From Switches to Transistors and Gates

1
From Switches to Transistors and Gates

• Prof. Sirer
• CS 303
• Koç University

2
Origins

• The most amazing and likely to be most long-lived
  invention of the 1800s was

3
Origins

• The most amazing and likely to be most long-lived
  invention of the 1800s was
• The steam engine?
• The lightning rod?
• The carbonated beverage?

4
Origins

• The most amazing and likely to be most long-lived
  invention of the 1800s was
• THE ELECTRIC SWITCH

5
A switch

• A switch is a simple device that can act as a
  conductor or isolator
• Can be used for amazing things

6
Switches

• Either (OR)

-
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Switches

• Either (OR)
• Both (AND)

-
-

• One can build basic devices, but operation
  requires mechanical force

8
Transistors

• What happens when switches are replaced with
  solid-state devices
• The most amazing invention of the 1900s
• Can be used for purposes other than switching
  (e.g. radios)
• Two types, PNP and NPN

-

P
P
N
collector
emitter
collector
collector
base

NPN
N
N
-
P
PNP
base
emitter
emitter
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P and N Transistors

• NPN Transistor
• Connect E to C whenbase 1

• PNP Transistor
• Connect E to C whenbase 0

E
E
B
B
C
C

• Can be used to build primitives from which
  electronic machines can be constructed

10
Then and Now

• The first transistor, on a workbench at ATT Bell
  Laboratories in 1947

• An Intel Pentium has approximately 125 million
  transistors

11
Inverter

• Function NOT
• Called an inverter
• Symbol
• Useful for taking the inverse of an input

in
out
gnd
in
out
In Out
0 1
1 0
Truth table
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NAND Gate

• Function NAND
• Symbol

a
a
b
out
out
b
gnd
A B out
0 0 1
1 0 1
0 1 1
1 1 0
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NOR Gate

out
b

• Function NOR
• Symbol

a
gnd
A B out
0 0 1
1 0 0
0 1 0
1 1 0
14
Building Functions

• NOT
• AND
• OR
• NAND and NOR are universal, can implement any
  function using just NAND or just NOR gates
• useful for manufacturing
• Can specify functions by describing gates, truth
  tables or logic equations

a
b
a
b
15
Logic Equations

• AND
• out ab
• OR
• out a b
• NOT
• out a

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Identities

• Identities are useful for manipulating logic
  equations
• For purposes of optimization ease of
  implementation
• a a 1
• a 0 a
• a 1 1
• aa 0
• a0 0
• a1 a
• a(bc) ab ac
• (a b) a b
• (a b) a b
• a a b a b

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Logic Manipulation

• Can manipulate logic equations algebraically
• Can also use a truth table to prove equivalence
• Example (ab)(ac) a bc

a b c ab ac LHS bc RHS
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 1 0 1
1 0 1 1 1 1 0 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
(ab)(ac) aa ab ac bc a a(bc)
bc a(1 (bc)) bc a bc
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Logic Minimization

• A common problem is how to implement a desired
  function most efficiently
• One can derive the equation from the truth table
• How does one find the most efficient equation?
• Manipulate algebraically until satisfied
• Use Karnaugh maps

for all outputs that are 1, take the
correspondingminterm, OR the minterms toobtain
the result in sum of products form
a b c minterm
0 0 0 abc
0 0 1 abc
0 1 0 abc
0 1 1 abc
1 0 0 abc
1 0 1 abc
1 1 0 abc
1 1 1 abc
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Karnaugh maps

• Encoding of the truth table where adjacent cells
  differ in only one bit

ab
00 01 11 10
a b out
0 0 0
0 1 0
1 0 0
1 1 1
0 0 1 0
Corresponding Karnaugh map
truth tablefor AND
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Bigger Karnaugh Maps
a
a
b
4-inputfunc
3-inputfunc
y
b
y
c
d
c
ab
c
00 01 11 10
ab
0
cd


00 01 11 10
00




1
01
11
10
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Minimization with Karnaugh maps (1)
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

• Sum of minterms yields
• abc abc abc abc

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Minimization with Karnaugh maps (2)
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

• Sum of minterms yields
• abc abc abc abc
• Karnaugh maps identify which inputs are
  (ir)relevant to the output

ab
c
00 01 11 10
0
0 0 0 1
1 1 0 1
1
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Minimization with Karnaugh maps (2)
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

• Sum of minterms yields
• abc abc abc abc
• Karnaugh map minimization
• Cover all 1s
• Group adjacent blocks of 2n 1s that yield a
  rectangular shape
• Encode the common features of the rectangle
• out ab ac

ab
c
00 01 11 10
0
0 0 0 1
1 1 0 1
1
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Karnaugh Minimization Tricks (1)
ab
c
00 01 11 10

• Minterms can overlap
• out bc ac ab
• Minterms can span 2, 4, 8 or more cells
• out c ab

0
0 1 1 1
0 0 1 0
1
ab
c
00 01 11 10
0
1 1 1 1
0 0 1 0
1
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Karnaugh Minimization Tricks (2)
ab
00 01 11 10
cd
00

• The map wraps around
• out bd
• out bd

0 0 0 0
1 0 0 1
1 0 0 1
0 0 0 0
01
11
10
ab
cd
00 01 11 10
00
1 0 0 1
0 0 0 0
0 0 0 0
1 0 0 1
01
11
10
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Karnaugh Minimization Tricks (3)
ab
00 01 11 10
cd
00

• Dont care values can be interpreted
  individually in whatever way is convenient
• assume all xs 1
• out d
• assume middle xs 0
• assume 4th column x 1
• out bd

0 0 0 0
1 x x x
1 x x 1
0 0 0 0
01
11
10
ab
cd
00 01 11 10
00
1 0 0 x
0 x x 0
0 x x 0
1 0 0 1
01
11
10
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Multiplexors

• A multiplexor selects between multiple inputs
• out a, if d 0
• out b, if d 1
• Build truth table
• Build Karnaugh map
• Derive logic diagram

a
0
b
d
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Multiplexor Implementation

• Build a truth table

a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
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Multiplexor Implementation

• Build the Karnaugh map

ab
d
00 01 11 10
0
0 0 1 1
0 1 1 0
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
1
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Multiplexor Implementation

• Derive minimal logic equation
• out ad bd

ab
d
00 01 11 10
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
0
0 0 1 1
0 1 1 0
1
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Multiplexor Implementation

• Draw the circuit
• out ad bd

ab
d
00 01 11 10
0
0 0 1 1
0 1 1 0
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
1
a
out
d
b
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Summary

• We can now implement any logic circuit
• Can do it efficiently, using Karnaugh maps to
  find the minimal terms required
• Can use either NAND or NOR gates to implement the
  logic circuit
• Can use P- and N-transistors to implement NAND or
  NOR gates