Boss Audio Systems BV8.5GA User Manual download pdf (Page 56)

entire

statement

is

true. The

NOT

func-

tion

simply reverses

a stazement

from

true

to false,

or

vice

versa.

Electronic

NOT

circuits

are commonly

referred

to

as

"inverters"

because

their

function

is

to

invert

the

polarity

of the

signal.

The

above

definitions

can

be tabu-

lated

into

a table

such

as

shown in

Fig.

1.

Such

a

table

is

useful in

showing

the

relationships

among

Boole's

three

logic

functions

and their

electronic

and

arithmetic

counterparts.

This

type

of table

is

sometimes

called

a "truth

table"

since it

sets forth

the

various

logic

conditions

for which

each

state-

ment

is

true.

Generally,

truth

tables

are

arranged

in

a

more

compact

form

simi-

lar

to those

shown for

the three

basic

logic

functions

in Fig.

2.

Truth

tables

can

be created

for

any

logic

function.

Specification

sheets

for

digital

logic

circuits

almost

always

in-

clude

a truth

table.

our

ON:

A

ABOUT.

AB

DUT.

O

O o o

0 0

o /

0

0

/

/

/

0

0

/

0

/

/

/

/

I

I /

I

A

O!/T.

0

/

/

O

Our.

Fig.

2.

AND,

OR,

and NOT

symbols

are

shown

with

truth

tables.

Logic

Symbols.

Boolean

logic

statements

can

be implemented

by

simply

writing

them

on paper,

using

alphabetic

symbols

to

correspond

to

"true"

and "false"

conditions.

Elec-

tronic

logic

diagrams,

however,

are

much

easier to

design

and

interpret

if

a

sort of

block

diagram

of the

circuit

is

presented.

For

this reason,

standard-

ized

logic

-block

symbols

have

been

devised

for the

three

basic logic

func-

tions.

They

are

shown

in Fig.

2.

Compound

Logic

Circuits.

Two

cir-

cuit

combinations

(the NOT -AND

and the

NOT -OR)

are used

so

frequently

that

they

are treated

as

basic logic

elements

and given

their

own

logic

symbols

and

truth

tables.

When

the

AND

function

is

followed

by

a NOT

statement,

the meaning

of the

AND

function

is reversed

to NOT

-AND,

commonly

called

a NAND

function.

Similarly, when

the

OR

function

is

fol-

lowed

by a NOT

statement,

the meaning

of the

OR

statement

is reversed

to

NOT

-OR, commonly

referred

to

as

a

NOR

function.

The

logic

symbols

and truth

tables

for the

NAND

and NOR

functions

are

shown in Fig.

3.

58

B

OUT

A B OUT.

A B OUT.

00/ 00/

D / I

0 /

O

/

0 /

/

0

0

/ / 0 /

/

O

Fig.

3. NAND

and NOR

symbols

with

associated

truth

tables.

DeMorgan's

Theorem.

About

the

same time

Boole

developed

his logic

theories,

Augustus

DeMorgan

was

also

developing

some

fundamental

theories

of logic.

His

most

important

contribution,

known

as DeMorgan's

Theorem,

relates

the AND,

OR,

and NOT

functions

as follows:

A +

B =

A

x

B.

A

x

B = A +

B.

The

arithmetic

symbols +

and

x

mean

OR

and AND,

respectively.

The

bar, or

vinculum,

over

a

letter

indicates

the

NOT

function.

Thus

A

means

NOT

A.

The

importance

of DeMorgan's

Theorem

is

that

an AND

circuit

contain-

ing

a NOT

at each

input

corresponds

to

an OR

circuit

followed

by

a

NOT.

Simi-

larly,

an

OR circuit

with

a NOT

at each

input

corresponds

to

an AND

circuit

fol-

lowed

by

a

NOT.

This

does

not

equate

the

NAND

and NOR

functions,

but it does

mean

that NAND

circuits

can

be

used to

implement

NOR

functions,

and vice

versa.

Complex

Logic

Systems.

Logic

systems

that

contain

three

or more

basic

logic

elements

are termed

"com-

plex."

One

of the

simplest

of the

com-

plex logic

systems

is

the EXCLUSIVE

OR

(sometimes

written

xoR)

function

shown

diagramatically

in

Fig.

4. From

the

truth

table,

note

that

this

function

is

identical

to

the

OR function

with

one

important

exception:

A true

condition

exists

only

when

one

or

the

other

con-

dition,

but not

both, is true.

The

EXCLUSIVE

OR

function

com-

pletes

the

connection

between

Boolean

logic,

the

binary number

sys-

tem,

and electronic

switching

circuits,

for it

can be

used to

add two

binary

bits.

To

see

how

this

is accomplished,

as-

sume

a

logic

1

at input

A and

a logic

Oat

A ..r..u

CARRY

B

Sv/4

Fig. 4.

Logic

array for

XOR

circuit.

input

B in

the EXCLUSIVE

OR

circuit

shown in

Fig. 4.

Since only

one

input

is

enabled

(input

A), AND

circuit

1

does

not

turn

on. Hence,

a 0 is

present

at

the

CARRY

output.

OR

circuit 1

does turn

on,

since

only

one input

need

be present.

Since the

NOT

circuit

inverts

the

0 from

AND

circuit 1

into

a

logic

1,

AND

circuit

1

has

two

input

signals

and

is therefore

turned

on. The

result

is

a logic

1

at the

suns

output. (The

circuit

has

added

0

+

1

to obtain

1.)

The

EXCLUSIVE

OR

circuit

is

often

called

a "half-

adder." Try

verifying

its

operation

yourself

by

adding 1

+ 1

in

binary.

Practical

Logic

Circuits. Figure

1

demonstrated

how

simple

switching

circuits

can

be used to implement

each

basic

logic

function.

However,

it

is

usually not

practical

to

employ

switches in

real

systems.

Instead,

transistors,

SCR's, tunnel

diodes,

or

other

solid -state

switches

are em-

ployed.

The

most

commonly

used

switch in

digital

electronics

is the

transistor.

Relatively

simple

circuits that

combine

diodes,

resistors,

and transistors

can

be

used to

implement

the

AND, OR,

and

NOT

functions.

Thanks

to integrated

circuit (IC)

technology,

several

or even

dozens

of

individual

logic

circuits

can

be placed

on

a single

compact

silicon

chip.

Resistor

-transistor

logic

(RTL)

was

once

the most

popular

type of

digi-

tal IC,

but it has

been largely replaced

by the more noise

-immune

transistor

-

transistor

logic (TTL)

type.

In

recent

years,

field-

effect

transistor

(FET)

technology

has

been

adapted

to

integrated

logic

circuits

of

amazing

complexity.

By insulating

the

gate

of

a

FET with

a layer

of

silicon dioxide,

ex-

tremely

high

impedances

are made

possible.

The

result

is

a logic

circuit

that

requires

microamperes

or

nanoamperes

of

operating

current

at

relatively

low voltages.

Insulated

-gate

fabrication

techni-

ques

are collectively

known

as MOS

(for

metal

oxide

semiconductor)

tech-

nology.

Since MOS

transistors

are

uni-

polar (p-

or

n -type)

and do not

require

separate

p

and n

sections

like

conven-

tional

bipolar pnp

and npn

transistors,

MOS

IC's

can have

a

much

higher

component

density

than most

conven-

tional

IC's. The

result

is

large

-scale

in-

tegration

(LSI).

So, the

next

time

you

read

or hear

the phrase

"MOS

LSI,"

you will know

that

it refers

to

a large

-

scale integrated

circuit

employing

metal

oxide

semiconductors.

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