Logic Gates

1
Logic Gates

• CS/APMA 202, Spring 2005
• Rosen, section 10.3
• Aaron Bloomfield

2
Review of Boolean algebra

• Just like Boolean logic
• Variables can only be 1 or 0
• Instead of true / false

3
Review of Boolean algebra

• Not is a horizontal bar above the number
• 0 1
• 1 0
• Or is a plus
• 00 0
• 01 1
• 10 1
• 11 1
• And is multiplication
• 00 0
• 01 0
• 10 0
• 11 1

_
_
4
Review of Boolean algebra
_ _ _

• Example translate (xyz)(xyz) to a Boolean
  logic expression
• (x?y?z)?(?x??y??z)
• We can define a Boolean function
• F(x,y) (x?y)?(?x??y)
• And then write a truth table for it

x y F(x,y)
1 1 0
1 0 0
0 1 0
0 0 0
5
Quick survey

• I understand the basics of Boolean algebra
• Absolutely!
• More or less
• Not really
• Boolean what?

6
Todays demotivators
7
Basic logic gates

• Not
• And
• Or
• Nand
• Nor
• Xor

8
Rosen, 10.3 question 1

• Find the output of the following circuit
• Answer (xy)y
• Or (x?y)??y

xy
__
9
Rosen, 10.3 question 2

• Find the output of the following circuit
• Answer xy
• Or ?(?x??y) x?y

10
Quick survey

• I understand how to figure out what a logic gate
  does
• Absolutely!
• More or less
• Not really
• Not at all

11
Rosen, 10.3 question 6

• Write the circuits for the following Boolean
  algebraic expressions
• xy

__
12
Rosen, 10.3 question 6

• Write the circuits for the following Boolean
  algebraic expressions
• (xy)x

_______
xy
13
Writing xor using and/or/not

• p ? q ? (p ? q) ? (p ? q)
• x ? y ? (x y)(xy)

x y x?y
1 1 0
1 0 1
0 1 1
0 0 0
____
xy
(xy)(xy)
xy
xy
14
Quick survey

• I understand how to write a logic circuit for
  simple Boolean formula
• Absolutely!
• More or less
• Not really
• Not at all

15
Converting decimal numbers to binary

• 53 32 16 4 1
• 25 24 22 20
• 125 124 023 122 021 120
• 110101 in binary
• 00110101 as a full byte in binary
• 211 128 64 16 2 1
• 27 26 24 21 20
• 127 126 025 124 023 022
• 121 120
• 11010011 in binary

16
Converting binary numbers to decimal

• What is 10011010 in decimal?
• 10011010 127 026 025 124 123
  
• 022 121 020
• 27 24 23 21
• 128 16 8 2
• 154
• What is 00101001 in decimal?
• 00101001 027 026 125 024 123
• 022 021 120
• 25 23 20
• 32 8 1
• 41

17
A bit of binary humor

• Available for 15 at http//www.thinkgeek.com/
  tshirts/frustrations/5aa9/

18
Quick survey

• I understand the basics of converting numbers
  between decimal and binary
• Absolutely!
• More or less
• Not really
• Not at all

19
How to add binary numbers

• Consider adding two 1-bit binary numbers x and y
• 00 0
• 01 1
• 10 1
• 11 10
• Carry is x AND y
• Sum is x XOR y
• The circuit to compute this is called a half-adder

x y Carry Sum
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
20
The half-adder

• Sum x XOR y
• Carry x AND y

21
Using half adders

• We can then use a half-adder to compute the sum
  of two Boolean numbers

0
0
1
1 1 0 0 1 1 1 0
0
1
0
?
22
Quick survey

• I understand half adders
• Absolutely!
• More or less
• Not really
• Not at all

23
How to fix this

• We need to create an adder that can take a carry
  bit as an additional input
• Inputs x, y, carry in
• Outputs sum, carry out
• This is called a full adder
• Will add x and y with a half-adder
• Will add the sum of that to the carry in
• What about the carry out?
• Its 1 if either (or both)
• xy 10
• xy 01 and carry in 1

x y c carry sum
1 1 1 1 1
1 1 0 1 0
1 0 1 1 0
1 0 0 0 1
0 1 1 1 0
0 1 0 0 1
0 0 1 0 1
0 0 0 0 0
24
The full adder

• The HA boxes are half-adders

25
The full adder

• The full circuitry of the full adder

26
Adding bigger binary numbers

• Just chain full adders together

27
Adding bigger binary numbers

• A half adder has 4 logic gates
• A full adder has two half adders plus a OR gate
• Total of 9 logic gates
• To add n bit binary numbers, you need 1 HA and
  n-1 FAs
• To add 32 bit binary numbers, you need 1 HA and
  31 FAs
• Total of 4931 283 logic gates
• To add 64 bit binary numbers, you need 1 HA and
  63 FAs
• Total of 4963 571 logic gates

28
Quick survey

• I understand (more or less) about adding binary
  numbers using logic gates
• Absolutely!
• More or less
• Not really
• Not at all

29
More about logic gates

• To implement a logic gate in hardware, you use a
  transistor
• Transistors are all enclosed in an IC, or
  integrated circuit
• The current Intel Pentium IV processors have 55
  million transistors!

30
Pentium math error 1

• Intels Pentiums (60Mhz 100 Mhz) had a
  floating point error
• Graph of z y/x
• Intel reluctantlyagreed to replace them in 1994
• Graph from http//kuhttp.cc.ukans.edu/cwis/units/I
  PPBR/pentium_fdiv/pentgrph.html

31
Pentium math error 2

• Top 10 reasons to buy a Pentium
• 10 Your old PC is too accurate
• 8.9999163362 Provides a good alibi when the IRS
  calls
• 7.9999414610 Attracted by Intel's new "You don't
  need to know what's
• inside" campaign
• 6.9999831538 It redefines computing--and
  mathematics!
• 5.9999835137 You've always wondered what it
  would be like to be a
• plaintiff
• 4.9999999021 Current paperweight not big enough
• 3.9998245917 Takes concept of "floating point"
  to a new level
• 2.9991523619 You always round off to the nearest
  hundred anyway
• 1.9999103517 Got a great deal from the Jet
  Propulsion Laboratory
• 0.9999999998 It'll probably work!!

32
Flip-flops

• Consider the following circuit
• What does it do?

33
Memory

• A flip-flop holds a single bit of memory
• The bit flip-flops between the two NAND gates
• In reality, flip-flops are a bit more complicated
• Have 5 (or so) logic gates (transistors) per
  flip-flop
• Consider a 1 Gb memory chip
• 1 Gb 8,589,934,592 bits of memory
• Thats about 43 million transistors!
• In reality, those transistors are split into 9
  ICs of about 5 million transistors each

34
Quick survey

• I felt I understood the material in this slide
  set
• Very well
• With some review, Ill be good
• Not really
• Not at all

35
Quick survey

• The pace of the lecture for this slide set was
• Fast
• About right
• A little slow
• Too slow

36
End of lecture on 27 January 2005