Ec (5)Digital Electronics

8/10/2019 Ec (5)Digital Electronics

1/31

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5. Digital electronics

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2/31

:

'..

i"

i

.-.

i-

l.-

ir:

t_.

I

iv

I

i.-

irz

I

t^

I

it-/

t-

i.u

I

i

I

t

I r-,,

I

I

l'

l.-

I

I.-

I

Iv

I

t

i\/

i

dktu

t

;[:

Page 1 of 172


3/31

Booteon

Al

-:'

A=

Booueeru

AuoeBE&

onre

[]-lt[ff-fTl

an4temenl

-'+

A

at

'

A'

Booleon

Alqebm

:-

(-)

Klhsn

na-

of

vAriabe

Cre

less

-

(

r'

2'

3

{.r

tt

rb

pretfered

cohen

outp4-S*-9-

L

1'

(.)

Klhen

no.

vorioHes

ore

31,t.,5

[opto

s

vorrabte)

G)

outPut

is

o,1

at,

a.

-

Tabulotion

mattrd

-

(.)

rt

,s

used

rl)hen

no'

ol

variobles

"ate

flflre

'

'

'

a=1

t

=A

AND

:-

A,O

=

O

A.A

A

A.

1

a,

l

=O

-_o

=l

a+o

=O

A+t

1+o

A8r

AB

4.0

70

A.A

AtA

A*l

**A

=A

At

A

-l

l+l

S

(B*a

(','

grg--4

PA6E

fT-n

-:*

iiitb-

Page 2 of 172


4/31

---

ABr

Adcr

A6c

oPtian.

V@,

O

G)2

DATT

filtrrT-n-l

the

min.

no.

NrlUD

Cnote

-

tb)

I

dte

af

ABr

Adc

+

AgC

=

AB

r

AE

(c*E)

NO

NAND

Advo

Minimi

znrhbn

g-

+No.o le

J,

,s

i

?aunr

drssrpohbn

*

A

-

ABr

A{

{','

c+c'

-t)

-

.'=

A(6r

e)

A+

g)

(A+c)

=

A

+8C

4

carnpleil

circtsil

{ess

-la

lan

in

+

(no

of inpuf

*

)

-

,

Coot

i,.

ftoblen'.t

Srnphfu

;-

toli

nE

+

ABd

+

edfo

--gf:-j_4qg

.

n

s

(

I

"

c,

o

)

=

AAC,

+

Ad

l

(8i

=6+c)

Sol

=

AB

+Ac

@+a)

(a+c)

A.A*,

A'C

+69

n

5t

A+

A(t+8)t

Bc

R(t+B+c)

*^8c

_9-

s)'

A

tBC

iftoa

Theoren

c-lpssr*te-

necr

[-[-l-l

Page 3 of 172


5/31

*rl

DArEf-Tlrnn-rn

Sr?nrlqrt

(A+Brc)

(A+6+c)

(ArA*e

)

toHe

A+B

:

X

=

(x+c;

1

R+6+c)

(x+d)

(x+ce)

(A+6+c)

x

(At

6tc)

(A+g(At

6+c)

A+

et6+c)

A+

OO-t6g

A+

Bc

(Af

Bi

(Af

B)

(At

B)

GA)

(A+

q)

ta+

p

)

(d*

ei

(At

e;

(nte6)*(A+sB)

=

m)(A)

X*Y)(i*=)

=

[+Yz

A+

BC =

(A+

B)

(

e-*c)

oistribution

theorern,

A+

AB

h

*E)

(A+

8)

I

(AtB)

=

Aae'

fi+

Ad

l(A+e)

=

A+8

ABt

Aq

+

A6'

A

(s*6)

r

Ae

-

A

+

Ag-

,

(A"A)(tu6)

At

B

An,

pece

f*T7T-l

Page 4 of 172


6/31

DArr

n-l

fil

rT-]-l-_l

A8*

AB

+

AE

6+

Ad

Iql4rlelgl

(r)

A+

B

An

ABG+,Q96+Asc

ABd

+ABC

+

AaC

{

A6G

ol":-

Atnr4)

AB

+ A


7/31

DATETTIIII|-T*rn

f X.*el

(iitc)

(8+c)

{A+

6)

1A+c)

,

":

(B-tc)

is

re&:dont

term.

{a+$1

1(q1c)

(A*c)

(A+B)(Brc)

rt6+

ee

+6d

Ta th,s

cose

orl

the voriobt

oye

omplemevrted ont

one

aYe

unepmplementd,.

then.

[':

.1$e

tevnt

rrlhiob.

is

crnCornplerer@

is

toxen)

A6+ao

AB

1 BC

+AC

gC

n

-fr,O

Ad

(Atd)

(6t6)

(Ar-d;

(6+c)(A+c)

fheorn.

Booleen

At

bra'.-

Vlinimization

Cornplgrrlsrl'

Expressbn

Tiufr't

tcble

stq{rneo}.

rnce

{-l-TJ

Page 6 of 172


8/31

DArr[-j-lmrTT-n

-

1

I

I

I

Mfn[mizatr'oo

;-

XY

f

XYruz

fi=

xY

and

=

A+

Ag

a

(A+A

)

(A+B)

A+B

2

XY+

WZ

tet

t(AlB)

=

Tnen

the

vatue

of

Lt

(y,rl

,

y)

,

z

@)

xY+2

cdl

(c)

7s+*z

rr,)

x{+z

x

+

Ltilrjr

-),tl

,

z

I

=

F L

x*

+

8Y

,

z

I

F

L

I,v

+

x7

*y

xi.

V

+z

ioi)i

+z

Yy.+

'Y/

+Z

I

Z--X*Y

lfien

lhe

(4)

x

(b)

1

Sol

i

I

Y\.+Z

nacr

J-T-[-l

Page 7 of 172


9/31

-;

tv

[-T-rTr

ArE

TTI r]-l

SOP

(Surn

of

ftoduat

Fovm

ABc

+AEc+ABE

mintertn

Th WP

FoYut

t

each

orcdw*

terrn

'S

l(nou)n

OS

Mfnterrn

or

ImpLioont

sop

Form

rs

used

urhh

ol

p

o

ressrbn

is

l.

(rneons

l->

A

and

o

-,

A

5

-r'

l0t

*.ABC

$

-+

lOOl

-'

nBdo

tur

the

iven

truth

tcHe

,

minirnize soP

et

soP

totrn

onl

t

toKeo.

Ad+

AB

A

[A+A)

Y

can

r,oritten

os

Y

(A,8)

=

2

rn

(O,:)

ed

the

el

ressron

Jot

Y(A,8)

=

=m{o,?,3

oo.

r(,

.

it.

ion

in

sp

.tovro

:-

y

=

Aa

+

AB-

+AB

A

B

=

6

(AtA)

+ AB

E+AB

(6*a)

(e+e)

ar6

ef*ssrrrte.

:*

A+

I

rncr

f-J-[7]

Page 8 of 172


10/31

soP

con

6

**

|otrn

,

(6)

P(

tn carrnia

+

*

n

Ag#:,

e,

(

rt

rs

a

minimal

fr-]

+

In

cOnd

'-fii4i1,,1orm,

eacb

term

rnus?

hove

all

tnioblg.

E'

HiJ

?:.

?

I

I

is'1003

&oblt

I

Aues'j

sotn:-

logrcql

Qt4

(b)

5

rasian

A+

ti-c

rs.

_(ct

6

{d)

7

A

+BC

-

l-

I

Gl_

P(_

=

{Aet

AA)(C+c)

+

AAC

+

ABG

+

AEC

t

"

nacr

[-T-[-l

Page 9 of 172


11/31

DArr

I-TJ

I

l

lLj-t-i

,l

ffis

Fo(m

I

Prodrct

of

(R+

s+c;

L

moa

term

A+6

sori''r

u)e

tot{

e

A +BrC

A.r

BT

G)

POs

form

used

.

rohon

o7p

is

togt6

'o1

fl-r

1->

Er

'.-

to

-.-:,

A

Bd

roO

.aecD

Fo,r

C

en

-'h

truth

toble

prinirnize

a

Y

I

o?

I

ore

A

:

A

FoS

eaPressfon

'

.:

^*n

Ol,

nnlu thot

voltrg

0-t

-:':)

-=

=-

(A+6)

(At

s)

urhich

olp

ts

R*B

O

I

col)

be

oritlen

fn

PoS

'|orm

QS,

Tr

M

(

1,3)

:E

v

(

A,B) =

and for

soP'--

)

(A,B)

-

Zm

La'z],

l.e.

Z

mLa,z>

=

Tr

M

(t,3J'

t+F

,"5

l6ere'

are

a

vorioblg

iqg

&

combinobbn

then

aye ,

2,3,5

,6'

n'lcrlt'

trn

rl

)

f*:

FIAro,c]

,

lmlAll'\'1)

TI'

M

{

2,3,

5,

6J

Page 10 of 172


12/31

DArrTTlt]]rl-rn

variable

mailmum

Wsside

minter

ms

Gr

z

tP)

msxrerms

oYe

?n.

l.>

lor,

fi

=

2.

f.e.

(A,

B)

Totol

,0.

ol

min

or

mot

ferms

aYe

2?

=

\.

(.)

ta,

,

Yl=

3

i.e

.

(A, B,c)

IAiol

fi4..

mia

or

mc,-n

terms

ore

2-

--

I

Ffir'n=q,

(A,8)

tuta-q

,6

erPresstoD

*+

t

Ag

.AB

AB

A+8

AB

+AB

4{B

ABt

6d

I+B

.A+

a

urith

'11'

o

I

,utth

n

vonoble

m\tlmum

possible

togicctt

erpression

odem'.

{cr'

a=

4

u:tro}

rs

the

iatal

rlo.

of

tr{icat

loqical erpresst'on

=

o4

J

_-

---_)--

recr

fT--tl

,lr,ss**t-e.

Page 11 of 172


13/31

ll,fnm|*l*l-n

Duol,

Form

:-

cqrresponds

tc

lC?tC

.O'_

C

1,,

'r,

-+

+5V

=

lcqrc

c

*1__

*::

aogfc

t

=

ov

C

O

*r

-Sv

Hrgher

voioe

at

vo[

togrc

L.

-.

-.

l.7V

-> -o'8

-0'8

Y

ls

to

value

fhon

'

t.tv

lt,ea

it

'd

+

tve

togic

.

ov

I

o

I

o

oo{'

ort

r10

-ive

togt'c

AND

I

I.

For

-ive

L

conwt

t

to

a

ond

o

fo

t.

fhqt

+rve

b d

qate

and

tO

1rv

oe

gote.

p

*c

-ive

elrssr.lte.

lc

AND

golte

:.[s

oo"r

[T]-l

Page 12 of 172


14/31

--** -* ,{

rEl-Tlmrrrn

Dml

eil

n

is

used

to

qnvert

+

ive

_toglb

-

inlo

=

iv6

sJic

_gr,

AB

Duol

-

.4+B

ornl

fs

nathi

bul--1y4qg/c.

At\rD

OR

oR

Ft'nd

D.,ol.

-ABc+

A*c+

Agc

solL

ouol:-

{_t_al;;_ {:

q

:i)

(a*

s;ij

drJof

i..i

qsed

SetJ

Duol

a

a

:48+'Bc

*4C

AND

q--.--+

be


15/31

trn

s et,

Dml

rPresston

,

rf

o

some

erPressron.

variabte

-''

self

Duo-l

=

A--'A

(')

AND

+-

oR

(.)sP+

canPle

ment

6J

eab

vqrbble

.

onrr

[*T-ifi

l

@

rrrn

esslon

lS

a(s

n

voriobles

tben

total no.oJ

6tf

duot e^W'

For

n'-

L.

Ifren

2

duoll xPresdlor,,

,{

--+

Self

dr.ral

}

A

Toiol

sel,

docl

er

eSSIOD

are

2.

tili

For n

--

2

?

=

k-

then

,,

dDal

'et?resston'

A

---+

A

8-+ B

A--A

6--

B

'a;U

For

fi-3

J2

=

16-

then

16

&tal

srPression'

g,A

,

8,

d,

c,

(,,

A6+ECttA

,

ABr

BC+oA

'

"'

Complerwnt

z-

il

I

=

ABC+

Aec

+

A8C

aamplement

iS,

t =

(A*6"c)(6*fltcJ

(

A+Bt6)

PA6r

[.-ltTl

Page 14 of 172


16/31

DATETTImTT-rn

s0l'?l

I

Y

=

Ad

+

46

+AB

I

=

etE+

A)

+

AB

For

ttr.ro

varto

bl

e

(

A,8)

.

for

o

Eirren

venn

diaqrorn

,

minimize

the

soP

expresslbn

=

g+AB

=

A+B

qq? 31E9r

|

=

AB+

AA

* AB

=

*rc-te)

+-AB

=

A+A.B

lsJr)

tt19-

=

A+B

r)

Page 15 of 172


17/31

DATTTItE

o

rrrn

ABtEgtABtAE

s(R+A)*

e(n+A)

B+

6

L.

A86

ABC

soP

Tarm

1or

shaded

porfion

Atc

'

j

ABC+

Agc

ragd

t

l6c

+

ABC

+ AB

8G(A1A)

r'

AB(dtc)

+Ac(gre-)

.(-'

eatr.ddpl'

=

ABr

BC+

cA

rncr

l-[fl

Page 16 of 172


18/31

DA,E

TTI

[I]

r-rri-l

For

Seyfes

:-

Tlufr1

table

:-

fruth table:-

Irl

ploce

for

Paratlel

:-

Y

a

fn

ploce

of

bllb

it

+here

is

resrstTr

then

ons$er

renmrrs

the

some

but

$rne

drap.

l

0

Y=AB

A

O

__l___/,

ll

eottch

C.)

For

A--

t.

trongslor

becomes short

.

circuil'

Fav

ttrlo

srrifch

A

and

B

3-

:'

cl*sy,rte.

rncr

[JJl

Page 17 of 172


19/31

@

ri-TT-l

I=i3

NAND+

Y=

A.(Brc)-o

(Rs+

Ac)

6

ABo

t

ACD

I

I

o

I

Y

o

I

t

T

Y

I

o

0

o

l=

AfB

6uesl

onc

l--[-[-lA

Page 18 of 172


20/31

_

= __

,._-

DATETT-ltI}rT_rn

(H).

STATEMENT

:-

G)

6ues:-

A

logic

civcuil

have

g

inqut

A,

B,c'

and

alP

is

Y-

o/p

v is r.

tor

the

loltwir\

e'ombinaklon.

rit'

B

and

c

are

fiue

___=-tO __

tru

A

and

e

ole

folse

=

Ei

Uti,

A

,Bqnd

c we

ltue

:

Agc

I

then

rninimize

the

o1p

1sr

Y

.

sol':-

alp

=1.

(At+e

min

ter@

Y

-

Bc

*

Ac

+ ABc +

e6e

_

=

gc(l

+n)

+

Ad(r+

6

)

:

4

alP

Y

=o

,

then

taXe

lhaa

terrn

(PoS

iorm).


21/31

DArE

[-T-]f]]

LffiIC

6ATES

:-

8cfu

,|,0T

-)

Sasic

oR

I{AND

universol

NOR

SxOR

Arrifi'rnetrb

et


22/31

Ct\t

ShOu:rr

iS

'o'

tov

a

3

tdt

to

M

Hz

f

=

2Ntp4

los

Ht

=

g+N

oerr

[--l-l

fi-lfffi_i

.atlh

NaT

re

wave,s

It

,s

}altic.

r

u)ove

tor.

iq;

As

olp

rs

not

stobte

Sornetime

t

and,

Sometime

O

Hence

tt

ts

otso cntld

ostioble

rnulhvt'batsv.

(rii,

l:lffiF.

lu,

&'nq

oscfl

lalor

-

lhol

ttne

perid

(T)

=

6TX

I

,t

I

I

t\J -

no.

of

inverters

in

leedboc\.

Tn

a

ct\t

shorvt

is wP

&c.

Ifien

1g6,tlz

|

6Hz

Lc)

tOO

t4HL

=

lGfiz

---]--

I

ehssnatc-

pncr

[-l--l-l

Page 21 of 172


23/31

o

rT-rn

*EfIlrn

is

gnsec. Tfien

*ime

wr*

@)

6ns

(G)

il

ns

t4

,2t\3

dt

6ns

Astoble

Mr.lt#vibrotor

e

lr)qve

'T

=

2 Nt

12

n396.

of

each

Nor

(,uclv8

k.

Thus

time perid.

ct

x

ed

y

is

sorre.

I

I

AN

D

EATE

i.

+

o/p

r's

too

tt af

the

Up

rs

lats

both rgrxrtutative

lao

ond

assoo.iotrve

lotr)

.

(.)

AB

.i.

AA.

nnce[lI

Page 22 of 172


24/31

DArr

[-i-i

(")

ABC

=-(AB)'C

=

A

(BC)

^o

-5*

o

cnnttol

rrp

disable'

AND

9o,t9

fs not

in"oorkinq

shrfe-

:i

AND

-l

_l

Note

i

+ben

I

i

*

I

-,n

-

-1-

*---

I

it

*,

l"g,r-fryb.--:flgo]rry

t lt-

u' -I

Disable

O

nab

remairs

ia'O'

dae

to

olp

is

oh

in

Enoded

=r

I

In

TTL

-3-

sryJlE-rs

oPen

-

@

499i--*j--

il

Ut

1'

u)fll

aot

os

r#

(openor

#o,ahr?

r{p)

i

I

I

--]--- ---.

:'

i

*

S,reShOO

-

1

T-L

'

io Exan-

-+

Urused

r/p's

unused

rlp

l Doruo'1

-in

turuttrpin

(I/p)

Aluo

gote

uoused

rlp

con

be

*irrrecEd

to

r,qic

L.

orn

Patt

4"

ue"-

e.fsssr,rEte-

can

W

canoected

noct

[-i-l_l

Page 23 of 172


25/31

r*i

@

m_T-T-t

ATE

l l-lft

A

.A.6

=AB

-AB

unused

r/p

can

be

connectect

b

crle

of

the

used

t/e.

(onlV

1or

rrl)

If

it

fs

rrL

ic

lcrnil

then

unused

L/

p

qn

be

open

(unconnected)

Becouse

of

uDneecessqr

t/p

attaahed, ta

B

,

1on

fn

rofll

be

dawn.

Besi

to

ennnecti

unused

pin

Lrlp)

in

AND

eannecti

ta

;

i:l

mcr

[-T*l-l

Page 24 of 172


26/31

rf)

c(

A-

B_

,*

o"ro"an*

lo*

\+B

=

B+A

ry

(A+B)+c

A+

(ts+c)

6

rlp

oR

Gote

.-

(lnctusive

oR)

9

I

+

ulhen

an\

of

'the

r7P rS

l+gh

io

oR

eote

then

olP

ig

Hrgh.

the

qote

is

enableol

,

------>

I

rncr

[-T-D

Page 25 of 172


27/31

DArr

[-Tlt]l

@

rrTTl

a

'l

o/p

is

ttrcd

or

not

Y,

A

L.

it

{ satd

to

be

drsoble.

Unused

Tlp's

'.-

Ln

OR

la

,

usused

rlP

fs eannectc{J

fo

la

,O

r_.pul

t

doun.'

Qanneot

to

ane af

the

used rlP.

W

it is

ect

then

unuBed

rfp

q,an

bc

open

or

)tmted

.

In

oR

,

conngctin

the

unusd

,/p

is

d

+a

conoect

ta

ic

'a'

fn

he

oKt

eho{.rln in

tn

rrL

,

AND,

oR

,

INVSRTER

oKi

the

en

j/

p

alp

ts

A

AB

l+AB

'-t

tb)

|

TC)

AB

tdt

dr3

In

TrL

,

rf

is

elnssvrltr.

CIff

rlP's

o{e

Jtmt

then

tr

PA6E

Page 26 of 172


28/31

DAT.

TT_JfI]TTN-I

for

tcl

a

o

lB)

I

te)

A8

.rNVERTER

.

1__

Page 27 of 172


29/31

tl

@

fTof6F-l

Etr

ATE

D

ffrE

:-

(Aubbled

oR)

A

g

A

ErE

f

I

I

wren

both

IlP

hi

the

a/p

istau.

Oa t

o

dr'sobts

(

ing

it

ae

1

6nsb|e.

Llp

iszera

\

NENO

C,ollGO

cammutat:ive

tarr:

bUt

not

talloto

assrcfative

lou

Y

o

the

tuo

nol

folloto

essofatrve

toCI

universal

te

NAND

NOR

qote.

, I

unused

T/P

NAND

gote

con

be

connaoted similar

ta

unused

rlp

io

AND

NOR

GATE

',-

(8ubbld

AND)

OR

te.

tooed

NOT

a

A

B

pecs

[T-I

Page 28 of 172


30/31

DAT.

TT_II

TT*I-TI

--r

unused

ilp

in

NoR

aote

con

be

conoected

simitar

l'o

O

4

u*ten

both

lte is

toto

lhe

o/P

ls

/t

-..

A-f\

.'=

l.-

dlsoble.

enable

and

dr'sble

Wth

ate

some

qs'

oR

gote'

ftOg

qote

{ottan:

cornrnutotrve

too

otd

,rot

Jolbe=-

ixsci-

(.)

AIETT

*

At e

c

1

uJhon A=

B

,

olp

fs

loru

r'.e

'o''

in

ctus

ive

oR

I-r-Tl

I

il

I

---i-

I

EX0R

cr,

xOR

i-

Exctusive

oR

OR

te

is

otsb

gate

)

Nhert

A*8

,

4P

is

High

i.e.

c

tl'.

Page 29 of 172


31/31

I

--------------+

DArETntlrrrn

I

.---------r'

o

o----^+

j.

-

inver

ter

-'

Buller

It

is

olso

cotted

controtled

iaverter.

Note

l

A@

AO

AO

O

-A

A@

|

=

A

rf

A@8

=

C

t*ren.

AOC=B

8OC

=

A

A@BAC

=

O

Since, A@A

=Q

Then

ul

od

no.oJ

sqrne

Up

ond

even

no.

Qrne

at/P

gives

szeY(

2A

ExoB

ffinQ

olP

adsoCIn---

B@

I

@80

n

B

,.

d

n

is

odd

ct6t

Sbo(l)n

,

conloios

easd'l

tlxts

the

lf

P *ho

o/

P

ts.

)x

olP

of

evn

exo8

gote

hcrue

sams

_: :

=X

AO

AOA

=

A

A@

A@AOA

=

A

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