theory lecture 7

7/27/2019 Theory Lecture 7

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MCT-212: DIGITAL LOGIC

DESIGN


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EXCLUSIVE OR GATE

The output of theXOR

gate is high whenever the two

inputs are different.

A B X = A B

0 0 0

1 0 1

0 1 1

1 1 0


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EXCLUSIVE NOR GATE

The output of the XNOR gate is high whenever the twoinputs are identical

A B X =

0 0 0

1 0 1

0 1 1

1 1 0


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BOOLEAN ALGEBRA AND LOGIC

DESIGN

The need forBoolean Algebra for logic design?

A B X

0 0 1

1 0 0

0 1 1

1 1 1

= + +


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BOOLEAN ALGEBRA AND LOGIC

DESIGN

The need forBoolean Algebra for logic design?

A B X

0 0 1

1 0 0

0 1 1

1 1 1

= + +

= +

= +


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BOOLEAN ALGEBRAzero one zero one zero


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BOOLEAN ALGEBRA

Mathematics of digital systems.

Important in the analysis of digital circuits.

Binary Variable : A symbol (letter) used to

represent a logic quantity.

Example - binary variables and their values:

A = 1

Z = 0

Complement : Inverse of a variable.

A = 1 , A= 0 / (A = 0)


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BOOLEAN ADDITION

A literal is a variable or its complement.

The addition is equivalent to OR operation.

The sum of terms is 1, if one ore more literals

are 1.

The sum is zero is each literal is 0.

Determine the value of A, B, C that will

make the sum term 0. + + = ?A = 0, B = 1, C = 0.


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BOOLEAN MULTIPLICATION

In Boolean algebra, multiplication is equivalent toAND operation.

The product will be 1 only if all of the literals are

1.What are the values of A, B, C that will

make the product term . . =

A = 1, B = 0, C = 1


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BOOLEAN OPERATIONS

Addition

0 + 0 = 0

1 + 0 = 1

0 + 1 = 1

1 + 1 = 1

Multiplication

0 * 0 = 0

1 * 0 = 0

0 * 1 = 0

1 * 1 = 1


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LAWS AND RULES OFBOOLEAN ALGEBRAWhat rules?


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LAWS AND RULES OF BOOLEAN

ALGEBRA

Copyright 2006 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458All rights reserved

Floyd

Digital Fundamentals, 9/e


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COMMUTATIVE LAWS

Applied to addition and multiplication.

Commutative law for addition: In terms of the

result, the order in which variables are ORed

makes no difference.

+ = +

Commutative law for Multiplication: In terms

of the result, the order in which variables are

ANDed makes no difference.

. = .


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ASSOCIATIVE LAWS

Also applied to addition and multiplication. Associative law for addition: When ORing

more than two variables, the result is the same

regardless of the grouping of the variables.

+ + = + +

Associative law for Multiplication: When

ANDing more than two variables, the result is the

same regardless of the grouping of the variables.

. . = . .


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ASSOCIATIVE LAWS

Associative law for addition: + + = + +

Associative law for Multiplication:

. . = . .


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VENN DIAGRAMS

The rules of Boolean algebra can be explained usingVenn Diagrams.

The variable A is shown as an area.

The rule + = can be illustrated easily with a

diagram. Add an overlapping area to represent thevariable B.

The overlap region

between A and B

represents AB


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VENN DIAGRAMS

The rules of Boolean algebra can be explained usingVenn Diagrams.

The diagram visually shows that + =


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VENN DIAGRAMS

What does this famous diagram say?? =D


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RULES OF BOOLEAN ALGEBRA

RULE 1: + =

A B X = A+B

0 0 0

1 0 10 1 1

1 1 1OR Truth Table


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RULE 2: + =

A B X = A+B

0 0 0

1 0 10 1 1

1 1 1OR Truth Table


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A B X = A.B

0 0 0

1 0 00 1 0

1 1 1


RULE 3:. =

AND Truth Table


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A B X = A.B

0 0 0

1 0 00 1 0

1 1 1


RULE 4:. =

AND Truth Table


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RULE 5: + =

A B X = A+B

0 0 0

1 0 10 1 1

1 1 1OR Truth Table


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RULE 6: + =

A B X = A+B

0 0 0

1 0 10 1 1

1 1 1OR Truth Table


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RULE 7:. =

A B X = A.B

0 0 0

1 0 00 1 0

1 1 1AND Truth Table


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RULE 9: =


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RULE 10: + =

Proof using Truth Table?


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RULE 11: + = +



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RULE 12: ( + )( + ) = +

Proof using Venn Diagram?



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RULE 12: ( + )( + ) = +


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DEMORGANS THEOREM

DeMorgans 1st Theorem: The complement ofa product of variables is equal to the sum of the

complemented variables.

= + A B +

0 0 1 1

1 0 1 1

0 1 1 1

1 1 0 0


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DEMORGANS THEOREM

DeMorgans 2st Theorem: The complement ofa sum of variables is equal to the product of the

complemented variables. + = .

Use DeMorgans Theorem to solve the following:

. +

A B + .

0 0 1 11 0 0 0

0 1 0 0

1 1 0 0

SIMPLIFICATION USING BOOLEAN


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SIMPLIFICATION USING BOOLEAN

ALGEBRA

Dont look here, simplify the expression writtenon the board..!


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STANDARD FORMS OF

BOOLEAN EXPRESSIONSBasssss hogyi hai ab to :(

STANDARD EXPRESSIONS FROM


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TRUTH TABLE

Truth tables show allpossible inputs of a

function and the values

that the output takes for

all those inputs.

Given n inputs, there are

2 possible input

combinations.

A B C F(A,B,C)

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 0



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TRUTH TABLE

Domain of a Boolean expression: The set ofvariables contained in the expression in either

uncomplemented and complemented form

The sum-of-product (SOP) form

X = AB + CD + EF

Domain: A,B,C,D,E,F

The product of sum (POS) form = ( + + )( + + )

In both forms, the over bar cannot extend over more than

one variable.


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SUM OF PRODUCTS STANDARD FORM

In SOP standard form, every variable in thedomain must appear in each term. This form is

useful for constructing truth tables or for

implementing logic in PLDs.

You can expand a nonstandard term to standard

form by multiplying the term by a term consisting

of the sum of the missing variable and its

complement.

TRUTH TABLES TO MIN TERM


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TRUTH TABLES TO MIN-TERM

EXPRESSIONS

NOT, AND , and OR

How would you create a min-term

expression from a truth table?

The need for making standardexpressions?


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PRODUCT OF SUMS FORM

Similarly, a product-of-sums is formed by the product ofsums in which all the sums are formed by single variables

only. Examples:

(A + B)(C + D + E)(A + C + E)

(A + B)(C + D +E)F

Each of the sums in the product-of-sums form is called a

maxterm. Thus the form is also called product-of-

maxterms. The following expressions are not sum-of-products:

(A+B)CD + EF

A + B + C + DE


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PRODUCT OF SUMS STANDARD FORM

In POS standard form, every variable in thedomain must appear in each sum term of the

expression.

You can expand a nonstandard POS expressionto standard form by adding the product of the

missing variable and its complement.

Making POS from truth table?


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INTEGRATED CIRCUITGATES


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DIGITAL LOGIC FAMILIES

TTL Transistor Transistor Logic ECL Emitter Coupled Logic

MOS Metal Oxide Semiconductor

CMOS Complementary Metal Oxide

Semiconductor

Two major Fixed Function Logic families are

TTLand CMOS.


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PERFORMANCE CHARACTERISTICS

AND PARAMETERS

Propagation delay Time DC Supply Voltage (VCC)

Power Dissipation

Input and Output Logic Levels Speed-Power product

Fan-Out and Loading


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PROPAGATION DELAY

Delay in the output change after the inputchanges.


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FAN-OUT AND LOADING

Maximum number of ICs that can be driven fromthe output of a single IC.


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ANY QUESTIONS?

Anyone willing to present?

Time allowed : 5 mins

Topic : Any

Bonus Points : +3


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REFERENCES

Chapter no 1: Binary Systems Digi tal Lo gic Designby Morris Mano

Chapter no 1: Digital Concepts

Digi tal Fundamentalsby Floyd

theory lecture 7

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