1. Introduction to Logic Circuits
Chapter 2
Introduction to
Logic Circuits

2. Objectives Know what Truth Tables are
Know the Truth Tables for the Basic Gates
AND, OR, NOT, NAND, NOR
Know how to Analyze Simple Logic Circuits
Via Timing Diagrams and Truth Tables
Be able to use Boolean Algebra to manipulate simple digital logic equations
Know what a Venn diagram is and how they apply to digital logic
Know how to implement a function in both Sum of Products and Product of Sums form
Know how to use Quartus II’s schematic entry tool to describe simple logic circuits
Know how to use Quartus II to simulate a logic circuit

3. Variables and Functions
Consider a Flashlight
L is the Function that represents the Flashlight
L = 0 when light is off, L = 1 when light is on
x is the Variable that represents the switch
Switch open x = 0, switch closed x = 1
x is an input Variable
When x = 0 L = 0
When x = 1 L = 1
L(x) = x

4. AND & OR Flashlight controlled by two switches in series = AND
L(x1,x2) = x1.x2
Flashlight controlled by two switches in parallel = OR
L(x1,x2) = x1+x2
AND and OR two very important building blocks

5. Inversion (NOT) Light on if switch open Light off if switch closed
L = 1 if x = 0
Light off if switch closed
L = 0 if x = 1
L(x) = x
x = x’ = !x = ~x = NOTx

6. Truth Tables
x1 x2 x1.x2 x1+x2 !x1
AND OR NOT

7. Logic Gates
AND OR
NOT (Inverter)

8. Logic Networks or Logic Circuits
Using AND, OR, and NOT logic functions of any complexity can be implemented
Number of gates/complexity of logic network is directly related to its cost
Reducing cost is always desirable
Reducing complexity is always desirable

9. Analysis of Logic Circuit
Determine how an existing Logic Circuit functions
Synthesis
Design a new network that implements a desired Logic Function

10. Logic Analysis
Truth Table
Timing Diagram

11. Truth Table X1 X2 A B C D E f(X1,X2) 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0
f(X1,X2) = !X1.!X2 + X1.X2 + X1.!X2

12. Timing Diagram X1 X2 A B C = A.B D = X1.X2 E = X1.B f(X1,X2) 1 1 1 1 11 1 1 1 1 1 1

14. Truth Table X1 X2 A B C D f(X1,X2) 0 0 1 1 1 0 1 0 1 1 0 0 0 0
f(X1,X2) = !X1.!X2 + X1

15. Timing Diagram X1 X2 A B
C = A.B
f(X1,X2) 1 1 1 1 1 1

16. Functionally Equivalent Circuits
= !X1.!X2 + X1.X2 + X1.!X2
= !X1.!X2 + X1
!X1.!X2 + X1.X2 + X1.!X2 = !X1.!X2 + X1
Two equations result in same output – Which is better?

17. Boolean Algebra
In Boolean Algebra elements can take on one of two values, 0 and 1
Axioms of Boolean Algebra
1a = 0
1b = 1
2a = 1
2b = 0
3a = = 0
3b = = 1
4a. If x = 0 then !x = 1
4b. If x = 1 then !x = 0

18. Single-Variable Theorems
5a. x . 0 = 0
5b. x + 1 = 1
6a. x . 1 = x
6b. x + 0 = 0
7a. x . x = x
7b. x + x = x
8a. x . !x = 0
8b. x + !x = 1
9 !!x = x
Validity easy to prove by substituting x = 0 and x = 1 and using the Axioms

19. Duality Dual of an expression is achieved by
Replacing all ANDs with ORs and all ORs with ANDs
Replacing all 1s with 0s and all 0s with 1s

20. Two and Three Variable Properties
Commutative
10a. X . Y = Y . X
10b. X + Y = Y + X
Associative
11a. X . (Y . Z) = (X . Y) . Z
11b. X + (Y + Z) = (X + Y) + Z
Distributive
12a. X . (Y + Z) = X . Y + X . Z
12b. X + Y . Z = (X + Y) . (X + Z)
Absorption
13a. X + X . Y = X
13b. X . (X + Y) = X
Combining
14a. X . Y + X . !Y = X
14b. (X + Y) . (X + !Y) = X
DeMorgan’s Theorem
15a. !(X . Y) = !X + !Y
15b. !(X + Y) = !X . !Y
16a. X + !X . Y = X + Y
16b. X . (!X + Y) = X . Y
Consensus
17a. X . Y + Y . Z + !X . Z = X . Y + !X . Z
17b. (X + Y) . (Y + Z) . (!X + Z) = (X + Y) . (!X + Z)

21. Algebraic Manipulation
!X1 . !X2 + X1 . X2 + X1 . !X2 =? !X2 + X1
!X1 . !X2 + X1 . (X2 + !X2)
Via 12a. Distributive (X . Y) + (X . Z) = X . (Y + Z)
!X1 . !X2 + X1 . (1)
Via 8b. X + !X = 1
!X1 . !X2 + X1 = !X1 . !X2 + X1
Via 6a. X . 1 = X
!X2 + X1
Via 16a. X + !X . Y = X + Y

22. Venn Diagram

23. !X1 . !X2 + X1 . X2 + X1 . !X2
!X2 + X1

24. Notation, Terminology, & Precedence
+ versus + called sum
. versus . called product
= AND
= OR
NOT, AND, OR Order of Precedence

25. Synthesis or Design
If the alarm is enabled and the window is open the alarm should sound
Assign variable names and functions
Create truth table
Write Sum of Products or Product of Sums
Create Schematic or HDL

26. Assign Variable Names and Functions
Alarm Enabled = En
Window Open = Wo
Alarm Sound = Al

27. Create Truth Table

28. Write Sum of Products or Product of Sums
f = En . Wo = Al En Al Wo

29. Cost
Total number of gates
Plus
Total number of inputs to all gates

30. Sum of Products Uses minterms to express the function
f(X1,X2) = Σ(m0,m2,m3) = !X1 . !X2 + X1 . !X2 + X1 . X2
Canonical Sum of Products
Note: Brown uses Σm(0,2,3) as a simple form
Using Boolean logic theorems and properties the Canonical Sum of Products expression can be manipulated to produce
f(X1,X2) = !X2 + X1
Which is a Minimum Cost Sum of Products expression of f
Row # X1 X2 f(X1,X2)
!X1 . !X2 - minterm
X1 . !X2 - minterm
X1 . X2 - minterm

31. Product of Sums Uses Maxterms to express function
f(X1,X2) = П(M1) = (X1 + !X2) . (!X1 + X2)
Canonical Product of Sums
Note: Brown uses П M(1) as a simple form
Also Minimum Cost Product of Sums in this case
Row # X1 X2 f(X1,X2)
X1 + !X2 – maxterm
!X1 + X2 – maxterm

32. Using De Morgan’s Theorem to generate Product of Sums
!f(X1,X2) = (!X1 . X2) + (X1 . !X2)
!(!f(X1,X2)) = !((!X1 . X2) + (X1 . !X2))
f(X1,X2) = !(!X1 . X2) . !(X1 . !X2)
f(X1,X2) = (X1 + !X2) . (!X1 + X2)
Row # X1 X2 f(X1,X2)

33. Minterm vs Maxterm Σ = Minterm – rows where f(x) = 1
П = Maxterm – rows where f(x) = 0
Row # X1 X2 f(X1,X2)
!X1 . !X2 - minterm
X1 + !X2 - maxterm
X1 . !X2 - minterm
X1 . X2 - minterm
f(X1,X2) = Σ(m0,m2,m3) = !X1 . !X2 + X1 . !X2 + X1 . X2
f(X1,X2) = П(M1) = X1 + !X2

34. Further Examples f(m0,m2,m3,m7) Truth table Sum of Products
Product of Sums
Minimization
Schematic
Timing Diagram

35. Sum of Products !X1!X2!X3 + !X1X2!X3 + !X1X2X3 + X1X2X3

36. Product of Sums (X1+X2+!X3)(!X1+X2+X3)(!X1+X2+!X3)(!X1+!X2+X3)
((X1 +!X3)+X2)((!X1+X3)+X2)((!X1+X3)+X2)((!X1+!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+!X2)
A B B C C D
A B C D
A C B D
((X1 +!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+X2)((!X1+X3)+!X2)
((X1 +!X3)+X2)((!X1+!X3)+X2)((!X1+X3)+X2!X2)

37. NAND and NOR Gates via DeMorgan’s Theorem
!(X1 . X2) = !X1 + !X = !X1 + !X2
!(X1 + X2) = !X1 . !X = !X1 . !X2

38. NAND gates in Sum of Products NOR gate in Product of Sums

39. Design Example Three-Way Light Control Multiplexer Circuit
Did you understand the book’s example?
Multiplexer Circuit

40. Laboratory Preparation
Reading
Chapter 2 – Omit section 2.10
Laboratory Preparation
Quartus II Introduction Using Schematic Design
ftp://ftp.altera.com/up/pub/Altera_Material/QII_9.0/Digital_Logic/DE2/Tutorials/tut_quartus_intro_schem.pdf

41. Homework Problems
, 2.6, 2.8, 2.9, 2.10
Create Truth Table, attempt to minimize, and write SOP & POS
Create Truth Table, attempt to minimize, and write required form
2.35 Create Truth Table
2.44, 2.45 Print or save as jpg and Schematic and Simulator Waveform

42. Schematic Design
Poor
Good