Upload
lamduong
View
257
Download
6
Embed Size (px)
Citation preview
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
2
SiliconValley&theSemiconductorIndustry
• Transistors:• Youtube video“Howdoesatransistorwork”
https://www.youtube.com/watch?v=IcrBqCFLHIY• Break:showsomeTransistor,Fab,Waferphotos
3
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
4
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
5
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.
6
iClicker Question
2-TransistorCombination:NOT• Logicgatesareconstructedbycombiningtransistorsincomplementaryarrangements
• Combinep&n transistorstomakeaNOTgate:
7
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
8
Function: NOTSymbol:
Truth Table:
in outin out
Vsupply (akalogic1)
(groundislogic0)
LogicGates• Digitalcircuitthateitherallows
signaltopassthroughitornot• Usedtobuildlogicfunctions• Sevenbasiclogicgates:
AND,OR,NOT,NAND (notAND),NOR (notOR),XORXNOR (notXOR)
9
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:
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
13
in out
Vdd
Vss
in out outadb
a
b
d out
WhichGateisthis?
A B out0 00 11 01 1
14
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:
15
ba
ba
a
Whatdoesthislogiccircuitdo?
16
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
GoalsforToday• FromSwitchestoLogicGatestoLogicCircuits• Transistors,LogicGates,TruthTables• LogicCircuits
§ FromTruthTablestoCircuits(SumofProducts)§ IdentityLaws
• LogicCircuitMinimization§ AlgebraicManipulations§ Karnaugh Maps
17
LogicImplementationHowtoimplementadesiredlogicfunction?
18
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.
19
NAND:(a⚫b)=!(a&b)=¬ (aÙ b)
NOR:(a+b) =!(a|b)=¬ (aÚ b)
XNOR:(a⨁ b) =ab+ab
Identities
20
Identitiesusefulformanipulatinglogicequations• Foroptimization&easeofimplementation
a+0=a+1=a+ā=
a·0=a·1=a·ā=
a11
0a0
Identitiesusefulformanipulatinglogicequations• Foroptimization&easeofimplementation
Identities
21
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
GoalsforToday• FromSwitchestoLogicGatestoLogicCircuits• Transistors,LogicGates,TruthTables• LogicCircuits
§ IdentityLaws§ FromTruthTablestoCircuits(SumofProducts)
• LogicCircuitMinimization–why?§ AlgebraicManipulations§ Karnaugh Maps
22
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
24
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
25
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
26
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
27
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
28
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
29
(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)
30
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"
31
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"
32
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.
33
Summary• Mostmoderndevicesmadeofbillionsoftransistors
§ Youwillbuildaprocessorinthiscourse!§ Moderntransistorsmadefromsemiconductormaterials
§ Transistorsusedtomakelogicgatesandlogiccircuits• Wecannowimplementanylogiccircuit
§ UseP- &N-transistorstoimplementNAND/NORgates
§ UseNANDorNORgatestoimplementthelogiccircuit
§ Efficiently: useK-mapstofindrequiredminimalterms34