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MassachusettsInstitute of TechnologyDepartment of Electrical Engineeringand Computer Science

6.002– Electronic Cir cuitsFall 2003

Final Exam

Pleasewrite your nameon eachpageof theexamin thespaceprovided,andcircle thenameof yourrecitationinstructorandthetimeof your recitationat thebottomof this page.

Pleaseverify thatthereare24pagesin yourexam.

Thereare8 majorquestionsin thisexam.Remember, if youfind thatyouarespendingtoomuchtimeonaspecificproblem,moveon to otherquestionsandreturnto this onelater.

To theextentpossible,doyourwork for eachquestionwithin theboundariesof thequestion,or onthebacksideof thepageprecedingthequestion.Extrapagesarealsoavailableat theendof yourexam.

Youmayusethreedouble-sidedpagesof notesandacalculatorwhile takingthis exam.

Final gradesin 6.002will not begivenout by phoneor by e-mail. Rather, they shouldbeavailablethroughWEBSISby December22. You mayreview andtake backyour final examat thebeginningof IAP in January2004from RoomNE43-624.

Goodluck!

Problem Points Score Grader

1 202 103 114 105 126 107 158 12

Total 100

Name:

Instructor: Perreault Antoniadis Chaniotakis Umans KolodziejskiTime: 10 11 11 11 12 12 1 2 3

Name: 2

Problem 1 – 20points

Thisproblemcontainsfour independentsubproblems.

(A) Write a booleanexpressionthat describesthe function of eachof the circuits below. Assumethat, ,

for the NFETs,

for the PFETs,and for

all the MOSFETs. Determinewhethereachcircuit satisfiesa staticdisciplinewith the following voltagethresholds:

"!$#&%' ( ) !*+%,.- )/!*+%,.- ) !* . (6 points)

VS

A

OUT1

BC

EN

RL

Satisfies static discipline: YES NO

VS

OUT2

A

B

Satisfies static discipline: YES NO

021436547 0214398:7

Name: 3

(B) Considerthefunction ;=<?> %A@B%DC=E specifiedin thetruth tablebelow.

> @ C ;<?> %D@B%AC=E0 0 0 10 0 1 00 1 0 00 1 1 01 0 0 11 0 1 11 1 0 01 1 1 1

Write a logic expressioncorrespondingto ;<F> %D@B%DCE . (3 points)

G 7

Implementthelogic function H<F> %D@B%DCEI > @JC,K > @JC

usingonly 2-inputlogic gates.(NotethatH

is notnecessarilythesamefunctionas ; ). Your implementationmayassumethat > %D@B%ACL% > % @ and

Careavailableasinputs.(3 points)

Name: 4

(C) Indicatewhethereachof thefilters shown below is a low passfilter (LPF), a high passfilter (HPF),abandpassfilter (BPF),or a bandstopfilter (BSF– A bandstopfilter passeslow andhigh frequencies,butattenuatesfrequenciesin between).Enteryouranswersin theboxesbelow eachfilter. (4 points)

vO+-

vI

-

+

vO+-

vI

-

+

vO

+-

vI

-

+

vO+-

vI

-

+

Name: 5

(D) A nonlineardevice M hasthecharacteristicsshown below.

iN+

-

vNN vN v( )0

iN

-1.5V 0.6V

It is connectedin an op-ampcircuit as indicatedbelow. Sketch the form of the voltage N and thecurrentOP for the input N - shown below, assumingthat theop-ampis ideal. Indicateclearly themaximumandminimumvaluesin yoursketches.(4 points)

+

-

iO

N

+

-

vI+

-vO

QΩ

ωt0

vI1

-1

π 2π 3π 4π

ωt0

vO

π 2π 3π 4π

ωt0

iO

π 2π 3π 4π

Name: 6

Problem 2 – 10points

YikesInc. hasmanufacturedfor an importantcustomerseveral million identicalresistive voltagedividerdevices,eachcontaininga pair of resistors4R and TS . Unfortunately, the designengineersuseda ratherlong wire to connectthe resistor S , which resultsin thepresenceof a non-zeroinductancein serieswiththeresistanceTS asillustratedin theequivalentcircuit for thedeviceshown on theleft.

L2

R2

R1

B

C

A

L2

R2

R1

B

C

A

vBvA

+

-

+

-

The figure on the right illustratesnormaluseof the voltagedivider device. TerminalC

is taken asacommonground.Theinput voltageNU is appliedacrosstheA andC terminals,andtheoutput NV is takenacrossthe

@andC

terminals.

(A) Supposethataninputof theformNU )WYX[Z]\Y^

is appliedto thecircuit. Determine_6<a`cb EdfeYgeYh , thefrequency responseof thecircuit. (Remember, for aninputof theform

WYX Z]\Y^, thesinusoidalsteadystateoutputwill taketheform

.iAX Zj\Y^, where

.iis thecomplex

amplitude). Furtherdeterminethe magnitudeof _6<a`cb E for a DC signal(b ), andfor b approachinginfinity. (4 points)

kml$no4p 7 eYgeYhrqs kml$no4p s

foro 7ut q s kml$no4p s

forowv x q

Name: 7

(B) Yikes’ customerlearnsof the frequency dependentbehavior of the voltagedivider and threatenstocanceltheir order. YikeshiresInduct-R-Us,a think-tankof ex-6.002gurus,to figureout a way to fix theirproblemwithout having to re-manufacturethedevices. Induct-R-Usquickly findsa solution,andsuggeststhatthefrequency dependenceof thevoltagedividercanbecancelledby addinganinductorof value yz atthe > terminalin serieswith R , asillustratedin theaugmentedcircuit below.

L2

R2

R1

B

C

A’

vA’

+

-

+

-

LX

vB

A

vA

+

-

DeterminethevoltageNV for N U| ) DC appliedacrossthe >d~ C terminals,in termsof oneor moreof R , S , yz , and y S , theparametersof theaugmentedvoltagedivider. Next, determinethevalueof yz thatmakes _6<a`cb E e.ge h | , the frequency responseof theaugmentedvoltagedivider, independentof frequency.(6 points)

V for U| 75c DCq z 7

Name: 8

Problem 3 – 11points

Thisprobleminvolvesop-ampcircuits.Assumethattheop-ampsareidealthroughouttheproblem.

(A) DetermineN <a E asa functionof N - <a E andN z < E in theop-ampcircuit below. (3 points)

+

-vO

vI

vX

R

R

2R

2R

RL

l$Dp 7

(B) DetermineNV<a E asa functionof NUd<a E in theop-ampcircuit below. (4 points)

+

-2R

vA

L

vB

V l$Dp 7

Name: 9

(C) Find thedifferentialequationrelating N< E to N - < E in thenetwork below. (You mayusetheresultsofParts(A) and(B)). (4 points)

+

-vO

vIR

2R

2R

L

vX +

-

R

2R

Differentialequation:

Name: 10

Problem 4 – 10points

This problemdealswith thetwo-stageamplifiercircuit shown below. AssumethattheMOSFETs, andw , operatein saturationthroughoutthis problem,andarecharacterizedby theequation

OP <N9'.E S

where > S)I

2K

+-

15V

vIN

90K 20K

10KC

M1 M2+

-

+

-

+

-

vO1 vI2

vO2

(A) Determinethevalueof N R whenN -?/ DC. (3 points)

R 7

Name: 11

(B) Determinethevalueof N - S , giventhat N -F)/ DC. Circleyouranswer. (2 points)

(1) 1.1V (2) 1.2V (3) 1.5V (4) 1.6V (5) 1.8V

(C) Supposethataninputof theform shown below is appliedto thecircuit.

N -F).-FK N )/K N¡ volts

Notice that the input comprisesa 3V DC componentand a small-signalAC componentN¡ . Draw anincrementalmodel for circuit. Clearly label the nodeswith small-signalvoltagesN , N¡S , N¢AR , and theoutputN¢PS . Substitutenumericalvalueswhereverpossible.(5 points)

Name: 12

Problem 5 – 12points

An op-ampcircuit incorporatinga resistoranda nonlineardevice calledthediodeis shown below. Assumetheop-ampis ideal.For theoperatingrangeof interestwewill modelthei-v characteristicof thediodeas:

OP )£? X9¤P¥¦?§¨ '

where,£?

and areknown constants.

+

-

iD

vO

+ -vI

R

vD

(A) Find the relationshipbetweentheoutputvoltageN andthe input voltageN - for this circuit. Assumethediodeactsasmodeledabove,andtheinput voltageis greaterthanzero.(4 points)

7

Name: 13

(B) Assumethat the input voltageN - is composedof a largepositive DC term-

plusa smallAC term N .In otherwords,

N -4)-©K N Theoutputvoltagecanbecorrespondinglyrepresentedas

N ) K N ¢Draw the small-signalcircuit model for the op-ampcircuit containingthe resistorand diode. Label allelementvaluesin termsof

-, ,£?

, and . (Rememberthat the small-signalcircuit equivalentof an

idealop-ampis theop-ampitself). (5 points)

(C) Find ª¬«ª¬­ , thesmall-signalgainof thecircuit asa functionof-

, ,£?

, and . (3 points)

ª¬«ª®­¯

Name: 14

Problem 6 – 10points

A 6.002student,FaeKwon, is looking to build a capacitorcharging circuit. Faefirst wiresup theresistivechargershown below, andoperatesit asfollows: thecapacitoris initially dischargedandtheswitchis open.At , Faeclosesthe ideal switch ° . Shere-opensit assoonasthe charging current Oj± reacheszeroagain.

vC

+-

V

R

C

iC

S

+

-

(Circuit A)

(A) For theresistive chargerin Circuit A, make a sketchof N± startingat . Your sketchshouldclearlyindicatethe initial value,thefinal value,andthetime constant.What is thefinal voltageon thecapacitor?How muchenergy is storedonthecapacitoraftertheswitchis re-opened,andhow muchenergy is dissipatedin thechargingprocess(betweenthetimetheswitchisclosed,to thetimetheswitchis re-opened)?(4points)

²³

´ µ

Final voltageoncap.7

Finalenergy oncap.7

Energy dissipatedwhencharging7

Name: 15

(B) Unhappy with theamountof energy dissipatedin Circuit A, Faereplacestheresistorwith aninductorasshown below in Circuit B, andoperatesit just like theresistivecharger: thecapacitoris initially discharged.At , sheclosesthe ideal switch ° . Shere-opensit assoonasthe charging current Oj± reacheszeroagain. (Moreprecisely, theswitchin Circuit B is openedat thefirst zerocrossingafter ) Y¶ .)

+-

V

C

iC

S

L

vC

+

-

(Circuit B)

For theinductivechargerin Circuit B, makesketchesof Oj± andN± . Thesketchesshouldclearlyindicateinitial, final andmaximumvalues,andthetime at which eachof theseoccurs.What is thefinal voltageonthe capacitor?How muchenergy is storedon the capacitorafter the switch is re-opened,andhow muchenergy is dissipatedin thecharging process(betweenthetime theswitchis closed,to thetime theswitchisre-opened)?(6 points)

·¸

¹ º

»¼

½ ¾

Final voltageoncap.7

Finalenergy oncap.7

Energy dissipatedwhencharging7

Name: 16

Problem 7 – 15points

Thefollowing circuit containingtwo MOSFETs( and ) actsasaninvertingamplifierwith inputsN RandNS , andoutput N S . Both MOSFETsareidentical,i.e., R S and

R S w. . Assumethatthesupplyvoltage

À¿¿).. Thisproblemwill explorethelargesignalresponseof thecircuit.

VS

RL

+-

+-

M1

M2

vO1

vO2

v1

D

S

G

S

GD

v2+

-

+

-

(A) Assumingthat both transistorsarein saturation,which of the following expressionsequalsN S , thegateto sourcevoltageof w . Circle theright answer. (2 points)

(1) N R (2) NS N R (3) N®S ' (4) N"R ' (5) NS

(B) Assumingthatbothtransistorsarein saturation,determineN R asa functionof N"R andNS . (5 points)

R 7

Name: 17

(C) Assumingthatbothtransistorsarein saturation,determineN S asa functionof N R . (4 points)

S 7

(D) Assumethat w is in saturationfor this question.For a givenvalueof NS , what is therangeof valuesof N"R sothat operatesin its saturationregion. (4 points)

Á R Á

Name: 18

Problem 8 – 12points

Minnie Delayrecentlydesignedacircuit for alight in herdormroomwhichturnsthelight brightergraduallywhenswitchedON, anddims it graduallywhenswitchedOFF. Not to beoutdone,her friend, Tom Driver,decidesto designa circuit that canmake a light flicker briefly whenturnedON or OFF. Tom decidestocreateaprototypeof thefollowing circuit in the6.002lab to seeif his ideawill work.

A

VS

RON

B

swi tch S

bulbC A

B

R

+

vAB

L

i

Tom picksa bulb which canbemodeledasa resistanceof value , anda switchwhich canbemodeledin its ON stateasaresistorof value ) ! . Tomlooksin thesupplyroomfor big inductorsandcapacitorsandpicks elementswith the following values:

C ; and y _ . Tom thenbuilds thecircuit andconnectsaDC supplywith

! .

(A) Whentheswitchis in theON state,whatis thesteadystatevalueof thevoltageacrossthebulb, NUV , inTom’scircuit? (2 points)

Steadystatevalueof UV 7

Name: 19

(B) After leaving theswitch in theON statefor a long time, Tom turnstheswitchOFF. To his dismay, thebulb doesnot flicker, ratherit dimsmonotonicallymuchlike Minnie’s circuit. Tom comesto you andasksfor your help in figuring out why his circuit doesnot flicker. Usingthepropertiesof secondordercircuits,explain thecauseto Tom in oneor two sentences.(3 points)

(C) Tom thenasksyou whetherthefollowing elementvalueswill causethebulb to flicker:CJÂ! ;

and y ) _ , keepingall othervaluesunchanged.(Note,themathis simplerif youuseC)Â! 4 S?ÃÄ Å; ). Assume,asbefore,thattheswitchis turnedOFFat afterbeingin theON statefor a long time. Determinethevaluesof O[< E , b ¢ , Æ , b&Ç , and È for thecircuit, andsketchtheform of NUÉVfor LÊ . Your sketchshouldclearlyindicateinitial value,final value,andtheapproximatetimesat whichzerocrossingsoccur. (7 points)

(Thenext pagehasspaceto fill in youranswers)

Name: 20

t0

vAB

Ë l t p 7 o ¢ 7

Ì 7 o Ç 7

Íu7

(End)

Name: 21

Name: 22

Name: 23

Name: 24