COMBINATIONAL LOGIC CIRCUITS

1. x1 Z x2 C.L. xn COMBINATIONAL LOGIC CIRCUITS • Z = F (x1, x2, ……., Xn) • F is a Binary Logic (BOOLEAN ) Function • Knowing F Allows Straight Forward Direct Implementation of the C.L. Circuit. • OBJECTIVES • LearnBinary Logic and BOOLEAN Algebra • Learn How To Manipulate Boolean Expressions and Simplify Them • Learn How to Map a Boolean Expression into Logic Circuit Implementation

2. Binary Logic & Logic Gates • Elements of Binary Logic / Boolean Algebra • 1- Constants, • 2- Variables, and • 3- Operators. • Constant Values are either0 or 1 • Binary Variables{ 0, 1} • 3 Possible Operators AND, OR, NOT • Physically • 1- Constants Power Supply Voltage ( Logic 1) •  Ground Voltage ( Logic 0) • 2- VariablesSignals (High = 1, Low = 0) • 2- Operators Electronic Devices (Logic Gates) • AND - Gate • OR - Gate • NOT - Gate (Inverter)

3. Logic Gates & Logic Operations

4. Multi-Input Gates Boolean Algebra • Boolean Expression • Combination of Boolean Variables, AND-operators, OR-operators, and NOT operators. • Boolean Expressions (Functions) Can be Expressed as a Truth Table

5. Boolean Algebra • ExampleF = X + Y . Z

6. Properties of Boolean Algebra Operator Precedence • 1- Parentheses • 2- Not operator (Complement) • 3- AND operator, • 4- OR operator

7. Algebraic Manipulation ExampleX + XY = X Proof:X + XY = X. (1 + Y) = X.1 = X ExampleX + X`Y = X + Y Proof:(1)X + X`Y = (X+ X`) (X + Y) = 1.(X + Y) = X + Y (2)X + X`Y = X.1 + X`Y = X.(1+Y) + X`Y = X +XY + X`Y (XY +X`Y) + X = X + Y Example ``Consensus Theory`` XY + X`Z + YZ = XY + X`Z Proof:XY + X`Z + YZ = XY + X`Z + YZ(X +X`)= XY(1 + Z) +X`Z(1 + Y) = XY + X`Z

8. Canonical & Standard Forms For 3- Variables X, Y, and Z Define the Following: (A) MinTerms • m0 =X`Y`Z` = 1 IFF XYZ = 000 • m1 =X`Y`Z = 1 IFF XYZ = 001 • m2 =X`Y Z` = 1 IFF XYZ = 010 • ………………………………………. • m7 =X Y Z = 1 IFF XYZ = 111 (A) MaxTerms • M0 = X + Y + Z = 0 IFF XYZ = 000 • M1 = X + Y + Z` = 0 IFF XYZ = 001 • M2 = X + Y` + Z = 0 IFF XYZ = 010 • ………………………………………. • M7 =X`+ Y`+ Z` = 0 IFF XYZ = 111 Mi = mi (DeMorgan / Truth Table) 2n minterms 2n MaxTerms

9. ORing F = m2+ m4 + m5 +m7 = (2, 4, 5, 7) = Sum of minterms F = m0+ m1 + m3 +m6 = (0, 1, 3, 6) Expressing Functions as a Sum of Minterms or a Product of MaxTerms Example: Consider the Function F Given By its Truth Table Complementary MinTerms

10. Expressing Functions as a Sum of Minterms or a Product of MaxTerms F = m2+ m4 + m5 +m7 = (2, 4, 5, 7) = Sum of minterms F = m0+ m1 + m3 +m6 = (0, 1, 3, 6) F = (F) = m0.m1.m3.m6 = M0.M1.M3.M6 = (0, 1, 3, 6) = Product of Maxterm F = M2.M4.M5.M7 = (2, 4, 5, 7) ANDing

11. F = (2, 4, 5, 7) = (0, 1, 3, 6) F = (0, 1, 3, 6) = (2, 4, 5, 7) Expressing Functions as a Sum of Minterms or a Product of MaxTerms Standard Forms:Sum of Products (SOP) andProduct of Sums (POS) Example Implement SOP F = XZ + Y`Z + X`YZ` Two-Level Implementation Level-1: AND-Gates ; Level-2: One OR-Gate

12. Expressing Functions as a Sum of Minterms or a Product of MaxTerms Example Implement POS F = (X+Z )(Y`+Z)(X`+Y+Z`) Two-Level Implementation Level-1: OR-Gates ; Level-2: One AND-Gate

13. Example Implement POS F = (X+Z) . (Y`+Z) . (X`+Y+Z`) Two-Level Implementation Level-1: OR-Gates ; Level-2: One AND-Gate