k map in electronics

1. Karnaugh Map Method

2. Karnaugh Map Technique
 K-Maps, like truth tables, are a way to show
the relationship between logic inputs and
desired outputs.
 K-Maps are a graphical technique used to
simplify a logic equation.
 K-Maps are very procedural and much
cleaner than Boolean simplification.
 K-Maps can be used for any number of
input variables, BUT are only practical for
fewer than six.

3. K-Map Format
 Each minterm in a truth table corresponds to a
cell in the K-Map.
 K-Map cells are labeled so that both horizontal
and vertical movement differ only in one
variable.
 Once a K-Map is filled (0’s & 1’s) the sum-of-
products expression for the function can be
obtained by OR-ing together the cells that
contain 1’s.
 Since the adjacent cells differ by only one
variable, they can be grouped to create simpler
terms in the sum-of-product expression.

4. Y
Y
X X
0
1
2
3
Truth Table -TO- K-Map
Y
0
1
0
1
Z
1
0
1
1
X
0
0
1
1
minterm 0 
minterm 1 
minterm 2 
minterm 3 
1
1
0
1

5. Y
Y
X X
0
0
1
0
X Y
Y
Y
X X
0
0
0
1 X Y
Y
Y
X X
1
0
0
0
X Y
Y
Y
X X
0
1
0
0 X Y
2 Variable K-Map : Groups of One

6. Adjacent Cells
X Y
Y
Y
X X
1
0
1
0
X Y
Y
Y
X X
1
0
1
0
Y = Z
Z = X Y + X Y = Y ( X + X ) = Y
1

7. Groupings
 Grouping a pair of adjacent 1’s eliminates the
variable that appears in complemented and
uncomplemented form.
 Grouping a quad of 1’s eliminates the two
variables that appear in both complemented
and uncomplemented form.
 Grouping an octet of 1’s eliminates the three
variables that appear in both complemented
and uncomplemented form, etc…..

8. Y
Y
X X
1
1
0
0
X
X
Y
Y
X X
1
0
1
0
Y
Y
2 Variable K-Map : Groups of Two
Y
Y
X X
0
1
0
1
Y
Y
X X
0
0
1
1

9. Y
Y
X X
1
1
1
1
1
2 Variable K-Map : Group of Four

10. Two Variable Design Example
S
S
R R
0
1
2
3
S
0
1
0
1
T
1
0
1
0
R
0
0
1
1
1
0
1
0
S
T = F(R,S) = S

11. 3 Variable K-Map : Vertical
minterm 0 
minterm 1 
minterm 2 
minterm 3 
minterm 4 
minterm 5 
minterm 6 
minterm 7 
C
0
1
0
1
0
1
0
1
Y
1
0
1
1
0
0
1
0
B
0
0
1
1
0
0
1
1
A
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
1
A A
B C
B C
B C
B C
0
1
4
5
3
2
7
6

12. 3 Variable K-Map : Horizontal
C
C
A B A B A B
A B
minterm 0 
minterm 1 
minterm 2 
minterm 3 
minterm 4 
minterm 5 
minterm 6 
minterm 7 
C
0
1
0
1
0
1
0
1
Y
1
0
1
1
0
0
1
0
B
0
0
1
1
0
0
1
1
A
0
0
0
0
1
1
1
1
1
0
1
1
1
0
0
0
0
1
2
3
6
7
4
5

13. 3 Variable K-Map : Groups of Two
C
C
A B A B A B
A B
1
0
1
0
0
0
0
0
A C
0
1
0
1
0
0
0
0
A C
0
0
0
0
1
0
1
0
A C
0
0
0
0
0
1
0
1
A C
0
0
1
0
1
0
0
0
B C
0
0
0
1
0
1
0
0
B C
1
0
0
0
0
0
1
0
B C
0
1
0
0
0
0
0
1
B C
1
1
0
0
0
0
0
0
A B
0
0
1
1
0
0
0
0
A B
0
0
0
0
1
1
0
0
A B
0
0
0
0
0
0
1
1
A B

14. 3 Variable K-Map : Groups of Four
C
C
A B A B A B
A B
1
1
1
1
0
0
0
0
A
0
0
0
0
1
1
1
1
A
0
0
1
1
1
1
0
0
B
1
1
0
0
0
0
1
1
B
1
0
1
0
1
0
1
0
C
0
1
0
1
0
1
0
1
C

15. 3 Variable K-Map : Group of Eight
C
C
A B A B A B
A B
1
1
1
1
1
1
1
1
1

16. Simplification Process
1. Construct the K-Map and place 1’s in cells corresponding
to the 1’s in the truth table. Place 0’s in the other cells.
2. Examine the map for adjacent 1’s and group those 1’s
which are NOT adjacent to any others. These are called
isolated 1’s.
3. Group any hex.
4. Group any octet, even if it contains some 1’s already
grouped, but are not enclosed in a hex.
5. Group any quad, even if it contains some 1’s already
grouped, but are not enclosed in a hex or octet.
6. Group any pair, even if it contains some 1’s already
grouped, but are not enclosed in a hex, octet or quad.
7. Group any single cells remaining.
8. Form the OR sum of all the terms grouped.

17. Three Variable Design Example #1
L
0
1
0
1
0
1
0
1
M
1
0
1
1
0
1
0
0
K
0
0
1
1
0
0
1
1
J
0
0
0
0
1
1
1
1
1
0
1
1
0
0
0
1
L
L
J K J K J K
J K
0
1
2
3
6
7
4
5
J L
J K J K L
M = F(J,K,L) = J L + J K + J K L

18. Three Variable Design Example #2
C
0
1
0
1
0
1
0
1
Z
1
0
0
0
1
1
0
1
B
0
0
1
1
0
0
1
1
A
0
0
0
0
1
1
1
1
1
0
0
0
0
1
1
1
C
C
A B A B A B
A B
0
1
2
3
6
7
4
5
B C
A C
Z = F(A,B,C) = A C + B C

19. Three Variable Design Example #3
C
0
1
0
1
0
1
0
1
F2
1
0
0
1
1
1
0
1
B
0
0
1
1
0
0
1
1
A
0
0
0
0
1
1
1
1
1
1
0
1
1
1
0
0
A
A
B C B C B C
B C
0 1 2
3
6
7
4 5
B C B C
A B
A C
F2 = F(A,B,C) = B C + B C + A B
F2 = F(A,B,C) = B C + B C + A C

20. Four Variable K-Map
minterm 0 
minterm 1 
minterm 2 
minterm 3 
minterm 4 
minterm 5 
minterm 6 
minterm 7 
minterm 8 
minterm 9 
minterm 10 
minterm 11 
minterm 12 
minterm 13 
minterm 14 
minterm 15 
Z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F1
1
0
0
0
1
1
0
1
1
1
0
0
0
1
1
1
Y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
X
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
W
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
4
5
12
13
8
9
3
2
7
6
15
14
11
10
W X W X W X
W X
Y Z
Y Z
Y Z
Y Z
0
0
1
0
1
1
0
0
1
0
1
1
0
1
1
1

21. Four Variable K-Map : Groups of Four
W X W X W X
W X
Y Z
Y Z
Y Z
Y Z
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
X Z
0
0
0
1
0
1
0
0
0
0
1
0
1
0
0
0
X Z
X Z
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0

22. Four Variable Design Example #1
Z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F1
1
0
1
0
1
0
1
0
0
0
1
0
1
1
0
0
Y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
X
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
W
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
4
5
12
13
8
9
3
2
7
6
15
14
11
10
W X W X W X
W X
Y Z
Y Z
Y Z
Y Z
0
1
0
1
0
0
0
1
1
0
1
0
1
1
0
0
W X Y
X Y Z
W Z
F1 = F(w,x,y,z) = W X Y + W Z + X Y Z
min 0 
min 15 

23. Four Variable Design Example #2
Z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F2
1
x
1
0
0
x
0
x
x
1
0
1
x
1
1
1
Y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
X
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
W
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
4
5
12
13
8
9
3
2
7
6
15
14
11
10
W X W X W X
W X
Y Z
Y Z
Y Z
Y Z
X
X
1
1
1
1
1
0
1
0
X
X
0
X
1
0
Y Z
F2 = F(w,x,y,z) = X Y Z + Y Z + X Y
X Y Z
X Y
min 0 
min 15 