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– 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

0 0 0 0 1

0 1 0 1 1

1 0 0 1 0

1 1 1 1 0

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

• Analysis– 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

1 0 0 1 0 0 1 1

1 1 0 0 0 1 0 1

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

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

Truth Table

f(X1,X2) = !X1.!X2 + X1

X1 X2 A B C D f(X1,X2)

0 0 1 1 1 0 1

0 1 1 0 0 0 0

1 0 0 1 0 1 1

1 1 0 0 0 1 1

Timing Diagram

X1

X2

A

B

C = A.B

f(X1,X2)

1

0

1

0

1

0

1

0

1

0

1

0

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 . 0 = 0

1b. 1 + 1 = 1

2a. 1 . 1 = 1

2b. 0 + 0 = 0

3a. 0 . 1 = 1 . 0 = 0

3b. 1 + 0 = 0 + 1 = 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 PropertiesCommutative

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

En Wo AlEn Wo Al0 00 11 01 1

En Wo Al0 0 00 1 01 0 01 1 1

Write Sum of Products or Product of Sums

f = En . Wo = Al

En Wo Al0 0 00 1 01 0 01 1 1

EnWo Al

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)

00 0 0 1

01 0 1 0

10 1 0 1

11 1 1 1

!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)

00 0 0 1

01 0 1 0

10 1 0 0

11 1 1 1

X1 + !X2 – maxterm

!X1 + X2 – maxterm

Using De Morgan’s Theoremto 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)

00 0 0 1

01 0 1 0

10 1 0 0

11 1 1 1

Minterm vs Maxterm• Σ = Minterm – rows where f(x) = 1

• П = Maxterm – rows where f(x) = 0

Row # X1 X2 f(X1,X2)

00 0 0 1

01 0 1 0

10 1 0 1

11 1 1 1

!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

!X1!X3(!X2 + X2) + X2X3(!X1 + X1)

!X1!X3 + X2X3

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 Gatesvia DeMorgan’s Theorem

!(X1 . X2) = !X1 + !X2 = !X1 + !X2

!(X1 + X2) = !X1 . !X2 = !X1 . !X2

NAND gates in Sum of ProductsNOR gate in Product of Sums

Sum of Products

Product of Sums

Design Example

• Three-Way Light Control– Did you understand the book’s example?

• Multiplexer Circuit

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• 2.1-2.4, 2.6, 2.8, 2.9, 2.10

• 2.28-2.30 Create Truth Table, attempt to minimize, and write SOP & POS

• 2.31-2.34 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 email Schematic and Simulator Waveform

Schematic Design

Poor

Good