1. THE K-MAP

2. Karnaugh Map (K-Map)
Karnaugh Mapping is used to minimize the number of logic gates that are required in a digital circuit.
This will replace Boolean reduction when the circuit is large.
Write the Boolean equation in a SOP form first and then place each term on a map.

3. Karnaugh Map (K-Map)
The map is made up of a table of every possible SOP using the number of variables that are being used.
If 2 variables are used then a 2X2 map is used
If 3 variables are used then a 4X2 map is used
If 4 variables are used then a 4X4 map is used
If 5 Variables are used then a 8X4 map is used

4. K-Map SOP Minimization

5. 2 Variables Karnaugh Map
B B A
Notice that the map is going false to true, left to right and top to bottom
B B
The upper right hand cell is A B if X= A B then put an X in that cell A 1
This show the expression true when A = 0 and B = 0

6. 2 Variables Karnaugh Map
B B
If X=AB + AB then put an X in both of these cells A 1 1
From Boolean reduction we know that A B + A B = B
B B
From the Karnaugh map we can circle adjacent cell and find that X = B A 1 1

7. 3 Variables Karnaugh Map
C C
Gray Code
00 A B
01 A B
11 A B
10 A B

8. 3 Variables Karnaugh Map (cont’d)
X = A B C + A B C + A B C + A B C
Gray Code
00 A B
01 A B
11 A B
10 A B
C C 1 1
Each 3 variable term is one cell on a 4 X 2 Karnaugh map 1 1

9. 3 Variables Karnaugh Map (cont’d)
X = A B C + A B C + A B C + A B C
Gray Code
00 A B
01 A B
11 A B
10 A B
C C
One simplification could be
X = A B + A B 1 1 1 1

10. 3 Variables Karnaugh Map (cont’d)
X = A B C + A B C + A B C + A B C
Gray Code
00 A B
01 A B
11 A B
10 A B
C C
Another simplification could be
X = B C + B C
A Karnaugh Map does wrap around 1 1 1 1

11. 3 Variables Karnaugh Map (cont’d)
X = A B C + A B C + A B C + A B C
Gray Code
00 A B
01 A B
11 A B
10 A B
C C
The Best simplification would be
X = B 1 1 1 1

12. On a 3 Variables Karnaugh Map
One cell requires 3 Variables
Two adjacent cells require 2 variables
Four adjacent cells require 1 variable
Eight adjacent cells is a 1

13. 4 Variables Karnaugh Map
Gray Code
00 A B
01 A B
11 A B
10 A B
C D C D C D C D

14. Simplify : X = A B C D + A B C D + A B C D + A B C D + A B C D + A B C D
Gray Code
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
Now try it with Boolean reductions 1 1 1 1 1 1
X = ABD + ABC + CD

15. On a 4 Variables Karnaugh map
One Cell requires 4 variables
Two adjacent cells require 3 variables
Four adjacent cells require 2 variables
Eight adjacent cells require 1 variable
Sixteen adjacent cells give a 1 or true

16. Simplify : Z = B C D + B C D + C D + B C D + A B C
Gray Code
00 A B
01 A B
11 A B
10 A B
C D C D C D C D 1 1 1 1 1 1 1 1 1 1 1 1
Z = C + A B + B D

17. Simplify using Karnaugh map
First, we need to change the circuit to an SOP expression

18. Simplify using Karnaugh map (cont’d)
Y= A + B + B C + ( A + B ) ( C + D)
Y = A B + B C + A B ( C + D )
Y = A B + B C + A B C + A B D
Y = A B + B C + A B C A B D
Y = A B + B C + (A + B + C ) ( A + B + D)
Y = A B + B C + A + A B + A D + B + B D + AC + C D
SOP expression

19. Simplify using Karnaugh map (cont’d)
Gray Code
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
Y = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20. K-Map POS Minimization

21. 3 Variables Karnaugh Map
Gray Code
0 0
0 1
1 1
1 0 C AB

22. 3 Variables Karnaugh Map (cont’d)
Example from text book - FLOYD (page 208)

23. 4 Variables Karnaugh Map
0 0
0 1
1 1
1 0
C D
A B

24. 4 Variables Karnaugh Map (cont’d)
Example from text book - FLOYD (page 208)

25. 4 Variables Karnaugh Map (cont’d)

26. Karnaugh Map - Example Mapping a Standard SOP expression Example:
Answer:
Mapping a Standard POS expression
Using K-Map, convert the following standard POS
expression into a minimum SOP expression
Y = AB + AC or standard SOP :

27. K-Map with “Don’t Care” Conditions
Example :
Input Output
3 variables with output “don’t care (X)”

28. K-Map with “Don’t Care” Conditions (cont’d)
4 variables with output “don’t care (X)”

29. K-Map with “Don’t Care” Conditions (cont’d)
Example:
Determine the minimal SOP using K-Map:
Answer:

30. Minimum SOP expression is
Solution : CD AB X
X X X X 00 01 11 10 AD BC CD
Minimum SOP expression is