Chap4_bekg1113
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Chap4_bekg1113 as PDF for free.
More details
- Words: 985
- Pages: 35
CHAPTER 4: DIGITAL ELECTRONIC
Basic Logic Gates
Logic gate is the most basic type of digital circuit, which consists of two or more inputs and one output.
It has only two states, either “1” or “0”.
A gate can be used alone to perform a logic function.
It can also be connected to several other gates to form a logic network.
Basic gates are the NOT, AND, OR, XOR, XNOR, NAND & NOR.
BASIC LOGIC GATES
NOT gate / Inverter
The output is the inverse of the input. A
Boolean equation: Truth table:
Y= A
YA Input, A
Output, Y
0
1
1
0
BASIC LOGIC GATES
OR gate The output is 1 if any or all of the inputs are 1. A F B
Boolean equation: Truth table:
F A B Input
Output
A
B
F
0
0
0
0
1
1
1
0
1
1
1
1
BASIC LOGIC GATES
AND gate
The output is 1 only if all the inputs are 1. A F B
F A B AB
Boolean equation: Truth table:
Input
Output
A
B
F
0
0
0
0
1
0
1
0
0
1
1
1
BASIC LOGIC GATES
NOR gate
The output is 1 only if all of the inputs are 0.
A
A
=
F B
F B
Boolean equation:
Truth table:
F A B Input
Output
A
B
F
0
0
1
0
1
0
1
0
0
1
1
0
BASIC LOGIC GATES
NAND gate The output is 1 if any or all the inputs are 0. A
=
F
B
F
B
F AB
Boolean equation: Truth table:
A
Input
Output
A
B
F
0
0
1
0
1
1
1
0
1
1
1
0
BASIC LOGIC GATES
X-OR GATE (Exclusive-OR gate)
The output is 1 only if an odd number of the inputs are 1. A B
F
Boolean equation:
F AB AB A B
Truth table:
Input
Output
A
B
F
0
0
0
0
1
1
1
0
1
1
1
0
BASIC LOGIC GATES
X-NOR GATE (Exclusive-NOR gate) The output is 1 only if an odd number of inputs are 0. A B
Boolean equation:
F
F AB AB A B
Truth table:
Input
Output
A
B
F
0
0
1
0
1
0
1
0
0
1
1
1
BOOLEAN EXPRESSIONS IMPLEMENTING CIRCUIT FROM BOOLEAN EXPRESSION
When a logic is defined by a Boolean expression, we can draw the circuit directly from the expression.
If an expression contains both AND and OR operation, the AND operations are performed first.
Unless there are parentheses in the expression, where the operation inside the parentheses is to be performed first.
BOOLEAN EXPRESSIONS
EXAMPLE: Given, F = ( A + B) • C so the diagram is performed as: A B
(A + B ) C
F= (A + B )·C
Given, F = ( A + B )( B + C ) the diagram is performed as:
BOOLEAN EXPRESSIONS
Evaluating Logic-Circuit Outputs Given A=0, B=1, C=1, D=1 and the Boolean expression as below. Determine the output for the expression.
F = ABC ( A + D ) Solution:
F ABC ( A D) 0 1 1(0 1) 1(0) 0
BOOLEAN EXPRESSIONS
Determining Output Level From A Diagram Determining the output level from a circuit diagram, where the inputs are A=0, B=1, C=1,D=1
BOOLEAN THEOREMS
Boolean theorems can be used to express logic circuit operations mathematically.
It can help us to simplify logic expression, therefore logic circuit.
There are two methods to simplify a complicated logic circuit.
Boolean algebra & DeMorgan’s theorem Karnaugh Map
The most important of Boolean algebra rules and laws are presented in the following section.
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Multivariable theorems (10a) (10b) (11) (12) (13a) (13b) (14) (15a) (15b)
A+B = B+A A•B = B•A A+(B+C) = (A+B)+C A(BC)=(AB)C A(B+C) = AB+AC (A+B)(C+D)=AC+BC+AD+BD A+AB = A A+AB=A+B A+AB = A+B
BOOLEAN THEOREMS
DeMorgan’s Theorem DeMorgan’s theorems are very useful in simplifying expressions for easy transfer back and forth from the product of variables to the sum of variables form.
It allows for elimination of overbar(s) that are over several variables.
The two theorems are:
First theorem The inverted of a product of variables is equal to the sum of the inverted individually variable.
A⋅ B = A + B
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Second theorem The inverted of a sum of variables is equal to the product of the inverted individual variable.
A + B = A⋅ B
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Applying Demorgans Theorems The following procedure illustrates the applications of DeMorgan’s theorems and Boolean Algebra to the specific expression:
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
SIMPLIFICATION USING BOOLEAN ALGEBRA
Apply the laws, rules and theorems of Boolean algebra to simplify general expressions.
SIMPLIFICATION USING BOOLEAN ALGEBRA
SIMPLIFICATION USING BOOLEAN ALGEBRA
UNIVERSALITY OF NAND GATES & NOR GATES
All Boolean expressions consists of various combinations of the basic operations of OR, AND, and INVERTER.
Therefore, any expression can be implement using combinations of OR gates, AND gates, and INVERTERs.
BOOLEAN THEOREMS
BOOLEAN THEOREMS
IEEE/ANSI STANDARD LOGIC SYMBOLS
The logic symbols used we have used so far in this chapter are the traditional standard symbols used in the digital industry for many, many years.
A newer standard for logic symbols was developed in 1984; it is called the IEEE/ANSI Standard 91-1984 for logic symbols.
The IEEE/ANSI standard uses rectangular symbols to represent all logic gates and circuits.
A special dependency notation inside the rectangular symbol indicates how the device outputs depend on the device inputs.
Basic Logic Gates
Logic gate is the most basic type of digital circuit, which consists of two or more inputs and one output.
It has only two states, either “1” or “0”.
A gate can be used alone to perform a logic function.
It can also be connected to several other gates to form a logic network.
Basic gates are the NOT, AND, OR, XOR, XNOR, NAND & NOR.
BASIC LOGIC GATES
NOT gate / Inverter
The output is the inverse of the input. A
Boolean equation: Truth table:
Y= A
YA Input, A
Output, Y
0
1
1
0
BASIC LOGIC GATES
OR gate The output is 1 if any or all of the inputs are 1. A F B
Boolean equation: Truth table:
F A B Input
Output
A
B
F
0
0
0
0
1
1
1
0
1
1
1
1
BASIC LOGIC GATES
AND gate
The output is 1 only if all the inputs are 1. A F B
F A B AB
Boolean equation: Truth table:
Input
Output
A
B
F
0
0
0
0
1
0
1
0
0
1
1
1
BASIC LOGIC GATES
NOR gate
The output is 1 only if all of the inputs are 0.
A
A
=
F B
F B
Boolean equation:
Truth table:
F A B Input
Output
A
B
F
0
0
1
0
1
0
1
0
0
1
1
0
BASIC LOGIC GATES
NAND gate The output is 1 if any or all the inputs are 0. A
=
F
B
F
B
F AB
Boolean equation: Truth table:
A
Input
Output
A
B
F
0
0
1
0
1
1
1
0
1
1
1
0
BASIC LOGIC GATES
X-OR GATE (Exclusive-OR gate)
The output is 1 only if an odd number of the inputs are 1. A B
F
Boolean equation:
F AB AB A B
Truth table:
Input
Output
A
B
F
0
0
0
0
1
1
1
0
1
1
1
0
BASIC LOGIC GATES
X-NOR GATE (Exclusive-NOR gate) The output is 1 only if an odd number of inputs are 0. A B
Boolean equation:
F
F AB AB A B
Truth table:
Input
Output
A
B
F
0
0
1
0
1
0
1
0
0
1
1
1
BOOLEAN EXPRESSIONS IMPLEMENTING CIRCUIT FROM BOOLEAN EXPRESSION
When a logic is defined by a Boolean expression, we can draw the circuit directly from the expression.
If an expression contains both AND and OR operation, the AND operations are performed first.
Unless there are parentheses in the expression, where the operation inside the parentheses is to be performed first.
BOOLEAN EXPRESSIONS
EXAMPLE: Given, F = ( A + B) • C so the diagram is performed as: A B
(A + B ) C
F= (A + B )·C
Given, F = ( A + B )( B + C ) the diagram is performed as:
BOOLEAN EXPRESSIONS
Evaluating Logic-Circuit Outputs Given A=0, B=1, C=1, D=1 and the Boolean expression as below. Determine the output for the expression.
F = ABC ( A + D ) Solution:
F ABC ( A D) 0 1 1(0 1) 1(0) 0
BOOLEAN EXPRESSIONS
Determining Output Level From A Diagram Determining the output level from a circuit diagram, where the inputs are A=0, B=1, C=1,D=1
BOOLEAN THEOREMS
Boolean theorems can be used to express logic circuit operations mathematically.
It can help us to simplify logic expression, therefore logic circuit.
There are two methods to simplify a complicated logic circuit.
Boolean algebra & DeMorgan’s theorem Karnaugh Map
The most important of Boolean algebra rules and laws are presented in the following section.
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Multivariable theorems (10a) (10b) (11) (12) (13a) (13b) (14) (15a) (15b)
A+B = B+A A•B = B•A A+(B+C) = (A+B)+C A(BC)=(AB)C A(B+C) = AB+AC (A+B)(C+D)=AC+BC+AD+BD A+AB = A A+AB=A+B A+AB = A+B
BOOLEAN THEOREMS
DeMorgan’s Theorem DeMorgan’s theorems are very useful in simplifying expressions for easy transfer back and forth from the product of variables to the sum of variables form.
It allows for elimination of overbar(s) that are over several variables.
The two theorems are:
First theorem The inverted of a product of variables is equal to the sum of the inverted individually variable.
A⋅ B = A + B
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Second theorem The inverted of a sum of variables is equal to the product of the inverted individual variable.
A + B = A⋅ B
BOOLEAN THEOREMS
BOOLEAN THEOREMS
Applying Demorgans Theorems The following procedure illustrates the applications of DeMorgan’s theorems and Boolean Algebra to the specific expression:
BOOLEAN THEOREMS
BOOLEAN THEOREMS
BOOLEAN THEOREMS
SIMPLIFICATION USING BOOLEAN ALGEBRA
Apply the laws, rules and theorems of Boolean algebra to simplify general expressions.
SIMPLIFICATION USING BOOLEAN ALGEBRA
SIMPLIFICATION USING BOOLEAN ALGEBRA
UNIVERSALITY OF NAND GATES & NOR GATES
All Boolean expressions consists of various combinations of the basic operations of OR, AND, and INVERTER.
Therefore, any expression can be implement using combinations of OR gates, AND gates, and INVERTERs.
BOOLEAN THEOREMS
BOOLEAN THEOREMS
IEEE/ANSI STANDARD LOGIC SYMBOLS
The logic symbols used we have used so far in this chapter are the traditional standard symbols used in the digital industry for many, many years.
A newer standard for logic symbols was developed in 1984; it is called the IEEE/ANSI Standard 91-1984 for logic symbols.
The IEEE/ANSI standard uses rectangular symbols to represent all logic gates and circuits.
A special dependency notation inside the rectangular symbol indicates how the device outputs depend on the device inputs.
More Documents from "am2030"
Chap4_bekg1113
April 2020 5
Chap6_bekg1113
April 2020 5
Chemistry 7 Thermochemistry
April 2020 8
Chap5_bekg1113
April 2020 3
Atom, Molecules And Ions
April 2020 5