# Introduction to Logic Circuits

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Introduction to Logic Circuits
Chapter 2 Introduction to Logic Circuits

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

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

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

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

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

Logic Gates AND OR NOT (Inverter)

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

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

Logic Analysis Truth Table Timing Diagram

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

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

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

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

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?

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

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

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

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)

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

Venn Diagram

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

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

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

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

Create Truth Table

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

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

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

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

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)

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

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

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

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)

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

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

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

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

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

Schematic Design Poor Good

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