Fundamentals oF IndustrIal electronIcs doc

Example 1.1: DC Circuit Analysis with Independent Sources For the circuit shown in Figure 1.5, apply Ohm’s law, KVL, and KCL to solve for the labeled voltages and currents.. 1.1.6 Voltag

The Industrial Electronics Handbook

S E c o n d E d I T I o n Fundamentals oF IndustrIal electronIcs

S E c o n d E d I T I o n Fundamentals oF IndustrIal electronIcs Power electronIcs and motor drIves control and mechatronIcs IndustrIal communIcatIon systems IntellIgent systems

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The Industrial Electronics Handbook

S E c o n d E d I T I o n Fundamentals oF IndustrIal electronIcs

Edited by

Bogdan M Wilamowski

J david Irwin

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Library of Congress Cataloging‑in‑Publication Data

Fundamentals of industrial electronics / editors, Bogdan M Wilamowski and J David Irwin.

p cm.

“A CRC title.”

Includes bibliographical references and index.

ISBN 978-1-4398-0279-3 (alk paper)

1 Industrial electronics I Wilamowski, Bogdan M II Irwin, J David III Title.

Acknowledgments xiii

Editorial.Board xv

Editors xvii

Contributors xxi

Part I Circuits and Signals

Carlotta A Berry and Deborah J Walter

Carlotta A Berry and Deborah J Walter

Stephen M Haddock and J David Irwin

Tina Hudson

Carlotta A Berry and Deborah J Walter

Thomas F Schubert, Jr and Ernest M Kim

Dalton S Nelson

Part II Devices

Bogdan M Wilamowski

Bogdan M Wilamowski and Guofu Niu

José M Quero, Antonio Luque, Luis Castañer, Angel Rodríguez,

Adrian Ionescu, Montserrat Fernández-Bolaños, Lorenzo Faraone,

Vishal Saxena and R Jacob Baker

Part III Digital Circuits

Part IV Digital and analog Signal Processing

The.field.of.industrial.electronics.covers.a.plethora.of.problems.that.must.be.solved.in.industrial.practice Electronic.systems.control.many.processes.that.begin.with.the.control.of.relatively.simple.devices.like.electric.motors,.through.more.complicated.devices.such.as.robots,.to.the.control.of.entire.fabrication.processes An.industrial.electronics.engineer.deals.with.many.physical.phenomena.as.well.as.the.sensors.that.are.used.to.measure.them Thus,.the.knowledge.required.by.this.type.of.engineer.is.not.only.tra-ditional.electronics.but.also.specialized.electronics,.for.example,.that.required.for.high-power.applica-tions The.importance.of.electronic.circuits.extends.well.beyond.their.use.as.a.final.product.in.that.they.are.also.important.building.blocks.in.large.systems,.and.thus.the.industrial.electronics.engineer.must.also.possess.knowledge.of.the.areas.of.control.and.mechatronics Since.most.fabrication.processes.are.relatively.complex,.there.is.an.inherent.requirement.for.the.use.of.communication.systems.that.not.only.link.the.various.elements.of.the.industrial.process.but.are.also.tailor-made.for.the.specific.industrial.environment Finally,.the.efficient.control.and.supervision.of.factories.require.the.application.of.intelli-gent.systems.in.a.hierarchical.structure.to.address.the.needs.of.all.components.employed.in.the.produc-tion.process This.is.accomplished.through.the.use.of.intelligent.systems.such.as.neural.networks,.fuzzy.systems,.and.evolutionary.methods The.Industrial.Electronics.Handbook.addresses.all.these.issues.and.does.so.in.five.books.outlined.as.follows:

1 Fundamentals of Industrial Electronics

2 Power Electronics and Motor Drives

3 Cont rol and Mechatronics

4 Industrial Communication Systems

5 Intelligent Systems

sible Thus,.this.book.closely.follows.the.current.research.and.trends.in.applications.that.can.be.found

The.editors.have.gone.to.great.lengths.to.ensure.that.this.handbook.is.as.current.and.up.to.date.as.pos-in.IEEE Transactions on Industrial Electronics This.journal.is.not.only.one.of.the.largest.engineering.

publications.of.its.type.in.the.world,.but.also.one.of.the.most.respected In.all.technical.categories.in.which.this.journal.is.evaluated,.its.worldwide.ranking.is.either.number.1.or.number.2.depending.on.category As.a.result,.we.believe.that.this.handbook,.which.is.written.by.the.world’s.leading.researchers.in.the.field,.presents.the.global.trends.in.the.ubiquitous.area.commonly.known.as.industrial.electronics

Fundamentals of Industrial Electronics deals with the fundamental areas that form the basis for.

the.field.of.industrial.electronics Because.of.the.breadth.of.this.field,.the.knowledge.required.spans.a.wide.spectrum.of.technology,.which.includes.analog.and.digital.circuits,.electronics,.electromagnetic

machines,.and.signal.processing The.knowledge.gained.here.is.then.applied.in.Power Electronics and

Motor Drives,.Control and Mechatronics,.Industrial Communication Systems,.and.Intelligent Systems,.

and.in.total.form.the.Industrial.Electronics.Handbook

For.MATLAB•.and.Simulink•.product.information,.please.contact

The.editors.wish.to.express.their.heartfelt.thanks.to.their.wives.Barbara.Wilamowski.and.Edie.Irwin.for.their.help.and.support.during.the.execution.of.this.project

John W Steadman

University.of.South.AlabamaMobile,.Alabama

Bogdan M Wilamowski.received.his.MS.in.computer.engineering.in.

1966,.his.PhD.in.neural.computing.in.1970,.and.Dr habil in.integrated.circuit.design.in.1977 He.received.the.title.of.full.professor.from.the.president.of.Poland.in.1987 He.was.the.director.of.the.Institute.of.Electronics.(1979–1981).and.the chair.of the solid state electronics.department (1987–1989) at the Technical University of Gdansk,.Poland He.was.a.professor.at.the.University.of.Wyoming,.Laramie,.from 1989 to 2000 From 2000 to 2003, he served as an associate.director at the Microelectronics Research and Telecommunication.Institute,.University.of.Idaho,.Moscow,.and.as.a.professor.in.the.elec-trical.and.computer.engineering.department.and.in.the.computer.sci-ence.department.at.the.same.university Currently,.he.is.the.director.of.ANMSTC—Alabama.Nano/Micro.Science.and.Technology.Center,.Auburn,.and.an.alumna.professor.in.the.electrical.and.computer.engineering.department.at.Auburn.University,.Alabama Dr. Wilamowski.was.with.the.Communication.Institute.at.Tohoku.University,.Japan.(1968–1970),.and.spent.one.year.at.the.Semiconductor.Research.Institute,.Sendai,.Japan,.as.a.JSPS.fellow.(1975–1976) He.was.also.a.visiting.scholar.at.Auburn.University.(1981–1982.and.1995–1996).and.a.visiting.professor.at.the.University.of.Arizona,.Tucson.(1982–1984) He.is.the.author.of.4.textbooks,.more.than.300.refereed.publications,.and.has.27.patents He.was.the.principal.professor.for.about.130.graduate.students His.main.areas.of.interest.include.semiconductor.devices.and.sensors,.mixed.signal.and.analog.signal.processing,.and.computa-tional.intelligence

Dr Wilamowski.was.the.vice.president.of.the.IEEE.Computational.Intelligence.Society.(2000–2004).and.the.president.of.the.IEEE.Industrial.Electronics.Society.(2004–2005) He.served.as.an.associate.edi-

tor.of.IEEE Transactions on Neural Networks,.IEEE Transactions on Education,.IEEE Transactions on

Industrial Electronics,.the.Journal of Intelligent and Fuzzy Systems,.the.Journal of Computing,.and.the International Journal of Circuit Systems and IES Newsletter He.is.currently.serving.as.the.editor.in.chief.

of.IEEE Transactions on Industrial Electronics.

Professor.Wilamowski.is.an.IEEE.fellow.and.an.honorary.member.of.the.Hungarian.Academy.of.Science In.2008,.he.was.awarded.the.Commander.Cross.of.the.Order.of.Merit.of.the.Republic.of.Poland.for.outstanding.service.in.the.proliferation.of.international.scientific.collaborations.and.for.achieve-ments.in.the.areas.of.microelectronics.and.computer.science.by.the.president.of.Poland

J David Irwin.received.his.BEE.from.Auburn.University,.Alabama,.

in 1961, and his MS and PhD from the University of Tennessee,.Knoxville,.in.1962.and.1967,.respectively

In.1967,.he.joined.Bell.Telephone.Laboratories,.Inc.,.Holmdel,.New.Jersey,.as.a.member.of.the.technical.staff.and.was.made.a.supervisor.in.1968 He.then.joined.Auburn.University.in.1969.as.an.assistant.professor.of.electrical.engineering He.was.made.an.associate.profes-sor.in.1972,.associate.professor.and.head.of.department.in.1973,.and.professor.and.head.in.1976 He.served.as.head.of.the.Department.of.Electrical.and.Computer.Engineering.from.1973.to.2009 In 1993,.he.was.named.Earle.C Williams.Eminent.Scholar.and.Head From.1982.to.1984,.he.was.also.head.of.the.Department.of.Computer.Science.and.Engineering He.is.currently.the Earle.C Williams.Eminent.Scholar.in.Electrical.and.Computer.Engineering.at.Auburn

Dr Irwin has served the Institute of Electrical and Electronic Engineers, Inc (IEEE) Computer

Society.as.a.member.of.the.Education.Committee.and.as.education.editor.of.Computer He.has.served.

as chairman of the Southeastern Association of Electrical Engineering Department Heads and the.National.Association.of.Electrical.Engineering.Department.Heads.and.is.past.president.of.both.the.IEEE.Industrial.Electronics.Society.and.the.IEEE.Education.Society He.is.a.life.member.of.the.IEEE.Industrial.Electronics.Society.AdCom.and.has.served.as.a.member.of.the.Oceanic.Engineering.Society

AdCom He.served.for.two.years.as.editor.of.IEEE Transactions on Industrial Electronics He.has.served.

on the Executive Committee.of the Southeastern.Center.for.Electrical Engineering Education,.Inc.,.and.was.president.of.the.organization.in.1983–1984 He.has.served.as.an.IEEE.Adhoc.Visitor.for.ABET.Accreditation.teams He.has.also.served.as.a.member.of.the.IEEE.Educational.Activities.Board,.and.was.the.accreditation.coordinator.for.IEEE.in.1989 He.has.served.as.a.member.of.numerous.IEEE.com-mittees,.including.the.Lamme.Medal.Award.Committee,.the.Fellow.Committee,.the.Nominations.and.Appointments.Committee,.and.the.Admission.and.Advancement.Committee He.has.served.as.a.mem-ber.of.the.board.of.directors.of.IEEE.Press He.has.also.served.as.a.member.of.the.Secretary.of.the.Army’s.Advisory.Panel.for.ROTC.Affairs,.as.a.nominations.chairman.for.the.National.Electrical.Engineering.Department.Heads.Association,.and.as.a.member.of.the.IEEE.Education.Society’s.McGraw-Hill/Jacob.Millman Award Committee He.has also served.as.chair.of the IEEE Undergraduate.and Graduate.Teaching.Award.Committee He.is.a.member.of.the.board.of.governors.and.past.president.of.Eta.Kappa.Nu,.the.ECE.Honor.Society He.has.been.and.continues.to.be.involved.in.the.management.of.several.international.conferences.sponsored.by.the.IEEE.Industrial.Electronics.Society,.and.served.as.general.cochair.for.IECON’05

Dr Irwin is the author and coauthor of numerous publications, papers, patent applications, and

presentations,.including Basic Engineering Circuit Analysis,.9th.edition,.published by.John Wiley.&.

Sons,.which.is.one.among.his.16.textbooks His.textbooks,.which.span.a.wide.spectrum.of.engineering.subjects,.have.been.published.by.Macmillan.Publishing.Company,.Prentice.Hall.Book.Company,.John.Wiley.&.Sons.Book.Company,.and.IEEE.Press He.is.also.the.editor.in.chief.of.a.large.handbook.pub-lished.by.CRC.Press,.and.is.the.series.editor.for.Industrial.Electronics.Handbook.for.CRC.Press.Dr Irwin.is.a.fellow.of.the.American.Association.for.the.Advancement.of.Science,.the.American.Society for Engineering Education, and the Institute of Electrical and Electronic Engineers He.received an IEEE Centennial Medal in 1984, and was awarded the Bliss Medal by the Society of.American.Military.Engineers.in.1985 He.received.the.IEEE.Industrial.Electronics.Society’s.Anthony.J Hornfeck.Outstanding.Service.Award.in.1986,.and.was.named.IEEE.Region.III.(U.S Southeastern.Region) Outstanding Engineering Educator in 1989 In 1991, he received a Meritorious Service.Citation from the IEEE Educational Activities Board, the 1991 Eugene Mittelmann Achievement.Award.from.the.IEEE.Industrial.Electronics.Society,.and.the.1991.Achievement.Award.from.the.IEEE.Education.Society In.1992,.he.was.named.a.Distinguished.Auburn.Engineer In.1993,.he.received.the.IEEE.Education.Society’s.McGraw-Hill/Jacob.Millman.Award,.and.in.1998.he.was.the.recipient.of.the

Editors xix

IEEE.Undergraduate.Teaching.Award In.2000,.he.received.an.IEEE.Third.Millennium.Medal.and.the.IEEE.Richard.M Emberson.Award In.2001,.he.received.the.American.Society.for.Engineering.Education’s.(ASEE).ECE.Distinguished.Educator.Award Dr Irwin.was.made.an.honorary.profes-sor,.Institute.for.Semiconductors,.Chinese.Academy.of.Science,.Beijing,.China,.in.2004 In.2005,.he.received.the.IEEE.Education.Society’s.Meritorious.Service.Award,.and.in.2006,.he.received.the.IEEE.Educational.Activities.Board.Vice.President’s.Recognition.Award He.received.the.Diplome.of.Honor.from.the.University.of.Patras,.Greece,.in.2007,.and.in.2008.he.was.awarded.the.IEEE.IES.Technical.Committee.on.Factory.Automation’s.Lifetime.Achievement.Award In.2010,.he.was.awarded.the.elec-trical.and.computer.engineering.department.head’s.Robert.M Janowiak.Outstanding.Leadership.and.Service.Award In.addition,.he.is.a.member.of.the.following.honor.societies:.Sigma.Xi,.Phi.Kappa.Phi,.Tau.Beta.Pi,.Eta.Kappa.Nu,.Pi.Mu.Epsilon,.and.Omicron.Delta.Kappa

Montserrat Fernández-Bolaños

Ecole.Polytechnique.Fédérale.de.LausanneLausanne,.Switzerland

Stephen M Haddock

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

James A Heinen

Department.of.Electrical.and.Computer

EngineeringMarquette.UniversityMilwaukee,.Wisconsin

Tina Hudson

Department.of.Electrical.and.Computer

EngineeringRose-Hulman.Institute.of.TechnologyTerre.Haute,.Indiana

Francisco Ibáñez

European.CommissionBrussels,.Belgium

Victor P Nelson

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

Russell J Niederjohn (deceased)

Department.of.Electrical.and.Computer

EngineeringMarquette.UniversityMilwaukee,.Wisconsin

Guofu Niu

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

Nam Pham

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

Arlen Planting

Department.of.Electrical.and.Computer

EngineeringBoise.State.UniversityBoise,.Idaho

José M Quero

Department.of.Electronic.EngineeringUniversity.of.Seville

Sevilla,.Spain

Sadasiva M Rao

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

Deborah J Walter

Department.of.Electrical.and.Computer

EngineeringRose-Hulman.Institute.of.TechnologyTerre.Haute,.Indiana

Buren Earl Wells

Department.of.Electrical.and.Computer

EngineeringThe.University.of.Alabama.in.HuntsvilleHuntsville,.Alabama

Edward Wheeler

Department.of.Electrical.and.Computer.Engineering

Rose-Hulman.Institute.of.TechnologyTerre.Haute,.Indiana

Bogdan M Wilamowski

Department.of.Electrical.and.Computer

EngineeringAuburn.UniversityAuburn,.Alabama

Tiantian Xie

Department.of.Electrical.and.Computer.Engineering

Auburn.UniversityAuburn,.Alabama

1.1 Introduction

Direct.current.(DC).circuit.analysis.is.the.study.of.circuits.with.a.constant.voltage.or.current.source The

most.popular.example.of.a.DC.circuit.is.a.battery.and.a.light.bulb A.DC.circuit.contains.an.active circuit.

element.(i.e.,.battery).capable.of.generating.electric.energy These.electric.sources.convert.nonelectric.energy.to.electric.energy.(i.e.,.a.voltage.or.current) Independent.electric.sources.produce.a.constant.voltage.or.current.in.the.circuit.regardless.of.the.current.through.or.voltage.across.the.source The.sym-bols.for.an.ideal.DC.voltage.and.current.source.are.shown.in.Figure.1.1 It.should.be.noted.that.an.ideal.voltage.and.current.source.can.deliver.or.absorb.power.to.an.electric.circuit An.example.of.an.ideal.voltage.source.absorbing.power.is.a.rechargeable.battery

Dependent.sources.establish.a.voltage.or.current.in.a.circuit.that.is.based.upon.the.value.of.a.voltage.or.current.elsewhere.in.the.circuit One.use.of.dependent.sources.is.to.model.operational.amplifiers.and.transistors Table.1.1.presents.a.summary.of.the.four.types.of.dependent.sources

A.passive.circuit.element.models.devices.that.cannot.generate.electric.energy.such.as.a.light.bulb The.most.common.passive.circuit.elements.are.inductors,.capacitors,.and.resistors The.voltage–current.relationships.for.these.devices.will.be.described.in.the.subsequent.section

1.1 Introduction 1-1

Ohm’s.Law • Inductors.and.Capacitors • Kirchhoff’s Current.Law • Kirchhoff’s.Voltage.Law • Series.and.Parallel Relationships • Voltage.and.Current.Divider.Rule • Delta–Wye (Δ–Y).Transformations

1.2 Systematic.Circuit.Analysis.Techniques 1-7

Node-Voltage.Method • Mesh-Current.Method • Superposition

1.3 Circuit.Modeling.Techniques 1-16

Source.Transformations • Thevenin.and.Norton.Equivalent Circuits • Maximum.Power.Transfer

1.4 Transient.Analysis 1-19

First-Order.Circuits • Second-Order.Circuits

1.5 Conclusions 1-36 Bibliography 1-36

1.1.2 Inductors and Capacitors

As previously stated, the other two passive circuit elements are inductors and capacitors Both the.inductor.and.the.capacitor.have.the.ability.to.store.energy Inductors.store.energy.in.the.form.of.current.and.capacitors.store.energy.in.the.form.of.voltage The.energy.stored.in.these.elements.is.released.back.into.the.circuit.when.a.DC.source.is.removed Therefore,.these.two.elements.exhibit.behavior.that.is.a.function.of.time The.analysis.of.these.types.of.circuits.is.transient.analysis.that.will.be.addressed.later.in.this.chapter Table.1.2.describes.the.current–voltage.relationship.for.inductors.and.capacitors.where

the.inductance.(L).is.in.henrys.(H),.capacitance.(C).is.in.farads.(F),.and.time.(t).is.in.seconds.(s).

1.1.3 Kirchhoff’s Current Law

ferred Another.way.to.state.this.law.is.for.any.electric.circuit,.the.total.power.delivered.by.the.elements.must.be.equal.to.the.total.power.absorbed.by.the.elements Kirchhoff’s.current.law.(KCL).is.based.upon.the.law.of.conservation.of.energy A.node.in.a.circuit.is.any.point.at.which.two.or.more.circuit.elements.are.connected KCL.states.that.the.sum.of.currents.entering.a.node.is.zero.(i.e.,.current.in.=.current.out) KCL.can.be.applied.to.any.node.in.a.closed.circuit The.circuit.in.Figure.1.3.has.three.branch.currents:

The.law.of.conservation.of.energy.states.that.energy.can.neither.be.created.nor.destroyed,.only.trans-I1,.I2,.and.I3 Since.all.of.these.currents.are.leaving.Node.A,.KCL.at.this.node.yields.Equation.1.6:

DC and Transient Circuit Analysis 1-31.1.4 Kirchhoff’s Voltage Law

C

i C dv dt

Example 1.1: DC Circuit Analysis with Independent Sources

For the circuit shown in Figure 1.5, apply Ohm’s law, KVL, and KCL to solve for the labeled voltages and currents

The first step in the analysis is to apply KCL at Node A and KVL at the left and right loop These tions are provided in Equations 1.8 through 1.10:

equa-KCL at Node A : − + + =I s I2 I3 0 (1.8)KVL at left loop : −120+ + =V V1 2 0 (1.9)KVL at right loop : − + + =V V V2 3 4 0 (1.10)Next, use Ohm’s law to rewrite Equations 1.9 and 1.10 in terms of the branch currents and resistor values These equations are shown in Equations 1.11 and 1.12:

KVL at left loop: 50I s+100I2=120 (1.11)KVL at right loop: −100I2+20I3+80I3=0 (1.12)Solving the simultaneous set of equations, (1.8), (1.11), and (1.12) yields

I s=1 2 A, I2=0 6 A, I3=0 6 A (1.13)The results in (1.13) and Ohm’s law can be used to find the unknown voltages:

DC and Transient Circuit Analysis 1-5

1.1.5 Series and Parallel relationships

At.times,.it.is.useful.to.simplify.resistive.networks.by.combining.resistors.in.series.and.parallel.into.an.equivalent.resistance Exactly.two.resistors.that.are.connected.at.a.single.node.share.the.same.current.and.are.said.to.be.connected.in.series It.is.important.to.note.that.the.equivalent.resistance.of.series.resis-tors.is.larger.than.each.of.the.individual.resistances Resistors.that.are.connected.together.at.a.pair.of.nodes.(“single.node.pair”).have.the.same.voltage.and.are.said.to.be.connected.in.parallel The.equivalent.conductance.of.resistors.in.parallel.is.the.sum.of.the.conductances.of.the.individual.resistors Therefore,.the.reciprocal.of.the.equivalent.resistance.is.the.sum.of.the.individual.conductances Note.that.the.equivalent.resistance.of.parallel.resistors.is.smaller.than.each.of.the.individual.resistances Figure.1.6a.provides.an.example.of.a.circuit.with.series.resistors.and.the.equivalent.resistance.seen.by.the.voltage.source Figure.1.6b.provides.an.example.of.a.circuit.with.parallel.resistors.and.the.equivalent.resistance.seen.by.the.current.source

Example 1.2: Analysis of Example  1.1 by Combining Resistors

It is possible to analyze the circuit in Example 1.1, to find the source current, I s The first step is to recognize that the 80 and 20 Ω resistors are in series and combine to yield 100 Ω This simplified circuit is shown

FIGURE 1.6 Resistors.in.(a).series.and.(b).parallel.

120 V

+ –

I s

50 Ω

FIGURE 1.7 Circuit.in.Example.1.1.simplified.by.putting.80.W.in.series.with.20.W.

Finally, the last step is to use Ohm’s law to solve I s, which yields

I s=120=

Note that this result is consistent with the answer to Example 1.1

1.1.6 Voltage and Current Divider rule

Example 1.3: Analysis of Example 1.1 Using Voltage and Current Divider

For the circuit in Figure 1.5, given that I s = 1.2 A, use the current divider to find I2 and the voltage divider to

find V4 The first step in the analysis is to recognize that the 100 Ω resistor is in parallel with the 80 and 20

Ω series combination The current divider relationship to find I2 is shown in Equation 1.19:

I s

100 Ω

FIGURE 1.9 Circuit in Figure 1.8.

simplified.by.putting.50.W.resistors.in series.

8 V +–

16 (16 + 12 + 4)8

16 (16 + 12 + 4)8

4 (16 + 12 + 4)8

(a)

96 || 120 || 80 96

I96Ω= 48 = 16 mA

96 || 120 || 80 120

96 || 120 || 80 80

DC and Transient Circuit Analysis 1-7

Ohm’s law can be used to find the voltage, V2, across the 100 Ω resistor, V2 = 100I2 = 60 V The voltage

divider can be used to find the voltage, V4, as shown in Equation 1.20:

V4 80 V2

80 20 48

=

Note that these results are consistent with the solution to Example 1.1

1.1.7 Delta–Wye (Δ–Y) transformations

tions.are.referred.to.as.delta.(“Δ”).or.wye.(“Y”).interconnections These.two.configurations.are.equiva-lent.based.upon.the.relationships.shown.in.Table.1.3 Equivalence.means.that.both.configurations

There.are.some.resistance.configurations.that.are.neither.in.series.or.parallel These.special.configura-have.the.same.voltage.and.current.characteristics.at.terminals.a,.b,.however.internal.to.the.network,.

the.values.may.not.be.the.same

1.2 Systematic Circuit analysis techniques

niques.are.the.node-voltage.method.based.on.KCL.and.the.mesh-current.method.based.on.KVL These.techniques.are.used.to.derive.the.minimum.number.of.linearly.independent.equations.necessary.to.find.the.solution

There.are.two.general.approaches.to.solving.circuits.using.systematic.techniques The.systematic.tech-1.2.1 Node-Voltage Method

The.node-voltage.method.is.a.general.technique.that.can.be.applied.to.any.circuit An.independent.KCL.equation.can.be.written.at.every.essential.node.(nodes.with.three.or.more.elements.connected).except.for.one The.standard.practice.is.to.choose.the.ground.node.as.the.reference.node.and.omit.the.ground

Example 1.4: Node-Voltage Method with Independent Sources

Given the circuit in Figure 1.11, use the node-voltage method to find the power delivered by each source

Recall that the first step in the analysis was to label the essential nodes The four essential nodes in

Figure 1.11 have already been labeled as V1, V2, V3, and ground (0 V) Since V1 is the voltage at that node

with respect to the reference node (“ground node”), it is tied to the 200 V source so V1 = 200 V The node voltages V2 and V3 are unknown, thus KCL must be performed to find these values In order to simplify analysis, the KCL equations are derived such that the current is drawn leaving the node if it is not given

The KCL equations at V2 and V3 are given in Equations 1.21 and 1.22:

Using the results of Equation 1.23, it is possible to find the power associated with the 1 A current source

Since the voltage across the current source is V3, and it is not in the passive sign convention, the power is

FIGURE 1.11 Node-voltage.method.circuit.

DC and Transient Circuit Analysis 1-9

P = −V3 (1) = −265 W or 265 W delivered In order to find the current through the 200 V source, it is necessary

to use KCL at V1 The KCL equation at V1 is given in Equation 1.24:

Example 1.5: Analysis of Example 1.4 with 𝚫-Y Transformations

For the circuit in Figure 1.11, use Δ-Y transformations to find the power associated with the 200 V source The first step in the analysis is to identify that the 500, 100, and 400 Ω resistors form a Δ configuration as

R a , R b , and R c, respectively This circuit can be simplified by converting the Δ configuration to a Y ration Equations 1.26 through 1.28 are used to find the resistor values in the Y configuration The simpli-fied circuit is shown in Figure 1.12

Using the result in Equation 1.30 to find the current through the 200 V source yields

I s=V A−2 =

00

Thus, the power absorbed by the 200 V source is 100 W, consistent with the prior solution

Example 1.6: Node-Voltage Method with Dependent Sources

The circuit in Figure 1.13 models an operational amplifier An operational amplifier is an active circuit element used to perform mathematical operations such as addition, subtraction, multiplication, divi-sion, differentiation, and integration This electronic unit is an integrated circuit that can be modeled as

a VCVS The gain of the op amp is the ratio of the output voltage to the input voltage, (V o  /V s ) Use KCL to determine the gain of the circuit in Figure 1.13

The KCL equations at Nodes A and B are shown in Equations 1.32 and 1.33:

Note that the dependent source introduces a constraint equation based upon the relationship between

the node voltage and the controlling voltage, V d This relationship is V A = −V d This produces two tions and two unknowns that can be solved for the gain shown in Equation 1.34:

equa-V V

–

–

FIGURE 1.13 DC.circuit.with.dependent.sources.(operational.amplifier.model).

DC and Transient Circuit Analysis 1-11

Example 1.7: Node-Voltage Method with Supernodes

Use the node-voltage method on the circuit in Figure 1.14 to find the current through the voltage source The first step in the analysis is to label the node voltages and supernode These have already been labeled

in the circuit in Figure 1.14 Next, KCL at the supernode yields Equation 1.35, and KVL at the supernode yields Equation 1.36:

500 100 125 0

+I Ω+I Ω+I Ω= + V + V + V = (1.35)KVL at supernode: − +V1 25 + =V2 0 (1.36)Solving these two equations and two unknowns yields

To find the current through the voltage source, it is necessary to perform KCL at V1 or V2 Since the

2 A current source is connected to V1, this selection will have one less term with a voltage variable

Assuming the current through the voltage source, I s , flows from right to left and applying KCL at V1 yields the following equation:

KCL at supernode: I s = 2 + I500Ω + I250Ω = 2 155 m + 100− m = 1.945 A (1.39)

1.2.2 Mesh-Current Method

The.goal.of.the.mesh-current.method.is.to.determine.all.of.the.unknown.mesh.currents.in.a.circuit A.mesh.is.a.loop.in.a.circuit.that.does.not.contain.any.other.loops The.mesh-current.method.is.only.applicable.to.planar.circuits,.circuits.that.can.be.drawn.on.a.plane.with.no.crossing.branches The.first.step.in.the.analysis.is.to.label.all.of.the.mesh.currents.in.a.circuit The.mesh.currents.are.fictitious.cur-rents.that.circulate.in.a.mesh Note.that.a.mesh.current.may.or.may.not.be.a.branch.current,.but.all.of.the.branch.currents.can.be.found.from.the.mesh.currents The.next.step.is.to.write.KVL.equations.summing

Example 1.8: Mesh-Current Method on Example 1.6

For the circuit in Figure 1.15, use the mesh-current method to determine the output voltage V o if the

constraint: V d = 2 (M I2−I1) (1.42)Solving these three simultaneous equations for the mesh currents yields

I1 = 299.997 µA, = 300.002 I2 µA, V d = 30.075 V− µ (1.43)

Using the mesh current value to find V o yields

V o = 50 + 2 10 = 6 VI 5V d

The reader should verify that this gain is consistent with Example 1.6

A special case of the mesh-current method occurs when a current source is shared between two meshes In this case, it is necessary to introduce another variable to describe the voltage across the current source in order to write the KVL equation An alternate approach is to define the two meshes

that include the current source and anything in series with it as a supermesh The 6 A current source

in series with the 1 Ω resistor in Figure 1.16 creates a supermesh denoted by the superimposed rectangle

In order to analyze a supermesh, it is necessary to perform KVL and KCL at the supermesh Lastly, write

a KVL equation for any other unknown mesh currents in the circuit and solve the simultaneous system of equations This method will be demonstrated on the circuit in Figure 1.16

DC and Transient Circuit Analysis 1-13

Example 1.9: Mesh-Current Method with a Supermesh

Use the mesh-current method to find the power associated with the 6 A current source The first step in the analysis is to label the supermesh and mesh currents These have already been labeled in the circuit

in Figure 1.16 The next step is to derive the KVL and KCL equations at the supermesh and these are shown in Equations 1.45 and 1.46:

KVL at supermesh: 2 + 3 +5− I1 I2−10 + 9 + 7 = 0I2 I1 (1.45)

KCL at supermesh: I1 = 6−I2 (1.46)

Solving this simultaneous set of equations yields

I1 = 4 , A I2 = 2 − A (1.47)

In order to determine the power associated with the 6 A current source, it is necessary to perform KVL

at the left or right mesh to find the voltage across the current source Assuming the voltage across the

current source, V s, is positive on top and applying KVL at the left mesh yields

same.time.is.equal.to.the.sum.of.the.same.quantity.due.to.each.source.acting.alone The.method.to.solve.for.an.unknown.variable.in.a.circuit.involves.solving.for.the.variable.of.interest.for.one.source.acting.alone.by.deactivating.all.the.other.independent.sources,.then.sum.the.results.for.each.source.act.ing.alone To.deactivate.an.independent.voltage.source,.replace.the.voltage.source.with.a.short.cir-cuit.(0.V) To deactivate.an.independent.current.source, replace.the current.source with an open.cir.cuit.(0.A) Dependent.sources.are.never.deactivated.(“turned.off ”) The.benefit.in.applying.the.principle.of.superposition.is.that.many.times,.the.circuit.with.the.deactivated.source.is.simpler.to.solve.for.the.unknown.value.

Example 1.10: Circuit Analysis Using Superposition

For the circuit in Figure 1.16, apply the principle of linear superposition to solve for the unknown branch

current I1 (see Figure 1.17)

The first step in the analysis is to disable the 6 A and 10 V sources and use KVL to calculate I1 The tion to this analysis is shown in Equation 1.49 The variable of interest is given a prime to denote that it is due to one source acting alone (see Figures 1.18 through 1.20)

1.3 Circuit Modeling techniques

Just.like.it.is.possible.to.model.the.behavior.of.multiple.resistors.connected.in.parallel.and.series.with.a.single.equivalent.resistance,.it.is.also.possible.to.model.resistive.circuits.containing.sources.and.resistors.as.either.a.Thevenin.or.Norton.equivalent.model These.models.are.useful.simplifying.techniques.when.only.the.circuit.behavior.at.a.single.port.is.of.interest

1.3.1 Source transformations

Source.transformations.are.another.simplifying.technique.for.circuit.analysis Source.transformations.are.based.upon.the.concept.of.equivalence.of.the.voltage.and.current.terminal.characteristics.at.a.single.port A.voltage.source.in.series.with.a.resistor.can.be.replaced.by.a.current.source.in.parallel.with.a.resis-tor.if.they.have.the.relationships.given.in.Figure.1.21

1.3.2 thevenin and Norton Equivalent Circuits

A.simple.resistive.circuit.can.be.simplified.to.an.independent.voltage.source.in.series.with.a.resistor.and.this

is.referred.to.as.the.Thevenin.equivalent.circuit The.voltage.source.is.referred.to.as.the.Thevenin.voltage,.V TH,

and.the.resistor.is.the.Thevenin.resistance,.R TH In.addition,.a.simple.resistive.circuit.can.be.simplified.to.an.independent.current.source.in.parallel.with.a.resistor.and.this.is.referred.to.as.the.Norton.equivalent.circuit

The.current.source.is.the.Norton.current,.I N,.and.the.resistance.is.the.same.as.the.Thevenin.resistance These.are.important.simplification.techniques.when.the.values.of.interest.are.the.port.characteristics.such.as.the.voltage,.current,.or.power.delivered.to.a.load.placed.across.the.terminals The.method.to.find.the.Thevenin

rent.is.to.find.the.short.circuit.current.between.terminals.a.and.b There.are.several.techniques.to.find.the.

voltage.is.to.determine.the.open.circuit.voltage.across.terminals.a.and.b The.method.to.find.the.Norton.cur-Thevenin.equivalent.resistance When.there.are.only.independent.sources,.one.of.the.more.popular.methods.is.to.deactivate.all.independent.sources.and.find.the.equivalent.resistance.of.the.network.across.terminals

a and.b Alternately,.the.Thevenin.resistance.can.be.calculated.by.using.the.following.formula:

V I

V I

oc sc

TH N

=

I p

FIGURE 1.21 Source transformation relationships (a) Series circuit, V s =I p R p , R s =R p and (b) parallel circuit,.

I p =V s /R s , R p =R s .

DC and Transient Circuit Analysis 1-17

Note.that.when.there.are.dependent.sources.in.the.circuit,.there.is.a.third.technique.based.upon.deacti-

vating.all.independent.sources.and.using.a.test.voltage.or.current.at.terminals.a.and.b.to.find.the.equiva-lent.resistance The.reader.is.encouraged.to.review.this.technique.for.future.study

Example 1.11: Thevenin Equivalent Resistance

For the circuit in Figure 1.22, determine the Thevenin equivalent resistance to the left of terminals

a and b.

In order to find the Thevenin equivalent resistance, deactivate the two independent sources and find

the equivalent resistance to the left of terminals a and b The circuit in Figure 1.22 is shown in Figure 1.23

with the sources deactivated

In the circuit in Figure 1.23, the 10 and 40 Ω resistors are in parallel This parallel combination is

in series with the 8 Ω resistor Equation 1.54 shows the derivation of the Thevenin equivalent

resis-tance, R TH:

R TH = 10 || 40 + 8 = 16Ω (1.54)

Example 1.12: Thevenin and Norton Equivalent Circuits

For the circuit in Figure 1.22, find the Thevenin and Norton

equivalent circuit to the left of terminals a and b The first step

in the analysis is to find the open circuit voltage between

terminals a and b (V TH = V oc = V a ) Either the node-voltage or mesh-current method would be an acceptable technique to find this value, however the node-voltage method was used by

writing the KCL equation at V 1 and V a and these are shown in Equation 1.55:

V1 = 80 V, = = = 112 VV TH V oc V a

The next step in the analysis is to find the short circuit current between terminals a and b (I N = I sc = I ab )

The mesh-current method will be used to determine short circuit current, I sc, as shown in Figure 1.24 The result of the analysis is shown in Equation 1.56:

KVL at : I1 −60 + 10( 4) + 40 ( ) = 0I1 − I1 − I N

(1.56)KVL at : 40 ( ) + 8( 4) = 0I N I N − I1 I N −

I1 = 7.6 A, I N = = = 7 AI sc I ab

The Thevenin equivalent resistance can also be found from

R V I

V I

TH oc sc TH N

Note that this Thevenin resistance is consistent with Example 1.11 The final step in the result is to

draw the Thevenin and Norton equivalent circuits to the left of terminals a and b These are shown

V R

L TH TH

= 2

Example 1.13: Maximum Power Transfer

For the circuit shown in Figure 1.22, determine the value of a load resistor placed across terminals a and b

for maximum power transfer and calculate the value of the power for the load selected (see Figure 1.26).Since the Thevenin equivalent resistance of this circuit is 16 Ω, select RL = R TH = 16 Ω for maximum power transfer Finally, the value of the power delivered to the 16 Ω is calculated as follows:

P V R

L TH TH

1.4.1 First-Order Circuits

tions.are.either.RL.circuits.or.RC.circuits.based.upon.whether.they.have.resistors.and.capacitors.or.resistors.and.inductors,.respectively RL.and.RC.circuits.are.known.as.first-order.circuits.because.the

FIGURE 1.26 Circuit.for.maximum.power.transfer.example.

Example 1.14: Natural Response of an RL Circuit

For the circuit in Figure 1.27, assume that the switch is in position a for a long time and moves to position

b at t = 0 Find the current through the inductor, i(t), and the voltage across the inductor, v(t), for t > 0.

a

b

10 V

i(t) v(t)

t = 0

1 Ω

+ –

+

–

100 mH

FIGURE 1.27 RL.circuit.for.Example.1.14.

DC and Transient Circuit Analysis 1-21

The first step in the analysis is to find the initial conditions for the circuit in Figure 1.27 In order to find

the initial conditions, redraw the circuit at t = 0− (before the switching occurs) under DC conditions The circuit to find the initial conditions is shown in Figure 1.28 Since an inductor under DC or steady-state

conditions is modeled as a short circuit, the initial voltage, v(0−), is 0 V It is modeled as a short circuit

because the current is constant with time; therefore, the time rate of change of the current (di L /dt) is zero and the voltage (v L = L(di L /dt)) over the inductor is 0 V Using Ohm’s law on the circuit in Figure 1.28, it is possible to find the initial current, i(0−) as shown in Equation 1.61 Note that because the inductor is a short circuit, the 20 Ω resistor is shorted out and has no affect on the circuit

i( )0 10

1 10

Since the voltage across an inductor can change instantaneously, the circuit must also be analyzed

immediately after switching occurs at t = 0+ For this analysis, model the inductor as a 10 A current source

because current cannot change instantaneously so i(0+) = i(0−) = 10 A Next, use KCL to find the voltage across the inductor The circuit is shown in Figure 1.29 and the analysis in Equation 1.62: