Karnaugh maps z 88
Transcript of Karnaugh maps z 88
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Karnaugh MapsKarnaugh Maps (K maps)(K maps)
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What are KarnaughWhat are Karnaugh11 maps? maps?
Karnaugh maps provide an alternative way of simplifying logic circuits.
Instead of using Boolean algebra simplification techniques, you can transfer logic values from a Boolean statement or a truth table into a Karnaugh map.
The arrangement of 0's and 1's within the map helps you to visualise the logic relationships between the variables and leads directly to a simplified Boolean statement.
1Named for the American electrical engineer Maurice Karnaugh. 3
Karnaugh mapsKarnaugh maps
Karnaugh maps, or K-maps, are often used to simplify logic problems with 2,
3 or 4 variables. BA
For the case of 2 variables, we form a map consisting of 22=4 cellsas shown in Figure
AB
0 1
0
1
Cell = 2n ,where n is a number of variables
00 10
01 11
AB
0 1
0
1
AB
0 1
0
1
BA
BA AB
BA+ BA +
BA+ BA +
Maxterm Minterm
0 2
1 3
4
Karnaugh mapsKarnaugh maps
3 variables Karnaugh map
CBA
ABABCC 00 01 11 10
0
1
CBA CAB CBA
CBA BCA ABC CBA
0 2 6 4
531 7
Cell = 23=8
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Karnaugh mapsKarnaugh maps
4 variables Karnaugh map
ABABCDCD 00 01 11 10
00
01
11
10
5
3
1
7
62
0 4
9
15
13
11
1014
12 8
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Karnaugh mapsKarnaugh maps
The Karnaugh map is completed by entering a '1‘(or ‘0’) in each of the appropriate cells.
Within the map, adjacent cells containing 1's (or 0’s) are grouped together in twos, fours, or eights.
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ExampleExample
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
2-variable Karnaugh maps are trivial but can be used to introduce the methods you need to learn. The map for a 2-input OR gate looks like this:
AB
0 1
0
1
1
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BB
AA
A+BA+B
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ExampleExample
AA BB CC YY
00 00 00 11
00 00 11 11
00 11 00 00
00 11 11 00
11 00 00 11
11 00 11 11
11 11 00 11
11 11 11 00
CA
CAB +B
ABABCC 00 01 11 10
0
1
1
1
1 1
1
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Truth Table to K-Map MappingTruth Table to K-Map Mapping
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Four Variable K-MapW X Y Z FWXYZ
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 1
1 0 0 0 1
1 0 0 1 0
1 0 1 0 0
1 0 1 1 0
1 1 0 0 1
1 1 0 1 0
1 1 1 0 0
1 1 1 1 1
V
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
X W
X W
X W
X W
Z Y Z Y ZY ZY
1 00 1
1 00 1
1 00 0
1 00 0
Only onevariable changes for every row
change
Only one variable changes for
every column change
Fwxyz= Y Z + X Y Z
Individual WorkIndividual Work
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Question(1Question(1((
A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a specific three-input logic circuit:
Out= A D + A C + B C
A
A
D
C
C
B
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Question(2Question(2((
A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out= B C
B
C
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Question(3Question(3((
A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out = A B
A
B
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Question(4Question(4((
A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out = B C D + B C D
B
C
D
B C
D15
steps work experiment:
1.Solution truth table by Karnaugh MapsKarnaugh Maps (K maps).(K maps).
2.2.Gate work by Karnaugh MapsGate work by Karnaugh Maps (K maps).(K maps).
3.3.Applied to the gate on test Applied to the gate on test board and make sure by truth board and make sure by truth table.table.
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Experiment :GROUP (AExperiment :GROUP (A((AA BB CC DD YY
00 00 00 00 11
00 00 00 11 00
00 00 11 00 00
00 00 11 11 00
00 11 00 00 11
00 11 00 11 00
00 11 11 00 11
00 11 11 11 00
11 00 00 00 00
11 00 00 11 11
11 00 11 00 00
11 00 11 11 11
11 11 00 00 00
11 11 00 11 00
11 11 11 00 11
11 11 11 11 00
11 00 00 00
11 00 00 11
00 00 00 11
00 11 11 00
C D
A B00
01
10
11
0100 1011
Y= A C D + A B D + B C D
A C D A B D B C D 17
Experiment :GROUP (BExperiment :GROUP (B((AA BB CC DD YY
00 00 00 00 00
00 00 00 11 00
00 00 11 00 11
00 00 11 11 00
00 11 00 00 00
00 11 00 11 11
00 11 11 00 11
00 11 11 11 00
11 00 00 00 00
11 00 00 11 00
11 00 11 00 00
11 00 11 11 00
11 11 00 00 00
11 11 00 11 11
11 11 11 00 00
11 11 11 11 00
00 00 00 11
00 11 00 11
00 11 00 00
00 00 00 00
AB
DC
00
1000 1101
01
11
10
Y= B D C + A D C
B
D
C
A D
C 18
Experiment :GROUP (CExperiment :GROUP (C((AA BB CC DD YY
00 00 00 00 00
00 00 00 11 00
00 00 11 00 00
00 00 11 11 00
00 11 00 00 11
00 11 00 11 11
00 11 11 00 00
00 11 11 11 00
11 00 00 00 00
11 00 00 11 00
11 00 11 00 11
11 00 11 11 11
11 11 00 00 11
11 11 00 11 11
11 11 11 00 00
11 11 11 11 00
00 00 00 00
11 11 00 00
11 11 00 00
00 00 11 11
AB
DC
00
00
01
01 11
11
10
10
Y=B D + A B D
A B
D
D
B
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22
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