Lecture_39_K-Map for minimization of Boolean expressions .pptx

kmap description for the solution of logic circuits

1. Course name: basic Electronics Course code : Ec 1001 lecture series no : 01(one) Credits : 3 Mode of delivery : online (Power point presentation) Faculty : Email-id : PROPOSED DATE OF DELIVERY: B.TECH FIRST YEAR ACADemic YEAR : 2022-2023

2. Session outcome “ A systematic way to minimize the given logic ”

3. Assessment criteria’S Assignment quiz mid term examination –II END TERM EXAMINATION

4. PROGRAM OUTCOMES MAPPING WITH CO4 [PO1] Demonstrate different number systems, Boolean expressions and different elements of communication systems and to promote different skills towards electronics Industries.

5. Karnaugh Map The Karnaugh map, also known as the K-map, is a method to simplify Boolean algebra expressions. The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. Karnaugh Map: A graphical technique for simplifying a Boolean expression into either form: minimal sum of products (MSP) minimal product of sums (MPS) Goal of the simplification. There are a minimal number of product/sum terms Each term has a minimal number of literals These terms can be used to write a minimal Boolean expression representing the required logic.

6. Karnaugh Maps - Rules of Simplification

7. The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones. 1.Groups may not include any cell containing a zero. 2.Groups may be horizontal or vertical, but not diagonal.

8. 3. Groups must contain 1, 2, 4, 8 cells or in general 2 n cells. That is if n = 1, a group will contain two 1's since 2 1 = 2. If n = 2, a group will contain four 1's since 2 2 = 4.

9. 4. Each group should be as large as possible. 5.Each cell containing a one must be in at least one group.

10. 6.Groups may overlap.

11. 7.Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.

12. 8. There should be as few groups as possible, as long as this does not contradict any of the previous rules.

13. Summmary: No zeros allowed. No diagonals. Only power of 2 number of cells in each group. Groups should be as large as possible. Every one must be in at least one group. Overlapping allowed. Wrap around allowed. Fewest number of groups possible.

14. E x amples

15. Two-Variable K-Map 1 1 a b

16. Karnaugh maps Karnaugh maps, or K - maps, are often used to simplify logic problems with 2, 3 or 4 variables . For the case of 2 variables, we form a map consisting of 2 2 =4 cells as shown in Figure 0 1 B 1 Cell = 2 n ,where n is a number of variables A A 0 1 B 1 A 1 B 1 A B AB A B AB A  B A  B A  B A  B 00 10 2 01 1 11 3 M a x term M in term

17. E x ample A B Y A B Y 1 1 1 1 1 1 1 2-variable Karnaugh maps are trivial but can be used to introduce the methods you need to learn . The map for a 2-input OR gate looks like this : A 0 1 B 1 1 1 1 B A A+B

18. Example 1: Consider the following map. The function plotted is: Z = f(A,B) = AB’ + AB Note that values of the input variables form the rows and columns. That is the logic values of the variables A and B (with one denoting true form and zero denoting false form) form the head of the rows and columns respectively. The map displayed is a one dimensional type which can be used to simplify an expression in two variables. There is a two-dimensional map that can be used for up to four variables, and a three-dimensional map for up to six variables.

19. Referring to the map above, the two adjacent 1's are grouped together. Through inspection it can be seen that variable B has its true and false form within the group. This eliminates variable B leaving only variable A which only has its true form. The minimized answer therefore is Z = A.

20. Karnaugh maps 3 variables Karnaugh map AB C 00 01 11 10 1 ABC 2 ABC 6 ABC 4 ABC 1 ABC 3 ABC 7 ABC 5 ABC Cell = 2 3 =8

21. Three-Variable K-Map ab c m m 2 m 6 m 4 m 1 m 3 m 7 m 5 00 01 11 10 1

22. Three-Variable K-Map ab c 00 01 11 10 1 Edges are adjacent

23. Terminology Implicant Product term that could be used to cover minterms of a function Prime Implicant An implicant that is not part of another implicant Essential Prime Implicant An implicant that covers at least one minterm that is not contained in another prime implicant

24. Guidelines for Simplifying Functions Group as many squares as possible. This eliminates the most variables. Make as few groups as possible. Each group represents a separate product term. You must cover each minterm at least once. However, it may be covered more than once.

25. E x ample A B C Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A C B  AC AB C 00 01 11 10 1 1 1 1 1 1 B

26. 1 1 1 1 00 01 11 10 c 1 m  1 , 3 , 5 , 6  F  a , b , c    Three-Variable K-Map Example Plot 1’s (minterms) of switching function ab

27. 00 01 11 10 F  a , b , c   a b  b c Three-Variable K-Map Example Plot 1’s (minterms) of switching function ab 1 1 1 1 ab c bc 1

28. Three-Variable K-Maps f   (0,4)  B C f   (4,5)  A B f   (0,1,4,5)  B f   (0,1,2,3)  A 1 1 1 BC A 00 01 11 10 1 1 1 BC A 00 01 11 10 1 1 1 1 1 BC A 00 01 11 10 1 1 1 1 1 BC A 00 01 11 10 f   (0,4)  A C f   (4,6)  A C f   (0,2)  A C f   (0,2,4,6)  C 1 1 1 BC A 00 01 11 10 1 1 1 BC A 00 01 11 10 1 1 1 1 1 BC A 00 01 11 10 1 1 1 BC A 00 01 11 10

29. Three-Variable K-Map Examples B C 1 00 01 11 10 A B C 1 00 01 11 10 A B C 1 00 01 11 10 A B C 1 00 01 11 10 A B C 1 00 01 11 10 A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B C 1 00 01 11 10 A

30. Four Variable Examples

31. Four-variable K-Map ab cd 00 01 11 10 m m 4 m 12 m 8 m 1 m 5 m 13 m 9 m 3 m 7 m 15 m 11 m 2 m 6 m 14 m 10 00 01 11 10

32. Four-variable K-Map ab 00 01 11 10 cd 00 01 11 10 Edges are adjacent Edges are adjacent

33. E x ample Use a K-Map to simplify the following Boolean expression m  , 2 , 3 , 6 , 8 , 1 2 , 1 3 , 1 5  F  a , b , c , d   

34. Four-variable K-Map ab 00 01 11 10 cd 00 01 11 10 F   m  , 2 , 3 , 6 , 8 , 1 2 , 1 3 , 1 5  1 1 1 1 1 1 1 1

35. Four-variable K-Map ab 00 01 11 10 cd 00 01 11 10 F  a b d  a b c  a c d  a b d  a c d 1 1 1 1 1 1 1 1

36. Four-Variable K-Maps f   (0,8)  B  C  D f   (5,13)  B  C  D f   (13,15)  A  B  D f   (4,6)  A  B  D f   (2,3,6,7)  A  C f   (4,6,12,14)  B  D f   (2,3,10,11)  B  C f   (0,2,8,10)  B  D 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10

37. Four-Variable K-Maps 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10 f   (4,5, 6, 7)  A  B f   (3, 7,11,15)  C  D f   (0, 3,5, 6, 9,10,12,15) f  A  B  C  D f   (1, 2, 4, 7,8,11,13,14) f  A  B  C  D f   (1, 3,5, 7, 9,11,13,15) f  D f   (0,2,4,6,8,10,12,14) f  D f   (4,5,6,7,12,13,14,15) f  B f   (0,1,2,3,8,9,10,11) f  B

38. Four-Variable K-Maps Examples CD 00 01 11 10 AB 00 01 11 10 CD 00 01 11 10 AB 00 01 11 10 CD 00 01 11 10 AB 00 01 11 10 CD 00 01 11 10 AB 00 01 11 10 CD 00 01 11 10 AB 00 01 11 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CD 00 01 11 10 AB 00 01 11 10

39. E x ample Use a K-Map to simplify the following Boolean expression m  2 , 3 , 6 , 7  F  a , b , c   

40. 1 1 1 1 00 01 11 10 c 1 Three-Variable K-Map Example Step 1: Plot the K-map ab m  2 , 4 , 5 , 7  F  a , b , c   

41. 1 1 1 1 00 01 11 10 c 1 Three-Variable K-Map Example Step 2: Circle Prime Implicants ab m  2 , 3 , 6 , 7  F  a , b , c    Wrong!! We really should draw A circle around all four 1’s

42. ab c 00 01 11 10 1 Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI EPI m  2 , 3 , 6 , 7  F  a , b , c    1 1 1 1 Wrong!! We really should draw A circle around all four 1’s

43. ab c 00 01 11 10 1 Three-Variable K-Map Example Step 4: Select Remaining Prime Implicants to complete the cover. EPI EPI 1 1 1 1 m  2 , 3 , 6 , 7  F  a , b , c   

44. ab c 00 01 11 10 1 Three-Variable K-Map Example Step 5: Read the map. ab ab 1 1 1 1 m  2 , 3 , 6 , 7  F  a , b , c   

45. Solution F  a , b , c   a b  a b  b Since we can still simplify the function this means we did not use the largest possible groupings.

46. 00 01 11 10 c 1 Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants ab EPI 1 1 1 1 m  2 , 3 , 6 , 7  F  a , b , c   

47. ab c 00 01 11 10 1 Three-Variable K-Map Example Step 5: Read the map. 1 1 1 1 b m  2 , 3 , 6 , 7  F  a , b , c   

48. Solution F  a , b , c   b

49. Special Cases

50. Three-Variable K-Map Example ab c 1 1 1 1 1 1 1 1 00 01 11 10 1 F  a , b , c   1

51. Three-Variable K-Map Example ab c 00 01 11 10 1 F  a , b , c  

52. Three-Variable K-Map Example ab c 1 1 1 1 00 01 11 10 1 F  a , b , c   a  b  c

53. E x ample Use a K-Map to simplify the following Boolean expression m  1 , 2 , 3 , 5 , 6  F  a , b , c   

54. 1 1 1 1 1 00 01 11 10 c 1 Three-Variable K-Map Example Step 1: Plot the K-map ab m  1 , 2 , 3 , 5 , 6  F  a , b , c   

55. 00 01 11 10 c 1 Three-Variable K-Map Example Step 2: Circle ALL Prime Implicants ab 1 1 1 1 m  1 , 2 , 3 , 5 , 6  F  a , b , c    1