1. CSI-2111 Structure of Computers Ipage 4-1 1X 00 4. Karnaugh Maps and Circuits v Objective: To know how to simplify switching functions by Karnaugh maps, v To understand what are the combinative and sequential circuits, v To know the characteristics of the integrated circuits.

2. CSI-2111 Structure of Computers Ipage 4-2 1X 00 4.1 Simplification of Switching Functions v Why simplify and optimize? –Constraints –Cost ($$$)! v How? –Algebraic method (still…) –Karnaugh maps (wow!)

3. CSI-2111 Structure of Computers Ipage 4-3 1X 00 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. CSI-2111 Structure of Computers Ipage 4-4 1X 00 Karnaugh Maps (I) v Simplification by algebraic method is DIFFICULT! v Method of simplification graphically suggested: Karnaugh maps v Usable with functions up to 6 variables

5. CSI-2111 Structure of Computers Ipage 4-5 1X 00 Example * v Diagram - 2 variables  f(A, B) =  m(0, 1) = A’ m0m0 m1m1 m3m3 m2m2 B B’ A A’ 11 00 A B

6. CSI-2111 Structure of Computers Ipage 4-6 1X 00 Karnaugh Maps (II) v Can be conceived from: –Truth tables –Canonical CSOP or SOP form –Canonical CPOS or POS form v Can give result like: –Minimal Sum of Products (SOP) form –Minimal Products of Sums (POS) form

7. CSI-2111 Structure of Computers Ipage 4-7 1X 00 Example *  f (A, B, C, D) =  m (0,1,2,5,8,9,10) v f SOP = v f POS = D C A B 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 (A' + B') (C' + D') (B' + D) B'D' + B'C' + A'C'D

8. CSI-2111 Structure of Computers Ipage 4-8 1X 00 Simplification * v Simplify starting from the SOP form: f (A, B, C, D) = CD’+A’D+ACD D C A B 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1

9. CSI-2111 Structure of Computers Ipage 4-9 1X 00 Simplification * v Simplify starting from the SOP form: f (A, B, C, D) = CD’+A’D+ACD D C A B 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 = C + A’D

10. CSI-2111 Structure of Computers Ipage 4-10 1X 00 Karnaugh Maps (III) v 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. CSI-2111 Structure of Computers Ipage 4-11 1X 00 Don’t-Care Values *  Simplify f (A, B, C, D) =  m (1, 2, 3, 7, 11, 15) X (0, 5) D C A B X 0 1 X 1 1 1 0 0 0 0 0 1 1 0 0 v f SOP = A’B’ + CD v The minterm 5 should not be included; it would not be minimal!

12. CSI-2111 Structure of Computers Ipage 4-12 1X 00 4.2 Circuits v Combinational: v Sequential: E 1 E n S 1 S m combinational circuit memory Input Variables Output Variables States E 1 E n S 1 S m combinational circuit input Variables Output Variables

13. CSI-2111 Structure of Computers Ipage 4-13 1X 00 Integrated Circuits (I) v The integrated circuits, material manufacture of logic gates and more complex functions, are characterized in several ways. v Why they used are? v Level of integretion? Quantity of transistors in a circuit.

14. CSI-2111 Structure of Computers Ipage 4-14 1X 00 Integrated circuits (II) v Manufacturing Technologies v Other characteristics

15. CSI-2111 Structure of Computers Ipage 4-15 1X 00 Complementary readings v In Mano and Kime: –Sections 2.4 and 2.5 u Simplification and Karnaugh maps –Section 2.8 (Optional) u Integrated circuits