mplitying Logic CivcuitS

ci iven a truth table , wha u the cowe>þevding

orolean Eguadiem.

Cen oulean euation , whel u thee

simplest logic civcuit er it 7

Digilal Casculs ae divided inte fwo bvoel Ccegce)

Combational ctsuts, evd 3. Seyueetial uls.

Fw Cabihatiomal cisuds, lhe cdbd et any mslevt

2.

tiwe depend epon e inputs broeo aHhd& Ihslnd tiwe. Thi means theve u no mewor il im these crruts . There ave dthes tyhes cs scuS Iwhuch he eudputs al any stnd S tiwe de poml bds PCn he breed v þud s a well as past Input/ctpuBs-Tw meas hene axe elements wwed to s past desmation. The>e elements nt bnowM o memoy. 5ueh culs ave b.mown Seguentrtal Ciscuts.

equC ments o combinetiona The destgmn

may be speed tn one ef he fellcuing w Sd o slelemeus

cleam þresim , amc

. A

Tuth table The ollowing

meihods cen be wed simp fy he Bulcan uncHen s 1. Algebric methed

2. KaY naugh - map fehniyue, Cutne- McClushey method, wnd Nantable 4 tescd mapin (NEM) chugur

Logical uncttens aE pre ss ed in desms

ogica N able, The vahut) asmecd : j1h

cga duncTic1 cewell as the loqical. vcie bles

CRE in he bina 1 esm Hy atbrt og logi

unctiom cam be pvessed in he Jelceg

S-j-bvodue dom(S OP) cerd

* Prcdud -e- Sum orm Pos).

Tuy doe> nd mean -Hhet hie legc duncdion

Cemnot bc wviHen in an hes psm. J4 can be

esms lem in Mou fos ms but he aore bw dosms

Conveniently sutel iw avihg astalard

metheds Joy designiy the ciyCut will bcOme

discuss io n.

clear Jrom he Followi'G dscSs Ic

Fundamental Preducts

9n digita systems, inpuds/ signals a avaa ble

in ethe COmplemented ox oncomblemented dem.

Fe t>tmce , i two Navabls A and esL

they complements emd noTmally ar also

PYeent -

Foux vale ossible woy t AND Jate

twc vnput ) si gued that are in comble mented

OS bn comple vmentec desm.

A

(C (d) (a) (6)

Eunalently , the cudp i 1 onl when FA= 0=o Fvom now on, we wil abb-seviqle the jeseYoVg inpu comdikem by

A, = 0, o

Fi (b). Y= A The oulbut can be equal to 1

only when A = o @vd :1

Fi e) = A8 equal a i cmly dex the pu

COnd om A- 6:o

Figd Y A eguat ca 1 only when Cam

A 1

The teble below Sumwasiz the deus passi ble.

AND AND Ig tw wpuds m com plementr and

Uncomplemevted dom. A

O

C

The ouy loqical vo ducls are A B, AB AB nd AG ave celld qunde mert al cduct, beceuse fhey, epveeds the Dac wny to cembine fuo

IMputs with en AND «t. hey

Fandcmedal pvoducl o hrre vaMaba Nanale: 3 =n, S, vavous ombtnafons

C Fundamerlal A

Dxadud o o AC

O A GC O

A SC

O O A BC

o A C

O A BC

AC

Funda mental product o ouT Naa ble.(n=4). NO o combinatins 2=I

Fundamelat pvoduct

A

O ABCD O

AGCD ABCO

ABCO

AGCO

A COO ABCD

O A Be D

A C OD

A GCOO

AGCO ABCD

ABCO

Sum c_ Product -

Cx bessttm fex the outft by ORing Ahe juuclce vwevial

soducts that breduce 1outputs. The jolleeivy amples

uve a tsuth tble cme cam tnd lie Dculeem

yo hcw

Funde ental Pro dt

Tth r ble

AD AG

ORg the dundamesdal beduts tos wuch tie outpui

e = AG +AG-Th Doelean ex þressien um ej Psoduct et oRAND

AG AND 0« A

logre c sCuit AND AND Qdc

So + oR gete.

AG AB4 Y- AB AB

xole hYee Navia ble)

loc scuil jo he gien Fundametl Phrduct

A

Tdh te-ble.

skp I- Fid the duudcam emal

reduct co epondang to eech i O

O O odput the truth table. seT- oR Hhe undawentel roduct

A BC -o Rged*

y @c+ABc+Agz¥ABc. O

D A ND att A BC

ABC A

AC

TMe bevleam expreoich SO obteined a

logical sum dunla mevtal preducts. Fer he

Teaon, th mthed oj getting The Acoken

*þressiem called the sum--bsduct" methd.

Sum a proluct loguc dcgyum AND-oR Metwcd .

ABC

A BC

D ACC

D

Cuve 2 orlem pres1on, wha w logAC.

Ccit AND

Y= ) *(9+ Ac) many way s locuc Cisu

cam be

ANO etc AND 9 te. obtun 6.C

A A FOC)]OR 9*tt

LGin

(A 40 ) 64AE)

wing AND, OR gett

AB

D D AC

AC

C i puts r

He we havC

as4med he

CNulyle

AND CR Teai saten oj he gien bcolean

expíeicn

i) Desigre the cicuit wth enly

LUnvssal 9al] NAND OG NoR

ybe od te Oe

(A+ Bc ) 6 + AC) Fst co west it into Sum Poduct (SoP) °7uatic.

E A COtATS + BC. 0t AC )

A-0+ A AC BC0+ BAC

A-+A A)-C 4 (@.B)- C + AB (C.c)

Y AB+c + AC ()

eguaken in sum oj bvodued dem. cun be Tealz«d sing AND-oR ealizaten

Thw hnown as two-Ieveleaizuhen. The Tst leve lev covs1s oj oR Gcte, as 5hown

Ccsists a) ND gedes and he SeIord levet

igu be loe.

A A

D D

A0+ B.C + A.C

C

Ievel- level-1

Re alizaton woln NA ND NA ND

2). APPly De Mosgan's thece m to equlien

Y A +c +AC

Tlang Complemewts om balh sid ep Navp Coat, a-13

Y iya13 NAND a. (3).

The seali zatn *7vcen

-.

A AC 3 level .

Level I

uing NOR te only L Product edSt. Sum egual n (1).

Y= (A+ Bc) + AT)

uS ny e bropesty (+C) = A+9 (A teS.

+Ac = t A) tc)

Egudium (1 en be Hew a

AO) (A ) (AAG) (t

(4) - A ND 0e

Pe . Sieii cJ y:tu) casn OP and rìtD 9a

io belo

A

A A+C -Y.

BT levd- AND 8 ett

eve-I

Demos auns Theoem, e cam w key. u oe

= A+O + Atc+0 t

YA+Ye + Nc wherA A+6

e A+C

Ta YA+Ne+Yc NoRga, Ic = B+ (5).

A10

+C = N2

-A A 3

eve- T level- I.

Pos e7alien cawn be de Srjhed usin NOR o es

nly Comy one (Tbe e gehe - vvev Scol gcehe No)

rnimig ahen/simpud* cuuon o logr a Boole an expsin

Cosides e 2. P.os] epiauen A+RC + AC

y Algebraie me thed.

A yAe t 1

A A AsAe R[I+*] + AC A

AGI+ ACI Bc

AB[e+C]4 AC+BC

ABC + ADC 1 Ac + BC

AI+ 0]+ B<[I+AJ

AC1 + Gc1

Y Ac + Bc. Is°P eyauo.cam be YeeUsed

b be m 8ms5

AND-oR

therem g cet cam be wn ed

NA NO

A AC +AC

Y y2 - 0 A

E. ). C(7)

Numbde Sod aie Tegurkd O F e 1 3 - ipt NAND os 3,-inbet NoR gay

(). Fes 7 3 3, 3 hpu NANo & I 3-input NAND.

Fa e S 3 n puB NoA ond L-3-1wpu NR

Cpos3 (3) Fox F1 (4) R AND elzedram

3-31h pul oR an i-3 tpu AD gate.

eeizuar SoP]J Fo e (2) AND- oR

3 a in put ANd gete and - 3 pud oR gad.

3- 2 bu AD g«tc

2- a-i put o A get cce uueh

ceees he pxpogercn celoy me, wuch

Opeten.

he Same decreaseF ú Sperd

F 325 Fa cGal Jane».n one ype od 9te

NANDNO ay Segu twhueh w c©n Neruev to

wwe when bec euwe a uwbe oj une CCs

sImilaa gate> a aywlable im *he Crme pacbog.

E7' Re qu mini mum humbey o cel e

leest wpogeaen delay

gebcwpuicatton xaple i

ACI D) Y A A A

Y= A ec+AoT +A8Sc.

x+X=x 1 ADc t AoC + ADCA

A4A A. =(A 4A ee+ Ae(c tc)

+A6.

Fple 2 ACABC ADC ASC.

sol-=KE (A4 A) C 4 Ac A C+Ac +C) BC 4 AGC+A BC

c+AC+ A GC B(A +c)+ AGC

=AG+ + A GC. A+CCet RBJ

AB+C A+9. = A+AC+ObC tAD (A+

o L