Logic and Integrated Circuits

Lin ZhongELEC101, Spring 2011

Key concepts

• Binary numeral system• Boolean logic

– Logic gates– Functional completeness

• CMOS gates• Integrated circuits

2

Binary computing

• Modern computing are based on binary states – Two values: HIGH vs. LOW, 1 vs. 0, true vs. false

• Why– Easy to implement– Robust against interference, noise,

3

Computing with binary states

• Binary numeral system– Represent numeric values using two values: 0 and 1– The more “natural” numeral system is decimal

• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

– One to one mapping between the two systems

4

Decimal Binary

0 0

1 1

2 10

3 11

4 100

Decimal Binary

5 101

6 110

7 111

8 1000

9 1001

Recall the single-input computer

• Inverter

5

In Out

How about a two-to-one computer

6

Binary “states” for input and output: HIGH or LOW (1 or 0)

How many different computers are there?

A

OutB

A B Out

0 0

0 1

1 0

1 1

How about a two-to-one computer

7

Binary “states” for input and output: HIGH or LOW (1 or 0)

Useless ones: Out always 1; Out always 0; Out=A; Out=B

InverterOut= Invert (A); Out= Invert (B)

Useful ones:????

A

OutB

Three basic logic operations• Inversion (NOT): Out = ¬ In

• AND: Out = A Λ B

• OR: Out = A V B

8

In Out

0 1

1 0

A B Out

0 0 0

0 1 0

1 0 0

1 1 1

A B Out

0 0 0

0 1 1

1 0 1

1 1 1

More gates

• NAND

• NOR

• XOR– A XOR B = [A Λ (¬B)] V [(¬A) Λ B]

9

A B Out

0 0 0

0 1 1

1 0 1

1 1 0

How about a two-to-one computer

10

Binary “states” for input and output: HIGH or LOW (1 or 0)

Useless ones:Out = 0; Out =1; Out =A; Out=B;Inverters:Out= ¬A; Out= ¬B;

Useful ones:Out = A Λ B(AND), Out= A V B(OR), Out= A XOR BOut = ¬(A Λ B) (NAND), Out= ¬ (A V B) (NOR), Out = ¬ ( A XOR B)

A Λ (¬B); (¬A) Λ B; A V (¬B); (¬A) V B;

A

OutB

Functional completeness

• NOT, AND and OR can be used to build ANY Boolean function– Functionally complete

• Can you prove the following?– NOR is functionally complete– NAND is functionally complete

11

CMOS gates implementation

• NOT

• AND

• OR

12

CMOS gates: NAND

13

Gnd

CMOS gates: NOR

14

Vdd

Gnd

A

B

Out

Lab: NAND gate

15

Adder

16

A B

Sum

Carry-out

A, B, and Sum are states that take value from 0 to 9

Carry-out is a state that take value from 0 to 1

A B Sum Carry-out

2 5 7 0

9 9 8 1

… … … …

The simplest adder

17

A B

Sum

Carry-out

A, B, and Sum are states that take value from 0 to 1

Carry-out is a state that take value from 0 to 1

A B Sum Carry-out

0 1 1 0

1 1 0 1

1 0 1 0

0 0 0 0

Truth table

The simplest adder (Contd.)

18

A B

Sum

Carry-out

A, B, and Sum are states that take value from 0 to 1

Carry-out is a state that take value from 0 to 1

A B Sum Carry-out

0 1 1 0

1 1 0 1

1 0 1 0

0 0 0 0

Truth table

Sum= A XOR BCarry-out = A AND B

How many transistors do you need?

19

Sum

Carry-out

Sum= A XOR B

Carry-out = A AND B

AND

20

XOR

Sum= A XOR B = [A Λ (¬B)] V [(¬A) Λ B]

= [A AND (NOT B)] OR [(NOT A) AND B]

=[A NAND (NOT B)] NAND [(NOT A) NAND B]

= [(A AND B) NOR (A NOR B)]

21

XOR (Contd.)

22

Out

XOR (Contd.)

23

Out

Vdd

Computing with binary states (Contd.)• Boolean logic

– Variables are binary (0 or 1)– Three operations on binary variables

• Inversion (¬), AND (Λ), and OR (V)

– Five axioms

24

George Boole

1815-1864

Integrated Circuit

25Shared Nobel Prize in Physics 2000

Photolithography

1969

Ivan Sutherland won Turing Award in 1988 for his Ph.D. work in 1963

http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-574.pdf

“Programmable” integrated circuit

30

A B

Sum

Carry-out

“Programmable” integrated circuit

31

A B

Output

Control

“Programmable” integrated circuit

32

A B

Output

Control

Storage

“Programmable” integrated circuit

33

A B

Output

Control

Storage

“Programmable” integrated circuit

34

A B

Output

Control

Storage

0110100010101Instruction in machine code

“Programmable” integrated circuit

35

A B

Output

Control

Storage

Default start instruction

36

Computing vs. human performance

Sources: intel.com and factmonster.com

1968 1972 1976 1980 1984 1988 1992 1996 2000 20041

10

100

1000

10000

100000

1000000

Olympic Gold Metal winner: 100m dash (men)Olympic God Metal winner: 100m dash (women)# of transistors for Intel processorProcessor performance measured in MIPS

Year

Tim

es

of

imp

rov

em

en

t

37

Computing vs. humanity

1965 1970 1975 1980 1985 1990 1995 2000 20050.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

USA Federal minimum wage in 2003 dollar

Average transistor price for Intel processors in contemporary dollar

US

A $

Source: Intel.com and dol.gov

38http://ftp.arl.army.mil/ftp/historic-computers/png/eniac4.png

Computing: 60 years agoWiring ENIAC with a new program