applied circuit analysis-chapter_2

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


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.


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


(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


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


Answer: 1.5 k

If I 8 mA in the circuit shown in Fig. 2.9, determine the value ofresistance R.


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


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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


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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Find the resistance of 1200 ft of AWG #10 copper wire.

Answer: 199


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


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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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.


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?



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%


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


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


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


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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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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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.


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?


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 2.37For 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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applied circuit analysis-chapter_2

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