ece 331 – digital system design basic logic functions, truth tables, and standard logic gates

ECE 331 – Digital System Design

Basic Logic Functions,Truth Tables,

andStandard Logic Gates

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Basic Logic Functions

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Basic Logic Functions

AND Logical multiplication (product) A . B = AB = “A and B”

OR Logical addition (sum) A + B = “A or B”

NOT Logical complement A‘ = “not A”

Order of Precedence:1. NOT2. AND3. OR- can be modified by parenthesis

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Truth Table

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x 1 x 2

x n

x 1 x 2 x n + + + x 1 x 2

x 1 x 2 +

x 1 x 2

x n

x 1 x 2

x 1 x 2 x 1 x 2 x n

(a) AND gates

(b) OR gates

x x

(c) NOT gate

Logic Gates

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Additional Logic Functions NAND

F = (A . B)‘ = (AB)’ F = not (A and B) Compound logic function

NOR F = (A + B)’ F = not (A or B) Compound logic function

Not the same as F = A’ . B’

Not the same as F = A’ + B’

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Truth Tables

Exercise:

Derive the Truth Table for the NAND and NOR gates.

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x 1 x 2

x n

x 1 x 2 x n + + + x 1 x 2

x 1 x 2 +

x 1 x 2

x n

x 1 x 2

x 1 x 2 x 1 x 2 x n

(a) NAND gates

(b) NOR gates

Logic Gates

shorthand for a NOT gate

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Additional Logic Functions

Exclusive-OR F = (A.B') + (A'.B) Output = 1 iff one of the inputs is 1 (but not both). Inclusive-OR: F = A + B

Exclusive-NOR F = (A.B) + (A'.B') Output = 1 iff both input are 0 or both inputs are

1. Also known as the Equivalence function.

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Truth Tables

Exercise:

Derive the Truth Table for the XOR and XNOR gates.

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Additional Logic Operations

XOR gate

XNOR gate

A B

F = A xor B

F = A xnor B

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Truth Tables

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Truth Tables Used to describe the functional behavior of a Boolean

expression and/or Logic circuit. Each row in the truth table represents a unique

combination of the input variables. For n input variables, there are 2n rows.

The output of the logic function is defined for each row. Each row is assigned a numerical value, with the rows

listed in ascending order. The order of the input variables defined in the logic

function is important.

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3-input Truth Table

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4-input Truth Table

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Standard Logic Gates

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Standard Logic Gates

Discrete components used to build logic circuits. 74xx08 AND gate 74xx32 OR gate 74xx04 NOT gate 74xx00 NAND gate 74xx02 NOR gate 74xx86 XOR gate

Logic Families (TTL, LS-TTL, F, HC)

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Standard Logic Gates: Data sheets

Data sheets provide essential information: Logic Function Truth Table Pin-out Electrical Characteristics Timing Characteristics Package Description(s)

This information is necessary when building logic circuits from discrete components.

Each Logic Family has a unique set of characteristics.

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Logic Families TTL

Low-Power TTL (“L”) High-Speed TTL (“H”) Schottky (“S”) Low-Power Schottky (“LS”) Advanced Schottky (“AS”) Advanced Low-Power Schottky (“ALS”) Fast (“F”)

CMOS High-Speed CMOS (“HC”) Advanced CMOS (“AC”)

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Table E.4 (p. 916 of textbook)

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Combinational Logic Circuits

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Combinational Logic Circuits

Composed of an interconnected set of logic gates. Also known as Switching Circuits Logic circuits can be designed from

Truth tables Boolean expressions

Logic circuits realized through Interconnection of discrete components Synthesis from a Hardware Description Language

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Example:

Derive the Boolean expression and draw the circuit diagram for the following truth table:

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f

x 1

x 2

Example

F = x’1.x’

2 + x’

1.x

2 + x

1.x

2

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Example:

Derive the Boolean expression and draw the circuit diagram for the following truth table:

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Wiring Diagrams

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(a) Dual-inline package

(b) Structure of 7404 chip

V DD

Gnd

Standard Logic Gate: Pin-out

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Standard Logic Gate: Pin-out

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Standard Logic Gate: Pin-out

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Example:

Draw the circuit diagram and the wiring diagram for the following Boolean expression:

F = x’2.x

3 + x

1.x

2

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V DD

x 1 x 2 x 3

f

7404

7408 7432

Example