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Combinational Logic Circuits

Combinational Logic Circuits. Chapter 2 Mano and Kime. Combinational Logic Circuits. Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits. Digital Logic Gates. *. Gates with More than Two Inputs.

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Combinational Logic Circuits

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  1. CombinationalLogic Circuits Chapter 2 Mano and Kime

  2. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  3. Digital Logic Gates *

  4. Gates with More than Two Inputs

  5. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  6. Basic Identities of Boolean Algebra

  7. Implementation of Boolean Function with Gates

  8. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  9. Minterms for Three Variables

  10. Sum of Products Design X Y minterms 0 0 m0 = !X & !Y 0 1 m1 = !X & Y 1 0 m2 = X & !Y 1 1 m3 = X & Y

  11. Sum of Products Design Design an XOR gate X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 m1 = !X & Y m2 = X & !Y Z = m1 + m2 = (!X & Y) + (X & !Y)

  12. Sum of Products: Exclusive-OR !X & Y Z = (!X & Y) + (X & !Y) X & !Y

  13. Maxterms for Three Variables

  14. Product of Sums Design Maxterms: A maxterm is NOT a minterm maxterm M0 = NOT minterm m0 M0 = m0’ =(X’ . Y’)’ = (X + Y)” = X + Y

  15. Product of Sums Design X Y minterms maxterms 0 0 m0 = !X . !Y M0 = !m0 = X + Y 0 1 m1 = !X . Y M1 = !m1 = X + !Y 1 0 m2 = X . !Y M2 = !m2 = !X + Y 1 1 m3 = X . Y M3 = !m3 = !X + !Y

  16. Product of Sums Design Design an XOR gate X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 Z is NOT minterm m0 AND it is NOT minterm m3

  17. Product of Sums Design Design an XOR gate X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 M0 = X + Y M3 = !X + !Y Z = M0 & M3 = (X + Y) & (!X + !Y)

  18. X Y X + Y Z !X + !Y X !X Y !Y Z = (X + Y) & (!X + !Y) Product of Sums: Exclusive-OR

  19. Three- Level and Two- Level Implementation

  20. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  21. Two-Variable Map

  22. Three-Variable Map

  23. Three- Variable Map: Flat and on a Cylinder to Show Adjacent Squares

  24. YZ 00 01 11 10 X 0 1 Three-variable K-Maps 1 1 1 1 F = !X & !Y + X & Z

  25. YZ 00 01 11 10 X 0 1 Three-variable K-Maps F = !X & !Y & !Z + !X & !Y & Z + X & !Y & Z + X & Y & Z 1 1 1 1 F = !X & !Y & (!Z + Z) + X & Z & (!Y + Y) = !X & !Y + X & Z

  26. YZ 00 01 11 10 X 0 1 Three-variable K-Maps 1 1 1 1 1 F = Y & !Z + X

  27. YZ 00 01 11 10 X 0 1 Three-variable K-Maps 1 1 1 1 1 1 F = !X & !Y + X & y + Z

  28. YZ 00 01 11 10 X 0 1 Three-variable K-Maps 1 1 1 1 F = X & Z + !X & !Z

  29. YZ 00 01 11 10 X 0 1 Three-variable K-Maps 1 1 1 1 1 1 F = Y + !Z

  30. YZ 00 01 11 10 X 0 1 3 2 0 4 5 7 6 1 Three-variable K-Maps 1 1 1 1 F = m0 + m2 + m5 + m7 = S(0,2,5,7)

  31. Four-Variable Map

  32. Four-Variable Map: Flat and on a Torus to Show Adjacencies

  33. YZ 00 01 11 10 WX 00 01 11 10 Four-variable K-Maps 1 2 0 3 6 7 4 5 12 15 13 14 11 10 9 8 Each square is numbered in the above K-map

  34. YZ 00 01 11 10 WX 0 1 3 2 00 4 5 7 6 01 12 13 15 14 11 8 9 11 10 10 Four-variable K-Maps F(W,X,Y,Z) = S(2,4,5,6,7,9,13,14,15)

  35. Four-variable K-Maps YZ 00 01 11 10 WX 00 1 F = !W & X + X & Y + !W & Y & !Z + W & !Y & Z 01 1 1 1 1 11 1 1 1 10 1

  36. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  37. Prime Implicants Each product term is an implicant F = XY’Z + X’Z’ + X’Y A product term that cannot have any of its variables removed and still imply the logic function is called a prime implicant.

  38. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  39. Digital Logic Gates >

  40. >

  41. Logical Operations with NAND Gates

  42. Alternative Graphics Symbols for NAND and NOT Gates

  43. Logical Operations with NOR Gates

  44. Two Graphic Symbols for NOR Gate

  45. Generalized De Morgan’s Theorem • NOT all variables • Change & to + and + to & • NOT the result • -------------------------------------------- • F = X & Y + X & Z + Y & Z • F = !((!X + !Y) & (!X + !Z) & (!Y + !Z)) • F = !(!(X & Y) & !(X & Z) & !(Y & Z))

  46. F = !(!(X & Y) & !(X & Z) & !(Y & Z))

  47. F = !(!(X & Y) & !(X & Z) & !(Y & Z)) NAND Gate

  48. F = X & Y + X & Z + Y & Z X Y X F Z Y Z

  49. CombinationalLogic Circuits • Binary Logic and Gates • Boolean Algebra • Standard Forms • Map Simplification • NAND and NOR Gates • Exclusive-OR Gates • Integrated Circuits

  50. Exclusive-OR Gate XOR X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 X Z Y Z = X $ Y X $ !Y = !(X $ Y) !X $ Y = !(X $ Y) A $ B = B $ A (A $ B) $ C = A $ (B $ C) = A $ B $ C X $ 0 = X X $ 1 = !X X $ X = 0 X $ !X = 1

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