Digital Electronics
Principles & Applications
Seventh Edition
Roger L. Tokheim
Chapter 4
Using Logic Gates
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
INTRODUCTION
• Logic Circuit from a Boolean expression
• Minterm and maxterm Boolean expressions
• Boolean expression from a truth table
• Truth table from a Boolean expression
• Simplifying Boolean expressions
• Karnaugh mapping
• NAND logic
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
INTRODUCTION (continued)
• Data selectors and their use
• Solving logic problems with data selectors
• Using computer simulations
• Variations in Boolean expressions
• DeMorgan’s theorem
• PLDs (programmable logic devices)
• BASIC Stamp® modules
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
“Tools of the Trade”
for Solving Logic Problems
• Gate symbols
• Truth tables
• Boolean expressions
Combinational logic circuits:
AND-OR pattern of gates
from
AND-OR pattern of gates
Sum-of-products Boolean expression such as:
AB + CD = Y
OR-AND pattern of gates
from
OR-AND pattern of gates
Product-of-sums Boolean expression such as:
(A+B) (C+D) = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Logic Circuit From
Boolean Expression
Example:
Draw the AND-OR logic diagram for the
Boolean expression: AB + CD = Y
Step 1: OR AB with CD
Step 2: Add top AND gate
Step 3: Add bottom AND gate
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Draw the OR-AND logic diagram for the Boolean
expression: (A+B) • (C+D) = Y
Step 1:
Step 2:
Step 3:
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Boolean Expressions
Sum-of-products form:
A•B + C•D =Y
Also called the minterm form
Product-of-sums form:
(A + B) • (C + D) = Y
Also called the maxterm form
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. The Boolean expression AB + BC = Y is
in __________ (product-of-sums, sum-ofproducts) form.
sum-of-products
2. The Boolean expression (A+B) (B+C)
+ Y is in __________ (product-ofsums, sum-of-products) form.
product-of-sums
3. The Boolean expression (A+B) (B+C) = Y is
in __________ (maxterm, minterm) form.
maxterm
4. The Boolean expression AB + BC = Y is in
__________ (maxterm, minterm) form.
minterm
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Boolean Expression from Truth Table
Write the Boolean expression that describes the logic in this truth table.
Truth Table
Input Output
ABC
Y
000
0
001
0
010
1
011
0
100
0
101
0
110
0
111
1
Step 1: Focus only on the truth table lines
with outputs of 1.
Step 2: AND the inputs for these two lines
and logically OR the ANDed groups.
A•B•C + A•B•C=Y
Minterm Boolean expression: A B C + A B C = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1
0
0
0
0
0
0
1
Write the Boolean expression that
describes the logic in this truth table.
A•B•C + A•B•C=Y
Minterm Boolean expression: A B C + A B C = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1. Write the sum-of-products Boolean
expression for this truth table.
______________________________
A’BC + ABC = Y
0
0
0
1
0
0
0
1
2. Write the minterm Boolean
expression for this truth table.
______________________________
A’BC + ABC = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Truth Table From Boolean Expressions
Fill in a truth table from a minterm Boolean Expression.
Minterm Boolean expression: A•B•C + A•B•C + A•B•C = Y
Step 1: Place three 1s
in output column.
Step 2: Place five 0s in
blanks in output column
of truth table.
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1
0
0
1
0
0
1
0
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Truth Table from Boolean Expressions
Fill in a truth table from a minterm Boolean Expression.
Minterm Boolean expression:
Step 1: Place single 1
output column for term
with three variables.
Step 2: Place two 1s in
output column for term
with two variables.
Step 3: Fill in 0s.
A•B + A•B•C =Y
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1
1
0
0
0
0
0
1
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1. Fill in the output column of the
truth table for the Boolean
expression A’B’C’ + ABC’ = Y.
1
0
0
0
0
0
1
0
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1. Fill in the output column of the
truth table for the Boolean
expression A’B’ + AB’C = Y.
1
1
0
0
0
1
0
0
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Simplifying Boolean Expressions
Truth Table
Input Output
ABC
Y
000
001
010
011
100
101
110
111
1
1
0
0
1
0
1
0
Unsimplified Boolean Expression
A•B•C + A•B•C + A•B•C + A•B•C = Y
Thisline
lineisisnot
nottotobebeconsidered
consideredininthe
theloop.
loop.
This
Simplified Expression: A•C + A•B = Y
RULE:
Eliminate term within loop that contains a term and its complement.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Simplify Boolean Expression
(Karnaugh map method)
Unsimplified Boolean expression (3 variables):
A•B•C + A•B•C + A•B•C + A•B•C = Y
Step 1: Plot 1s
_ _
AB
Step 2: Loop groups
Step 3: Eliminate variables
Step 4: Form simplified
minterm expression
_
C
C
C
1
1
_
AB
AB
1
_
AB
1
B
C
Simplified Expression: A C + A B = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Simplify Boolean Expression (Karnaugh map method)
Unsimplified Boolean expression (4 variables):
ABCD + ABCD + ABCD + ABCD = Y
Step 1: Plot 1s
__
CD
Step 2: Loop groups
_ _
AB
Step 3: Eliminate variables
_
AB
Step 4: Form simplified
minterm expression
AB
1
_
AB
1
B
_
CD
CD
_
CD
1
1
D
Simplified Expression: ACD + ABC = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Developing a NAND Logic Diagram
Minterm expression: AB + AB = Y
Step 1: Draw AND-OR
logic diagram from
minterm expression.
Step 2: Substitute NAND
gates for each inverter,
AND, and OR gate.
NOTE: Both logic diagrams will generate the same truth table.
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1-OF-8 Data Selector
Logic Symbol:
Data Selector
Data
Inputs
0
1
2
3
4
5
6
7
W
Output
C
B A
Data Select Inputs
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QUIZ
What is the output from the data selector?
1-of-8
Data Selector
Data
Inputs
1
0
0
1
0
1
1
0
Data Select Inputs:
0
1
2
3
4
W
5
6
7 C B A
?HIGH
HIGH
LOWH
10 10 10
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Use the data selector to perform the logic described in the truth table
Truth Table
1-of-8
Data Selector
C B A Y
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
0
1
2
3
4
W
5
6
7 C B A
?
HIGH
LOW
10 10 01
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Circuit Simulation - Using the Logic Converter
Conversion
options
Truth table
area
Boolean expression area
Converts back and forth from
Boolean expression,
truth table, and
Screen from Electronics Workbench® or MultiSIM®
logic diagram
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Truth Table to Boolean Expression
Step1: Click number of
variables.
Step2: Fill in truth table.
Step3: Click truth table to
Boolean expression
conversion option.
Unsimplified
Boolean Expression
appears here
A’B’C+A’BC+AB’C’+AB’C+ABC
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Simplifying the Boolean Expression
Step1: Click number of
variables.
Step2: Fill in truth table.
Step3: Click truth table to
simplified Boolean
expression conversion
option.
Simplified
Boolean Expression
appears here
AB’ + C
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Boolean Expression to Logic Diagram
Step1: Type the Boolean
expression.
Step2: Click Boolean expression
to logic diagram
conversion option.
Step3: Close Logic Converter
window.
AB’+C=Y is converted
into this AND-OR logic diagram
by the simulation software
AB’ + C
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. A circuit simulator such as Electronic
Workbench® or MultiSIM® contain a
useful instrument called a __________
(logic converter, logic probe) used to
convert back and forth from truth table
to Boolean expression.
2. The logic converter instrument from
Electronic Workbench® or MultiSIM®
can convert back and forth from logic
diagram to Boolean expression. (True
or False)
logic converter
True
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Variations in Boolean Expressions
Boolean expression (textbook style):
A BA++CBD==YY
A B=Y
Boolean expression (keyboard style):
A’B
CD’
=
Y
(A+
+
B)’
=
A’ B’ = YY
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. The Boolean expression A B + C D = Y
is written as _______ in keyboard style.
A B’ + C’D = Y
2. The Boolean expression A+B+C = Y is
written as _______ in keyboard style.
(A+B+C)’ = Y
3. The Boolean expression A B C = Y is
written as _______ in keyboard style.
A’ B’ C’ = Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
DeMorgan’s Theorems
First theorem:
A+B=A
B
Both used
to eliminate
Second
long overbars
theorem:
A B=A+B
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
DeMorgan’s Theorems
First theorem:
Second theorem:
A+B=A
B
A B=A+B
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
DeMorgan’s Theorems
START >
A B=Y
Step 1: Change all ORs to ANDs and all ANDs to ORs.
A
B=Y
Step 2: Complement each individual variable
(short overbar).
A+B=Y
Step 3: Complement the entire function
(long overbar).
A+B=Y
Step 4: Eliminate all groups of double overbars.
FINISHED (alternative NAND expression)
(click to see final Boolean expression)
A+B=Y
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. The statement A B = A + B is DeMorgan’s
__________ (first, second) theorem and
suggests converting from a NAND to an OR
situation.
2. The statement A + B = A B is DeMorgan’s
__________ (first, second) theorem and
suggests converting from a NOR to an AND
situation.
second
first
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PLDs
• Programmable Logic Device (PLD) - The generic
name for an IC that can be programmed by the
user to execute a complex logic function.
• PLDs have many inputs and outputs. PLDs can
be used to implement minterm Boolean
expressions using AND-OR logic. More
complex PLDs have latches and other memory
devices.
• PLDs have several advantages. PLDs are
inexpensive, can be programmed by the user,
and very reliable.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
PLDs (continued)
PLD is generic for a programmable logic device, but
are also known by many other names including:
• PAL - programmable array logic
• GAL - generic array logic
• FPL - fuse-programmable logic
• PLA - programmable logic array
• PEEL - programmable electrically erasable logic
• FPGA - field programmable gate array
• CPLD - complex programmable logic device
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. PLD is the generic name for a __________
(peripheral logic driver, programmable logic device)
which is an IC that can solve logic problems.
programmable
logic device
2. PLDs commonly have many inputs but always
have only one output. (True or False)
False
3. PLDs have several advantages including they can
be programmed in the local lab or school, are very
reliable, and inexpensive. (True or False)
True
4. Various manufacturers might refer to PLDs by
acronyms like TTLs, CMOSs, PALs, ASCIIs,
ASVABs, CPLSs, FPGAs, PEELs, or GALs. (True
or False)
False
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Programming Logic Functions
using the BASIC Stamp® Module
•One trend in electronics is the development of
programmable ICs.
•A second trend in electronics is the use of billions of
inexpensive imbedded devices called microcontrollers.
•A popular microcontroller used by schools is the BASIC
Stamp® module, by Parallax, Inc.
•BASIC Stamp® modules are programmable using a
student-friendly version of B.A.S.I.C.
•Solving a logic problem using a microcontroller (BASIC
Stamp® 2 module) is detailed on the next slide.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Using the BASIC Stamp® Module
Truth Table
Step 3: Disconnect PC from BS2 module
INPUTS
RED
GREEN YELLOW
Step 1: Write program on PC in PBASIC.
AKey
B lines
C of code
Y1 are:
Y2
Y3
0 0Y1=(A&B&C)|(~A&~B&~C)
0
1
1
1
0 0Y2=(~A&~B)|(A&C)
1
0
1
1
0 1Y3=(~A)|(~B&C)
0
0
0
1
0 1 1
0
0
1
1 0 0
0
0
0
1 0 1
0
1
1
1 1 0
0
0
0
1 1 1
1
1
0
Logic Expressions in PBASIC:
Y1= (A&B&C)|(~A&~B&~C)
Step
Download
program
from
Step 4: Power
OFF
and ON
(BS2 module).
Y2= 2:
(~A&~B)|(A&C)
PC
BS2 module.
Manipulate
andto
observe
outputs.
Y3=inputs
(~A)|(~B&C)
Digital Inputs
A
Input
AA==LOW
Input
LOW
Input
A
=LOW
HIGH
Input
B
=
Input
B
=
LOW
Input
= LOW
Input
CCB
==LOW
Input
Input C =HIGH
LOW
B
C
Digital Outputs
Y1
HIGH
LOW
LOW
Y2
HIGH
HIGH
LOW
Y3
HIGH
HIGH
LOW
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. The BASIC Stamp® module contains an inexpensive
programmable IC called a _____.
microcontroller
2. The BASIC Stamp® module is programmed using
a version of B.A.S.I.C. on a _____ (modem, PC)
and then downloaded into the BS2 module.
PC (Personal Computer)
3. A BASIC Stamp® (such as the BS2 module) will always
have many inputs but only one output. (True or False)
4. Programming the BASIC Stamp® module is done using
an easy-to-use language called _____ (PBASIC, MICRO).
False
PBASIC
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
REVIEW
• Logic Circuit from a Boolean expression
• Minterm and maxterm Boolean expressions
• Boolean expression from a truth table
• Truth table from a Boolean expression
• Simplifying Boolean expressions
• Karnaugh mapping
• NAND logic
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
REVIEW (continued)
• Data selectors and their use
• Solving logic problems with data selectors
• Using computer simulations
• Variations in Boolean expressions
• DeMorgan’s theorem
• PLDs (programmable logic devices)
• BASIC Stamp® modules
©2008 The McGraw-Hill Companies, Inc. All rights reserved.