K map.

Go trough the slides i assure you, get some basic ideas about K-map.

1. KARNAUGH MAP By, Chethan N I MCA.

2. Karnaugh Map :-  Karnaugh Map is also called K-Map. K-Maps are graphical representation of Boolean function.  It provides a systematic method for the simplification of Boolean expression.  It is composed of adjacent cells.  Adjacent cells are those which differ by a single variable. Each cell represents a combination of variables in product/sum form.

3. Ordering Of Variables :- It means cell of adjacency(side by side). The cells in a K-Map are arranged so that there is only a single variable change b/w adjacent cells. Adjacency is defined by a single variable change. For Ex: In the 3 variable map the 010 cell is adjacent to the 000 cell but it is not adjacent to the 101 cell.

4. Adjacent Cell :- Adjacent cells on a K-Map are those that differ by only one variables. Arrow point b/w adjacent cells.

5. 2 -Variable K-Map :- The 2 – variable K-Map is an array of 4 cells, as shown in the below (2^2=4). Binary values of A/B along the left side & the values of B/A are across the top. The values of a given cell is the binary values of A/B at the left in the same row combined with the binary values of B/A at the top in the same column.

6. Format Of 2-Variable :-

7. K-Map SOP Minimization :- For an SOP expression in standard form, a 1 is p laced on the K-Map for each product term in Expression. Each 1 is placed in a cell corresponding to the value of product term. For Ex: Now we studying the two variable map.

8. Simplification Of Boolean Expression Using K-Map For 2/3/4 Variables :- A given expression can be simplified on to K-Map with the help of the following steps ; 1.Plotting the expression on to the K-Map. 2.Grouping of cells. 3.Simplification.

9. 1.Plotting The Expression On To The K-Map :- A given expression can be plotted on to K-Map by placing ‘1’ in each cell corresponding to a product term present in the expression. For Ex: Z=f(A,B)=A’B’+AB’+A’B

10. Grouping :- Adjacent cells can be combined together to form a group using the following rules , Adjacent cells are those which differ by a single variable. Adjacent cells can be combined in groups of 1,2,4,8,16,…. Each group is extended by merging with the adjacent groups so that it includes as many adjacent cells as possible. Each group represents a product term.

11. 3-Variable K-Map :- * The 3-variable K-Map is an array of eight cells as shown in below. In this case, binary values A&B/B&C are along the left side and the values of C/A are across the top. The value of a given cell is the binary values of A & B at the left in the same row combined with the value of C at the top in the same column.

12. Format Of 3-Variable :-

13. Example Of 3-Variable :- 1.Plotting The Expression On To The K-Map . 2.Grouping . Consider the equation ; Z=f(A,B,C)=A’B’C’+A’BC’+AB’C’ +AB’C 3.Simplification . Z=A’B’C’.A’BC’+AB’C’.A’B’C The simplified equation is, Z= A’C’+A’B’

14. 4-Variable K-Map :- The 4-variable K-Map is an array of sixteen cells, as shown in below. Binary values of A&B/C&D are along the left side and the values of C&D/A&B are across the top. The values of given cell is the binary values of A&B at the left in the same row combined with the binary values of C&D at the top in the same column.

15. Format Of 4-Variable :-

16. Simplification :- Simplification involves the following steps , Each group represents a product term composed of variables in normal or in complemented form. In a term, if a variable presents both in normal and in complemented form then it is discarded. The final simplified expression is the sum of all the product terms. Z=A’B’A’B+AB’A’B’ The simplified equation is Z=A’+B’.

17. Example Of 4-Variable :- 1.Plotting The Expression On To The K-Map. 2.Grouping. Consider a equation , Z=f(A,B)=A’BC’D+A’BCD+A’BCD’+ABCD’+ABCD’+ABC’D

18. 3.Simplification. Z=A’BC’D.A’BCD+A’BCD’.ABCD’+ABC’D.ABC’D’ The Simplified Equation Is , Z=A’BD+BCD’+ABC’

19. “ Don’t Care “ Condition :- Functions that have unspecified output for some input c0mbinations are called incompletely specified functions. Unspecified minterms of a functions are called “don’t care” conditions. We simply don’t care whether the value of ‘zero’ or ‘one’ is assigned to Z for particular minterm . Don’t care conditions are represented by X in the K-Map. NOTE :Don’t care conditions play a central role in the specification and optimization of logic circuits as they represent the degrees of freedom of transforming a network into a functionally equivalent one.

20. Example :- Simplify the Boolean equation ; Z = f(A,B,C,D) = €(1,3,7,11,15) Without Using Don’t Care , Z = A’B’C’D.A’B’CD + A’B’CD.A’BCD.ABCD.AB’CD The simplified equation is , Z = A’B’D+CD With Using Don’t Care , Z = A’B’C’D’.A’B’C’D.A’B’CD.A’B’CD’ + A’B’CD.A’BCD.ABCD.AB’CD The simplified equation is , Z = A’B’+CD

21. THANK YOU