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Analysis of Clocked Sequenial Ci!cuis
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«T"e Sae
● Sae # Values of all $li%&$lo%s
E'a(%le
A B # ) )
D Q
Q
CLK
D Q
Q
A
B
y
x
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«Sae Equaions
D Q
Q
CLK
D Q
Q
A
B
y
x A*t +,- # DA
# A*t - x*t -+ B*t - x*t -
# A x + B x
B*t +,- # DB
# A’*t - x*t -
# A’ x
y*t - # . A*t -+ B*t -/ x’*t - # * A + B- x’
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«Sae Ta0le *T!ansiion Ta0le-
D Q
Q
CLK
D Q
Q
A
B
y
x
A*t +,- # A x + B x
B*t +,- # A’ x
y*t - # * A + B- x’
1!esenSae
In%uNe'Sae
Ou%u
A B x A B y
) ) )
) ) ,) , )
) , ,
, ) )
, ) ,, , )
, , ,
t +, t t
) ) )
) , )) ) ,
, , )
) ) ,
, ) )) ) ,
, ) )
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«Sae Ta0le *T!ansiion Ta0le-
D Q
Q
CLK
D Q
Q
A
B
y
x
A*t +,- # A x + B x
B*t +,- # A’ x
y*t - # * A + B- x’
1!esenSae
Ne' Sae Ou%u
x # ) x # , x # ) x # ,
A B A B A B y y
) ) ) ) ) , ) )) , ) ) , , , )
, ) ) ) , ) , )
, , ) ) , ) , )
t +, t t
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« Sae Dia2!a(
D Q
Q
CLK
D Q
Q
A
B
y
x
) ) , )
) , , ,
) 3 )
) 3 ,
, 3 )
, 3 )
, 3 )
, 3 ) ) 3 ,
) 3 ,
A4 input 3 output
1!esen
Sae
Ne' Sae Ou%u
x # ) x # , x # ) x # ,
A B A B A B y y
) ) ) ) ) , ) )
) , ) ) , , , )
, ) ) ) , ) , ), , ) ) , ) , )
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« D $li%&$lo%s
Example5 D Q
Q
x
CLK
y A
1!esenSae
In%uNe'Sae
A x y A
) ) )
) ) ,
) , )
) , ,
, ) ), ) ,
, , )
, , ,
)
,
,
)
,)
)
,
) ,))6,, ))6,,
),6,)
),6,)
A*t +,- # DA # A 7 x 7 y
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8
« JK $li%&$lo%s
Example5
J Q
Q K
CLK
J Q
Q K
x
A
B
JA = B KA = B x’
JB = x’ KB = A 7 x
A*t +,- # JA Q’A + K’A QA # A’B + AB’ + Ax
B*t +,- # JB Q’B + K’B QB
# B’x’ + ABx + A’Bx’
1!esenSae
I31Ne'Sae
$li%&$lo%In%us
A B x A B JA KA JB KB
) ) )
) ) ,
) , )
) , ,
, ) )
, ) ,
, , )
, , ,
) ) , )
) ) ) ,
, , , )
, ) ) ,
) ) , ,
) ) ) )
, , , ,
, ) ) )
) ,
) )
, ,
, )
, ,
, )
) )
, ,
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« JK $li%&$lo%s
Example5
J Q
Q K
CLK
J Q
Q K
x
A
B1!esen
SaeI31
Ne'Sae
$li%&$lo%In%us
A B x A B JA KA JB KB
) ) )
) ) ,
) , )
) , ,
, ) )
, ) ,
, , )
, , ,
) ) , )
) ) ) ,
, , , )
, ) ) ,
) ) , ,
) ) ) )
, , , ,
, ) ) )
) ,
) )
, ,
, )
, ,
, )
) )
, ,
) ) , ,
) , , )
, ) ,
)
,
) )
,
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«T $li%&$lo%s
Example5
TA = B x TB = x
y = A B A*t +,- # TA Q’A + T’A QA
# AB’ + Ax’ + A’Bx
B*t +,- # TB Q’B + T’B QB
# x 7 B
A
B
T Q
Q R
T Q
Q R
CLK Reset
x y
1!esenSae
I31Ne'Sae
$8$In%us
O31
A B x A B TA TB y
) ) )
) ) ,
) , )
) , ,
, ) )
, ) ,
, , )
, , ,
) )
) ,
) )
, ,
) )
) ,
) )
, ,
) )
) ,
) ,
, )
, )
, ,
, ,
) )
)
)
)
)
)
)
,
,
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«T $li%&$lo%s
Example5
A
B
T Q
Q R
T Q
Q R
CLK Reset
x y
1!esenSae
I31Ne'Sae
$8$In%us
O31
A B x A B TA TB y
) ) )
) ) ,
) , )
) , ,
, ) )
, ) ,
, , )
, , ,
) )
) ,
) )
, ,
) )
) ,
) )
, ,
) )
) ,
) ,
, )
, )
, ,
, ,
) )
)
)
)
)
)
)
,
,
) ) ) ,
, , , )
) 3 )
, 3 )
) 3 )
, 3 )
, 3 )
, 3 ,
) 3 )) 3 ,
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9ealy and 9oo!e 9odels www.rejinpaul.com
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«T"e 9ealy (odel5 "e ou%us a!e funcions of
0o" "e %!esen sae and in%us *$i28 :&,:-8● T"e ou%us (ay c"an2e if "e in%us c"an2e du!in2 "e
clock %ulse %e!iod8
♦ T"e ou%us (ay "a;e (o(ena!y false ;alues unless "ein%us a!e sync"!oni
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Fig. 5.21 Block diagram of Mealy and Moore sae mac!ine
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1!esenSae
I31 Ne'Sae
O31
A B x A B y
) ) ) ) ) )
) ) , ) , )
) , ) ) ) ,) , , , , )
, ) ) ) ) ,
, ) , , ) )
, , ) ) ) ,
, , , , ) )
Mealy
For the same state ,the output han!es "ith the input
1!esenSae
I31 Ne'Sae
O31
A B x A B y
) ) ) ) ) )
) ) , ) , )
) , ) ) , )) , , , ) )
, ) ) , ) )
, ) , , , )
, , ) , , ,
, , , ) ) ,
Moore
For the same state ,the output #oes not han!e "ith the input
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9oo!e Sae Dia2!a( www.rejinpaul.com
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Sae 3 Ou%u
) ) 3 ) ) , 3 )
, , 3 , , ) 3 )
)
,
,
,
))
)
,
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Sae Reducion and Assi2n(enwww.rejinpaul.com
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«Sae Reducion
Reducions on "enu(0e! of fli%&flo%s and"e nu(0e! of 2aes8
● A !educion in "enu(0e! of saes (ay!esul in a !educion in"e nu(0e! of fli%&flo%s8
● An e'a(%le saedia2!a( s"o=in2 in $i28
:8>:8
Fig. 5.25 "ae diagram
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Sae Reducion www.rejinpaul.com
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● Only "e in%u&ou%u
sequences a!e i(%o!an8● T=o ci!cuis a!e
equi;alen
♦ Ha;e idenical ou%us fo!all in%u sequences?
♦ T"e nu(0e! of saes isno i(%o!an8
Fig. 5.25 "ae diagram
State: a a b c d e f f g f g a
Input: 0 1 0 1 0 1 1 0 1 0 0
Output: 0 0 0 0 0 1 1 0 1 0 0
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«Equi;alen saes
● T=o saes a!e said o 0e equi;alen♦ $o! eac" (e(0e! of "e se of in%us6 "ey 2i;e e'acly "e
sa(e ou%u and send "e ci!cui o "e sa(e sae o! o anequi;alen sae8
♦ One of "e( can 0e !e(o;ed8
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«Reducin2 "e sae a0le
● e # ! *!e(o;e !-?● # # $ *!e(o;e $ -?
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● T"e !educed finie sae (ac"ine
State: a a b c d e d d e d e a
Input: 0 1 0 1 0 1 1 0 1 0 0
Output: 0 0 0 0 0 1 1 0 1 0 0
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● T"e c"eckin2 of eac" %ai!of saes fo! %ossi0le
equi;alence can 0e donesyse(aically usin2I(%licaion Ta0le8
● T"e unused saes a!e!eaed as don@&ca!e
condiion fe=e!co(0inaional 2aes8
Fig. 5.26 #ed$ced "ae diagram
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I(%licaion Ta0le www.rejinpaul.com
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« %!e sae&red$cion 'roced$re for com'leely s'ecified sae
a(les is (ased on !e algori!m !a )o saes in a saea(le can (e com(ined ino one if !ey can (e s!o)n o (ee*$i+alen. %!ere are occasions )!en a 'air of saes do no!a+e !e same ne, saes- ($- none!eless- go o e*$i+alenne, saes. onsider !e follo)ing sae a(le/
« a- b imply c- d and c- d imply a- b. Bo! 'airs of saesare e*$i+alen i.e.- a and b are e*$i+alen as )ell as c and d.
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I(%licaion Ta0le www.rejinpaul.com
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«%!e c!ecking of eac! 'air of saes for 'ossi(le
e*$i+alence in a a(le )i! a large n$m(er of saescan (e done sysemaically (y means of animplication table. %!is a c!ar !a consiss of s*$ares-one for e+ery 'ossi(le 'air of saes- !a 'ro+ide
s'aces for lising any 'ossi(le im'lied saes.onsider !e follo)ing sae a(le/
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I(%licaion Ta0le www.rejinpaul.com
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«%!e im'licaion a(le is/
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I(%licaion Ta0le
« ! l f id l g ! i l li d ll !
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« n !e lef side along !e +erical are lised all !e saesdefined in !e sae a(le e,ce' !e las- and across !e (oom!orionally are lised all !e saes e,ce' !e las.
« %!e saes !a are no e*$i+alen are marked )i! a , in !ecorres'onding s*$are- )!ereas !eir e*$i+alence is recorded)i! a .
« "ome of !e s*$ares !a+e enries of im'lied saes !a m$s
(e f$r!er in+esigaed o deermine )!e!er !ey aree*$i+alen or no.
« %!e step-by-step 'roced$re of filling in !e s*$ares is asfollo)s/
1. lace a cross in any s*$are corres'onding o a 'air of saes )!oseo$'$s are no e*$al for e+ery in'$.
2. ner in !e remaining s*$ares !e 'airs of saes !a are im'lied (y!e 'air of saes re'resening !e s*$ares. :e do !a (y saring from!e o' s*$are in !e lef col$mn and going do)n and !en 'roceeding)i! !e ne, col$mn o !e rig!.
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I(%licaion Ta0le
3 M k i ! ! ! (l d i ! !
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3. Make s$ccessi+e 'asses !ro$g! !e a(le o deermine )!e!er anyaddiional s*$ares s!o$ld (e marked )i! a ,. ; s*$are in !e a(le iscrossed o$ if i conains a leas one im'lied 'air !a is no e*$i+alen.
4. Finally- all !e s*$ares !a !a+e no crosses are recorded )i! c!eckmarks. %!e e*$i+alen saes are/ a- b- d- e- d- g- e- g.
:e no) com(ine 'airs of saes ino larger gro$'s of e*$i+alen saes.%!e las !ree 'airs can (e com(ined ino a se of !ree e*$i+alen saes
d- e-g (eca$se eac! one of !e saes in !e gro$' is e*$i+alen o !eo!er )o. %!e final 'ariion of !ese saes consiss of !e e*$i+alensaes fo$nd from !e im'licaion a(le- oge!er )i! all !e remainingsaes in !e sae a(le !a are no e*$i+alen o any o!er sae/
a- b c d- e- g f
%!e red$ced sae a(le is/
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Sae Assi2n(en www.rejinpaul.com
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«Sae Assi2n(en
«To (ini(in m-
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● Any 0ina!y nu(0e! assi2n(en is saisfaco!y as lon2as eac" sae is assi2ned a unique nu(0e!8
● Use 0ina!y assi2n(en ,8
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Desi2n 1!ocedu!e www.rejinpaul.com
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«Desi2n 1!ocedu!e fo! sequenial ci!cui
● T"e =o!d desc!i%ion of "e ci!cui 0e"a;io! o 2e asae dia2!a(?
● Sae !educion if necessa!y?
● Assi2n 0ina!y ;alues o "e saes?
● O0ain "e 0ina!y&coded sae a0le?
● C"oose "e y%e of fli%&flo%s?
● De!i;e "e si(%lified fli%&flo% in%u equaions and
ou%u equaions?● D!a= "e lo2ic dia2!a(?
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Desi2n of Clocked Sequenial Ci!cuis
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« Example5
Deec B o! (o!e consecui;e ,s
S) 3 ) S, 3 )
SB 3 , S> 3 )
)
,
,
))
,
)
,
Sae A B
S) ) )
S, ) ,
S> , )SB , ,
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Desi2n of Clocked Sequenial Ci!cuis
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« Example5
Deec B o! (o!e consecui;e ,s
1!esenSae
In%uNe'Sae
Ou%u
A B x A B y
) ) )) ) ,
) , )
) , ,
, ) ), ) ,
, , )
, , ,
) ) )
) , )
) ) )
, ) )
) ) ), , )
) ) ,
, , ,
S) 3)
S, 3)
SB 3
,
S> 3
)
)
,
,
))
,
)
,
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Desi2n of Clocked Sequenial Ci!cuiswww.rejinpaul.com
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« Example5
Deec B o! (o!e consecui;e ,s
1!esenSae
In%uNe'Sae
Ou%u
A B x A B y
) ) )) ) ,
) , )
) , ,
, ) ), ) ,
, , )
, , ,
) ) )
) , )
) ) )
, ) )
) ) ), , )
) ) ,
, , ,
A*t +,- # DA * A6 B6 x-
# *B6 :6 -
B*t +,- # DB * A6 B6 x-
# *,6 :6 -
y * A6 B6 x- # *F6 -
Syn"esis usin2 D $li%&$lo%s
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Desi2n of Clocked Sequenial Ci!cuis =i" D $8$8
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« Example5
Deec B o! (o!e consecui;e ,s
DA * A6 B6 x- # *B6 :6 - # A x + B x
DB * A6 B6 x- # *,6 :6 -
# A x + B’ x
y * A6 B6 x- # *F6 -
# A B
Syn"esis usin2 D $li%&$lo%s
B
) ) , )
A ) , , )
x
B
) , ) )
A ) , , )
x B
) ) ) )
A ) ) , ,
xGet useful study materials from www.rejinpaul.com
Desi2n of Clocked Sequenial Ci!cuis =i" D $8$8
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« Example5
Deec B o! (o!e consecui;e ,s
DA # A x + B x DB # A x + B’ x
y # A B
Syn"esis usin2 D $li%&$lo%s
D Q
Q
A
CLK
x
B D Q
Q
y
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$li%&$lo% E'ciaion Ta0les
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1!esenSae
Ne'Sae
$8$8In%u
Q*t - Q*t +,- D
) )
) ,
, )
, ,
1!esenSae
Ne'Sae
$8$8In%u
Q*t - Q*t +,- J K
) )
) ,
, )
, ,
) ) *No c"an2e-) , *Rese-
) '
, '
' ,
' )
)
,
)
,
, ) *Se-, , *To22le-
) , *Rese-, , *To22le-
) ) *No c"an2e-, ) *Se-
Q*t - Q*t +,- T
) )) ,
, )
, ,
),
,
)
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Desi2n of Clocked Sequenial Ci!cuis =i" JK $8$8
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« Example5
Deec B o! (o!e consecui;e ,s
1!esenSae
In%uNe'Sae
$li%&$lo%In%us
A B x A B JA KA JB KB) ) ) ) )
) ) , ) ,
) , ) ) )
) , , , )
, ) ) ) )
, ) , , ,
, , ) ) )
, , , , ,
) '
) '
) '
, '
' ,
' )
' ,
' )
JA * A6 B6 x- # *B-
#JA * A6 B6 x- # *G6:6F6-
KA * A6 B6 x- # *G6 F-
#KA * A6 B6 x- #
*)6,6>6B- JB * A6 B6 x- # *,6 :-
#JB * A6 B6 x- #
*>6B6F6-
KB *A6 B6 x- # *>6 B6 F-
Syn"esis usin2 JK $8$8
) '
, '
' ,
' ,
) '
, '
' ,
' )Get useful study materials from www.rejinpaul.com
Desi2n of Clocked Sequenial Ci!cuis =i" JK $8$8
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« Example5
Deec B o! (o!e consecui;e ,s
JA # B x KA # x’
JB # x KB # A’ + x’
Syn"esis usin2 JK $li%&$lo%s
B
) ) , )
A ' ' ' '
x
B
' ' ' '
A , ) ) ,
x B
) , ' '
A ) , ' '
x
B
' ' , ,
A ' ' ) ,
x
CLK
J Q
Q K
x
A
B
J Q
Q K y
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Desi2n of Clocked Sequenial Ci!cuis =i"T $8$8
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« Example5
Deec B o! (o!e consecui;e ,s
1!esenSae
In%uNe'Sae
$8$8In%u
A B x A B TA TB
) ) ) ) )) ) , ) ,
) , ) ) )
) , , , )
, ) ) ) ), ) , , ,
, , ) ) )
, , , , ,
))
)
,
,)
,
)
Syn"esis usin2 T $li%&$lo%s
),
,
,
),
,
)
TA * A6 B6 x- # *B6 G6 F-
TB * A6 B6 x- # *,6 >6 B6 :6
F-
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« Example5
Deec B o! (o!e consecui;e ,s
TA # A x’ + A’ B x
TB # A’ B + B 7 x
Syn"esis usin2 T $li%&$lo%s
B
) ) , )
A , ) ) ,
B
) , , ,
A ) , ) ,
A
B
y
T Q
Q
x
T Q
Q
