EENG 2710 Chapter 2Algebraic Methods For The Analysis and Synthesis of Logic circuits

Chapter 2 Homework2.1c, 2.2c, 2.3, 2.4, 2.5, 2.6b, 2.7a, 2.10b, 2.16b, 2.18b, 2.25, 2.29a

Logic Function or Gate RepresentationLogic functions or gates can be represented:algebraicallyusing truth tablesusing electronic circuits.

Basic Logic FunctionsThe three basic logic functions are:

AND Gate

OR Gate

NOT GateY = AB

Algebraic RepresentationUses Boolean algebra.Boolean variables have two states (binary).Boolean operators include AND, OR, and NOT gates.

Truth Table RepresentationDefines the output of a function for every possible combination of inputs.A system with n inputs has 2n possible combinations.

Electronic Circuit RepresentationUses logic gates to perform Boolean algebraic functions.Gates can be represented by schematic symbols.Symbols can be either distinctive-shape or rectangular-outline.

Distinctive Shape Schematic SymbolsUses different graphic representations for different logic functions.Uses a bubble (a small circle) to indicate a logical inversion.

Rectangular-Outline Schematic SymbolsAll functions are shown in rectangular form with the logic function indicated by standard notation inside the rectangle.The notation specifying the logic function is called the qualifying symbol.Inversion is indicated by a 1/2 arrowhead.

NOT FunctionOne input and one output.The output is the opposite logic level of the input.The output is the complement of the input.

NOT Function Boolean RepresentationInversion is indicated by a bar over the signal to be inverted.

NOT Function Electronic CircuitCalled a NOT gate or, more usually, an INVERTER.Distinctive-shape symbol is a triangle with inversion bubble.Rectangular-shape symbol uses 1 and the inversion 1/2 arrowhead.

NOT Function Electronic Circuit

AND FunctionTwo or more inputs, one output.Output is HIGH only when all of the inputs are HIGH.Output is LOW whenever any input is LOW.

AND Function

AND Boolean RepresentationAND symbol is or nothing at all.

AND Function Electronic CircuitCalled an AND gate.Distinctive-shape symbol uses AND designation.Rectangular-shape symbol use & as designator.

AND Function Electronic Circuit

AND Function Electronic Circuit

AND Function Electronic Circuit

OR FunctionTwo or more inputs, one output.Output is HIGH whenever one or more input is HIGH.Output is LOW only when all of the inputs are LOW.

OR Function

OR Boolean RepresentationOR symbol is +.Y = A + B

OR Function Electronic CircuitCalled an OR gate.Distinctive-shape symbol uses OR designation.Rectangular-shape symbol uses as designator.

OR Function Electronic Circuit

Active LevelThe logic level defined as ON for a circuit.When a logic HIGH is ON, the signal is active-HIGH.When a logic LOW is ON, the signal is active-LOW.

NAND FunctionGenerated by inverting the output of the AND function.Output is HIGH whenever any input is LOW.Output is LOW only when all inputs are HIGH.

NAND Function

NAND Boolean RepresentationUses AND with an inversion overbar.

NAND Function Electronic CircuitCalled a NAND gate.Uses the AND symbol with inversion on.

NAND Function Electronic Circuit

NOR FunctionGenerated by inverting the output of the OR function.Output is HIGH only when all inputs are LOW.Outputs is LOW whenever any input is HIGH.

NOR Function

NOR Boolean RepresentationUses OR with an inversion overbar.

NOR Function Electronic CircuitCalled a NOR gate.Uses OR symbol with inversion on the output.

NOR Function Electronic Circuit

3 Input NOR and NAND FunctionTruth Tables3 Input NAND:

3 Input NOR:

3 Input NOR and NAND FunctionTruth Tables

Exclusive OR GateTwo inputs, one output.Output is HIGH when one, and only one, input is HIGH.Output is LOW when both inputs are equal both HIGH or both LOW.

Exclusive OR Gate

Exclusive OR Gate

Exclusive NOR GateTwo inputs, one output.Output is HIGH when both inputs are equal both HIGH or both LOW.Output is LOW when one, and only one, input is HIGH.

Exclusive NOR Gate

Exclusive NOR Gate

Gate Equivalence NANDA NAND gate can be represented by an AND gate with inverted output.A NAND gate can be represented by an OR gate with inverted inputs.

Gate Equivalence NAND

Gate Equivalence NORA NOR gate can be represented by an OR gate with inverted output.A NOR gate can be represented by an AND gate with inverted inputs.

Gate Equivalence DeMorgan FormsChange an AND function to an OR function and an OR function to an AND function.Invert the inputs.Invert the outputs.

DeMorgans Theorem Break the line and change the sign

DeMorgans Theorem The following are two common errors associated with DeMorgans Theorem:

Active Logic LevelsAny INPUT or OUTPUT that has a BUBBLE is considered as active LOW.Any INPUT or OUTPUT that has no BUBBLE is considered as active HIGH.

Active Logic Levels - NOR At least one input HIGH makes the output LOW. All inputs LOW make the output HIGH.

Active Logic Levels - NOR

Postulates of Boolean AlgebraP1: + = OR, . = ANDP2(a): a + 0 = a

Postulates of Boolean AlgebraP2(b): = a 1 = a

Postulates of Boolean AlgebraP3(a): a + b = b + aP3(b): b + a = a + b

P4(a): a + (b + c) = (a + b) + cP4(a): a(b c) = (a b)c

Postulates of Boolean AlgebraP5(a): a + bc = (a + b)(a + c)

Postulates of Boolean AlgebraP5(b): a (b + c) = ab + ac

Postulates of Boolean AlgebraP6(a): a + a = 1 where a = not a

Postulates of Boolean AlgebraP6(b): a a = 0

Postulates of Boolean AlgebraPostulates not in your book: x 0 = x

Postulates of Boolean AlgebraPostulates not in your book: x 1 = x

Theorems of Boolean AlgebraT1(a): a + a = aShort Proof: If a = 1, 1 + 1 = 1 or If a = 0, 0 + 0 = 0Long proof: (shown in book)

Theorems of Boolean AlgebraT1(b): a a = aShort Proof: If a = 1, 1 x 1 = 1 or If a = 0, 0 x 0 = 0T2(a): a + 1 = 1Short Proof: If a = 1, 1 + 1 = 1 or If a = 0, 0 + 1 = 1T2(b): a 0 = 0Short Proof: If a = 1, 1 x 0 = 0 or If a = 0, 0 x 0 = 0

Theorems of Boolean AlgebraT3(a): (a) = aShort Proof: If a = 1, then a =0, Thus (0) = 1 If a = 0, then a =1, Thus (1) = 0

T4(a): a + ab = aProof: a + ab = a(1 + b) = a(1) = a

Theorems of Boolean AlgebraT4(b): a (a + b) = aProof: a (a + b) = aa + ab) = a + ab = a(1 + b) = a(1) = a

T4(b): a (a + b) = aProof: a (a + b) = aa + ab) = a + ab = a(1 + b) = a(1) = a

Theorems of Boolean AlgebraT5(a): a + ab = a + bProof: a + ab = (a + a)( a + b) = 1(a + b) = a + b

T5(b): a (a+b) = ab

Theorems of Boolean AlgebraT6(a): ab + ab = aProof: ab + ab = a(b + b) = a(1) = a

T6(b): (a + b)(a + b) = aProof: (a + b)(a + b) = aa + ab +ab +bb = a + ab = a(1 + b) = a(1) =a

Theorems of Boolean AlgebraT7(a): ab + abc = ab + acProof: ab + abc = a(b + bc) = a (b + b)( b + c) = a(1)(b + c) = ab + ac

Theorems of Boolean AlgebraT7(b): (a + b)(a + ab + c) = (a + b) + (a + c)Proof: a + ab + c a + b aa + aab + ac ab + abb + bc a + ab + ac + ab + 0 + bc a(1 + b) + ac + ab + bc a + ac + ab + bc a(1 + c) + ab + bc a + ab + bc a(1 +b) + bc a + bc = (a + b) + (a + c) Same as P5(a)

Theorems of Boolean Algebra (DeMorgans Theorem)T8(a): (a + b) = ab

T8(b): (ab) = a + b

Theorems of Boolean AlgebraT9(a): ab + ac + bc = ab + acProof: ab + ac + bc = ab +ac + (1)(bc) = ab +ac + (a + a)(bc) = ab +ac + (abc + abc) = (ab +abc) + (ac + acb) = ab(1+c) + ac(1 + b) = ab + ac

Theorems of Boolean AlgebraT9(b): (a + b)(a + c)(b + c) = (a + b)(a + c)

Operations with Logic 0 & 1

Operations with the Same Variable & Complement of a Variable

Simplifying an Expression

Simplifying an Expression

Problem 6aSimplify the following switching function:

Algebraic Forms of switching functions Product term:Part of a Boolean expression where one or more true or complement variables are ANDed.Sum term:Part of a Boolean expression where one or more true or complement variables are ORed.

Algebraic Forms of Switching FunctionsSum-of-products (SOP):A Boolean expression where several product terms are summed (ORed) together.Product-of-sum (POS):A Boolean expression where several sum terms are multiplied (ANDed) together.

Algebraic Forms of Switching Functions(Examples of SOP and POS Expressions)

Algebraic Forms of Switching FunctionsSOP and POS UtilitySOP and POS formats are used to present a summary of a switching circuit operation.

Algebraic Forms of Switching Functions(Canonical SOP Minterms)

Algebraic Forms of Switching Functions(Canonical SOP Minterms)

Algebraic Forms of Switching Functions(Canonical SOP Minterms)Example

Algebraic Forms of Switching Functions(Canonical SOP Minterms)

Algebraic Forms of Switching Functions(Canonical POS Maxterms)

Algebraic Forms of Switching FunctionsTruth Table: POS = 0s and SOP = 1s

M(0,1, 4, 5) = m(2, 3, 6, 7)

Problem 16a & 18a

M(1) = m(0, 2,3)

Problem 18a (Using Boolean Algebra)

Universality of NAND/NOR GatesAny logic gate can be implemented using only NAND or only NOR gates.

NOT from NANDAn inverter can be constructed from a single NAND gate by connecting both inputs together.

NOT from NAND

AND from NANDThe AND gate is created by inverting the output of the NAND gate.

AND from NAND

OR and NOR from NAND

OR from NAND

NOR from NAND

NOT from NORAn inverter can be constructed from a single NOR gate by connecting both inputs together.

NOT from NOR

OR from NORThe OR gate is created by inverting the output of the NOR gate.

OR from NOR

AND and NAND from NOR

AND from NOR

NAND from NOR

Simplest Switching Expression From A Timing Diagram

Simplest Switching Expression From A Timing Diagram

Simplest Switching Expression From A Timing Diagram

Simplest Switching Expression From A Timing Diagram

Problem 29b

Venn Diagrams

Venn Diagrams