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4. Karnaugh Maps and Circuits

4. Karnaugh Maps and Circuits

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4. Karnaugh Maps and Circuits

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  1. 4. Karnaugh Maps and Circuits • Objective: To know how to simplify switching functions by Karnaugh maps, • To understand what are the combinative and sequential circuits, • To know the characteristics of the integrated circuits.

  2. 4.1 Simplification of Switching Functions • Why simplify and optimize? • Constraints • Cost ($$$)! • How? • Algebraic method (still…) • Karnaugh maps (wow!)

  3. Algebraic Handling * Canonical form: L = A’B’C’+A’BC’+AB’C’+AB’C+ABC’ 9 NOT (* 1) + 5 AND (* 3) + 1 OR (* 5) = 29 Simplified Form: L = AB’ + C’ 2 NOT (* 1) + 1 AND (* 2) + 1 OR (* 2) = 6

  4. Karnaugh Maps (I) • Simplification by algebraic method is DIFFICULT! • Method of simplification graphically suggested: Karnaugh maps • Usable with functions up to 6 variables

  5. B B’ A’ m0 m1 m2 m3 A 1 1 0 0 A Example * • Diagram - 2 variables • f(A, B) = m(0, 1) = A’ B

  6. Karnaugh Maps (II) • Can be conceived from: • Truth tables • Canonical CSOP or SOP form • Canonical CPOS or POS form • Can give result like: • Minimal Sum of Products (SOP) form • Minimal Products of Sums (POS) form

  7. 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 Example * • f (A, B, C, D) = m (0,1,2,5,8,9,10) • f SOP= • fPOS = C B'D' + B'C' + A'C'D B A (A' + B') • (C' + D') • (B' + D) D

  8. C 0 1 1 1 0 1 1 1 B 0 0 1 1 A 0 0 1 1 D Simplification * • Simplify starting from the SOP form:f (A, B, C, D) = CD’+A’D+ACD

  9. C 0 1 1 1 0 1 1 1 B 0 0 1 1 A 0 0 1 1 D Simplification * • Simplify starting from the SOP form:f (A, B, C, D) = CD’+A’D+ACD = C + A’D

  10. Karnaugh Maps (III) • Don’t-Care values (X) • Certain switching functions are known as incompletely defined: certain combinations of their variables of inputs are never supposed to occur or not to have an effect on the result. One calls these combinations don’t-care values and one indicates them as ' X' in the truth tables. • In the Karnaugh maps, one considers them like 1 (SOP) or of the 0 (POS) only to make larger groupings, but it is not necessary to gather them.

  11. C X 1 1 1 0 X 1 0 B 0 0 1 0 A 0 0 1 0 D Don’t-Care Values * • Simplify f (A, B, C, D) = m (1, 2, 3, 7, 11, 15) X (0, 5) • f SOP =A’B’ + CD • The minterm 5 should not be included; it would not be minimal!

  12. S 1 S m S 1 S m 4.2 Circuits • Combinational: • Sequential: E 1 Output Variables input Variables combinational E n circuit E 1 Output Variables Input Variables combinational E n circuit States memory

  13. Integrated Circuits (I) • The integrated circuits, material manufacture of logic gates and more complex functions, are characterized in several ways. • Why they used are? • Level of integretion? Quantity of transistors in a circuit.

  14. Integrated circuits (II) • Manufacturing Technologies • Other characteristics

  15. Complementary readings • In Mano and Kime: • Sections 2.4 and 2.5 • Simplification and Karnaugh maps • Section 2.8 (Optional) • Integrated circuits