Gates and Logic: From Transistorsto Logic Gates and Logic ... · PDF fileGates and Logic: From...

GatesandLogic:FromTransistors toLogicGatesand

LogicCircuits

Prof.AnneBracyCS3410

ComputerScienceCornellUniversity

The slides are the product of many rounds of teaching CS 3410 by Professors Weatherspoon, Bala, Bracy, and Sirer.

GoalsforToday• FromSwitchestoLogicGatestoLogicCircuits• Transistors,LogicGates,TruthTables• LogicCircuits

§ IdentityLaws§ FromTruthTablestoCircuits(SumofProducts)

• LogicCircuitMinimization§ AlgebraicManipulations§ Karnaugh Maps

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SiliconValley&theSemiconductorIndustry

• Transistors:• Youtube video“Howdoesatransistorwork”

https://www.youtube.com/watch?v=IcrBqCFLHIY• Break:showsomeTransistor,Fab,Waferphotos

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Transistors101

N-TypeSilicon:negativefree-carriers(electrons)P-TypeSilicon:positivefree-carriers(holes)P-Transistor:negativechargeongategenerateselectricfieldthat

createsa(+charged)p-channelconnectingsource&drainN-Transistor:workstheoppositewayMetal-OxideSemiconductor(Gate-Insulator-Silicon)• ComplementaryMOS=CMOStechnologyusesbothp- &n-type

transistors

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N-type

Off

Insulator

P-type P-type

Gate DrainSource

+++++++++

++

----- ------ --

- --

-

--

-

--

- --

--

+++N-type

On

Insulator

P-type P-type

Gate DrainSource

++++++++

----- ------ --

- --

-

--

-

--

- --

--

+ +P-typechannelcreated+ ++ ++

—

CMOSNotationN-type

P-type

Gate input controls whether current can flow between the other two terminals or not.

Hint: the “o” bubble of the p-type tells you that this gate wants a 0 to be turned on

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gate

Off/Open

0

On/Closed

1

Off/Open

1

On/Closed

0gate

Whichofthefollowingstatementsisfalse?

(A) P- andN-typetransistorsarebothusedinCMOSdesigns.

(B) Astransistorsgetsmaller,thefrequencyofyourprocessorwillkeepgettingfaster.

(C) Astransistorsgetsmaller,youcanfitmoreandmoreofthemonasinglechip.

(D) Puresiliconisasemiconductor.(E) ExpertsbelievethatMoore’sLawwillsoonend.

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iClicker Question

2-TransistorCombination:NOT• Logicgatesareconstructedbycombiningtransistorsincomplementaryarrangements

• Combinep&n transistorstomakeaNOTgate:

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p-gatecloses

n-gate stays open

p-gatestays open

n-gate closes

CMOS Inverter :

ground (0)

power source (1)

input output

p-gate

n-gate

power source (1)

ground (0) ground (0)

power source (1)

1 00

—

—

+

+

1

Inverter

In Out0 11 0

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Function: NOTSymbol:

Truth Table:

in outin out

Vsupply (akalogic1)

(groundislogic0)

LogicGates• Digitalcircuitthateitherallows

signaltopassthroughitornot• Usedtobuildlogicfunctions• Sevenbasiclogicgates:

AND,OR,NOT,NAND (notAND),NOR (notOR),XORXNOR (notXOR)

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Didyouknow?GeorgeBooleInventoroftheideaoflogicgates.HewasborninLincoln,Englandandhewasthesonofashoemaker.

George Boole,(1815-1864)

NOT:

AND:

OR:

XOR:.

LogicGates:Names,Symbols,TruthTables

A B Out0 0 00 1 11 0 11 1 1

A B Out0 0 00 1 01 0 01 1 1

A Out0 11 0

AB

AB

A

A B Out0 0 00 1 11 0 11 1 0

AB

A B Out0 0 10 1 01 0 01 1 0

A B Out0 0 10 1 11 0 11 1 0

AB

AB

NAND:

NOR:

A B Out0 0 10 1 01 0 01 1 1

AB

XNOR:

NORGate

A B out0 0 10 1 01 0 01 1 0

Function: NORSymbol:

Truth Table:

ba out

A

out

Vsupply

B

BA

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WhichGateisthis?

A B out0 00 11 01 1

12

Function:Symbol:

Truth Table:

A

out

Vsupply

B

BA

Vsupply

iClicker Question

(A) NOT(B) OR(C) XOR(D) AND(E) NAND

Abstraction• Hidecomplexitythroughsimpleabstractions

§ Simplicity• Boxdiagramrepresentsinputsandoutputs

§ Complexity• HidesunderlyingNMOS- andPMOS-transistorsandatomicinteractions

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in out

Vdd

Vss

in out outadb

a

b

d out

WhichGateisthis?

A B out0 00 11 01 1

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Function:Symbol:

Truth Table:

iClicker Question

(A) NOT(B) OR(C) XOR(D) AND(E) NAND

ab

Out

UniversalGates• NANDandNOR:

§ Canimplementany functionwithNANDorjustNORgates§ usefulformanufacturing

• NOT:

• AND:

• OR:

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ba

ba

a

Whatdoesthislogiccircuitdo?

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Function:Symbol:Truth Table:

a

b

dOut

a b d Out

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

Multiplexing Like a Boss


§ FromTruthTablestoCircuits(SumofProducts)§ IdentityLaws

• LogicCircuitMinimization§ AlgebraicManipulations§ Karnaugh Maps

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LogicImplementationHowtoimplementadesiredlogicfunction?

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a b c out0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 0

1) Writeminterms2) Writesumofproducts:

ORofallmintermswhereout=1

out=abc +abc +abc

mintermabcabcabcabcabcabcabcabc

c

out

ba

Any combinationalcircuitcanbeimplementedintwolevelsoflogic(ignoringinverters)

LogicEquationsNOT: =ā =!a=¬a

AND: =a·b=a&b=aÙ b

OR: =a+b=a|b=aÚ b

XOR: =aÅ b=ab+āb

LogicEquations§ Constants:true=1,false=0§ Variables:a,b,out,…§ Operators(above):AND,OR,NOT,etc.

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NAND:(a⚫b)=!(a&b)=¬ (aÙ b)

NOR:(a+b) =!(a|b)=¬ (aÚ b)

XNOR:(a⨁ b) =ab+ab

Identities

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Identitiesusefulformanipulatinglogicequations• Foroptimization&easeofimplementation

a+0=a+1=a+ā=

a·0=a·1=a·ā=

a11

0a0

Identitiesusefulformanipulatinglogicequations• Foroptimization&easeofimplementation

Identities

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A

B

A

B↔

A

B

A

B↔

a+ab=a

a(b+c)=ab +ac

(a+ b) = a • b

(ab) = a+ b

a(b+ c) = a+ b • c


§ IdentityLaws§ FromTruthTablestoCircuits(SumofProducts)

• LogicCircuitMinimization–why?§ AlgebraicManipulations§ Karnaugh Maps

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23

(a+b)a+(a+b)c=aa+ba +ac+bc=a+a(b+c) +bc=a +bc

Minimizethislogicequation:

(a+b)(a+c) =

a+0=aa+1=1a+ā =1a·0=0a·1=aa·ā =0

a+ab=aa(b+c)=ab +ac

(a+ b) = a • b(ab) = a+ ba(b+ c) = a+ b • c

MinimizationExample

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a+0=aa+1=1a+ā =1a·0=0a·1=aa·ā =0

a+ab=aa(b+c)=ab +ac

(a+b)(a+c) à a +bc

How many gates were required before and after?

(a+ b) = a • b(ab) = a+ ba(b+ c) = a+ b • c

iClicker Question

BEFORE AFTER(A) 2OR 1OR(B) 2OR,1AND 2OR(C) 2OR,1AND 1OR,1AND(D) 2OR,2AND 2OR(E) 2OR,2AND 2OR,1AND

CheckingEqualityw/TruthTablescircuits↔truthtables↔equations

Example:(a+b)(a+c)=a+bc

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a b c (a+b) LHS (a+c) RHS bc

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

CheckingEqualityw/TruthTablescircuits↔truthtables↔equations

Example:(a+b)(a+c)=a+bc

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a b c (a+b) LHS (a+c) RHS bc

0 0 0 0 0 0 0 0

0 0 1 0 0 1 0 0

0 1 0 1 0 0 0 0

0 1 1 1 1 1 1 1

1 0 0 1 1 1 1 0

1 0 1 1 1 1 1 0

1 1 0 1 1 1 1 0

1 1 1 1 1 1 1 1

MinimizationinPracticeHowdoesonefindthemostefficientequation?• Manipulatealgebraicallyuntil…?• UseKarnaughMaps(optimizevisually)• Useasoftwareoptimizer

Forlargecircuits• Decomposition&reuseofbuildingblocks

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BuildingaKarnaugh Map

a b c out0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 11 0 1 11 1 0 01 1 1 0

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Sum of minterms yieldsout =

K-maps identify which inputs are relevant to the output0 0 0 1

1 1 0 1

00 0111100

1

cab

abc+ abc+ abc+ abc

0 0 0 11 1 0 1

MinimizationwithK-Maps

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(1) Circle the 1’s (see below)(2) Each circle is a logical component of the final equation

= ab" + a"c

00 0111100

1

cab

Rules:• Usefewestcirclesnecessarytocoverall1’s• Circlesmustcoveronly1’s• Circlesspanrectanglesofsizepowerof2(1,2,4,8…)• Circlesshouldbeaslargeaspossible(allcirclesof1?)• CirclesmaywraparoundedgesofK-Map• 1maybecircledmultipletimesifthatmeansfewer

circles

Karnaugh MinimizationTricks(1)

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Minterms can overlapout = bc" + ac" + ab

Minterms can span 2, 4, 8 or more cells

out = c" + ab

0 1 1 10 0 1 0

00 01 1110

0

1

cab

1 1 1 10 0 1 0

00 01 1110

0

1

cab

KarnaughMinimizationTricks(2)

• Themapwrapsaroundout=b"d

out=b" d"

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1 0 0 10 0 0 00 0 0 01 0 0 1

00 011110

00

01

ab

cd

11

10

0 0 0 01 0 0 11 0 0 10 0 0 0

00 01 1110

00

01

ab

cd

11

10

Don’tCares“Don’tcare”valuescanbeinterpretedindividuallyinwhateverwayisconvenient

• assumeallx’s=1à out=d

• assumemiddlex’s=0(ignorethem)

• assume4th columnx=1à out=b" d"

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1 0 0 x0 x x 00 x x 01 0 0 1

00 011110

00

01

ab

cd

11

10

0 0 0 01 x x x1 x x 10 0 0 0

00 011110

00

01

ab

cd

11

10

Takeaway

• Binary—twosymbols:true andfalse—isthebasisofLogicDesign

• MorethanoneLogicCircuitcanimplementsameLogicfunction.UseAlgebra(Identities)orTruthTablestoshowequivalence.

• Anylogicfunctioncanbeimplementedas“sumofproducts”.Karnaugh Mapsminimizenumberofgates.

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Summary• Mostmoderndevicesmadeofbillionsoftransistors

§ Youwillbuildaprocessorinthiscourse!§ Moderntransistorsmadefromsemiconductormaterials

§ Transistorsusedtomakelogicgatesandlogiccircuits• Wecannowimplementanylogiccircuit

§ UseP- &N-transistorstoimplementNAND/NORgates

§ UseNANDorNORgatestoimplementthelogiccircuit

§ Efficiently: useK-mapstofindrequiredminimalterms34

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