the karnaugh map
TRANSCRIPT
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?