ELECTRICITY 2 devices circuit and mateials

Where direct current isrequired, alternating current is generated at a power station as shown in Figure 1-1,transmitted some distance, and then converted to direct current at the point w

ELECTRICITY 2

E L E C T R I C I T Y

2

DEVICES, CIRCUITS, AND MATERIALS

NINTH EDITION

THOMAS KUBALA

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Thomas Kubala

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Printed in the United States of America

1 2 3 4 5 XX 11 10 09 08

PREFACE / vii

1 ALTERNATING-CURRENT PRINCIPLES / 1

2 INDUCTANCE AND INDUCTIVE REACTANCE / 11

3 CAPACITANCE AND CAPACITIVE REACTANCE / 19

4 SERIES CIRCUIT: RESISTANCE AND INDUCTANCE / 29

5 SERIES CIRCUIT: RESISTANCE AND CAPACITANCE / 37

6 SERIES CIRCUIT: RESISTANCE, INDUCTANCE, AND CAPACITANCE / 45

7 AC PARALLEL CIRCUITS CONTAINING INDUCTANCE / 55

8 AC PARALLEL CIRCUITS CONTAINING INDUCTANCE

AND CAPACITANCE / 63

9 AC POWER, POWER FACTOR, AND POWER FACTOR

CORRECTION / 73

v

10 SUMMARY REVIEW OF UNITS 1–9 / 85

11 INSTALLATION OF SINGLE-PHASE, THREE-WIRE

ENTRANCE FOR A SINGLE-FAMILY RESIDENCE / 91

12 INSTALLATION OF A SINGLE-PHASE, THREE-WIRE

SERVICE ENTRANCE FOR AN APARTMENT BUILDING / 105

13 INSTALLATION OF A THREE-PHASE, THREE-WIRE

SERVICE ENTRANCE / 115

14 INTRODUCTION TO FLUORESCENT LIGHTING / 125

15 INSTALLATION OF FLUORESCENT LIGHTING / 133

16 SUMMARY REVIEW OF UNITS 11–15 / 145

APPENDIX / 149

GLOSSARY / 151

INDEX / 153

The ninth edition of ELECTRICITY 2 has been updated to reflect current materials and

tech-niques in electrical applications, while maintaining the features that have made the text so popular through previous editions Summary statements are found at the end of each unit, and several new problems have been included in the Achievement Review sections.

ELECTRICITY 2 helps the student achieve a basic understanding of the characteristics of

alternating-current circuits and the devices contained in the circuits The knowledge obtained by a study of this text permits the student to progress to further study It should be realized that both the development of the subject of electricity and the study of the subject are continuing processes The electrical industry constantly introduces new and improved devices and materials, which in turn often lead to changes in installation techniques Electrical codes undergo periodic revisions to upgrade safety and quality in electrical installations.

The text is easy to read and the topics are presented in a logical sequence The problems vided in the text require the use of simple algebra and simple trigonometry for their solutions The student is advised that electron movement (from negative to positive) is used in this text to define cur- rent direction.

pro-Each unit begins with objectives to alert students to the learning that is expected as a result of ing the unit An Achievement Review at the end of each unit tests student understanding to determine if the objectives have been met Following selected groups of units (Units 1–9 and 11–15), a Summary Review unit contains additional questions and problems to test student comprehension of a block of information This combination of reviews is essential to the learning process required by this text All students of electricity will find this text useful, especially those in electrical apprenticeship programs, trade and technical schools, and various occupational programs.

study-The most recent edition of the National Electrical Code® (published by the National Fire Protection Association [NFPA]) should be available for reference as the student uses ELECTRICITY 2.

Applicable state and local regulations should also be consulted when executing actual installations Features of the ninth edition include

• Summary statements in all units

• Up-to-date content reflecting current methods and materials of the trade

• Currency with the most recent edition of the National Electrical Code®

• Detailed problem solutions in most units

• Achievement Reviews that reinforce concepts

• Practical problems to test student learning

• Numerous new problems for student practice

Instructor’s Guides for ELECTRICITY 1 through ELECTRICITY 4 are available The guides

include the answers to the Achievement Reviews and Summary Reviews for each text and additional test questions covering the content of each text Instructors can use these questions to devise tests to evaluate student learning.

vii

ABOUT THE AUTHOR

Dr Thomas Kubala received an AAS degree in Electrical Technology from Broome Community College, Binghamton, New York; a BS degree in Electrical Engineering from the Rochester Institute of Technology, Rochester, New York; and an MS degree in Vocational-Technical Education from the State University of New York at Oswego, New York He earned his doctoral degree from the University of Maryland, College Park, Maryland.

Dr Kubala has served as a full-time faculty member at two community colleges and a ment head supervising a vocational-technical program In addition to his extensive background in technological education, Dr Kubala has industrial experience with responsibilities in the fields of aerodynamics, electrical drafting, electrical circuit design, equipment testing, and systems evaluation.

depart-ACKNOWLEDGMENTS

The revision of ELECTRICITY 2 was based on information and recommendations submitted by the

following instructors:

Phillip Serina, Kaplan Career Institute, Brooklyn, Ohio

Silas Qualls, Mountain Empire Community College, Big Stone Gap, Virginia

ELECTRICAL TRADES

The Delmar series of instructional material for the electrical trades includes the texts, text workbooks, and related information workbooks listed below Each text features basic theory with practical applications and student involvement in hands-on activities.

E = PI

E = IR

I = ER

R = EI

Equations based on Ohm’s law.

NOTE: In this text, E is used to denote a voltage source, and V is used for a voltage drop

1

ALTERNATING-CURRENT PRINCIPLES

OBJECTIVES

After studying this unit, the student should be able to

• discuss the characteristics of alternating current

• describe the generation of alternating current

• define the terminology related to alternating current

Most of the electrical energy used in the United States is generated as ing current (AC) This is not because alternating current is superior to direct current(DC) in industrial or residential applications In fact, there are many instances wheredirect-current energy is necessary for industrial purposes Where direct current isrequired, alternating current is generated at a power station as shown in Figure 1-1,transmitted some distance, and then converted to direct current at the point where itwill be used

alternat-The reasons for generating nearly all electrical energy as alternating current are asfollow:

1 Alternators (AC generators) have no commutators Therefore, units with higherpower ratings and the resultant heavier current ratings can be used without theproblem of brush arcing and heating

2 Because alternators lack commutators, they are capable of generating tively high voltages, such as 11,000 to 13,800 volts

compara-3 Alternating-current energy can be transmitted economically over great distances.Therefore, alternating current can be generated in large quantities in a single stationand distributed over a large territory

4 For constant-speed work, the alternating-current, squirrel-cage induction motor isless expensive than the direct-current motor, both in initial cost and maintenance

GENERATING ALTERNATING VOLTAGE

The volt (V) is the unit of electromotive force (EMF) One volt is developed by ting 100 million magnetic lines of force in 1 second The simplest method of generatingEMF is by turning a coil of wire between two magnetic field poles

cut-Figure 1-1 Alternators (Courtesy of New York Power Authority)

Figure 1-2 illustrates a simple alternating-current generator The left-hand rule

is used to determine the direction of the current in the coil and in the external circuitcreated by the generated EMF

Figure 1-3 is a more convenient form of representing the simple generator inFigure 1-2 A front-view section from the slip ring side of the generator is shown

CONDUCTOR MOTION

CONDUCTOR MOTION

INDUCED CURRENT LINES

OF FORCE

BRUSHES

EXTERNAL RESISTANCE

LEFT-HAND RULE SHOWING

INDUCED CURRENT (ELECTRON

MOVEMENT) IN A-B PORTION OF

INDUCED CURRENT (ELECTRON MOVEMENT) IN C-D PORTION OF COIL

I

I D

C S

CURRENT (ELECTRON MOVEMENT) IN DIRECTION AWAY FROM SLIP RINGS

CURRENT (ELECTRON MOVEMENT) IN DIRECTION TOWARD SLIP RINGS LINES OF FORCE

GRAPH OF GENERATION CURRENT COIL POSITIONS

270 360 +

–

Figure 1-4 Start of cycle.

Figure 1-7 Three-quarter turn.

180 DEGREES (ONE-HALF TURN)

270 360 +

MECHANICAL AND ELECTRICAL DEGREES

In Figure 1-8, the complete turn represents

When a coil or a conductor makes one complete revolution, it passes through 360

mechanical degrees When either an EMF or an alternating current passes through one

cycle, it passes through 360 electrical degrees.

If a single-loop coil is placed in a generator that has four magnetic poles, two completecycles of alternating current or 720 electrical degrees are generated in one mechanical rev-olution because one cycle is generated when each side of a coil or loop passes two poles

FREQUENCY

Frequency is the number of electrical cycles that occur in 1 second If the coil inFigure 1-8 turns at the rate of 3,600 revolutions per minute (3,600 r/min), or 60 revolutionsper second, 60 electrical cycles are generated in 1 second The electrical frequency is

60 cycles per second, or 60 hertz (Hz) With the four-pole generator in Figure 1-9, ing the same speed of 3,600 r/min, the number of electrical cycles generated in 1 second

assum-is 120 Thassum-is assum-is true because two electrical cycles are generated during each mechanicalrevolution The frequency is 120 Hz

Formulas commonly used to find frequency, speed, and number of poles follow:

Where F = frequency in hertz

270 360 +

–

Figure 1-8 One full turn completed.

Example: At what speed must a four-pole generator rotate to develop a frequency

of 50 Hz?

EFFECTIVE VALUE OF ALTERNATING CURRENT

A 60-Hz, 120-volt line contains current and EMF varying from zero to maximumpositive and negative values 60 times per second Instantaneous values are of little value.The effective values, shown in Figure 1-10, have been accepted as practical workingquantities When values of current and voltage are specified in AC circuits, they are under-stood as effective values, unless otherwise specified The values indicated by AC amme-ters and voltmeters are effective values

P

120(50)4

MAXIMUM VALUE = 170 VOLTS

EFFECTIVE VALUE = 120 VOLTS

Figure 1-11 Current in phase

with the voltage.

Figure 1-12 Current out of phase with the voltage (lagging current).

The effective value of alternating current produces the equivalent amount of heat

at a load as the same numerical value of direct current

Example: An electric heater is rated at 10 amperes (A) AC or DC Either current

produces the same amount of heat in a given amount of time

SINGLE PHASE

Voltage generated in a single winding of a generator is called single-phase voltage

IN PHASE

When both the voltage wave and the current wave reach their corresponding zeros,

maxima, and intermediate values at exactly the same time, they are said to be in phase,

as shown in Figure 1-11

OUT OF PHASE (LAG)

When current is supplied to an induction motor or any circuit with inductance, it

lags the voltage This current is out of phase with the voltage, as shown in Figure 1-12.

OUT OF PHASE (LEAD)

A synchronous motor or capacitor connected to a line causes current to lead the

voltage by as much as 90 electrical degrees Figure 1-13 shows a current out of phase

with the voltage

SUMMARY

The left-hand rule is a handy way to predict the direction of induced current in a erator The graph describing the induced current is alternating; hence, alternating current.There is a positive maximum current, and a negative maximum Frequency of alternations

gen-is expressed as hertz The alternating current supplied to residences has a frequency of

60 Hz Because the current changes from positive to negative, an effective value is used

to described a steady-state value The effective value is used for electrical calculations

ACHIEVEMENT REVIEW

1 A four-pole, single-loop generator revolves at the rate of 3,600 r/min, and themaximum generated value of voltage is 3.0 volts

a Determine the generated voltage when the loop conductors are located in front

of the poles (see Figure 1-5)

b Determine the generated voltage when the loop conductors are located betweenthe poles (see Figure 1-6)

c How many electrical cycles are generated in one mechanical revolution? _

d Calculate the electrical frequency

e Calculate the effective value of voltage. _

2 State four advantages of generating alternating current as compared to direct current

a _

b _

c _

d _

3 An ammeter indicates 15 amperes in an AC induction motor line

a What value is measured: effective, instantaneous, or average?

Figure 1-13 Current out of phase with

the voltage (leading current).

4 Calculate the electrical and mechanical degrees in one complete mechanical lution for each of the specified generators.

b Frequency

c Effective value

6 A ten-pole alternator revolves at 600 r/min What is the value of the electrical quency generated?

fre-7 To produce a frequency of 25 Hz, at what r/min must a six-pole alternator be driven? _

8 To produce a frequency of 60 Hz, how many poles does an alternator have if thespeed is 150 r/min?

9 Find the effective value of an alternating current if the maximum value is 20 amperes. _

10 The source voltage of an AC circuit has a maximum value of 100 volts Find theeffective value of the voltage

_

11 If the effective voltage of a 60-Hz source is 400 volts, what is the maximum voltage? _

12 The total current in a single-phase AC circuit has an effective value of 12.73 amperes.Find the maximum value

13 A six-pole alternator revolves at 450 r/min and develops a maximum voltage of

198 volts Find the effective value of the voltage.

2

INDUCTANCE AND INDUCTIVE REACTANCE

OBJECTIVES

After studying this unit, the student should be able to

• describe an inductive circuit

• describe self-induction and mutual induction

• define inductive reactance

• demonstrate the relationship between voltage and current in various inductivecircuits by the use of vectors

A coil of wire is an important part of many pieces of electrical equipment A netic field is produced when current exists in the coil As the strength of the magneticfield changes, an induced electromotive force (EMF) is created across the coil Theinduced voltage opposes the source voltage As the opposition becomes greater, lesscurrent exists in the circuit

mag-The coil has a property that opposes change in the current This property is called

inductance (L) The amount of opposition to current change is called inductive tance, and is a function of frequency and inductance.

reac-LENZ’S LAW

According to Lenz’s law, the induced voltage in a coil always flows in the

opposite direction of the effect that produces it

Self-Inductance

When the varying lines of magnetic force induce an EMF in the coil itself, the coil

has self-inductance.

Mutual Inductance

When the varying lines of magnetic force from a coil induce an EMF in an

adja-cent coil, the coils have mutual inductance Figure 2-1 illustrates a transformer

contain-ing a primary coil and a secondary coil The primary coil contains a current that creates

a magnetic field Part of the field links the secondary coil.

Because the field is changing, a voltage is induced in the

sec-ondary This is a step-up transformer because the secondary

voltage is greater than the primary voltage An actual

trans-former is shown in Figure 2-2

Measurement of Inductance

The unit of inductance is the henry (H) A circuit or coil has an inductance of 1 Hwhen current varying at the rate of 1 ampere per second induces an EMF of 1 volt acrossthe terminals of the circuit or coil The inductance can be varied by varying the amount

of magnetic flux or the number of turns in the coil.

Effect of Inductance

Example: Connect a lamp in series with a coil having a movable iron core Connect

the combination to an AC source Note the following conditions:

Core out of coil – Lamp will be bright Core in coil – Lamp will be dimWhen the core is out of the coil, few lines of magnetic force are produced by thecoil because air is a poor magnetic conductor The induced EMF is weak, and little oppo-sition is offered to the line voltage Therefore, a normal quantity of current exists in thelamp

When the iron core is inserted in the coil, a better magnetic path is provided InducedEMF is higher, and consequently, there is less current as indicated by the dim lamp

Figure 2-1 Transformer showing location of

self-inductance and mutual inductance.

Figure 2-2 A transformer.

(Courtesy of General Electric Company)

Inductive Reactance

The opposition in coils having inductance can be measured in ohms (Ω) If the

fre-quency and inductance are known, the opposition, or inductive reactance (XL), can becalculated

XL = 2πfL Where XL = inductive reactance in ohms

π = 3.14

f = frequency in hertz

L = inductance in henrysExamples:

inductive reactance varies directly with inductance and frequency

Current Lag Due to Inductance

Tests show that if a coil with negligible resistance is connected to an AC line, thecurrent lags the voltage by 90°, as shown in Figure 2-3

VECTOR REPRESENTATION

The relationship between voltage and

current in an inductive circuit is shown more

conveniently by the use of vectors

A vector is a line representing quantity or

magnitude and direction

The vectors shown in Figure 2-4

repre-sent 110 volts with a current of 10 amperes

lagging by 90° These vectors may be

visual-ized as clock hands rotating in a

The lengths of the vectors in

Fig-ure 2-4 depend on the scale used The scale

for the voltage vector is 1 inch = 50 volts

The length of the vector is 110/50 = 2.2

in-ches The scale for the current vector is

1 inch = 10 amperes

FINDING CURRENT

Figure 2-5 and the following

exam-ple show how current is determined in an

AC inductive circuit

First, find XL: XL = 2πfL

= 2 ×3.14 ×60 ×0.3

= 113.1 ΩUsing Ohm’s law, I =

In a circuit containing only inductance, the current will lag behind the voltage by 90°, asseen in the waveforms

= 120113.1

Figure 2-5 Inductive circuit.

1400.7

=502.4 =200

ACHIEVEMENT REVIEW

In items 1 through 10, select the best answer to make the statement true Place the

letter of the selected answer in the space provided

a the same as reactance

b the property of a coil

c magnetic field strength

b increasing source voltage

c inserting an iron core

d increasing current

e decreasing inductance

5 In a purely inductive circuit (no resistance),

a current lags voltage

b voltage lags current by 90°

c current and voltage are in phase

d voltage leads current

e current lags voltage by 90°

6 The inductive reactance of a 0.06-H coil connected

to a 120-V, 60-Hz source is

a 2.26 Ω c 7.2 Ω e 432 Ω

b 3.6 Ω d 22.62 Ω

7 A purely inductive circuit contains a voltage source of 280 V

at 40 Hz The total inductive reactance of the circuit is 20 Ω

The value of the total current, in amperes, is

c 2.0

8 A current of 5 A exists in a purely inductive circuit connected to

a 120-V, 60-Hz source The total inductive reactance of the circuit

120 V

60 Hz E

L X

5 A

Figure 2-7 Inductive reactance.

120 V

60 Hz E

L 0.16 H

Figure 2-8 Finding current

12 If the inductance in problem 11 (Figure 2-8) is changed to 2.0 H, what is thecircuit current?

13 Determine the inductance of the coil shown in Figure 2-9 if the circuit current is

2 amperes

14 In problem 13 (Figure 2-9), if the frequency is changed to 200 Hz, find L

_

15 What is the circuit frequency in Figure 2-10 if the circuit current is 20 amperes?

16 Using the circuit in problem 15 (Figure 2-10), change the current to 50 amperes,and find the frequency

_

400 V

Figure 2-9 Inductive circuit

Figure 2-10 Finding frequency.

3

CAPACITANCE AND CAPACITIVE REACTANCE

OBJECTIVES

After studying this unit, the student should be able to

• discuss the characteristics of capacitance

• describe the effect of capacitance in an alternating-current circuit

• use vectors to show the voltage and current relationship in a capacitor

Practically all electrical equipment contains a combination of resistors or coils.Some industrial equipment, such as capacitor motors, capacitor banks, and automaticswitch gear, use capacitors Transmission lines have capacitance between the wires

A capacitor consists of two plates of electrical conducting material separated by aninsulating material The plates are commonly aluminum, tin, or any other nonmagnetic

substance The insulating material, called the dielectric, may be any of a large variety of

substances, such as air, mica, glass, wax, paper, fiber, rubber, or oil as per Figure 3-1.When electric potential is connected to the plates, an electrical charge is stored inthe capacitor In an AC circuit, the alternating voltage causes the capacitor to charge anddischarge during every cycle Although current cannot pass through the capacitor, anammeter connected in the line will measure current resulting from the alternating chargeand discharge

Figure 3-1 Oil-filled paper capacitor.

Capacitance is the property of a capacitor, and is defined as the amount of

electri-cal charge that a capacitor receives for each volt of applied potential The unit for

capac-itance is the farad (F) However, the farad is a very large unit in terms of the charges that are normally present, so the microfarad (µF) is generally used

CAPACITIVE REACTANCE

A capacitor in a circuit limits the current therein just as resistance and inductivereactance limit current

The opposition from a capacitor is called capacitive reactance (XC)

If the capacity (in microfarads) and the frequency are known, the reactance inohms can be calculated

Where XC = capacitive reactance in ohms

π = 3.14

f = frequency in hertz

C = number of microfaradsExamples:

1

= 1,000,000or

1 farad microfarads

CAPACITOR f (Hz) C (µF) X C (Ω)

Capacitive reactance varies indirectly with capacitance and frequency

Capacitive reactance may be decreased by increasing either the capacitance or the

frequency

DANGER: A capacitor holds a charge for a long period of time following use in a

circuit Discharge a capacitor before handling The proper method for discharging is

shown in Figure 3-2

SHORT CIRCUIT TO DISCHARGE THE CAPACITOR (BE SURE MAIN SWITCH IS OPEN) CAPACITOR

Figure 3-2 Discharging

a capacitor.

CURRENT LEADS THE VOLTAGE IN A CAPACITOR

A capacitor connected to an AC line causes the current to lead the voltage by90°, as shown in Figure 3-3 Oscilloscope pictures of the current and voltage wave-forms show this relationship Figure 3-4 uses vectors to show the same information asFigure 3-3

When capacitors are connected in parallel, their combined capacitance may befound using the method by which the combined resistance of series-connected resistors

is found:

Ct= C1+ C2+ C3

When capacitors are connected in series, their combined capacitance may be foundusing the method by which the combined resistance of parallel-connected resistors isfound:

When only two capacitors are connected in series, their combined capacitance isfound by the product over the sum method:

2 × 3.14 × 60 × 88.5

= 1,000,00033,330

VOLTAGE (E) CURRENT (I)

Using Ohm’s law:

= 4 A

CALCULATING CAPACITANCE

AND CURRENT

In Figure 3-6, the capacitance of

the circuit must be found to determine

the total circuit current Find the

com-bined capacitance of C2and C3

C2,3= 25 µF + 35 µF = 60 µF

C2,3 is the series equivalent of

the two parallel capacitors Therefore,

the circuit becomes a series circuit as

shown in Figure 3-7

The total capacitance is found

with the following formulas:

120

t t

1

=+C

C

X

E

=30120

µ

C

60 F 2,3

Figure 3-7 Series equivalent circuit.

= + = 0.025 + 0.0166640

160

1

t

C1

The alternate solution for Ctis: Ct = =24 µF

+

×40

406060

Figure 3-8 An AC electrolytic capacitor.

SUMMARY

Capacitors are built to store an electrical charge Capacitance is the property of acapacitor, and capacitive reactance results from the amount of capacitance and the fre-quency of the current As with inductive resonance, capacitive reactance is similar to resis-tance in terms of its opposition to the establishment of current in a circuit The difference

is that the current in a capacitive circuit leads the voltage In a circuit containing onlycapacitance, the current will lead by 90° A wide variety of capacitors exists to provideunique characteristics for circuits and systems One such capacitor is shown in Figure 3-8

ACHIEVEMENT REVIEW

In problems 1 through 10, select the best answer to make the statement true Place

the letter of the selected answer in the space provided

1 The most generally used unit of capacitive reactance is the

2 Before a capacitor is handled, be sure it

a is large enough to do the job

b has clean plates

c has proper polarity

4 Capacitive reactance of a circuit may be increased by

a decreasing total capacitance

b increasing total capacitance

c increasing the number of farads

d increasing source voltage

e increasing frequency

a current leads voltage

b voltage leads current

c current is in phase with voltage

d current lags voltage

e voltage is in phase with current

6 The total capacitance of two 10-µF capacitors connected

8 The total capacitance of a 40-µF capacitor connected in series with an 80-µF capacitor is

12 Change C to 40 µF in problem 11, and calculate the current

13 Determine the total capacitive reactance in the circuit shown in Figure 3-10 andthe value of C2in microfarads if the total current equals 3 amperes.

14 Using the circuit for problem 13 (Figure 3-10), change C1 to 50 µF, and thetotal current to 5 amperes Find the total Xc and the value of C2

15 Find the total current in the circuit shown in Figure 3-11

16 For the circuit in problem 15 (Figure 3-11), change the frequency to 60 Hz, andfind the total current.

17 Find XCtfor the circuit shown in Figure 3-12

18 Using the circuit shown in problem 17 (Figure 3-12), change the 8-µF capacitor

to 10 µF, and find the total capacitive reactance

After studying this unit, the student should be able to

• explain the current–voltage relationship in an AC series circuit containing tance and inductance

resis-• apply vectors to the analysis of an RL AC series circuit

RESISTANCE AND INDUCTANCE IN SERIES

Resistors, coils, and capacitors may be connected in series in several combinations forspecialized purposes One such purpose is shown in Figure 4-1 regarding automobile pro-duction This unit focuses on a circuit containing a resistor connected in series with a coil.Resistors and coils offer opposition to alternating current The voltage and current

in a resistor are in phase In a coil, however, the current lags the voltage drop across thecoil by 90°

The combined opposition of resistors and coils is called impedance and

is measured in ohms

Circuit A: Series Circuit Containing Two Resistors

As shown in Figure 4-2, the current in a series circuit is the same throughout Thetotal resistance is the sum of all the resistances in the circuit The total voltage is the sum

of the voltage across each resistor

Circuit B: Series Circuit Containing a Resistor and a Coil

For the 5-ohm resistor in Figure 4-3, the current is in phase with the voltage Thecurrent in the coil lags the coil voltage by 90° The line current lags the total or line volt-age by less than 90° depending on the values of R and XL

The resistance and the reactance must, therefore, be added vectorially to

obtain the value of impedance