ee - 4 kmap logic simplification

8/3/2019 EE - 4 Kmap Logic Simplification

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Slide 1

Electronic Engineering 1

BENGEEE-SHU-1B


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Slide 2

Digital Electronics

Boolean Algebra and Logic SimplificationBoolean operations and expressions

Laws and rules of Boolean algebra

DeMorgans TheoremKarnaugh mapSOP/POS simplification

Dont care conditions

5 variables


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Slide 3

Converting product terms to STD SOP

Step 1: Multiply each non std product term by a termmade up of the sum of a missing variable and itscomplement.

Step 2: Repeat Step 1 until all terms contain all literals.The number product terms is doubled for eachmissing variable.

CBACABABC

CBACCABCBAAB

Literal C ismissing

The product

term is doubled

Equivalent to 1(always TRUE)


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Slide 4

Converting sum terms to STD POS

CBACBACBACBACCBACBABA

Literal C is

missing

The sum term is doubled

Equivalent to 0(always FALSE)

Step 1: Add to each non std sum term with a term madeup of the product of a missing variable and itscomplement.

Step 2: Apply Rule 12; A + BC = (A+B) (A+C)

Step 3: Repeat Step 1 until all terms contain all literals.

The number sum terms is doubled for eachmissing variable.


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FloydDigital Fundamentals, 9/e

Copyright 2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Slide 5

The Karnaugh Map


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Slide 6

Karnaugh map

Simplification of Combinational Logic Circuits usingKarnaugh Map

The Karnaugh map provides a systematic method forsimplifying Boolean expressions.

If properly used, will produce the simplest SOP or POSexpression possible, known as the minimum expression.


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Karnaugh map

A Karnaugh map is an array of cells in which each cell

represents a binary value of the input variables.

The cells are arranged in a way so that simplification ofa given expression is simply a matter of properlygrouping the cells.

Karnaugh maps can be used for expressions with two,three, four, and five variables.

The number of cells in a K-map is equal to the total

number of possible input variable combinations, i.e. thenumber of rows in a truth table.


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2-variable Karnaugh map

The 2-variable K-map is an array of 4 cells as shown below. In

part (b), it shows the standard products terms that arerepresented by each cell in the K-map.

BA

0 1

0

1

BA BA

BA AB


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Karnaugh map

The 3-variable K-map is an array of 8 cells as shown. In

part (b), it shows the standard products terms that arerepresented by each cell in the K-map.

CBA CBA

CBA CBA

CBA ABC

CBA CBA

C

AB0 1

00

01

11

10


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Karnaugh map

The 4-variable K-map is an array of 16 cells as shown. In part(b), it shows the standard products terms that are

represented by each cell in the K-map.

Take note ofthe order of

the inputbinary values


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Karnaugh map

Cell Adjacency

The cells in a K-map are arranged so that there is only asingle-variable change between adjacent cells. Adjacencyis defined by a single-variable change as shown.

Cells that differ by only one variable are adjacent.

Cells with values that differ by more than one variable arenot adjacent.


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Karnaugh mapExample of a 3 variable K Map

CAB 0 1

00 000 001

01 010 011

11 110 111

10 100 101

Identify the cells adjacent to the 010 cell.

The 010 cell is adjacent to the 000 cell, the 011 cell andthe 110 cell.


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Mapping a Standard SOP Expression

For an SOP expression in standard form, a 1 is placed

on the K-map for each product term in the expression.

Each 1 is placed in a cell corresponding to the value of aproduct term.


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Karnaugh mapExample 2

Map the following truth tables onto a K-map

Input Output

A B X

0 0 1

0 1 0

1 0 0

1 1 1

B

A 0 1

0 1 0

1 0 1


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Karnaugh map

C

AB0 1

00 0 0

01 1 0

11 0 0

10 1 0

A B C Output, X

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 0

Example 2 (continue)



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Karnaugh map

A B C Output, X

0 0 0 1

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 01 1 0 1

1 1 1 0

CAB 0 1

00 1 1

01 1 011 1 0

10 0 0

Example 2 (continue)



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K-Map Simplification of SOP Expression

To obtain a minimum SOP expression using K-map:

Grouping the 1s

A group must contain either 1, 2, 4, 8 etc, which are allpower of two.

The goal is to maximize the size of the groups and tominimize the number of groups. Always include the largestpossible number of 1s in a group.

Each 1 on the map must be included in at least one group.

The 1s already in a group can be included in another groupas long as the overlapping groups include non common 1s.

Stop when all 1s has been designated to at least a group.


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Karnaugh map

2. Determining the min SOP expression from K-map

Each group of cells containing 1s creates one productterm composed of all variables that occur in only oneform (either uncomplemented or complemented) withinthe group.

Variables that occur both uncomplemented andcomplemented within the group are eliminated.

These are called contradictory variables.


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Karnaugh map

2. Determining the min SOP expression from K-map

(continue)Determine the minimum product term for each group.

For a 3-variable map:

A 1-cell group yields a 3-variable product term

A 2-cell group yields a 2-variable product term

A 4-cell group yields a 1-variable term

An 8-cell group yields a value of 1 for the expression

3. When all the minimum product terms are derivedfrom the K-map, they are summed to form theminimum SOP expression.


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Karnaugh map

Example 3

Determine the product terms for the K-map and write theresulting minimum SOP expression.

CBA BC

AB

B

CA

AC

K-map iscontiguous


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Karnaugh map

Example 4

Group the following K maps and write the simplified expression

C

AB 0 1

00 1 1

01 1 0

11 0 0

10 1 1

C

AB 0 1

00 1 101 0 0

11 1 1

10 1 1

C

AB 0 1

00 0 101 0 1

11 0 1

10 1 0

CAB BA CBABCCA


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Slide 23

Karnaugh mapExample 5

Use a K map to minimize the following standard SOP

expression

CBACBACBABCACBA

BCA CAB 0 1

00 1 1

01 0 1

11 0 0

10 1 1


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Slide 24

Karnaugh map

Example 6

The table shows truth table for output X

A B C X

0 0 0 1

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 1

Write the STANDARDsum ofproducts, SOP, logic expression for

the output X in terms of A, B & C.

ABCCBACBABCACBAX


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Slide 26

Mapping a Standard POS Expression

For an POS expression in standard form, a 0 is placed

on the K-map for each sum term in the expression.


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Slide 27

Mapping a Standard POS Expression

Example 7Map the following standard POS expression on a K-map

DCBADCBADCBADCBADCBA


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Slide 28

Karnaugh map

Example 8Use a Karnaugh map to minimize the following standard POS

expression:

CBACBACBACBACBA

CBA:POS

s1thegroupingbysametheisresultthe,Similarly

lawvedistributi

ACBA:SOP


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Slide 30

Karnaugh mapDont care conditionsExample 9

Map the following truth table onto a K-map and produce the

minimum SOP espression


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Slide 31

Karnaugh map 5 variables


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Slide 32

Karnaugh map 5 variables example

Yellow group:

Orange group:

Red group:

Grey cell:

ED

ECB

DBA

EDCB

EDCBDBACEBEDx


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Slide 35

Notice

Next lesson Combinational logic circuit

Notes # 5


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Floyd

Digital Fundamentals, 9/e

Copyright 2006 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458All rights reserved.

Slide 36

The Karnaugh Map

3-Variable Karnaugh Map3-Variable Example


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Floyd




Slide 37

The Karnaugh Map

4-Variable Karnaugh Map

4-Variable Example


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Floyd




Slide 38

The Karnaugh Map

5-Variable Karnaugh Mapping


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Floyd




Slide 40

VHDL

VHDL Operators

and

ornot

nand

nor

xor

xnor

VHDL Elements

entity

architecture


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Floyd




Slide 41

VHDL

Entity Structure

Example:

entity AND_Gate1 is

port(A,B:in bit:X:out bit);

end entity AND_Gate1


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Floyd




Slide 42

VHDL

Architecture

Example:

architecture LogicFunction of AND_Gate1 is

begin

X


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Floyd Copyright 2006 by Pearson Education, Inc.

Hardware Description Languages (HDL)

Boolean Expressions in VHDLAND X

ee - 4 kmap logic simplification

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