 # Combinational Logic Circuits Chapter 2 Mano and Kime.

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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

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

Basic Identities of Boolean Algebra

Implementation of Boolean Function with Gates

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

Minterms for Three Variables

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

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

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

Maxterms for Three Variables

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

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

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

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

Product of Sums: Exclusive-OR

Three- Level and Two- Level Implementation

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

Two-Variable Map

Three-Variable Map

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

Three-variable K-Maps X YZ 00011110 0 1 11 11 F = !X & !Y + X & Z

Three-variable K-Maps X YZ 00011110 0 1 11 11 F = !X & !Y & !Z + !X & !Y & Z + X & !Y & Z + X & Y & Z F = !X & !Y & (!Z + Z) + X & Z & (!Y + Y) = !X & !Y + X & Z

Three-variable K-Maps X YZ 00011110 0 1 1 1 11 F = Y & !Z + X 1

Three-variable K-Maps X YZ 00011110 0 1 11 111 1 F = !X & !Y + X & y + Z

Three-variable K-Maps X YZ 00011110 0 1 11 11 F = X & Z + !X & !Z

Three-variable K-Maps X YZ 00011110 0 1 11 11 1 1 F = Y + !Z

Three-variable K-Maps X YZ 00011110 0 1 0123 4567 11 11 F = m0 + m2 + m5 + m7 =  (0,2,5,7)

Four-Variable Map

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

Four-variable K-Maps WX YZ 00011110 00 01 11 10 0 1 3 2 4 5 7 6 8 9 13 15 14 11 10 12 Each square is numbered in the above K-map

Four-variable K-Maps WX YZ 00011110 00 01 11 10 0123 4567 89 11 12131415 F(W,X,Y,Z) =  (2,4,5,6,7,9,13,14,15)

Four-variable K-Maps 111 1 1 WX YZ 00011110 00 01 11 10 111 1 F = !W & X + X & Y + !W & Y & !Z + W & !Y & Z

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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

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 >

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Logical Operations with NAND Gates

Alternative Graphics Symbols for NAND and NOT Gates

Logical Operations with NOR Gates

Two Graphic Symbols for NOR Gate

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))

NAND Gate

X Y X Z Y Z F F = X & Y + X & Z + Y & Z

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

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

Exclusive-OR Constructed with NAND gates X & (!X + !Y) + Y & (!X + !Y) = X & !X + X & !Y + Y & !X + Y & !Y = X & !Y + Y & !X = X & !Y + !X & Y = X \$ Y

Parity Generation and Checking

Combinational Logic Circuits Binary Logic and Gates Boolean Algebra Standard Forms Map Simplification NAND and NOR Gates Exclusive-OR Gates Integrated Circuits

Fully Complementary CMOS Gate Structure and Examples An Integrated circuit (IC) is a silicon semiconductor crystal, containing the components for the digital gates. The various gates are connected on the chip to form the IC.