Gates and Logic: From Transistors to Logic Gates and Logic Circuits - - PowerPoint PPT Presentation

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Gates and Logic: From Transistors to Logic Gates and Logic Circuits - - PowerPoint PPT Presentation

Mar 23, 2024 166 likes •437 views

Gates and Logic: From Transistors to Logic Gates and Logic Circuits CS 3410: Computer System Organization and Programming [K. Bala, A. Bracy, E. Sirer, and H. Weatherspoon] Goals for Today From Switches to Logic Gates to Logic Circuits

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SLIDE 1

Gates and Logic: From Transistors to Logic Gates and Logic Circuits

CS 3410: Computer System Organization and Programming

[K. Bala, A. Bracy, E. Sirer, and H. Weatherspoon]

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SLIDE 2

Goals for Today

  • From Switches to Logic Gates to Logic Circuits
  • Transistors, Logic Gates, Truth Tables
  • Logic Circuits

§ Identity Laws § From Truth Tables to Circuits (Sum of Products)

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SLIDE 3

Silicon Valley & the Semiconductor Industry

  • Transistors:
  • Youtube video “How does a transistor work”

https://www.youtube.com/watch?v=IcrBqCFLHIY

  • Break: show some Transistor, Fab, Wafer photos

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SLIDE 4

Transistors 101

N-Type Silicon: negative free-carriers (electrons) P-Type Silicon: positive free-carriers (holes) P-Transistor: negative charge on gate generates electric field that creates a (+ charged) p-channel connecting source & drain N-Transistor: works the opposite way Metal-Oxide Semiconductor (Gate-Insulator-Silicon)

  • Complementary MOS = CMOS technology uses both p- & n-type

transistors

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N-type

Off

Insulator P-type P-type Gate Drain Source + + + + + + + + + + +

  • +

+ + N-type

On

Insulator P-type P-type Gate Drain Source + + + + + + + +

  • +

+

P-type channel created

+ + + + +

—

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SLIDE 5

CMOS Notation

N-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 On/Closed 1 Off/Open 1 On/Closed gate

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SLIDE 6

Which of the following statements is false? (A) P- and N-type transistors are both used in CMOS designs. (B) As transistors get smaller, the frequency of your processor will keep getting faster. (C) As transistors get smaller, you can fit more and more of them on a single chip. (D) Pure silicon is a semi conductor. (E) Experts believe that Moore’s Law will soon end.

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iClicker Question

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SLIDE 7

2-Transistor Combination: NOT

  • Logic gates are constructed by combining transistors

in complementary arrangements

  • Combine p&n transistors to make a NOT gate:

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p-gate closes n-gate stays open p-gate stays open n-gate closes

CMOS Inverter :

ground (0) power source (1) input

  • utput

p-gate n-gate power source (1) ground (0) ground (0) power source (1) 1 — — + + 1

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SLIDE 8

Inverter

In Out 1 1

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Function: NOT Symbol: Truth Table:

in

  • ut

in

  • ut

Vsupply (aka logic 1) (ground is logic 0)

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SLIDE 9

Logic Gates

  • Digital circuit that either allows

signal to pass through it or not

  • Used to build logic functions
  • Seven basic logic gates:

AND, OR, NOT, NAND (not AND), NOR (not OR), XOR XNOR (not XOR)

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Did you know?

George Boole Inventor of the idea

  • f logic gates. He was born in

Lincoln, England and he was the son

  • f a shoemaker.

Ge George Boole,(1815-1864) 1864)

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SLIDE 10

NOT: AND: OR: XOR:.

Logic Gates: Names, Symbols, Truth Tables

A B Out 1 1 1 1 1 1 1 A B Out 0 0 0 1 1 0 1 1 1 A Out 1 1

A B A B A

A B Out 0 0 0 1 1 1 0 1 1 1

A B

A B Out 1 1 1 1 1 A B Out 0 0 1 0 1 1 1 0 1 1 1

A B A B

NAND: NOR:

A B Out 0 0 1 0 1 1 0 1 1 1

A B

XNOR:

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SLIDE 11

NOR Gate

A B out 0 0 1 0 1 1 0 1 1 Function: NOR Symbol: Truth Table:

b a

  • ut

A

  • ut

Vsupply B B A

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SLIDE 12

Which Gate is this?

A B out 0 0 0 1 1 0 1 1

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Function: Symbol: Truth Table:

A

  • ut

Vsupply B B A Vsupply

iClicker Question

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

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SLIDE 13

Abstraction

  • Hide complexity through simple abstractions

§ Simplicity

  • Box diagram represents inputs and outputs

§ Complexity

  • Hides underlying NMOS- and PMOS-transistors and atomic

interactions

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in

  • ut

Vdd Vss in

  • ut
  • ut

a d b a b d

  • ut
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SLIDE 14

Which Gate is this?

A B out 0 0 0 1 1 0 1 1

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Function: Symbol: Truth Table:

iClicker Question

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

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SLIDE 15

Universal Gates

  • NAND and NOR:

§ Can implement any function with NAND or just NOR gates § useful for manufacturing

  • NOT:
  • AND:
  • OR:

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b a b a a

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SLIDE 16

What does this logic circuit do?

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Function: Symbol: Truth Table: a b d Out

a b d Out 1 1 1 1 1 1 1 1 1 1 1 1 Multiplexing Like a Boss

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SLIDE 17

Goals for Today

  • From Switches to Logic Gates to Logic Circuits
  • Transistors, Logic Gates, Truth Tables
  • Logic Circuits

§ From Truth Tables to Circuits (Sum of Products) § Identity Laws

  • Logic Circuit Minimization

§ Algebraic Manipulations § Karnaugh Maps

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SLIDE 18

Logic Implementation

How to implement a desired logic function?

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a b c out 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 1 1) Write minterms 2) Write sum of products: OR of all minterms where out=1

  • ut = abc + abc + abc

minterm a b c a b c a b c a b c a b c a b c a b c a b c

c

  • ut

b a

Any combinational circuit can be implemented in two levels of logic (ignoring inverters)

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SLIDE 19

Logic Equations

NOT: = ā

= !a = ¬a

AND: = a · b = a & b = a Ù b OR:

= a + b = a | b = a Ú b

XOR: = a Å b = ab+ āb Logic Equations

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

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NAND:

(a b) = !(a & b) = ¬ (a Ù b)

NOR:

(a + b) = !(a | b) = ¬ (a Ú b)

XNOR:

(a ⨁ b) = ab + ab

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SLIDE 20

Identities

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Identities useful for manipulating logic equations

  • For optimization & ease of implementation

a + 0 = a + 1 = a + ā = a · 0 = a · 1 = a · ā = a 1 1 a

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SLIDE 21

Identities useful for manipulating logic equations

  • For optimization & ease of implementation

Identities

21 A B A B

↔

A B A B

↔

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

(a + b) = a • b

(ab) = a + b

a(b + c) = a + b • c

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SLIDE 22

Goals for Today

  • From Switches to Logic Gates to Logic Circuits
  • Transistors, Logic Gates, Truth Tables
  • Logic Circuits

§ Identity Laws § From Truth Tables to Circuits (Sum of Products)

  • Logic Circuit Minimization

§ No details this semester, but quick discussion of what

this is and why you’d do it.

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SLIDE 23

Checking Equality w/Truth Tables

circuits ↔ truth tables ↔equations Example: (a+b)(a+c) = a + bc

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a b c (a+b) LHS (a+c) RHS bc 1 1 1 1 1 1 1 1 1 1 1 1

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SLIDE 24

Checking Equality w/Truth Tables

circuits ↔ truth tables ↔equations Example: (a+b)(a+c) = a + bc

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a b c (a+b) LHS (a+c) RHS bc 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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SLIDE 25

Summary

  • Most modern devices made of billions of transistors

§ You will build a processor in this course! § Modern transistors made from semiconductor

materials

§ Transistors used to make logic gates and logic circuits

  • We can now implement any logic circuit

§ Use P- & N-transistors to implement NAND/NOR

gates

§ Use NAND or NOR gates to implement the logic

circuit

§ Efficiently: use K-maps to find required minimal terms

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