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 1800’s was…

3. Origins
The most amazing and likely to be most long- lived invention of the 1800’s 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 1800’s 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) -

7. 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

9. P and N Transistors PNP Transistor Connect E to C when base = 0
NPN Transistor
Connect E to C when base = 1 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 AT&T 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 1
Truth table

12. NAND Gate +
Function: NAND
Symbol: a a b
out
out b
gnd A B
out 1

13. NOR Gate +
out b
Function: NOR
Symbol: a
gnd A B
out 1

14. Building Functions NOT: AND: a OR: b
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

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

17. Logic Manipulation Can manipulate logic equations algebraically
Can also use a truth table to prove equivalence
Example: (a+b)(a+c) = a + bc
(a+b)(a+c)
= aa + ab + ac + bc
= a + a(b+c) + bc
= a(1 + (b+c)) + bc
= a + bc a b c
a+b
a+c
LHS bc
RHS 1

18. Logic Minimization for all outputs that are 1,
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 a b c
minterm
abc 1
for all outputs
that are 1,
take the corresponding minterm,
OR the minterms to obtain the result in
“sum of products” form

19. Corresponding Karnaugh map
Karnaugh maps
Encoding of the truth table where adjacent cells differ in only one bit ab a b
out 1 1
Corresponding Karnaugh map
truth table for AND

20. Bigger Karnaugh Maps a a b 4-input func 3-input func y b y c d c ab c ab cd 00 1 01 11 10

21. Minimization with Karnaugh maps (1)b c
out 1
Sum of minterms yields
abc + abc + abc + abc

22. Minimization with Karnaugh maps (2)b c
out 1
Sum of minterms yields
abc + abc + abc + abc
Karnaugh maps identify which inputs are (ir)relevant to the output ab c 1 1

23. Minimization with Karnaugh maps (2)b c
out 1
Sum of minterms yields
abc + abc + abc + abc
Karnaugh map minimization
Cover all 1’s
Group adjacent blocks of 2n 1’s that yield a rectangular shape
Encode the common features of the rectangle
out = ab + ac ab c 1 1

24. Karnaugh Minimization Tricks (1)ab c
Minterms can overlap
out = bc + ac + ab
Minterms can span 2, 4, 8 or more cells
out = c + ab 1 1 ab c 1 1

25. Karnaugh Minimization Tricks (2)ab cd 00 1
The map wraps around
out = bd 01 11 10 ab cd 00 1 01 11 10

26. Karnaugh Minimization Tricks (3)ab cd 00 1 x
“Don’t care” values can be interpreted individually in whatever way is convenient
assume all x’s = 1
out = d
assume middle x’s = 0
assume 4th column x = 1
out = bd 01 11 10 ab cd 00 1 x 01 11 10

27. 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 b d

28. Multiplexor Implementationb d
Build a truth table a b d
out 1

29. Multiplexor Implementationb d
Build the Karnaugh map ab d 1 a b d
out 1 1

30. Multiplexor Implementationb d
Derive minimal logic equation
out = ad + bd ab d a b d
out 1 1 1

31. Multiplexor Implementationb d
Draw the circuit
out = ad + bd ab d 1 a b d
out 1 1 a
out d b

32. 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