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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