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Prof.AnneBracy
CS3410
ComputerScience
CornellUniversity
The slides are the product of many rounds of teaching CS 3410 by
Professors Weatherspoon, Bala, Bracy, and Sirer.
See:P&HAppendix B.2andB.3(Also,seeB.1)
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FromSwitchestoLogicGatestoLogicCircuits
Transistors(electronicswitch)
LogicGates
• TruthTables
LogicCircuits
• IdentityLaws
• FromTruthTablestoCircuits(SumofProducts)
LogicCircuitMinimization
• AlgebraicManipulations
• Karnaugh Maps
2
Transistors:
• Youtube videofromlastweek
https://www.youtube.com/watch?v=IcrBqCFLHIY
3
Source
Gate
Insulator
Drain
Source
Gate
Drain
—
Insulator
- + + + +
+ + + + - + + + - - - + + + P-type
P-type
N-type
- - - - - - - - - -
+ + channelcreated
+ + + + + + P-type
- + +
+ - +- - +
++
P-type
P-type
N-type
- - - - - - - - - -
Off
On
-
N-TypeSilicon:negativefree-carriers(electrons)
P-TypeSilicon:positivefree-carriers(holes)
P-Transistor:negativechargeongategenerateselectricfieldthat
createsa(+charged)p-channelconnectingsource&drain
N-Transistor:workstheoppositeway
Metal-OxideSemiconductor(Gate-Insulator-Silicon)
ComplementaryMOS=CMOStechnologyusesbothp- &n-type
transistors
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N-type
Off/Open
0
gate
P-type
Off/Open
gate
1
On/Closed
1
On/Closed
0
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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Logicgatesareconstructedbycombiningtransistorsin
complementaryarrangements
Combinep&n transistorstomakeaNOTgate:
CMOS Inverter :
power source (1)
power source (1)
—
p-gate
input
output
n-gate
ground (0)
power source (1)
+
p-gate
closes
0
1
n-gate
stays open
—
ground (0)
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p-gate
stays open
0
1
+
n-gate
closes
ground (0)
Vsupply (akalogic1)
in
out
• Function: NOT
• Symbol:
out
in
• Truth Table:
(ground islogic0)
In
0
1
Out
1
0
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Vsupply
A
Vsupply
B
B
A
out
• Function:
• Symbol:
a
b
out
• Truth Table:
A
0
1
0
1
B out
0
0
1
1
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Vsupply
• Function: NOR
• Symbol:
A
B
out
A
B
a
b
out
• Truth Table:
A B out
0 0
1
1 0
0
0 1
0
1 1
0
NANDandNORareuniversal
• Canimplementany functionwithNANDorjustNORgates
• usefulformanufacturing
NOT:
a
AND:
a
b
OR:
a
b
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Hidecomplexitythroughsimpleabstractions
• Simplicity
– Boxdiagramrepresentsinputsandoutputs
• Complexity
– HidesunderlyingNMOS- andPMOS-transistorsandatomic
interactions
Vdd
in
a
out
out
b
Vss
in
d
out
a
d
b
out
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FromSwitchestoLogicGatestoLogicCircuits
Transistors(electronicswitch)
LogicGates
• TruthTables
LogicCircuits
• IdentityLaws
• FromTruthTablestoCircuits(SumofProducts)
LogicCircuitMinimization
• AlgebraicManipulations
• Karnaugh Maps
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• Digitalcircuitthateitherallows
signaltopassthroughitornot
• Usedtobuildlogicfunctions
• Sevenbasiclogicgates:
AND,
OR,
NOT,
George Boole,(1815-1864)
NAND (notAND),
NOR (notOR),
Didyouknow?
XOR
GeorgeBooleInventoroftheidea
XNOR (notXOR)
oflogicgates.Hewasbornin
Lincoln,Englandandhewastheson
ofashoemakerinalowclassfamily.
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NOT:
A
A
Out
0
1
1
0
A B Out
AND:
OR:
XOR:.
A
B
A
B
A
B
A B Out
NAND: A
B
0 0
1
0 1
1
0
1 0
1
1
1 1
0
0 0
0
0 1
0
1 0
1 1
A
B Out
0
0
1
0
1
0
1
1
0
0
1
1
1
0
A
B Out
0
0
0
0
1
1
1
0
1
1
A B Out
NOR:
A
B
A B Out
XNOR: A
B
0 0
1
0 1
0
1
1 0
0
0
1 1
1
0 0
0
0 1
1
1 0
1 1
Fillinthetruthtable,giventhefollowingLogic
CircuitmadefromLogicAND,OR,andNOTgates.
Whatdoesthelogiccircuitdo?
a
b
Out
a
b
Out
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Fillinthetruthtable,giventhefollowingLogic
CircuitmadefromLogicAND,OR,andNOTgates.
Whatdoesthelogiccircuitdo?
a
b
d
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Out
a
d
Out
b
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FromSwitchestoLogicGatestoLogicCircuits
Transistors(electronicswitch)
LogicGates
• TruthTables
LogicCircuits
• FromTruthTablestoCircuits(SumofProducts)
• IdentityLaws
LogicCircuitMinimization
• AlgebraicManipulations
• Karnaugh Maps
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GivenaLogicfunctionà createaLogicCircuitthat
implementstheLogicFunction…
22
Howtoimplementadesiredlogicfunction?
a
0
0
0
0
1
1
1
1
b
0
0
1
1
0
0
1
1
c out minterm 1) Write minterms
0 0
abc 2) Write sumofproducts:
ORofallmintermswhereout=1
1 1
abc
out=abc +abc +abc
0 0
abc
1 1
abc
a
b
0 0
abc
c
out
1 1
abc
0 0
abc
1 0
abc
Any combinationalcircuitcanbe implementedin
twolevelsoflogic(ignoringinverters)
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NOT: =ā
=!a=¬a
AND: =a·b=a&b=a∧ b
NAND:
OR:
NOR:
=a+b=a|b=a∨ b
XOR: =a⊕ b=ab+āb
(a • b) =!(a&b)=¬ (a∧ b)
(a + b) =!(a|b)=¬ (a∨ b)
XNOR:
(a
b)
=ab +ab
LogicEquations
• Constants:true=1,false=0
• Variables:a,b,out,…
• Operators(above):AND,OR,NOT,etc.
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Identitiesusefulformanipulatinglogicequations
• Foroptimization&easeofimplementation
a+0= a
a+1= 1
a+ā= 1
a·0= 0
a·1= a
a·ā= 0
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Identitiesusefulformanipulatinglogicequations
• Foroptimization&easeofimplementation
A
(a + b) = a • b
(ab) = a + b
B
A
B
↔
↔
A
B
A
B
a+ab=a
a(b+c)=ab +ac
a(b + c) = a + b • c
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FromSwitchestoLogicGatestoLogicCircuits
Transistors(electronicswitch)
LogicGates
• TruthTables
LogicCircuits
• IdentityLaws
• FromTruthTablestoCircuits(SumofProducts)
LogicCircuitMinimization
• AlgebraicManipulations
• Karnaugh Maps
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ImplementtheLogicFunction…
withtheminimumnumberoflogicgates
Fewergates:Acheaper($$$)circuit!
Howtostandardizethisminimizing?
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a+0=a
a+1=1
a+ā =1
a·0=0
a·1=a
a·ā =0
Minimizethislogicequation:
(a+b)(a+c) =
a+ab=a
a(b+c)=ab +ac
(a + b) = a • b
(ab) = a + b
a(b + c) = a + b • c
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circuits↔truthtables↔equations
• Example:(a+b)(a+c)=a+bc
a
b
c
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
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Howdoesonefindthemostefficientequation?
– Manipulatealgebraicallyuntil…?
– UseKarnaugh Maps (optimizevisually)
– Useasoftwareoptimizer
Forlargecircuits
– Decomposition&reuseofbuildingblocks
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c
a b
c
out
0 0
0
0
0 0
1
1
0 1
0
0
0 1
1
1
1 0
0
1
1 0
1
1
1 1
0
0
1 1
1
0
out =
abc + abc + abc + abc
ab
00 011110
0 0 0 0 1
1 1
1
0
Sum of minterms yields
K-maps identify which inputs
are relevant to the output
1
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c
ab
00 011110
0 0 0 0 1
1 1
1
0
1
(1) Circle the 1’s (see below)
(2) Each circle is a logical
component of the final equation
= ab" + a"c
Rules:
• Usefewestcirclesnecessarytocoverall1’s
• Circlesmustcoveronly1’s
• Circlesspanrectanglesofsizepowerof2(1,2,4,8…)
• Circlesshouldbeaslargeaspossible(allcirclesof1?)
• CirclesmaywraparoundedgesofK-Map
• 1maybecircledmultipletimesifthatmeansfewer
circles
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c
c
ab
00 01 1110
0
0
1
1
1
1
0
0
1
0
Minterms can overlap
out = bc" + ac" + ab
ab
00 01 1110
0
1
1
1
1
1
0
0
1
0
Minterms can span 2, 4, 8 or
more cells
out = c" + ab
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ab
cd
00 01 1110
00
0
0
0
0
01
1
0
0
1
11
1
0
0
1
10
0
0
0
0
Themapwrapsaround
out=b"d
ab
cd
00 011110
00
1
0
0
1
01
0
0
0
0
11
0
0
0
0
10
1
0
0
1
out=b" d"
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ab
cd
00 011110
00
0
0
0
0
01
1
x
x
x
11
1
x
x
1
10
0
0
0
0
ab
“Don’tcare”valuescanbe
interpretedindividuallyin
whateverwayisconvenient
• assumeallx’s=1
à out=d
00
1
0
0
x
01
0
x
x
0
• assumemiddlex’s=0
(ignorethem)
• assume4th columnx=1
11
0
x
x
0
10
1
0
0
1
" d"
à out=b
cd
00 011110
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• Binary—twosymbols:true andfalse—isthebasis
ofLogicDesign
• MorethanoneLogicCircuitcanimplementsame
Logicfunction.UseAlgebra(Identities)orTruth
Tablestoshowequivalence.
• Anylogicfunctioncanbeimplementedas“sumof
products”.Karnaugh Mapsminimizenumberof
gates.
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Mostmoderndevicesmadeofbillionsoftransistors
• Youwillbuildaprocessorinthiscourse!
• Moderntransistorsmadefromsemiconductormaterials
• Transistorsusedtomakelogicgatesandlogiccircuits
Wecannowimplementanylogiccircuit
• UseP- &N-transistorstoimplementNAND/NORgates
• UseNANDorNORgatestoimplementthelogiccircuit
• Efficiently: useK-mapstofindrequiredminimalterms
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