 600
OFF
Analog Multimeter
AC Volts O
hms
DC
Vol
ts D
C m
A
OHMS OHMS
DCAC
ACdBm
dBm
AC
ACDC
AMPS AMPS
1
2345
10
100204
150306
200408 25050
103006012
20
50
00
0
102
50
200
1k
0
410
2
1563
20
20
104 2
2 46 8
1011
0
84 2510
53012
6
5
00
0
21
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