EG1108 Part 2 - MAE CUHKbmchen/courses/EG1108_Part_2.pdf · 2020. 8. 27. · EG1108 PART2~ PAGE6...

EG1108PART 2 ~PAGE 1EG1108PART 2 ~PAGE 1 BEN M.CHEN,NUSECEBEN M.CHEN,NUSECE

EG1108:Electrical Engineering

Part2:ApplicationExamples

EG1108:Electrical Engineering

Part2:ApplicationExamples

BenM.ChenProfessorofElectricalandComputerEngineering

NationalUniversityofSingaporeOffice:E4‐06‐08Phone:6516‐2289

Email:[email protected]~ http://www.bmchen.net

BenM.ChenProfessorofElectricalandComputerEngineering

NationalUniversityofSingaporeOffice:E4‐06‐08Phone:6516‐2289

Email:[email protected]~ http://www.bmchen.net


Whattobecoveredinthis2ndpart?

MagneticCircuitsand

Transformers

MagneticCircuitsand

Transformers

DigitalLogicCircuits

DigitalLogicCircuits

DCPowerSupply

DCPowerSupply

ElectricGeneratorElectricGenerator

DCMotorDCMotor

EG1108Part2EG1108Part2


CourseOutline

1. IntroductiontoElectricalEngineeringIntroductiontosomepracticalelectricalengineeringexamples.

2. MagneticCircuitsandTransformersPrinciplesofmutualinductanceandtransformers.

3. DCPowerSupplyDiodecharacteristics.Rectifiercircuits.Bridgerectifiers.

4. BriefIntroductiontoDCMotorsandElectricGeneratorsBasicprinciplesofoperationofDCmotorsandelectricgenerators.

5. DigitalLogicCircuitsDigitallogic.Booleanalgebra.Combinationallogic.Logiccircuitdesign.


Textbook&References

1. BasicCircuitAnalysisforElectricalEngineering

ByC.C.KoandB.M.Chen,2ndEd.,PrenticeHall,1998

2. ElectricalEngineering

ByS.Elangovan andD.Srivivasan,PrenticeHall,2005

3. PrinciplesofElectricalEngineering

ByP.Z.Peebles,Jr.andT.A.Giuma,McGrawHill,1991

4. ElectricalEngineeringPrinciplesandApplications

ByA.R.Hambley,PrenticeHall,2011

5. PrinciplesandApplicationsofElectricalEngineering

ByG.Rizzoni,McGrawHill,2007


Attendanceisessential

Anyquestionatanytimeduringthelecture

iswelcome!

Attendanceisessential

Anyquestionatanytimeduringthelecture

iswelcome!

Lectures


Tutorials

AsinPart1,therewillbe4tutorialsetsforthispart.

• Week8:Freetutoringsession.Youshouldtaketheopportunitytoclarifyanydoubtsandproblemsyouhave.

• Week9: Freetutoringsession.Youshouldtaketheopportunitytoclarifyanydoubtsandproblemsyouhave.

• Week10:Tutorial5

• Week11: Tutorial6



Youarefreetodiscussamongyourclassmatesand/ortoconsultmeoryourtutorsifyouhavemetanydifficultiesinattemptingthetutorialproblems.


OfficeHours

4:30–7:00pm

EveryThursdayuptoReadingWeek

@MyOffice

4:30–7:00pm

EveryThursdayuptoReadingWeek

@MyOffice


IntroductiontoElectricalEngineeringDisciplinesandExamples

IntroductiontoElectricalEngineeringDisciplinesandExamples


ElectricalEngineeringDisciplines…1. PowerSystems

Theoldestspecialtywithinthefielddealswiththegenerationandtransmissionofelectricityfromonelocationtoanother.

2. ElectricMachinery

Itdealswithconversionofenergytoandfromelectricalform,andstudiesthedesignandoperationofdevicessuchasmotorsandgenerators.

3. Electronics

Thiscoversstudyandapplicationofmaterials,devicesandcircuitsusedinamplifyingandswitchingelectricalsignals.

4. ComputerSystems

Theseprocessandstoreinformationindigitalform.Itincludesdesignanddevelopmentofcomputerhardwaresystemsandthecomputerprograms(software)thatcontrolthem.


ElectricalEngineeringDisciplines…

5. ControlSystems

Theseareaveryimportantclassofsystemsthatgatherinformationwithsensorsanduseelectricalenergytocontrolaphysicalprocess.

6. CommunicationsSystems

Thesesystemstransportinformationinelectricalformbyencodinginformationonanelectricalsignal.Someexamplesofsuchsystemsincludecellularphone,radio,satellitetelevision,andtheInternet.

7. InstrumentationSystems

Theyincludesensorsandinstrumentscommonlyusedinengineeringsystems.Moderninstrumentationsystemstypicallyuseelectronicamplifiersandconverters.


ElectricalEngineeringDisciplines…


Example:APassengerAutomobile…


AnElectricCircuitinanAutomobile

HeadlightSystem

ElectricCircuit


Example:UnmannedHelicopters…

HELION K-LIONK-LION

U-LIONU-LIONQ-LIONQ-LION


ActualFlightTests

mechanicalengineering?mechanicalengineering?

materialengineering?materialengineering?

engineeringandarts?engineeringandarts?


Basic PrinciplesBasic Principles

MagneticCircuits&TransformersMagneticCircuits&Transformers


ExamplesofTransformers


Inelectrostatic,anelectricfieldisformedbystaticcharges.Itisdescribedinterms

oftheelectricfieldintensity.Thepermittivity isameasureofhoweasyitisforthe

fieldtobeestablishedinamediumgiventhesamecharges.

Thepermittivityoffreespaceorvacuumpermittivityorelectricconstantisgiven

by

MagneticFieldandMaterial

120 8.8542 10 F(arad) m

Iftheinsulatorisfreespace(orapieceofpaper),theresultingcapacitance

0Q ACV d

~~


Similarly,amagneticfield isformedbymovingchargesorelectriccurrents.ItisdescribedintermsofthemagneticfluxdensityB,whichhasaunitoftesla(T)=(N/A)m.Thepermeability isameasureofhoweasyitisforamagneticfieldtobeformedinamaterial.Thehigherthe,thegreatertheB forthesamecurrents.

Infreespace, isandtherelativepermeabilityr isdefinedas

70 4 10 H m

0r

Most“non‐magnetic”materialssuchasairandwoodhavear 1.However,“magneticmaterials”suchasironmayhaver 5000.

Permeability

pureiron rμ 5,000

siliconGO steel rμ 40,000

supermalloy rμ 1,000,000


Considerthefollowingmagneticsystem:

N turns

Magnetic material

i

Permeability

Cross sectional area A

Average length l

If islarge,almosttheentiremagneticfieldwillbeconcentratedinsidethematerialandtherewillbenofluxleakage.

MagneticFlux

distributionoffluxdensity:

iTotal flux

Field linesform closed paths

N turns

Sincethefieldlinesformclosedpathsandthereisnoleakage,thetotalfluxpassingthroughanycrosssectionofthematerialisthesame.

righthandrule…

righthandrule…


FluxDensity

AssumingthefluxtobeuniformlydistributedsothatthefluxdensityB havethesame

valueovertheentirecrosssectionalareaA:

Total flux

Cross sectional area A

Same flux densityB = A

AB withunits

2mWbweberTtesla NikolaTesla

SerbianAmerican1856–1943


Ampere’sLaw

Thevaluesof orB canbecalculatedusingAmpere'slaw:

Lineintegralof alonganyclosedpath=currentenclosedbypathB

Lineintegralof alongdottedpath

AlBl

= CurrentenclosedbydottedpathNi

lNiA

B

Andre‐MarieAmpereFrench

1775–1836


Considerthemagneticcircuit:

N turns

Permeability

Area A

Average length l

i )(t

Assumingnofluxleakageanduniformfluxdistribution,thereluctanceandtheflux

linkingorenclosedbythewindingare

Al

Ni tt

and

Inductance


SideNote– Notationforabranchvoltage

Forthissecondpart,thefollowingsymbolsareidentical

Thearrowpointstothe‘positive’potentialofthecircuitelementeven

thoughitcanbephysicallynegativedependingontheactualvalueofv.

1.5 V 1.5 V+

v v≡


i )( tN= )( t

Flux linking winding

N turns

i )( t

Voltage induced

N=v )( t )( tdd t =

i )( tdd t

N 2

FromFaraday'slawofinduction,avoltagewillbeinducedinthewindingifthefluxlinkingthewindingchangesasafunctionoftime.Thisinducedvoltage,calledthebackemf(electromotiveforce)willattempttoopposethechangeandisgivenby

2Nd t di tv t N

dt dt

Faraday’sLawofInductance

2d t di tNv t Ndt dt

Anequivalentinductor:

inductance

MichaelFaradayEnglish

1791–1867


Reluctance

N2N1

Flux (t)

i1(t) i2 (t)

v2 (t)v1(t)

Primarywinding

Secondarywinding

Thetwodots are

associatedwiththe

directions ofthe

windings.Thefields

producedbythe

twowindingswill

beconstructive if

thecurrents going

intothedotshave

thesamesign.

1 1 2 2N i t N i tt

Totalflux

dt

tdiNNdt

tdiNdt

tdNtv 22112

111

dt

tdiNdt

tdiNNdt

tdNtv 222121

22

2 21 2 1 2N N N N

MutualInductor


Inductanceofprimarywindingonitsown2

11

NL

Inductanceofsecondarywindingonitsown2

22

NL

1 2 1 21 1 1 2 1

di t di t di t di tv t L L L L M

dt dt dt dt

1 2 1 22 1 2 2 2

di t di t di t di tv t L L L M L

dt dt dt dt

where iscalledthemutualinductancebetweenthetwowindings.1 2M L L

MutualInductance


ACEnvironment

Why?How?......


Why?How?......Itfollowsfromthephasortechnique

Byusingphasors,atime‐varyingacvoltage

2 cos Re 2j j tv t r t re e

becomesasimplecomplextime‐invariantnumber/voltage

jV r e r

r V magnitudeofV and =Arg[V]=phaseofV.where

Usingphasors,alltime‐varyingACquantitiesbecome

complexDCquantitiesandallDCcircuitanalysis

techniquescanbeemployedforACcircuitswithout

virtuallyanymodification!



1 21 1 1 1 1 2 2 2 2 22 cos 2 cos j ji t r t I r e i t r t I r e Let

1 1 2 21 21 1 1

1 1 1 2 2

1 1 1 2 2

2 cos 2 cos

2 sin 2 sin

2 cos 2 cos2 2

d r t d r tdi t di tv t L M L M

dt dt dt dtL r t M r t

r L t r M t

1 21 22 2 2 2

1 1 1 2 1 1 2

j j j jj jV r L e r M e r L e e r Me e

2 cos sin2 2

j

e j j

1 21 1 2 1 1 2 j jj L r e j M r e j L I j M I



Similarly,onecanderive

Thus,foranACenvironment,

2 1 2 2 V j M I j L I


2

2

1

2 2

11

2

2

Lj L Zj M

j Lj M

L LL

I

nL L

I

1 2 2 1 2 1 2 2

1 1 2 1 1 1

2

2 2

2 1 1 2 2

1 1 1 2 2

22 2

22

1

1

11

j MI L I L L I L Ij L I MI L I L L I

L L I

VV

N nN

L I

L L I L I

L NL

, turn ratio

TransformerNowconsiderconnectingamutualinductortoaloadwithimpedanceZL

if 2| |LZ j L

2 2 1 2 2 20 L LV Z I j M I j L I Z I

byKVL


equivalent!

EquivalentLoad

I1

V1

1 : n

V1n

I1n

Voltagesandcurrentsoftheprimary

andsecondarywindingsoftheideal

transformerwith

EquivalentLoad:Aloadconnected

tothesecondaryofatransformer

canbereplacedbyanequivalent

loaddirectlyconnectedtotheprimary.

I1

V1

1 : n

V1n

I1n

ZL

I1

V1ZL

n2

2 Lj L Z

1 112

1

LL

V Z InV ZI n n


Application:ImpedanceMatching

Averyimportantapplicationoftransformersisasanimpedancematchingdeviceusingtheconceptofequivalentload. Recallthatthemaximumpowertransfertheoremstatesthatapowersourcedeliversmaximumpowertotheloadwhentheloadresistanceisequaltotheinternalresistanceofthesource.Thiscanbeaccomplishedbyusingatransformertomatchthetworesistances.

Bychoosinganappropriatetransformerturnration,theeffectiveloadresistanceRL (oreffectiveloadimpedanceZL)canbemadeequaltotheinternalresistance(orimpedance)ofthesource.Suchaprocessiscalledimpedancematching.

1 : n


Example1

Asoundsystemwithaloudspeakercanberepresentedinacircuitdiagrambelow.

Iftheinternalresistanceofthesourceis75,andtheresistanceoftheloadis

300,findanappropriatetransformerturnsratio,whichresultsinimpedance

matching.

Solution:Theequivalent

loadresistanceseenbythe

source(ortheprimary

winding)isgivenby

equivalent load 2

300Rn

Tomatchitwiththesourceinternalresistance,weset

2equivalent load 2

300 30075 4 275

R n nn

therequiredturnsratio

equivalent loadR


Example2

Forthegivencircuit,

findthephasor

currentsand

voltages,andthe

power

deliveredto

theload.

1 1000R 1I 2I

1V2V1000 0 VsV 10 20LZ j

Solution:Theequivalentloadimpedanceseenbytheprimarysideisgivenby

, 22

10 20 100 10 201/10

LL equiv

Z jZ jn

Thetotalimpedanceseenbythesourceis:

1 ,

1000 1000 2000 2828 45total L equivZ R Z

j

,L equivZsV

1 1000R

1V

1I

imim

rerexx

yy

z = x + j yz = x + j y

im

rex

y

z = x + j y

tan tany yθ θx x

1

tan=

yjxx y e

12 2

1: n


,L equivZsV

1 1000R

1V

1I

Example2(cont.)

Theprimarysidecurrentandvoltagecanbe

calculatedas:

and

11000 0 0.3536 45 A2828 45

s

total

VIZ

1 1 , 0.3536 45 (1000 2000) 790.6 18.43 VL equivV I Z j

Thesecondarysidecurrentandvoltagecanbecalculatedas:

12

0.3536 45 3.536 45 A1/10

IIn

2 11 790.6 18.43 79.06 18.43 V

10V nV

Thepowerdeliveredtotheload: 2 22 3.536 10 125 WL LP I R


TransformersUsedinPowerTransmission

Transformersusedtoraise/lowervoltagesinelectricalenergydistribution


Application2:PowerSupply

TransformerusedinDCpowersupply


DCPowerSupply

RectifierCircuits

DCPowerSupply

RectifierCircuits


ExamplesofDCPowerSupply


LearningObjectives

Themainlearningobjectivesforthistopicareasfollows:

1. Tounderstandvoltage‐currentcharacteristicsofadiode

2. Tounderstandoperationofhalf‐waveandfull‐waverectifiercircuits

3. Todeterminationofoutputvoltagesandcurrents

4. Toanalyzeoperationofrectifiercircuitwithcapacitorfilter

5. Tocalculatepeakinversevoltagesforrectifiercircuits


DiodeDiodesallowelectricitytoflowinonlyonedirection.Diodesaretheelectricalversionofavalveandearlydiodeswereactuallycalledvalves.Theschematicsymbolofadiodeisshownbelow.Thearrowofthecircuitsymbolshowsthedirectioninwhichthecurrentcanflow.Thediodehastwoterminals,acathodeandananodeasshowninthefigures.

Characteristicsofanidealdiode

─


CharacteristicsofSiliconDiodes

Theactualcharacteristicsofasilicondiodeisslightlydifferentfromtheidealone.Inparticular,ifthevoltagevD ismorenegativethantheReverseBreakdownvoltage,thediodeconductsagain,butinareversedirection.

─

─


DiodeRectifierCircuits

OneoftheimportantapplicationsofasemiconductordiodeisinrectificationofACsignalstoDC.DiodesareverycommonlyusedforobtainingDCvoltagesuppliesfromthereadilyavailableACvoltage.

Therearemanypossiblewaystodesignrectifiercircuitsusingdiodes,e.g.,

• HalfWaveRectifier

• FullWaveRectifier

• BridgeRectifier

XX


SimpleHalfWaveRectifier

─ TheACpowersupplyisgivenby

1sin sin 2 sin 2s m m mv V t V f t V tT

WeareinterestedinobtainingDCvoltage

acrossthe“loadresistance”RL.

Duringthepositivehalfcycle,the

diodeison.Thesourcevoltageis

directlyconnectedacrosstheload.

Duringthenegativehalfcycle,the

diodeisoff.Thesourcevoltageis

disconnectedfromtheload.The

waveformsforvs andvo areshown

infigureontheright…

¤ vo variesbetweenthepeakVm and0ineachcycle.Thisvariationiscalledripple,andthecorrespondingvoltageiscalledthepeak‐to‐peakripplevoltage,Vp‐p.


AverageLoadVoltage&Current

IfaDCvoltmeterisconnectedtomeasuretheoutputvoltageofthehalf‐waverectifier

(i.e.,acrosstheloadresistance),thereadingobtainedwouldbetheaverageload

voltageVave,alsocalledtheDCoutputvoltage.The meteraveragesoutthepulsesand

displaysthefollowingaverage:

Averageloadcurrent:

m

L

VR

¤ Trulyspeaking,the

outputvoltage&current

arenotso‘DC’…

12 2fT

/ 2

0 0

1 1( ) sin 0 cos 0 cos cos 0 cos2 2

T Tm m m

ave o mV V VTV v t dt V t dt

T T T


ReverseVoltageoftheDiode

─

Dv

Clearly,themaximumreversevoltageacrossthediodeisVm.Notethatthediodewill

functionproperlysolongasVm issmallerthanitsreversebreakdownvoltage(i.e.,Vz).

ByKVL: S D O D S Ov vv vv v


Example1

A50Ωloadresistanceisconnectedacrossahalfwaverectifier.Theinputsupply

voltageis230V(rms)at50Hz.DeterminetheDCoutput(average)voltage,peak‐to‐

peakrippleintheoutputvoltage(Vp‐p),andtheoutputripplefrequency(fr).

Solution:Peakamplitudeofthesourcevoltageis:

OutputDCvoltage:

Peak‐to‐peakripplevoltageisthedifferencebetweenmaxandmininvo.

Percentageripple=(Vp‐p/Vave)× 100=314%

Theripplefrequencyisalso50Hz,i.e.,fr=50 Hz.

230 2 325 V.3mV

325.3 V3.14

103.5avemVV

- max min 3 V.30 25p p mV V V V

Notethatthepeak‐to‐peakrippleandthepercentageripplevaluearethemeasureofthesmoothnessoftheoutput‘DC’voltage…


HalfWaveRectifierwithCapacitor

WehaveobservedthatthesimplehalfwaverectifierdoesnotreallyprovidesDCvoltageandcurrent.Togetabetterloadvoltageandcurrent,weaddanadditionalcapacitortothecircuit…

positivehalfcycle

negativehalfcycle

Outputloadvoltage

¤ Theoutputvoltageispretty‘DC’…


OperationalAnalysisDuringeachpositivehalfcycle,thecapacitorchargesduringtheintervalt1 tot2.Duringthisinterval,thediodewillbeON.Duetothischarging,thevoltageacrossthecapacitorvowillbeequaltotheACpeakvoltageVm att2.

ThecapacitorwillsupplycurrenttoloadresistorRLduringtimeintervalt2 tot3.Duringthisinterval,diodewillOFFsincetheACvoltageislessthanvo.Duetothelargeenergystoredinthecapacitor,thecapacitorvoltagewillnotreducemuchduringt2 tot3,andvowillremainclosetothepeakvalue.

discharging

2ndv+

–

+ –

ov

1 2nd o Dv v v

2ndv

ov


ReverseVoltageoftheDiode

2ndv

ov

ov

+

+

–

–

2ndv+

–ov

ByKVL:

1 12nd o 2nd oDD v v vv v v

m2V

0V1Dv

Thepeakvoltageacrossthediodeis

1 m m m2Dv V V V

mV

mV

Themaximumreversediodevoltageisapproximately2Vm.

mV



Inpracticalapplications,averylargecapacitorisusedsothattheoutputvoltageis

closetothepeakvalue.Theaveragevoltage(alsocalledDCoutputvoltage)across

theloadcanthereforebeapproximatedto:

Theaverageloadcurrentisthengivenby

ave mV V

mL

L

VIR


CapacitorandCapacitance

Acapacitor isanelectricalcomponentusedtostoreenergyinanelectricfield.Theformsofpracticalcapacitorsvary,butallcontainatleasttwoelectricalconductorsseparatedbyadielectric(insulator).Thecapacitanceisdefinedas

Forgeneralsituations,wehavecapacitorvoltage

QCV

( )( ) q tv tC

Notingthat(thedefinitionofelectriccurrent)

( )( ) dq ti tdt

wehave

( ) 1 ( ) 1 ( )( ) ( )dv t dq t dv ti t i t Cdt C dt C dt

QVC

QIT


ThevoltagewaveformsshowasmallACcomponentcalled“ripple”presentintheoutputvoltage.Thisripplecanbeminimizedbychoosingthelargestcapacitancevaluethatispractical.Wecancalculatetherequiredcapacitance asfollows.

Thechargeremovedfromthecapacitorduringthedischargecycle(i.e.,t2 tot3)is:

whereIL istheaverageloadcurrentandT istheperiodoftheACsource.Astheintervalt1 tot2 isverysmall,thedischargetimecanbeapproximatedbyT.

IfVp‐p isthepeak‐to‐peakripplevoltageandC1 isthecapacitance,Thechargeremovedfromthecapacitorcanalsobeexpressedas

Requiredcapacitanceisgivenby

Peak‐to‐PeakRippleVoltage&RequiredCapacitance

m1

- -

F (Farads)L

p p L p p

I T V TCV R V

LQ I T

- 1 -1

p p p pQQ V C V

C

QIT

QVC


Duringthecapacitordischargingperiod(thegreen curves),thegreen portionofthecircuitbelowisgovernedby

1( ) ( ) 0o

L odv tR C v t

dt

ov

1( )odv tC

dt

1( )o

Ldv tR C

dt

AnAlternativeWayforFinding RequiredCapacitance

AsthedischargetimecanbeapproximatedbyT,wehave

1m m -( ) L

TR

o pC

pv T V V Ve

1m m -1 p

Lp

TR C

V V V

1m( ) L

tR C

ov t V e

TransientResponse…

m-

1p p

L

V T VR C

m1

- -

L

p p L p p

I T V T CV R V

21 2!

x xe x

RequiredCapacitance

ov

1( )

odv tCdt


InthecircuitofExample1,a10000μFfiltercapacitorisaddedacrosstheloadresistor.Thevoltageacrossthesecondaryterminalsofthetransformeris230V(rms).DeterminetheDCoutputvoltage(i.e.averagevoltage),loadcurrent,peak‐to‐peakrippleintheoutputvoltage,andtheoutputripplefrequency.

Solution:OutputDCvoltage:

Averageloadcurrent:.Thiscurrentdischargesthecapacitor

duringtheintervalt2 tot3.

ThetimeperiodoftheACsource=20ms(forfrequencyof50Hz).Thus,thepeak‐to‐

peakrippleintheoutputvoltageisgivenby

Theripplevoltageisonly4%(=13.02/325.3)now,itsfrequencyremainsat50Hz.

Example2

25.3 V3maveV V

325.3 6.51 A50L

ave

L

VR

I

3

61

m1 -

- -

6.51 20 10 V10000 10

13.02

L

p pp p L

L

p p

I T V TV

IC VCR

TV


FullWaveRectifier

Whatistheshortagewiththehalf‐waverectifier?

??

?? ?? ??


Solution…

Thefullwaverectifier…

Circuitoperationduringthepositivehalfcycle…

Circuitoperationduringthe

negativehalfcycle



2 mave

VV

2ave mL

L L

V VIR R


Example3

Thetransformerinthefull‐waverectifiercircuithasaturnsratioof1:2.ItsprimarywindingisconnectedacrossanACsourceof230V(rms),50Hz.Theloadresistoris50Ω.DeterminetheDCoutputvoltage,peak‐to‐peakrippleintheoutputvoltage,andoutputripplefrequency.

Solution: Thermsvalueofthesecondaryvoltage=460 V.Thus,thermsvalueofv2(andv3)=230V.Thepeakvalueofv2 (andv3):

DCoutputvoltage(averageloadvoltage):

Peak‐to‐peakripplevoltage:.Ripplefrequency=100Hz(why?)

2 230 325. V3mV

2 V207mave

VV

- 0 V325.3mp pV V

¤ Therippleisstillverylarge,butitspercentageisreducedto157%...


Full WaveRectifierwithCapacitor

Therequiredcapacitancecanbefoundtobe(halfofthevaluerequiredforthehalf‐waverectifier):

-

F 2

L

p p

I TCV


Equivalentcircuitforpositivehalfcycle:(WhatisthereversevoltageofD2 andD4?)

FullWaveBridgeRectifier

Circuitoperationduringthepositivehalfcycle…

== D2 D4


OperationalAnalysis

Circuitoperationduringthenegativehalfcycle…

Equivalentcircuitfornegativehalfcycle:(WhatisthereversevoltageofD1 andD3?)

== D1 D3


Asusual,inordertohaveasmootheroutputvoltage,wecanaddacapacitor…

FullWaveBridgeRectifierwithCapacitor

D1 &D3 on D2 &D4 on

vo


Video:HowtoMakeaRealDCPowerSupply?


DCMotorsandElectricGeneratorsAVeryBriefIntroduction

DCMotorsandElectricGeneratorsAVeryBriefIntroduction


ElectromechanicalEnergyConversion

ElectricMotor

ElectricGenerator


ComponentsofaDCMotor

DCmotorsconsistofonesetofcoils,thearmaturewinding,insideanothersetofcoilsorasetofpermanentmagnets,calledthestator.Applyingavoltagetothecoilsproducesatorqueinthearmature,resultinginmotion.

Stator

• Thestatoristhestationaryoutsidepartofamotor.• Thestatorofapermanentmagnetdcmotoriscomposedoftwoormorepermanentmagnetpolepieces.

• Themagneticfieldcanalternativelybecreatedbyanelectromagnet.Inthiscase,aDCcoiliswoundaroundamagneticmaterialthatformspartofthestator.

Rotor

• Therotoristheinnerpartwhichrotates.• Therotoriscomposedofwindings(calledarmaturewindings)whichareconnectedtotheexternalcircuitthroughamechanicalcommutator.

• Bothstatorandrotoraremadeofferromagneticmaterials.


DCMotorBasicPrinciplesIfelectricenergyissuppliedtoaconductorlyingperpendiculartoamagneticfield,theinteractionofcurrentflowingintheconductorandthemagneticfieldwillproducemagneticforce,whichattemptstomovetheconductorinadirectionperpendiculartothemagneticfield,andisgivenby

F= B I L(Newton)

Fleming’sLeftHandRule

B :magneticfluxdensity;L :conductorlengthI :currentflowingintheconductor

JohnA.FlemingEnglish

1849–1945


DCMotorOperationalPrinciple

AnactualDCmotormightcontainmanyturnsofarmaturewindingsinitsrotor…


ElectricGeneratorsAnelectricgenerator isadeviceconvertingmechanicalenergytoelectricalenergy.Electricmotorsandgeneratorshavemanysimilarities.Infact,manymotorscanbemechanicallydriventogenerateelectricity…

ElectricMotor

ElectricGeneratorF B I L

F :theforce;B :fluxdensityL :conductorlength

I :currentflowingintheconductor


Video:MotorsandGenerators


ExamplesofElectricGenerators

Thesourceofmechanicalenergyusedinrealsituationsforgeneratingelectricitymaybeareciprocatingorturbinesteamengine,waterfallingthroughaturbineorwaterwheel,aninternalcombustionengine,awindturbine,ahandcrank,compressedairoranyothersourceofmechanicalenergy.

powerplantandsteamturbinegenerator

hydropowerandwaterdam

windpower


DigitalLogicCircuitsDigitalLogicCircuits


Introduction

Manyscientific,industrialandcommercialadvanceshavebeenmadepossiblebythe

adventofcomputers.DigitalLogicCircuitsformthebasisofanydigitalsystem,such

ascomputers,laptopsandsmartphones.Inthistopic,wewillstudytheessential

featuresofdigitallogiccircuits,whichareattheheartofdigitalcomputers.

DigitalLogicCircuitsmaybesubdividedintoCombinationalLogicCircuitsand

SequentialLogicCircuits.InEG1108,ourfocuswillbeonCombinationalLogic

Circuits.Thesecircuitscanbeeasilyanalyzed/designedusingBooleanAlgebra,which

isthemathematicsassociatedwithbinarysystems.

WewillseehowCombinationalLogicCircuitscanbedesignedandusedfor

interestingpracticalproblems.CircuitminimizationtechniquessuchasKarnaugh

mapsforsimplificationofcombinatoriallogiccircuitswillalsobecovered.


SomeDevicesInvolvedDigitalLogicCircuits

DigitalWatchDigitalWatch PersonalComputerPersonalComputer

SuperComputerSuperComputer

TrafficLightTrafficLight

AircraftAircraft

SmartPhonesSmartPhones


LearningObjectives

Themainlearningobjectivesforthistopicareasfollows:

1. Tounderstandbasicterminology,typesoflogicgates

2. Tounderstandthebasicoperationsusedincomputersandother

digitalsystems

3. TostudybasicrulesofBooleanalgebra,DeMorgan’slaws

4. TostudyKarnaughmapsforcircuitminimization


Implementthelogicalexpressionusinglogicgates

Determinealogicalexpressioncharacterizingtheinput‐output

relationship

Obtainarelationshipbetweeninputandoutputvariables

Identifynecessaryinputandoutputvariables

ProductorDesignSpecificationsExample: DesignanLEDpaneltodisplay

LettersE orCExample: DesignanLEDpaneltodisplay

LettersE orC

TwoLEDbarsasshowninRedandBlackonleftaresufficientTwoLEDbarsasshowninRedandBlackonleftaresufficient

TodisplayLetterE,bothR andB arerequiredtobeon;

TodisplayLetterC,R isrequiredtobeonandB istobeoff

TodisplayLetterE,bothR andB arerequiredtobeon;

TodisplayLetterC,R isrequiredtobeonandB istobeoff

E= R·B;C= R·E= R·B;C= R·B

FlowchartforDesigningaLogicCircuit


FlowchartforDesigningaLogicCircuit(cont.)



Determinealogicalexpressioncharacterizingtheinput‐outputrelationship

Determinealogicalexpressioncharacterizingtheinput‐outputrelationship





ProductorDesignSpecificationsProductorDesignSpecifications

ActualDesignProcedure…

TopicsCoveredinLectures…TopicsCoveredinLectures…

K‐mapsimplificationK‐mapsimplification

SOP/POSexpressionsSOP/POSexpressions

BasiclogicgatesBasiclogicgates

BooleanalgebraBooleanalgebra

TruthtablesTruthtables


NumberSystemsAnumbersystemisanorderedsetofsymbols(digits)withrelationshipsdefinedforaddition,subtraction,multiplicationanddivision.Thebaseofthenumbersystemisthetotalnumberofdigitsinthesystem.Forexample,fortheusualdecimal system,thesetofdigitsis{0,1,2,3,4,5,6,7,8,9} andhencethebaseisten (B =10).Inthebinary system,thesetofdigits(bits)is{0, 1}andhencethebaseistwo (B =2).

Therearetwopossiblewaysofwritinganumberinagivensystem:positionalnotationandthepolynomialrepresentation.

PositionalNotation:AnumberN canbewritteninpositionalnotationasfollows:

PolynomialRepresentation:Theabovenumbercanalsobewrittenasapolynomialoftheform

Forexample, .

n n m BN b b b b b b b b N1 2 2 1 0 1 2 10

forexample 2536.47

1 2 2 1 0 1 21 2 2 1 0 1 2

n n mn n mN b B b B b B b B b B b B b B b B

3 2 1 0 1 210

2536.47 2 10 5 10 3 10 6 10 4 10 7 10N


Binary‐NumberSystem

Thebinarynumberisabase2systemwithtwodistinctdigits(bits),1and0.Itisexpressedasastringof0sand1sandabinarypoint,ifafractionexists.Toconvertfromthebinarytodecimalsystem,expressthebinarynumberinthepolynomialformandevaluatethispolynomialbyusingdecimal‐systemaddition.

Thefollowingaremappingsofsomebinarynumberstotheirdecimalcounterparts:

Similarly,

000 = 0 22 +0 21 +0 20 = 0001 = 0 22 +0 21 +1 20 = 1

111 = 1 22 +1 21 +1 20 = 7

010 = 0 22 +1 21 +0 20 = 2011 = 0 22 +1 21 +1 20 = 3100 = 1 22 +0 21 +0 20 = 4101 = 1 22 +0 21 +1 20 = 5110 = 1 22 +1 21 +0 20 = 6

Binary Polynomial Representation Decimal

1000 = 81001 = 9

1111 = 15

1010 = 101011 = 111100 = 121101 = 131110 = 14

Binary Decimal


Booleanalgebra isthemathematicsofdigitallogicandisparticularlyusefulindealingwiththebinary‐numbersystem.Booleanalgebraisusedintheanalysisandsynthesisoflogicalexpressions.Logicalexpressionsareconstructedusinglogical‐variablesandoperators.Thevalueofanylogicalexpressionboilsdowntoanyoneofthetwologicalconstants:true and false.

InBooleanalgebra,alogicalvariableXmaytakeanyoneofthetwopossiblevalues1and0.X =1andX =0,whichmayrepresentrespectively

• truth orfalsehood ofastatement• on oroff statesofaswitch• high orlow ofavoltagelevel

A logicalexpressionisafinitecombinationoflogicalvariables(whicharealsocalledinput variables)thatarewell‐formedaccordingtotherulesofBooleanalgebra.Thevalueofthelogicalexpressionisreferredtoastheoutput variable(whichitselfisalsoalogicalvariable).

BooleanAlgebraandLogicGates

GeorgeBooleEnglish

1815–1864


TruthTables

Atruthtable isamathematicaltableusedinlogictocomputethefunctionalvaluesoflogicalexpressionsoneachoftheirfunctionalarguments,thatis,oneachcombinationofvaluestakenbytheirlogicalvariables.Inparticular,truthtablescanbeusedtotellwhetherapropositionalexpression(output)istrueorfalseforallpossiblecombinationsofinputvalues.Thefollowingaretheexamples ofthetruthtablesfor1,2,3and4inputvariables,respectively:

Input Output

A Z0 11 0

Input Variables Output

A B W0 0 0 11 0 1 12 1 0 13 1 1 0

A B C R0 0 0 0 11 0 0 1 02 0 1 0 13 0 1 1 04 1 0 0 15 1 0 1 06 1 1 0 07 1 1 1 1

A B C D Y0 0 0 0 0 11 0 0 0 1 12 0 0 1 0 13 0 0 1 1 04 0 1 0 0 15 0 1 0 1 06 0 1 1 0 07 0 1 1 1 08 1 0 0 0 19 1 0 0 1 010 1 0 1 0 011 1 0 1 1 012 1 1 0 0 013 1 1 0 1 014 1 1 1 0 015 1 1 1 1 0


LogicCircuits

Electricalcircuitsdesignedtorepresentlogicalexpressionsarepopularlyknownaslogiccircuits.Suchcircuitsareextensivelyusedinindustrialprocesses,householdappliances,computers,communicationdevices,trafficsignalsandmicroprocessorstomakeimportantlogicaldecisions.LogiccircuitsareusuallyrepresentedbylogicoperationsinvolvingBooleanvariables.

Therearethreebasiclogicoperationsaslistedbelow:

• NOT operation• AND operation• OR operation

WewillillustratethesebasiclogicaloperationsinthefollowingsectionsusingBooleanvariablesAandB.Alogicgateisanelectroniccircuit/devicewhichmakesthelogicaldecisionsbasedontheseoperations.Logicgateshaveoneormoreinputsandonlyoneoutput.Theoutputisactiveonlyforcertaininputcombinations.Logicgatesarethebuildingblocksofanydigitalcircuit.Logicgatesarealsocalledswitches.


NOTOperation

NOT operationisrepresentedby

C=

TheNOTgatehasonlyoneinputwhichistheninvertedbythegate.Here,isthe

'complement'ofA.Thesymbolandtruthtablefortheoperationareshownbelow:

A

A

TruthTableforNOTGate

A C

SwitchCircuitforaNOTGate


ANDOperation

AND operationisrepresentedbyC=A• B

ItsassociatedTRUTHTABLEisshownbelow.Atruthtablegivesthevalueofoutputvariable(hereC)forallcombinationsofinputvariablevalues(hereAandB).ThusinanAND operation,theoutputwillbe1 (True)onlyifalloftheinputsare1 (True).

TruthTableforANDGate

Thefollowingrelationshipscanbeeasilyderived:

A• A=A 1• A=A 0• A=0

A• Ā=0 A• B=B• A

Note:The• signcanbeomittedwhenindicatinganANDoperation.Thus,C=A•BandC=ABmeanthesameoperation.

A

C

B


OROperation

OR operationisrepresentedbyC=A+B

HereA,BandCarelogical(Boolean)variablesandthe+signrepresentsthelogicaladdition,calledanORoperation.Thesymbolfortheoperation(calledanORgate)isshownbelow.ItsassociatedTRUTHTABLEisshownbelow.ThusinanORoperation,theoutputwillbe1 (True)ifeitheroftheinputsis1 (True).Ifbothinputsare0(False),onlythentheoutputwillbe0 (False).Noticethatthoughthesymbol+isused,thelogicaladditiondescribedabovedoesnotfollowtherulesofnormalarithmeticaddition.

TruthTableforORGate

Thefollowingrelationshipscanbeeasilyderived:

A+Ā=1 A+A=A 0+A=A1+A=1

A

C

B

SwitchCircuitforanORGate


MultivariableLogicGatesManylogicgatescanbeimplementedwithmorethantwoinputs,andforreasonsofspaceincircuits,usuallymultipleinput,complexgatesaremade.

A

Z

B C A

B

CZ

3‐InputORGateTruthTable

3‐InputANDGateTruthTable

Z=ABC Z=A+B+C

A•(B•C)=(A•B)•C=A•B•C (A+B)+C=A+(B+C)=A+B+C


NAND Gate

WecouldcombineANDandNOToperationstogethertoformaNANDgate.Thusthe

logicalexpressionforaNANDgateis.

Thesymbolandtruthtablearegiveninthefollowingfigure.TheNANDgatesymbol

isgivenbyanANDgatesymbolwithacircleattheoutputtoindicatetheinverting

operation.

C = A B

TruthTableforNANDGate


NOR Gate

Similarly,ORandNOTgatescouldbecombinedtoformaNORgate.

C = A + B

TruthTableforNORGate


RealizationofLogicGatesbyIdealTransistors⋆

Logicgatescanberealizedusingidealtransistors,whichhavetheproperties:

x

x

V=VO +VR VO =VwhenVR=0,i.e.,I=0

V

VR

VO

I

VVO

VR I

A B

A B

A+B

OV A+B


PropertiesofPracticalTransistors⋆

transientresponse

x


ElementsofBooleanAlgebraAsymbolicbinarylogicexpressionconsistsofbinaryvariablesandtheoperatorsAND,ORandNOT(e.g.A+B· ).ThepossiblevaluesforanyBooleanexpressioncanbetabulatedinatruthtable.

BooleanalgebraexpressionscanbeimplementedbyinterconnectionofANDgates,ORgates,andinverters.

C

Ascanbeseen,thenumberofsimplegatesneededtoimplementanexpressionisequaltothenumberofoperationsintheBooleanexpression.WecouldusetherulesofBooleanAlgebraorKarnaughMapstosimplifyagivenBooleanexpression.Thiswouldallowthegivenexpressiontobeimplementedusinglessnumberofgates.


BasicLawsofBooleanAlgebra

SomeofthebasicrulesofBoolean

algebrathatmaybeusedto

simplifytheBooleanexpressions

areshownontheright.These

rulesmaybeprovedusingthe

truthtables.Essentially,we

considerallcombinationsofinputs

andshowthatinallcasestheLHS

expressionandRHSexpression

leadtothesameresult.Sucha

methodofprovinglogical

equationsisknownasproofby

perfectinduction.

Rules:


Example1

Showbyperfectinductionthat[SeeRule18].A A B A B

Proof: LetuscreatethefollowingtableandshowtheLHS=RHSforallthevaluesof

AandB.

identical


DeMorgan’sLaws

TheselawsareveryusefulinsimplifyingBooleanexpressions.AccordingtoDeMorgan'stheorem,wehave

NoticethattheDeMorgan'sLawsgivethelinkbetweentheORoperationandtheANDoperation.ApplicationofDeMorgan'stheoremmakesiteasytodesignlogiccircuitsusingNANDandNORlogicgateswhichwewillsoonsee.

Becauseoftheaboverelationships,anylogicalfunctioncanbeimplementedbyusingonly(i)ANDandNOTgatesor(ii)ORandNOTgates.Thus,anORgatecanbeimplementedwithANDandNOTgatesasshownontheright.

A B A B

A B A B

A B A B A B

A B A B A B

A B C A B C

A B C A B C

AugustusDeMorganEnglish

1806–1871


SomeSideNotes onBooleanAlgebra

InBooleanalgebra,theequalityofA+B=A+CdoesnotonlyimplythatB=C.

Thiscanbeshownbyaspecifictheexample:LetA=1,B=1andC=0.Then,wehave

A+B=A+C=1

Obviously,B C.

Similarly,inBooleanalgebra,theequalityofA B=A CdoesnotonlyimplythatB=Ceither.

ThiscanbeshownbylettingA=0,B=1andC=0.Then,wehave

A B=A C=0

Again,itisobviousthatB C.

Lastly,wenotethatA+B C (A+B) C.

Note:InBooleanalgebra,wedonotdefinelogical‘subtraction’and‘division’.


UniversalityofNANDandNORGates

UniversalityofNANDGates

NANDgatescanbeusedto

implementthefunctionsofa

NOTgate,anANDgate,and

anORgate,asillustrated

fromthefiguresontheright.

Thus,anygivenlogic

functioncanbeimplemented

byusingNANDgatesalone.

Forthisreason,NANDgateis

saidtobelogicallycomplete.

A B A B

A B A B

A

A

A

AB

B

A A A A A

A BA B


Example2

ImplementusingonlyNANDgates.Z A B C D

Solution:

CircuitimplementationusingNANDgates:

Z A B C D A B C D A B C D

A B A B

A B A B


UniversalityofNORGates

Likewise,itcanbeshownthat

anylogicfunctioncanbe

implementedusingNORgates

alone.ThisissobecauseNOR

gatescanbeusedtoimplement

thefunctionsofaNOTgate,an

ANDgateandanORgate,as

showninthefiguresonthe

right.

ANORgateisfunctionally

completebecauseAND,OR,and

NOTgatescanbeimplemented

usingNORgatesalone.

A B A B

A B A B

A A A

A BA B


Example3

ImplementusingonlyNORgates.Z A B C D

Solution:

CircuitimplementationusingNORgates:

Z A B C D A B C D A B C D

A B A B

A B A B


ExclusiveORandExclusiveNORGates

ExclusiveORoperationisdefinedas

C A B

TruthTableforexclusiveORGate

(ThisistobeproveninExample5)

A B A B

ExclusiveNORoperationisdefinedas

C A B

TruthTableforexclusiveNORGate

( ) ( )

( ) ( )

A B A B

A B A B

A B A B


Example4

FigurebelowshowsalogicalcircuitthatmaybeusedtoachieveexclusiveORoperation.DeterminetheoutputC.

Solution: Theoutputofeachgateis

labeleddirectlyinthefigure.Itis

easytoseethat

A B

A B

( ) ( )C A B A B

TruthTableforORGate TruthTableforNANDGate TruthTableforexclusiveORGate

A B

?

checked

·· · ·· · ·· · ·· ·

X Y


Example5

SimplifytheexpressionforCinExample4anddrawanequivalentlogicalcircuitforexclusiveORoperation.

Solution:

( ) ( ) ( ) ( )

0 0

A B A B A B A B

A A A B B A B B

A B B A

C A B

A B B A

A B A B

A B A B

A B




relationship



ProductorDesignSpecificationsAmajoritydetector

hasfourinput

variablesA,B,C

andD andthree

outputlight

indicators.Redlight

willbeonif

majorityofthe

inputvariablesare

equalto0.Design

anappropriatelogic

circuitforthis

application.

DesignProcedure…

LogicCircuitDesign– AnExample

truetabletruetable

??


LogicCircuitDesign– AnExampleConstructa correspondingtruthtablefortheredlightoutput,forexample…

Inputs

A B C D0 0 0 00 0 0 10 0 1 00 0 1 10 1 0 00 1 0 10 1 1 00 1 1 11 0 0 01 0 0 11 0 1 01 0 1 11 1 0 01 1 0 11 1 1 01 1 1 1

# 0s>#1s

# 0s>#1s

# 0s>#1s

{Output

R1110100010000000


LogicCircuitDesignIndesigningdigitalcircuits,thedesigneroftenbeginswithatruthtabledescribing

whatthecircuitshoulddo.Thedesigntaskistodeterminewhattypeofcircuit

willperformthefunctiondescribedinthetruthtable.Although,itmaynotalways

beobviouswhatkindoflogiccircuitwouldsatisfythetruthtable,twosimple

methodsfordesigningsuchacircuitarefoundinstandardformofBoolean

expressioncalledtheSum‐Of‐Products(orSOP)formandProduct‐Of‐Sums (or

POS)forms.

Basedonthedescriptionoftheproblem,naturallanguageisfirsttranslatedintoa

truthtableandBooleanexpressionsarefoundmethodicallyusingoneofthese

twomethods.TheBooleanexpressionisthensimplifiedusingrulesofBoolean

algebraorKarnaughMaps(whichwewillstudylater),sothatitcanbe

implementedusingminimumnumberoflogicgatesforpracticalimplementation.


Sum‐of‐ProductsImplementation

Asyoumightsuspect,aSum‐Of‐ProductsBooleanexpressionisliterallyasetofBooleantermsadded(summed)together,eachtermbeingamultiplicative(product)combinationofBooleanvariables:

{sum‐of‐products‐expression} = {productterm} + + {productterm}

Producttermsthatincludealloftheinputvariables(ortheirinverses)arecalledminterms.Inasum‐of‐productsexpression,weformaproductofalltheinputvariables(ortheirinverses)foreachrowofthetruthtableforwhichtheresultislogic1.Theoutputisthelogical“sum”ofthesemintermsSum‐Of‐ProductsexpressionsareeasytogeneratefromtruthtablesasshowninExample6,bydeterminingwhichrowsofthetablehaveanoutputof1,writingoneproducttermforeachrow,andfinallysummingalltheproductterms.ThiscreatesaBooleanexpressionrepresentingthetruthtableasawhole.

Sum‐Of‐ProductsexpressionslendthemselveswelltoimplementationasasetofANDgates(products)feedingintoasingleORgate(sum).


Example6.a

ObtainC fortheexclusiveORoperationfromthetruthtablebelowintheSumofProducts(SOP)form.

A BA B

Thistermhasavalueof1ifandonlyifThistermhasavalueof1ifandonlyifThistermhasavalueof1ifandonlyif 1, 1 A B

Thistermhasavalueof1ifandonlyifThistermhasavalueof1ifandonlyifThistermhasavalueof1ifandonlyif 1, 1 A B

C X Y A B A B

X Y

TheoutputvariableC istrueifandonlyifX istrueorY istrueTheoutputvariableC istrueifandonlyifX istrueorY istrue

Solution:……

XY

WhentheoutputC istrue?


Example6

ObtainWfromthetruthtableintheSumofProducts(SOP)form.Drawthelogicalcircuittoimplementit.

Solution:

A B C

A B C

W A B C A B C A B C A B C A B C

A B C

A B C

A B C

Thistermwouldhaveavalueof1ifandonlyifThistermwouldhaveavalueof1ifandonlyifThistermwouldhaveavalueof1ifandonlyif 1, 1, 1 A B C






W A B C A B C A B C A B C A B C

Example6(cont.)


Example7

SimplifythelogicalexpressionWobtainedinExample6andshowitsimplementation.

Solution:TheBooleanexpressioncanbesimplifiedasfollows

A B C A B C

A B C A

A B C

A B C

A B C

A B C B A A

W A B C A B C

A B C A

C B A C

B C

A B C C

A B

B

B C

A A B C

B

C B A B C

C B C B C

Rule18A A B A B

Rule18

Implementation

ofWafter

simplification:


Product‐of‐SumsImplementation

AnalternativetogeneratingaSum‐Of‐Productsexpressiontoaccountforallthe

“high”(1)outputconditionsinthetruthtableistogenerateaProduct‐Of‐Sums,or

POS,expression,toaccountforallthe“low”(0)outputconditionsinstead.POS

Booleanexpressionscanbegeneratedfromtruthtablesquiteeasily,bydetermining

whichrowsofthetablehaveanoutputof0,writingonesumtermforeachrow,and

finallymultiplyingallthesumterms.ThiscreatesaBooleanexpressionrepresenting

thetruthtableasawhole.

{product‐of‐sums‐expression} = {sumterm}· · {sumterm}

These“sum”termsthatincludealloftheinputvariables(ortheirinverses)are

calledmaxterms.ForPOSimplementation,theoutputvariableisthelogicalproduct

ofmaxterms.Product‐Of‐Sumsexpressionslendthemselveswelltoimplementation

asasetofORgates(sums)feedingintoasingleANDgate(product).


Z

Example8.a

FindZintermsofA,BandCinproduct‐of‐sum(POS)

formfromthefollowingtruthtable.

Solution:WefindSOPfor…

A B C

A B C

A B C

Z A B C A B C A B C

Z

Z Z A B C A B C A B C

A B C A B C A B C

A B C A B C A B C

A B C A B C

A B C A B C

POSformPOSform


Example8

FindZintermsofA,BandCinproduct‐of‐sum(POS)formfromthefollowingtruth

table.Solution:Alternatively,welookfor0 inthetruthtable…

( )A B C

( )A B C

( )A B C

Z A B C A B C A B C

Thistermwouldhaveavalueof0ifandonlyifThistermwouldhaveavalueof0ifandonlyifThistermwouldhaveavalueof0ifandonlyif 0, 0, 0A B C



ComparethiswiththatinExample8.a


KarnaughMapsFromthepreviousexampleswecanseethatrulesofBooleanalgebracanbeappliedinordertosimplifyexpressions.Apartfrombeinglaborious(andrequiringustorememberallthelaws)thismethodcanleadtosolutionswhich,thoughtheyappearminimal,arenot.TheKarnaughmap (orKmap)providesasimpleandstraightforwardmethodofminimizingBooleanexpressions.WiththeKmapBooleanexpressionshavinguptofourandevensixvariablescanbesimplifiedeasily.

AKmapprovidesapictorialmethodofgroupingtogetherexpressionswithcommonfactorsandthereforeeliminatingunwantedvariables.Thevaluesinsidethesquaresarecopiedfromtheoutputcolumnofthetruthtable,thereforethereisonesquareinthemapforeveryrowinthetruthtable.AroundtheedgeoftheKmaparethevaluesofthetwoinputvariable.

MauriceKarnaughAmerican1924–

Thesimplifiedlogicalexpressionisthenusedsothatminimumhardwareisutilizedintheimplementationoflogicalcircuits.


SimplificationProcess

1. Drawoutthepatternofoutput1’sand0’sinamatrixofinputvalues

2. ConstructtheKmapandplace1sand0sinthesquaresaccordingtothetruth

table.

3. Grouptheisolated1swhicharenotadjacenttoanyother1s(singleloops).

4. Groupanypairwhichcontainsa1adjacenttoonlyoneother1(doubleloops).

5. Groupanyquadthatcontainsoneormore1sthathavenotalreadybeen

grouped,makingsuretousetheminimumnumberofgroups.

6. Groupanypairsnecessarytoincludeany1sthathavenotyetbeengrouped,

makingsuretousetheminimumnumberofgroups.

7. FormtheORsumofallthetermsgeneratedbyeachgroup.

WeillustratetheconceptofKMapsinexamples…


KMapfor2Variables

Considerthefollowingtruthtable.KMap

ThelogicalexpressionX isgivenby X A B A B

A B

A B

Consider Y A B A B

KMap

Y B

Consider Y A B A B A B

KMap

Y A B

Group2n numberof1swhichareadjacenttoeachother,withn beingthelargestpossibleinteger.n =0,20 =1n =1,21 =2n =2,22 =4n =3,23 =8


KMapfor3Variables

Considerthefollowingtruthtable.

ThecorrespondingBooleanexpression

usingSOPis

X A B C A B C A B C A B C

KMap

A B

B C

X A B B C

Group2n numberof1swhichareadjacenttoeachother,withn beingthelargestpossibleinteger.


MoreExamplesforKMapswith3Variables…

Consider

Y A B C A B C A B C A B C

KMap

Y C

Consider

W A B C A B C A B C A B C

KMap

W B

ItisobviousthatKmapsisanexcellenttoolforsimplifyingBooleanexpressions…



KMapfor4Variables

KnowinghowtogenerateGraycodeshouldallowustobuildlargermaps.Actually,

allweneedtodoislookatthelefttorightsequenceacrossthetopofthe3‐variable

map,followasimilarsequencefortheothertwovariablesandwriteitdownonthe

leftsideofthe4‐variablemap.

KmapoffourvariablesA,B,CandDisshowninthefollowingfigure.Aswehave

showninthepreviousexamples,wemayeasilyprovethat:

Combiningeightadjacentsquares

inKmapeliminatesthreevariables

fromtheresultingBoolean

expressionofthecorresponding

squares.


Example9

Consider X A B C D A B C D A B C D A B C D

KMap

X A D

B isdon’tcareand

C isalsodon’tcare…



Example10

Consider Y A B C D A B C D A B C D

A B C D A B C D A B C D

A B C D A B C D A B C D

KMap

C isdon’tcare, D isdon’tcare,C isdon’tcare, D isdon’tcare,

B isdon’tcare, D isdon’tcare,B isdon’tcare, D isdon’tcare,

B isdon’tcare, C isdon’tcare,B isdon’tcare, C isdon’tcare,

A isdon’tcare, B isdon’tcare,A isdon’tcare, B isdon’tcare,Y A B A C A D C D

A B

A C

A D

C D



Example11:KMapSimplificationforPOSFromthetableshownbelow,findZintermsofA,BandCusingProduct‐of‐sums(POS)form.UseKmaptosimplifytheresultingexpression.

A B C Z0 0 0 10 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 0

Solution: FollowtheusualKmapsimplificationfor:Z

1 0 1 0

1 1 0 1

C

C

A B A B A B A B

A A B CB CB AZ C

A C

A C

A B C

A B C

A B

Z Z B C

B C

B C

A B

AB CC

A B

A

DeMorgan'sLaw

DeMorgan'sLaw

Then,thePOSsimplificationcanbeobtainedbythefollowingmanipulations:

Z

0

0

0

11

1

11



Summary:KMapSimplificationProcess

1. Drawoutthepatternofoutput1’sand0’sinamatrixofinputvalues

2. ConstructtheKmapandplace1sand0sinthesquaresaccordingtothetruth

table.

3. Grouptheisolated1swhicharenotadjacenttoanyother1s(singleloops).

4. Groupanypairwhichcontainsa1adjacenttoonlyoneother1(doubleloops).

5. Groupanyquadthatcontainsoneormore1sthathavenotalreadybeen

grouped,makingsuretousetheminimumnumberofgroups.

6. Groupanypairsnecessarytoincludeany1sthathavenotyetbeengrouped,

makingsuretousetheminimumnumberofgroups.

7. FormtheORsumofallthetermsgeneratedbyeachgroup.


Design Example

Amajoritydetectorhas

fourinputvariablesA,B,C

andD andthreeoutput

lightindicators.Green

lightwillbeonifmajority

oftheinputvariablesare

equalto1.Redlightwill

beonifmajorityofthe

inputvariablesareequal

to0.Yellowlightwillbe

onifthereisatie.Design

anappropriatelogic

circuitforthisapplication.Implementthelogicalexpression

usinglogicgates


relationship



ProductorDesignSpecifications

DesignProcedure…


Solution: Constructthecorrespondingtruthtable…

Inputs Outputs

A B C D G R Y0 0 0 0 0 1 00 0 0 1 0 1 00 0 1 0 0 1 00 0 1 1 0 0 10 1 0 0 0 1 00 1 0 1 0 0 10 1 1 0 0 0 10 1 1 1 1 0 01 0 0 0 0 1 01 0 0 1 0 0 11 0 1 0 0 0 11 0 1 1 1 0 01 1 0 0 0 0 11 1 0 1 1 0 01 1 1 0 1 0 01 1 1 1 1 0 0

TruthTable

EG1108PART 2 ~PAGE 129EG1108PART 2 ~PAGE 129

A B D A C DR B C D A B C

Exercise: ConstructKmapforYandshowthatExercise: ConstructKmapforYandshowthat Y G R G R

A B A B A B A B

CD

CD

CD

CD

A B A B A B A B



LogicCircuitImplementation

A A B B C C D D

·

·

····

·

······

··

A

B

C

D

· · ·· ·· ·· ·· ·· ···

G

R

Y


LogicCircuitTrainer IntegratedCircuitChip

ActualLogicCircuitImplementation


Video:3‐bitBinaryCounter


FinalExam&FinalGrade

FinalExamination:

• Therewillbeafinalexamforthismodule.

• Thefinalexamwillhave4questions(2fromPart1and2fromPart2).

• Thefinalexamwillhaveadurationof2hours.

• Thefinalexamisofclosedbook.

• StudentsareallowedtobringanA4sheet(doublesided)offormulaewiththemintotheexaminationvenue.

FinalGrade:

• YourFinalGrade=80%ofFinalExamMarks+20%LabMarks


Acknowledgement…

… SpecialthankstoProfDiptiSrinivasanofNUSECEforsharingpartsofmaterials

coveredinthissecondpartofEG1108…

… ToUAVResearchGroupofNUSECEformaterialsonunmannedaerialsystems…

… ToYouTubeforvideoclipson3‐bitBinaryCounter,DCPowerSupply,and

MotorsandGenerators…


BugsBunnyHollywoodMovieStar

1940–?

That’sall,folks!

ThankYou!

That’sall,folks!

ThankYou!

EG1108 Part 2 - MAE CUHKbmchen/courses/EG1108_Part_2.pdf · 2020. 8. 27. · EG1108 PART2~ PAGE6...

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Transcript of EG1108 Part 2 - MAE CUHKbmchen/courses/EG1108_Part_2.pdf · 2020. 8. 27. · EG1108 PART2~ PAGE6...