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## 4. Karnaugh Maps and Circuits

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**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.**4.1 Simplification of Switching Functions**• Why simplify and optimize? • Constraints • Cost ($$$)! • How? • Algebraic method (still…) • Karnaugh maps (wow!)**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**Karnaugh Maps (I)**• Simplification by algebraic method is DIFFICULT! • Method of simplification graphically suggested: Karnaugh maps • Usable with functions up to 6 variables**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**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**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**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**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**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.**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!**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**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.**Integrated circuits (II)**• Manufacturing Technologies • Other characteristics**Complementary readings**• In Mano and Kime: • Sections 2.4 and 2.5 • Simplification and Karnaugh maps • Section 2.8 (Optional) • Integrated circuits