The Karnaugh Map

DIGITAL LOGIC DESIGN LAB GROUP-05

NAME ID

Islam, Md. Arafatul 14-26696-2Islam, Mohammad Kamrul 14-26664-2

THE KARNAUGH MAP

(DEVELOPED BY MAURICE KARNAUGH)

DEFINITION The Karnaugh Map provides a simple and straight forward method of minimizing Boolean expression.

MAPPING SOP AND POS EXPRESSION ON TO A K-MAP

Map the following expression into a k-map:• A’BC+AB’C’+ABC• (P+Q+R)(P’+Q’+R)(P’+Q+R)

STEP1: DETERMINE HOW MANY VARIABLE ARE PRESENT AND IF THE PROVIDED SOP/POS EXPRESSION IS IN STANDARD FORM OR NOT.

STEP2: DETERMINE HOW MANY CELLS WILL BE PRESENT IN THE K-MAP DEPENDING ON THE NUMBER OF VARIABLE

STEP3: NAME THE ROWS AND COLUMN USING THE INPUT VARIABLES.

If there are 3 variables A,B and C then A and B should be used to name the row and C to name the columns.

CAB

STEP4: WRITE THE CORRESPONDING INPUT COMBINATIONS FOR THE VARIABLES IN EACH ROW AND COLUMN BUT ENSURE THAT WITH EVERY ROW/COLUMN CHANGE ONLY “ONE”. THE CELLS ARE ORDERED IN GRAY CODE.

Only one variable changes from R1 to R2, R2 to R3 and R3 to R4.

CAB 0 100

01

11

10

STEP 5: FIND INPUT VARIABLE STATES FROM THE GIVEN SOP/POS EXPRESSION AND PUT 1 (FOR SOP EXPRESSION OUTPUTS) AND 0 (FOR POS EXPRESSION OUTPUTS) IN THE CORRESPONDING CELLS.

1) A’BC + AB’C’ + ABC

You can put 0’s in the remaining cells or leave them blank.

CAB 0 1

00

0111

10

1

1

1

0 1 1 1 0 0 1 1 1

2) (P+Q+R)(P’+Q’+R)(P’+Q+R)

You can put 1’s in the remaining cells or leave them blank.

CAB 0 1

00

0111

10

0

00

0 0 0 1 1 0 1 0 0

ADVANTAGE OF K-MAP GROUPING

Karnaugh maps reduce logic functions.Most simplified equationsIts fast and easier compared to Boolean algebra.

Lowest cost

K-MAP GROUPINGMinimize the following expression using K-MapAB’C+A’BC+A’B’C+A’B’C’+AB’C’

Circuit Design Before K-MAP Grouping

5 AND Gates and

3 NOT Gates

STEP 1: The 1’s in adjacent cells can be grouped together

STEP 2: The no. of cells in a group can be 1,2,4… i.e 2^n (n=0,1,2…)STEP 3: The GOAL is to always try to maximize the group size and minimize no. of groups. STEP 4: Overlapping can be done if it helps to increase group sizeSTEP 5: Each 1 in the K-Map must be in at least one group or the other

HOW TO DO K-MAP GROUPING?

Simplified SOP expression =

CAB 0 1

00

0111

10

1

1

1 1

1

HOW TO DO K-MAP GROUPING?

A’C B’ +

AB’C+A’BC+A’B’C+A’B’C’+AB’C’ 1 0 1 0 1 1 0 0 1 0 0 0 1 0 0

K-MAP GROUPING

Circuit Design After K-MAP Grouping

1 NAND Gate 1 NOR Gate

and 1 NOT Gate

DISADVANTAGE OF K-MAP?