THE KMAP Karnaugh Map KMap Karnaugh Mapping is

THE K-MAP

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.

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 2 X 2 map is used • If 3 variables are used then a 4 X 2 map is used • If 4 variables are used then a 4 X 4 map is used • If 5 Variables are used then a 8 X 4 map is used

K-Map SOP Minimization

2 Variables Karnaugh Map B B A 0 1 A 2 3 Notice that the map is going false to true, left to right and top to bottom B The upper right hand cell is A B if X= A B then put an X in that cell A B 1 A This show the expression true when A = 0 and B = 0

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

3 Variables Karnaugh Map Gray Code 0 1 C C 00 AB 0 1 01 AB 2 3 11 AB 6 7 10 AB 4 5

3 Variables Karnaugh Map (cont’d) X=ABC+ABC+ABC Gray Code 00 AB 01 AB 10 AB 0 1 C C 1 1 Each 3 variable term is one cell on a 4 X 2 Karnaugh map

3 Variables Karnaugh Map (cont’d) X=ABC+ABC+ABC Gray Code 00 AB 01 AB 10 AB 0 1 C C 1 1 One simplification could be X=AB+AB 1 1

3 Variables Karnaugh Map (cont’d) X=ABC+ABC+ABC Gray Code 0 1 C 1 Another simplification could be 00 AB 01 AB X=BC+BC 11 AB 10 AB A Karnaugh Map does wrap around 1 1

3 Variables Karnaugh Map (cont’d) X=ABC+ABC+ABC Gray Code 0 1 00 AB C 1 01 AB The Best simplification would be 11 AB X =B 10 AB 1 1

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

4 Variables Karnaugh Map Gray Code 0 0 01 CD CD 11 10 CD CD 00 AB 0 1 3 2 01 AB 4 5 7 6 11 AB 12 13 15 14 10 AB 8 9 11 10

Simplify : X=ABCD+ABCD+ABCD Gray Code 00 01 11 CD CD 00 AB 1 01 AB 1 11 AB 10 AB 1 10 1 Now try it with Boolean reductions 1 1 X = ABD + ABC + CD

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

Simplify : Z=BCD+CD+BCD+ABC Gray Code 00 AB 01 AB 10 AB 00 01 11 10 CD CD 1 1 1 Z=C +AB +BD

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

Simplify using Karnaugh map (cont’d) Y= A + B C + ( A + B ) ( C + D) Y = AB + BC + AB(C+D) Y=AB+BC+ABC +A B D Y=AB+BC+ABCABD Y = A B + B C + (A + B + C ) ( A + B + D) Y = A B + B C + A B + A D + B D + AC + C D SOP expression

Simplify using Karnaugh map (cont’d) Gray Code 00 01 11 10 CD CD 1 1 00 AB 1 01 AB 1 1 1 1 10 AB 1 1 Y=1

K-Map POS Minimization

3 Variables Karnaugh Map Gray Code C 0 AB 0 00 01 11 10 1 1 2 3 6 7 4 5

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

4 Variables Karnaugh Map CD AB 00 01 1 1 10 00 0 1 3 2 01 4 5 7 6 11 12 13 15 14 10 8 9 11 10

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

4 Variables Karnaugh Map (cont’d)

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

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

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

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

Solution : CD AB 00 BC 00 0 01 1 11 X 10 0 0 4 12 13 01 1 X X 5 13 0 8 10 1 11 1 1 X 1 9 3 7 15 10 0 0 2 6 X 14 AD 0 11 Minimum SOP expression is CD

Extra Exercise • Minimize this expression with a Karnaugh map ABCD + ACD + BCD + ABCD