1. L14: Boolean Logic and Basic Gates
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L14: Boolean Logic and Basic Gates
Fall 2005

2. L14: Boolean Logic and Basic Gates
LECTURE 14:
Transistors to logic gates
Boolean Logic
Basic logic gates
Introduction to switching logic
Turning logic circuits into switching expressions
Simplifying switching expressions, Karnaugh Maps (no time)
L14: Boolean Logic and Basic Gates
Fall 2005

3. From Transistors to Computer Chips
L14: Boolean Logic and Basic Gates
Fall 2005

4. Transistors - Where it all starts
Transistors are physical structures
Bipolar Junction Transistors
Field Effect Transistors
For digital logic, you can think of a transistor as a switch that has two states: “ON” and “OFF”
Wiring transistors together, we can create logic functions and storage elements
L14: Boolean Logic and Basic Gates
Fall 2005

5. Example: MOSFET (Metal Oxide Semiconductor Field Effect Transistor)
Source
Gate
Drain
Two different types of MOSFET
p channel
n channel
L14: Boolean Logic and Basic Gates
Fall 2005

6. L14: Boolean Logic and Basic Gates
Transistor as a Switch
Drain
Drain
Gate
Gate
Source
Source
The voltage on the gate decides if the switch will be open (i.e., “OFF”) or closed (i.e., “ON”)
L14: Boolean Logic and Basic Gates
Fall 2005

7. CMOS - Complementary Metal Oxide Semiconductors
Today, CMOS is the most common technology used for manufacturing digital computer chips
Combines p-channel and n-channel MOSFETS
High levels of integration are possible
Low power requirements
L14: Boolean Logic and Basic Gates
Fall 2005

8. when Gate voltage is high, when Gate voltage is low,
MOSFETs
n-Channel
p-Channel
Source
Drain
Gate
Gate
Source
Drain
Designed so that
when Gate voltage is high,
switch is “ON”
Designed so that
when Gate voltage is low,
switch is “ON”
L14: Boolean Logic and Basic Gates
Fall 2005

9. Transistors and Logic Gates
Transistors can be connected together with wires
Connect transistors to create simple, logical functions - these functions are called logic gates
In modern computer chips, the transistors and wires are very, very small
L14: Boolean Logic and Basic Gates
Fall 2005

10. L14: Boolean Logic and Basic Gates
CMOS Inverter
Vdd = “High”
Input
Output
Input
Output
Low
High
High
Low
Ground = “Low” = 0V
L14: Boolean Logic and Basic Gates
Fall 2005

11. L14: Boolean Logic and Basic Gates
NOT Logic Gate
Input A
Output
Input A
Output 1 1
The NOT logic gate is an inverter
Logical function: A’ or A
L14: Boolean Logic and Basic Gates
Fall 2005

12. L14: Boolean Logic and Basic Gates
CMOS NAND Logic Gate
Vdd = “High”
Output
Input A
Input A
Input B
Output
Input B 1 1 1 1 1 1 1
L14: Boolean Logic and Basic Gates
Ground = “Low” = 0V
Fall 2005

13. L14: Boolean Logic and Basic Gates
NAND Logic Gate
Input A
Input B
Output
Input A
Output 1 1 1
Input B 1 1 1 1
Logical function “NAND”
AB alternatively (AB)’
L14: Boolean Logic and Basic Gates
Fall 2005

14. L14: Boolean Logic and Basic Gates
AND Logic Gate
Input A
Input B
Output
Input A
Output 1
Input B 1 1 1 1
Logical function “AND” AB
L14: Boolean Logic and Basic Gates
Fall 2005

15. L14: Boolean Logic and Basic Gates
OR Logic Gate
Input A
Input B
Output
Input A
Output 1 1
Input B 1 1 1 1 1
Logical function “OR”
A + B
L14: Boolean Logic and Basic Gates
Fall 2005

16. L14: Boolean Logic and Basic Gates
NOR Logic Gate
Input A
Input B
Output
Input A
Output 1 1
Input B 1 1 1
Logical function “NOT OR”
(A + B) alternatively (A + B)’
L14: Boolean Logic and Basic Gates
Fall 2005

17. L14: Boolean Logic and Basic Gates
XOR Logic Gate
Input A
Input B
Output
Input A
Output 1 1
Input B 1 1 1 1
Logical function “Exclusive OR”
A B + A B alternatively A’B + AB’
L14: Boolean Logic and Basic Gates
Fall 2005

18. L14: Boolean Logic and Basic Gates
Truth Tables
Truth Tables are used to define the output for any given combination of inputs
Input A
Input B
Input A
Input B
Input C
Output
Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
L14: Boolean Logic and Basic Gates
Fall 2005

19. Using Truth Tables to Create Digital Logic Functions
Input x
Input y
Input z
Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
L14: Boolean Logic and Basic Gates
Fall 2005

20. L14: Boolean Logic and Basic Gates
A Generalized Procedure for Creating a Digital Logic Design from a Truth Table
Canonical Sum of Products form
Create an input line for each input variable, and branch off another line with a NOT gate to form the complement of the variable.
You will need as many AND gates as there are “1”s in the truth table. The inputs to each AND gate are the input variables or their complements - as indicated by the truth table entries.
For example if x=0, y=1, and z=0 and OUTPUT=1, you will need x’ , y, and z’ as inputs into one of the AND gates
You will need one OR gate. All of the outputs of the AND gates will go into the OR gate. The output of the OR gate is the output function for the circuit
L14: Boolean Logic and Basic Gates
Fall 2005

21. L14: Boolean Logic and Basic Gates
A Generalized Procedure for Creating a Digital Logic Design from a Truth Table
Canonical Product of Sums form
Create an input line for each input variable, and branch off another line with a NOT gate to form the complement of the variable.
You will need as many OR gates as there are “0”s in the truth table. The inputs to each OR gate are the input variables or their complements - as indicated by the truth table entries.
For example if x=0, y=0, and z=1 and OUTPUT=0, you will need x , y, and z’ as inputs into one of the OR gates
You will need one AND gate. All of the outputs of the OR gates will go into the AND gate. The output of the AND gate is the output function for the circuit
L14: Boolean Logic and Basic Gates
Fall 2005

22. Canonical Sum of Products
Using sum of products is one way to get a correct circuit design from a truth table
Approach doesn’t always give you the most efficient circuit design!
L14: Boolean Logic and Basic Gates
Fall 2005

23. L14: Boolean Logic and Basic Gates
Boolean Algebra
George Boole, 19th Century mathematician
Developed a mathematical system (algebra) involving logic
later known as “Boolean Algebra”
Primitive functions: AND, OR and NOT
The power of BA is there’s a one-to-one correspondence between circuits made up of AND, OR and NOT gates and equations in BA
+ means OR,• means AND, x means NOT
L14: Boolean Logic and Basic Gates
Fall 2005

24. Switching Algebra Identities
Identity Law
1 A = A 0 + A = A
Null Law
0 A = A = 1
Idempotent Law
AA = A A + A = A
Inverse Law
AA = 0 A + A = 1
L14: Boolean Logic and Basic Gates
Fall 2005

25. More Switching Algebra Identities
Commutative Law
AB = BA A + B = B + A
Associative Law
(AB)C = A(BC) (A + B) + C = A + (B + C)
Absorption Law
A(A + B) = A A + AB = A
Distributive Law
A + BC = (A + B)(A + C) A(B + C) = AB + AC
L14: Boolean Logic and Basic Gates
Fall 2005

26. One Other Theorem: Involution
A = A
(A’)’ = A
L14: Boolean Logic and Basic Gates
Fall 2005

27. L14: Boolean Logic and Basic Gates
DeMorgan’s Law
A B = A + B alternatively (AB)’ = A’ + B’
A + B = A B alternatively (A + B)’ = A’ B’
Very useful for simplifying functions!
L14: Boolean Logic and Basic Gates
Fall 2005

28. Turning a Switching Expression into A Digital Logic Circuit
The expression has the directions “and’s” turn into “and” gates; “or’s” turn into “or” gate, complements can be “not” gates….
Example: xy + z’ = output x y
output z
L14: Boolean Logic and Basic Gates
Fall 2005

29. Working with Switching Algebra Quickie Quiz 1 (Draw this circuit)
f (A,B,C) = A’ + B’ + A’ B C
A B C A’ B’
A’ B C
F(A,B,C)
L14: Boolean Logic and Basic Gates
Fall 2005

30. Working with Switching Algebra
Complicated functions can be simplified using Boolean Algebra
L14: Boolean Logic and Basic Gates
Fall 2005

31. L14: Boolean Logic and Basic Gates
Another Example
Input A
Output
Input B
What’s the logic function?
((A + B’)’ B’)’
Simplify using DeMorgan’s Law and identities
= (A + B’) + B = A + (B’ + B) = A + 1 = 1
L14: Boolean Logic and Basic Gates
Fall 2005

32. Working with Switching Algebra QUICKIE QUIZ (Simplify this function)
f (w,x,y,z) = x y + w’ x y z’ + x’ y
= y (x + x’ + w’ x z’)
= y ( 1)
= y
L14: Boolean Logic and Basic Gates
Fall 2005

33. Simplified solutions aren’t always unique
f(x,y,z) = x’ y’ z’ + x’ y z’ + x’ y z + x y’ z’ + x y’ z + x y z
= x’ z’ + x y’ + y z
= x’ y + y’ z’ + x z
= x’ z’ + y’ z’ + y z + x z
L14: Boolean Logic and Basic Gates
Fall 2005

34. Karnaugh Maps A Tool for Simplifying Logic Functions
x y
0 0
0 1
1 1
1 0 z 1
Karnaugh map for three variables –
a truth table turned on its side
L14: Boolean Logic and Basic Gates
Fall 2005

35. L14: Boolean Logic and Basic Gates
Karnaugh Maps
Karnaugh maps are tools to help you visually/graphically group the “1s” in a truth table to develop simplified expressions
From the logic expression, you can develop a logic- gate-level design implementing the function
L14: Boolean Logic and Basic Gates
Fall 2005

36. Representing Functions Using a Karnaugh Map
x y
0 0
0 1
1 1
1 0 z 1
L14: Boolean Logic and Basic Gates
Fall 2005

37. Karnaugh Map Guidelines (1 of 3)
Prepare the map using the truth table inputs
Each “1” on the Karnaugh map represents a min-term for a “sum of products” expression of the truth table’s function
You can simplify the logic expression by combining the product terms (the “min terms”) that are grouped on the Karnaugh map
Groups of two, four, eight, (any power of two) “1” ‘s on the map can be used to develop a simpler expression of fewer variables
L14: Boolean Logic and Basic Gates
Fall 2005

38. Karnaugh Map Guidelines (2 of 3)
Groups can “wrap around” the sides of the Karnaugh map
Top and bottom rows
Right-most and left-most columns
Determine which variables can be eliminated from the min-terms using the row and column entries of the Karnaugh map
Any variable that appears as both a “1” and a “0” in the grouping can be eliminated from the product term
For a complete expression, you need to have expressions for every “1” on the Karnaugh map and combine the terms in a “sum of products” expression
L14: Boolean Logic and Basic Gates
Fall 2005

39. Karnaugh Map Guidelines (3 of 3)
You can have a “1” included in more than one simplified expression
Redundancy is okay if you end up with the same or fewer terms in your sum of products expression
Combine the simplified expressions into a sum of products form
Translate the sum of products form into a logic gate design
L14: Boolean Logic and Basic Gates
Fall 2005

40. L14: Boolean Logic and Basic Gates
Another Example
Input x
Input y
Input z
Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
L14: Boolean Logic and Basic Gates
Fall 2005

41. Another Example with Three Variables
x y z 1
0 0
1 0
1 1
0 1
f(x,y,z) = z’ + x y z
L14: Boolean Logic and Basic Gates
Fall 2005