Lecture 3 ETR 271 Fall 2014 Boolean Logic
Transcript of Lecture 3 ETR 271 Fall 2014 Boolean Logic
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September 2014 Johnny Cornett
ETR 271 – MicrocomputerElectronics I
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Homeor!
Ch"pter ##$1% #$2% #$&% #$'% #$(% #$1&% #$##%#$#'% #$#)% #$44
*ue 10 September
Ch"pter 4
4$4% 4$&% 4$)% 4$12% 4$14% 4$20% 4$22%4$#1*ue 17 September
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in"ry Sitches
• *iscreet sitches "re " +oo "n"lo+y to i+it"l6bin"ry sitches$ Tr"nsistors oper"te in " simil"rm"nner to sitches
open
Two states of a switch (open/closed)
close
open close
o8 on
9"lse true
0 1
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in"ry Sitches
• i+ht sitches "re " +oo "n"lo+y to i+it"l6bin"ry sitches
Sitch
"ttery i+ht x
St"tus o9 sitch 6: etermines i9 i+ht is -; or -,,
I9 : < 0% sitch is -5E;% i+ht is -,, 60
I9 : < 1% sitch is C-SE*% i+ht is -; 61
L(x) = x
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Series Connection 63;*• To sitches "re pl"ce in series% so -THmust be close 9or li+ht to turn on
St"tus o9 2 sitches 6:1
% :2
etermines i9 i+ht is -; o9
-,,I9 :1 < 1 3;* :2 < 1 sitches "re both C-SE*% i+ht is
-;-therise% i+ht is -,, 6i9 either sitch is 0
S
"ttery
S
i+ht x 1 x 2
L(x1,x2) = x1 •
x2
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5"r"llel Connection 6-R
= To sitches "re pl"ce in parallel% so i9 EITHER isclose the li+ht ill turn on
St"tus o9 2 sitches 6:1
% :2
etermines i9 i+ht is -; o9
-,,I9 :1 < 1 -R :2 < 1 sitches "re C-SE*% i+ht is -;
i+ht is -,, only i9 -TH sitches "re open
S
5oersupply S
i+ht
x 1
x 2
L(x1,x2) = x1
+ x2
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Combin"tion o9 Connections
= Three sitches "re pl"ce in series "n parallel
L(x1,x2) = (x1 + x2) = x3
S
5oersupply S i+ht
S X
1
X 2
X
#
>hen ill the li+ht turn on? 6:1% :2% :# < ?
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In/ersion@Complement
•
>h"t i9 e "nt the li+ht -; hen : < 0?
L(x) = x
>h"t ill the circuit loo! li!e?
S i+ht
5oersupply
R
x
ETR 271 *i+it"l Circuits ,"ll 2014)
L(x) = x’ = !x = ~x = T x
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The in/erter per9orms the oole"n T oper"tion$ >hen the input is ->% the outputis HIHA hen the input is HIH% the output is
->$
The In/erter A X
Input
A X
Output
-> 60 HIH61HIH 61 ->
60
The T oper"tion 6complement is shonith "n o/erb"r$ Thus% the oole"n e:pression
9or "n in/erter is X < A.
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The In/erter
E:"mple"/e9ormsB A
X
A X
3 +roup o9 in/erters c"n be use to 9orm the1s complement o9 " bin"ry numberBin"ry number
1s complement
1 0 0 0 1 1
0 1
0 1 1 1 0
0 1 0
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ETR 271 *i+it"l Circuits ,"ll2014
E:"mple"/e9ormsB A
X
The 3;* oper"tion is use in computerpro+r"mmin+ "s " selecti/e m"s!$ I9 you "nt
to ret"in cert"in bits o9 " bin"ry number butreset the other bits to 0% you coul set " m"s!ith 1s in the position o9 the ret"ine bits$
The 3;* "te
A
B
X
B
00000011I9 the bin"ry number 10100011 is3;*e ith the m"s! 00001111%
h"t is the result?
& A
B
X
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ETR 271 *i+it"l Circuits ,"ll2014
The % $ate prouces " HIH output i9 "nyinput is HIHA i9 "ll inputs "re ->% the outputis ->$ ,or " 2Dinput +"te% the truth t"ble is
The -R "te
The % oper"tion is shon ith " plus si+n6 beteen the /"ri"bles$ Thus% the -Roper"tion is ritten "s X < A + B.
Inputs
A B X
Output
00
01
10
11
0111
A
B
X A
B
X F1
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ETR 271 *i+it"l Circuits ,"ll2014
E:"mple"/e9ormsB A
X
The -R oper"tion c"n be use in computerpro+r"mmin+ to set cert"in bits o9 " bin"rynumber to 1$
The -R "te
B
A
B
X A
B
X F1
3SCII letters h"/e " 1 in the bit &
position 9or loer c"se letters "n " 0 inthis position 9or c"pit"ls$ 6it positions"re numbere 9rom ri+ht to le9t st"rtin+ith 0$ >h"t ill be the result i9 you -R"n 3SCII letter ith the (Dbit m"s!00100000?
The resultin+ letter ill be loer c"se$
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ETR 271 *i+it"l Circuits ,"ll2014
The "# $ate prouces " -> outputhen "ll inputs "re HIHA otherise% theoutput is HIH$ ,or " 2Dinput +"te% the trutht"ble is
The ;3;* "te
Inputs
A B X
Output
0 00 11 01 1
1110
A
B
X A
B X
&
The "# oper"tion is shon ith " otbeteen the /"ri"bles "n "n o/erb"rco/erin+ them$ Thus% the ;3;* oper"tion is
ritten "s X < A.
B 63ltern"ti/ely% X = AB.)
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ETR 271 *i+it"l Circuits ,"ll2014
E:"mple"/e9ormsB A
X
The ;3;* +"te is p"rticul"rly use9ul bec"useit is " Guni/ers"l +"te – "ll other b"sic +"tesc"n be constructe 9rom ;3;* +"tes$
The ;3;* "te
B
Ho oul you connect " 2Dinput;3;* +"te to 9orm " b"sic in/erter?
AB
X AB
X &
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ETR 271 *i+it"l Circuits ,"ll2014
The % $ate prouces " -> output i9"ny input is HIHA i9 "ll inputs "re HIH%the output is ->$ ,or " 2Dinput +"te% thetruth t"ble is
The ;-R "te
Inputs
A B X
Output
00
0
110
11
100
0
AB
X AB
X F1
The % oper"tion is shon ith " plus si+n6 beteen the /"ri"bles "n "n o/erb"rco/erin+ them$ Thus% the ;-R oper"tion isritten "s X < A + B.
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E:"mple"/e9ormsB A
X The ;-R oper"tion ill prouce " -> i9 "nyinput is HIH$
The ;-R "te
B
>hen is the E* is -; 9or thecircuit shon?
The E* ill be onhen "ny o9 the 9ourinputs "re HIH$
A
C B
D
X
330Ω
+5.0 V
AB
X AB
X F1
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ETR 271 *i+it"l Circuits ,"ll2014
The &% $ate prouces " HIH output onlyhen both inputs "re "t opposite lo+ic le/els$ The truth t"ble is
The -R "te
Inputs
A B X
Output
00
01
10
11
0110
AB
X AB
X = 1
The &% oper"tion is ritten "s X = AB + AB$ 3ltern"ti/ely% it c"n be ritten ith "circle plus si+n beteen the /"ri"bles "s
X = A + B.
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Complement
= >h"t i9 e h"/e more th"n one /"ri"ble?
>h"t is the complement o9B 6:1% :2 < :1
:2?L(x1,x2) = (x1 +
x2)
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Truth T"bles
= Cre"te " t"ble to etermine h"t the outcomeis 9or " /"riety o9 input /"lues
1$ ,irst columns "re input /"ri"bles 6:1% :2% etc$
2$ ;e:t columns "re outcomes@outputs
:
1
:
2
:1=
:2
:1
:2
3;* -R
0 0
0 11 01 1
0
001
0
111
1
000
;-T
;-T
6:1 :2
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Truth T"bles
Cre"te truth t"bles 9or the 9olloin+ 9unctionsB
,6:1% :2 < :1 :2
5r"ctice
6:1% :2% :# < 6:1 :2 = :#H6:% y < 6: = y
J6:% y < : y
6: < : 1
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Selecte ey Terms
Inverter
Truth table
Timingdiagram
AND gate
OR gate
3 lo+ic circuit th"t in/erts orcomplements its inputs$
3 t"ble shoin+ the inputs "ncorresponin+ output6s o9 " lo+iccircuit$3 i"+r"m o9 "/e9orms shoin+ theproper time rel"tionship o9 "ll o9 the"/e9orms$ 3 lo+ic +"te th"t prouces " HIHoutput only hen "ll o9 its inputs "reHIH$
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3 lo+ic +"te th"t prouces " HIH outputhen one or more inputs "re HIH$
S l t T
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Selecte ey Terms
NAND gate
NOR gate
Exclusive-OR gate
Exclusive-NOR gate
3 lo+ic +"te th"t prouces " -> outputonly hen "ll o9 its inputs "re HIH$
3 lo+ic +"te th"t prouces " -> outputhen one or more inputs "re HIH$
3 lo+ic +"te th"t prouces " HIH outputonly hen its to inputs "re "t oppositele/els$
3 lo+ic +"te th"t prouces " -> outputonly hen its to inputs "re "t oppositele/els$
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1$ The truth t"ble 9or " 2Dinput 3;* +"te is
0 00 1
1 01 1
Inputs
A B X
Output
0 00 1
1 01 1
10
00
Inputs
A B X
Output
0 00 11 01 1
Inputs
A B X
Output
Inputs
A B X
Output
0 00 11 01 1
0111
a. b.
c. d.
01
10
0001
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KuiL
K i
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2$ The truth t"ble 9or " 2Dinput ;-R +"te is
0 00 1
1 01 1
Inputs
A B X
Output
0 00 1
1 01 1
Inputs
A B X
Output
0 00 11 01 1
Inputs
A B X
Output
Inputs
A B X
Output
0 00 11 01 1
a. b.
c. d.
01
10
0001
10
00
0111
ETR 271 *i+it"l Circuits ,"ll 2014 2(
KuiL
K i
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#$ The truth t"ble 9or " 2Dinput -R +"te is
0 00 1
1 01 1
Inputs
A B X
Output
0 00 1
1 01 1
Inputs
A B X
Output
0 00 11 01 1
Inputs
A B X
Output
Inputs
A B X
Output
0 00 11 01 1
a. b.
c. d.
01
10
0001
10
00
0111
ETR 271 *i+it"l Circuits ,"ll 2014 2)
KuiL
K i
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'$ 3 lo+ic +"te th"t prouces " HIH outputonly hen "ll o9 its inputs "re HIH is "6n
"$ -R +"te
b$ 3;* +"te
c$ ;-R +"te
$ ;3;* +"te
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KuiL
K i
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($ 3 2Dinput +"te prouces the output shon$6 X represents the output$ This is "6n
"$ -R +"te
b$ 3;* +"te
c$ ;-R +"te
$ ;3;* +"te
A
X
B
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KuiL
K i
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)$ 3 2Dinput +"te prouces " HIH outputonly hen the inputs "+ree$ This type o9 +"teis "6n"$ -R +"te
b$ 3;* +"te
c$ ;-R +"te
$ ;-R +"te
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KuiL
o+ic Symbols
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o+ic Symbols
= >hen r"in+ " circuit% it is help9ul to h"/e
symbols to escribe the lo+ic"l 9unctions 63;*%-R% ;-T
x1 x2
x1 x2. x1 x2
x1 x2 x x
3;*+"te
-R
+"te
;-T
+"te
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3n"lysis o9 o+ic ;etor!
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3n"lysis o9 o+ic ;etor!
= 3 lo+ic netor! c"n be "n"lyLe by trutht"ble
f 6:1%:2 < :1 6:1 =
:2
2$ *r" the netor!% usin+ lo+ic symbols
1$ Cre"te the truth t"ble 9or f
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3n"lysis o9 o+ic ;etor!
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3n"lysis o9 o+ic ;etor!
f 6:1%:2 < :1 6:1 = :2
x1
x2
f3
• 3 "n "re intermei"te points in the netor! –helps us unerst"n 6"n /eri9y h"t is h"ppenin+ithin the netor!
x1 x2 f x1 x2%6
0101
0011
1101
3
10
10
00
01
ETR 271 *i+it"l Circuits ,"ll 2014 #&
Eui/"lent ;etor!s
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Eui/"lent ;etor!s
g 6:1%:2 < ? x1
x2
f x1 x2
%6
0
1
0
1
0
0
1
1
1
1
0
1
3
10
10
00
01
= Is there "nother netor! th"t ill output the s"
result?
1 1 0 0→ → → 0 0 1 1→ → →
1 1 0 1→ → → 0 1 0 1→ → → g
x1
x2
= >hich netor! is better? >hy?
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ETR 271 *i+it"l Circuits ,"ll 2014#(
oole"n 3l+ebr"
oole"n 3ition
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ETR 271 *i+it"l Circuits ,"ll2014
In oole"n "l+ebr"% " 'aiale is " symbol
use to represent "n "ction% " conition% or"t"$ 3 sin+le /"ri"ble c"n only h"/e " /"lueo9 1 or 0$
oole"n 3ition
The co*ple*ent represents the in/erse o9 "/"ri"ble "n is inic"te ith "n o/erb"r$
Thus% the complement o9 A is A$3 liteal is " /"ri"ble or its complement$
3ition is eui/"lent to the -R oper"tion$ The sum term is 1 i9 one or more i9 the liter"ls"re 1$ The sum term is Lero only i9 e"ch liter"lis 0$ *etermine the /"lues o9 A, B, "n C th"t
m"!e the sum term o9 the e:pression A +B + C < 0?E"ch liter"l must < 0A there9ore A < 1% B < 0 "n C < 1$
oole"n Multiplic"tion
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ETR 271 *i+it"l Circuits ,"ll2014
In oole"n "l+ebr"% multiplic"tion is
eui/"lent to the 3;* oper"tion$ The proucto9 liter"ls 9orms " prouct term$ The prouctterm ill be 1 only i9 "ll o9 the liter"ls "re 1$
oole"n Multiplic"tion
>h"t "re the /"lues o9 the A% B "nC i9 the prouct term o9 A.B.C < 1?
E"ch liter"l must < 1A there9ore A <1% B < 0 "n C < 0$
C t ti
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ETR 271 *i+it"l Circuits ,"ll2014
Commut"ti/e "s
n te*s of the eslt, the ode inwhich 'aiales ae %ed *a-es no
di.eence
The co**tati'e laws "re "pplie to"ition "n multiplic"tion$ ,or "ition% thecommut"ti/e l" st"tes
A + B = B+ A
n te*s of the eslt, the ode inwhich 'aiales ae "#ed *a-es nodi.eence
,or multiplic"tion% the commut"ti/e l"st"tes
AB =BA
3ssoci"ti/e "s
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ETR 271 *i+it"l Circuits ,"ll2014
3ssoci"ti/e "s
0hen %in$ *oe than two'aiales, the eslt is the sa*ee$adless of the $opin$ of the
'aiales
The associati'e laws "re "lso "pplie to
"ition "n multiplic"tion$ ,or "ition% the"ssoci"ti/e l" st"tes
A + 6B +C = 6 A + B +C
,or multiplic"tion% the "ssoci"ti/e l" st"tes
0hen "#in$ *oe than two'aiales, the eslt is the sa*ee$adless of the $opin$ of the'aiales A6BC = 6 ABC
*istributi/e "
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ETR 271 *i+it"l Circuits ,"ll2014
*istributi/e "
The distiti'e law is the 9"ctorin+ l"$ 3common /"ri"ble c"n be 9"ctore 9rom "ne:pression .ust "s in orin"ry "l+ebr"$ Th"t is
AB + AC =
A6B+ C The istributi/e l" c"n be illustr"te itheui/"lent circuitsB
B+ C C
A X
B AB
B
X
A
C
A AC
AB + AC A6B+ C
oole"n 3l+ebr"
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oole"n 3l+ebr"
= 3:ioms 6"sic 3ssumptions
1"$ 0 = 0 < 0
1b$ 1 1 < 1
2"$ 1 = 1 < 12b$ 0 0 < 0
#"$ 0 = 1 < 1 = 0 < 0
#b$ 1 0 < 0 1 < 1
4"$ i9 : < 0% then ;-T : < 14b$ i9 : < 1% then ;-T : < 0
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oole"n 3l+ebr"
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oole"n 3l+ebr"
= Sin+le N"ri"ble Theorems
&"$ : = 0 < 0
&b$ : 1 < 1
'"$ : = 1 < :
'b$ : 0 < :
7"$ : = : < :
7b$ : : < :
("$ : = ;-T : < 0(b$ : ;-T : < 1
)$ ;-T 6;-T : < :
ETR 271 *i+it"l Circuits ,"ll 20144&
oole"n 3l+ebr"
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oole"n 3l+ebr"
= MultiDN"ri"ble Theorems
10"$ : = y < y = :
10b$ : y < y :
11"$ : = 6y = L < 6: = y = L
11b$ : 6y L < 6: y L12"$ : = 6y L < 6: = y 6: = L12b$ : 6y = L < 6: y =6: L1#"$ : 6: = y < :
1#b$ : = 6: y < :
14"$ 6: = y 6: = ;-T y <:14b$ 6: y = 6: ;-T y <
:1a T (x • ) = T x +T 1 T (x + ) = T x• T 1'"$ : 6;-T : = y < : y
1'b$ : = 6;-T : y < : = y
17"$ 6: = y 6y = L 6;-T : = L < 6: = y 6;-T := L17b$ 6: y = 6y L = 6;-T : L < 6: y = 6;-T
: L4'
ETR 271 *i+it"l Circuits ,"ll 2014
oole"n 3l+ebr"
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oole"n 3l+ebr"
Reuce the 9olloin+ e:pressions to simplest9orm% usin+ oole"n "l+ebr"
6: = ;-T L 6;-T y = ;-T L 6: = L 6;-T y =
L
5r"ctice
6;-T : = ;-T y 6: = y 6: = ;-T y
ETR 271 *i+it"l Circuits ,"ll 2014
47
Nenn *i"+r"ms
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Nenn *i"+r"ms= >h"t oes the li+ht blue "re" represent in the
i"+r"ms belo?
y x x y x y
x y
x y
z
ETR 271 *i+it"l Circuits ,"ll 2014
4(
Terminolo+y
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Terminolo+y
= o+ic"l SumD "nother term 9or -R oper"tions
= o+ic"l 5rouct
D "nother term 9or 3;* oper"tions
6: = y 6y = L 6;-T : = L is the sum o9 threeproucts 6S-5
6: y = 6y L = 6;-T : L is the prouct o9three sums 65-S
ETR 271 *i+it"l Circuits ,"ll 2014
4)
5receence@3ppe"r"nce
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@ pp
= 5"rentheses ill "l"ys etermine preceence 6li!e"rithmeticD too m"ny p"rentheses c"n c"use con9usion
= >ithout p"rentheses% orer isB ;-T% 3;*% -R
: = y y = L 1st +ener"te proucts% thensum o9 proucts
= C"n "lso simpli9y "ppe"r"nce% by remo/in+
= 6i9 ob/ious
: = y y = L :y yL
ETR 271 *i+it"l Circuits ,"ll 2014
&0
Synthesis o9 o+ic ;etor!
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Synthesis o9 o+ic ;etor!
x1 x2 f (x1, x2)
0 0
0 1
1 0
1 1
1
1
0
1
= SynthesiLe "n "rbitr"ry 9unction usin+ 3;*% -R%;-T +"tes
ETR 271 *i+it"l Circuits ,"ll 2014
&1
Synthesis o9 o+ic ;etor!
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y +
x1 x2 f (x1, x2)
0 0
0 1
1 0
1 1
1
1
0
1
= -ption 1 – ,in Sum o9 5roucts (AB + CD + DE
+ etc.)
= -nly use terms here f 6:1% :2 < 1
f (x1, x2) = x1’x2’ +
x1’x2 + x1x2
ETR 271 *i+it"l Circuits ,"ll 2014 &2
Synthesis o9 o+ic ;etor!
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y +x1 x2
0 0
0 1
1 0
1 1
1
1
0
1
= *r" the circuit% usin+ 3;*% -R% ;-T +"tesO
f (x1, x2) = x1’x2’ + x1’x2 + x1x2
f
x1
x2
ETR 271 *i+it"l Circuits ,"ll 2014 &#
Synthesis o9 o+ic ;etor!
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y +x1 x2
0 0
0 1
1 0
1 1
1
1
0
1
= Is this the simplest circuit? C"n it be reuce?Ho?
f (x1, x2) = x1’x2’ + x1’x2 + x1x2
ETR 271 *i+it"l Circuits ,"ll 2014 &4
Synthesis o9 o+ic ;etor!
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y += Psin+ oole"n "l+ebr"% reuce this 9unction
f (x1, x2) = x1’x2’ +
x1’x2 + x1x2 f (x1, x2) = x1x2 + x1’x2’ +
x1’x2 +x1’x2
f (x1, x2) = x2(x1 + x1’) + x1’ (x2
+ x2’) f (x1, x2) = x2(1) + x1’ (1)
f (x1, x2) = x2 + x1’= *r" the circuit% usin+ 3;*% -R% ;-T +"tesO
ETR 271 *i+it"l Circuits ,"ll 2014 &&
f (x1, x2) = x1x2 + x1’x2 + x1’x2 +x1’x2’
Synthesis o9 o+ic ;etor!
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= Psin+ oole"n "l+ebr"% reuce this 9unction
f (x1, x2) = x1’x2’ + x1’x2 +
x1x2 f (x1, x2) = x2 + x1’
= *r" the circuit% usin+ 3;*% -R% ;-T +"tesO
f x2
x1
ETR 271 *i+it"l Circuits ,"ll 2014 &'
Synthesis o9 o+ic ;etor!
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y +
= 5r"ctice
= *r" the minimum circuit% usin+ 3;*% -R% ;-T+"tesO
x1 x2
11 1 1
1
11
f (x1, x2)
ETR 271 *i+it"l Circuits ,"ll 2014 &7
Synthesis o9 o+ic ;etor!
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= 5r"ctice
= *r" the minimum circuit% usin+ 3;*% -R% ;-T+"tesO
:
1
:
20 00 0
0 10 1
01
00
f (x1, x2):
#01
011 01 01 11 1
0101
1110
ETR 271 *i+it"l Circuits ,"ll 2014 &(
St"n"r ,orms
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59
St"n"r ,orms
There "re to st"n"r 9orms in hich "lloole"n e:pressions c"n be rittenB
1$ Sum o9 5roucts 6S-5
2$ 5rouct o9 Sums 65-S
ETR 271 *i+it"l Circuits ,"ll 2014
Sum o9 5roucts 6S-5
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60
Sum o9 5roucts 6S-5
The istributi/e l"s "re use to multiplyout " +ener"l oole"n e:pression to obt"inthe sum o9 proucts 6S-5 9orm
The istributi/e l"s "re "lso use to
con/ert " oole"n e:pression in 5-S 9ormto one in S-5 9orm
3 S-5 e:pression is re"liLe usin+ " set o93;* +"tes 6one 9or e"ch prouct term
ri/in+ " sin+le -R +"te 69or the sum
ETR 271 *i+it"l Circuits ,"ll 2014
Sum o9 5roucts 6S-5
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61
Sum o9 5roucts 6S-5
5rouct Term− o+ic"l prouct < 3;* oper"tion
− 3 prouct term is the 3;*in+ o9 liter"ls
− E:"mplesB 3Q% 3QQC% 3QC% QCQ*%
3QQCQ*
GSum o9− o+ic"l sum < -R oper"tion
− The sum o9 proucts is the -Rin+ o9prouct terms$
ETR 271 *i+it"l Circuits ,"ll 2014
Minterms 6S-5
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= More system"tic metho – Minterms
ETR 271 *i+it"l Circuits ,"ll 2014 '2
Minterm
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'
#
Minterm In +ener"l% " minterm o9 n /"ri"bles is " prouct
63;*in+ o9 n liter"ls in hich e"ch /"ri"ble"ppe"rs e:"ctly once in either true orcomplemente 9orm% but not both
− 3 liter"l is " /"ri"ble or its complement
,or " +i/en ro in the truth t"ble% thecorresponin+ minterm is 9orme by
− Incluin+ the true 9orm " /"ri"ble i9 its /"lueis 1$
− Incluin+ the complemente 9orm o9 "/"ri"ble i9 its /"lue is 0$
ETR 271 *i+it"l Circuits ,"ll 2014
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5rouct o9 Sums 65-S
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'&
5rouct o9 Sums 65-S
Sum Term− o+ic"l sum < -R oper"tion
− 3 sum term is the -Rin+ o9 liter"ls
− E:"mplesB 3% 3C% 3C% C*
G5rouct o9− o+ic"l prouct < 3;* oper"tion
− The prouct o9 sums is the 3;*in+ o9sum terms$
ETR 271 *i+it"l Circuits ,"ll 2014
M":terms 65-S
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= Minterms "re here f < 1% M":terms "re here f <
0
ETR 271 *i+it"l Circuits ,"ll 2014 ''
M":term
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'
7
M":term In +ener"l% " m":term o9 n /"ri"bles is " sum
6-Rin+ o9 n liter"ls in hich e"ch /"ri"ble"ppe"rs e:"ctly once in either true orcomplemente 9orm% but not both
− 3 liter"l is " /"ri"ble or its complement$
,or " +i/en ro in the truth t"ble% thecorresponin+ m":term is 9orme by
− Incluin+ the true 9orm " /"ri"ble i9 its /"lueis 0$
− Incluin+ the complemente 9orm o9 "/"ri"ble i9 its /"lue is 1$
ETR 271 *i+it"l Circuits ,"ll 2014
Synthesis o9 o+ic ;etor!
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= ThreeD/"ri"ble 9unction
0ite the Cannical sum-f- !rducts
:
1
:
20 00 0
0 10 1
01
00
:
#01
01
1 01 0
1 11 1
01
01
11
10
f 6:1% :2% :#
ETR 271 *i+it"l Circuits ,"ll 2014 '(
Synthesis o9 o+ic ;etor!
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= ThreeD/"ri"ble 9unction
f (x1, x2, x3) = 4 (*1, *5, *, *6) = 4
*(1, 5, , 6)
:
1
:
20 00 0
0 10 1
01
00
:
#01
01
1 01 0
1 11 1
01
01
11
10
f 6:1% :2% :#
ETR 271 *i+it"l Circuits ,"ll 2014 ')
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C"nonic"l 5rouctDo9DSums
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= 3;* e"ch minterm ith the /"lue o9 f’ 9or th"tminterm
:
1
:
2
f 6:1%
:20 00 1
1 01 1
11
01
f ’’ = f = (x1x2’)’
De#rgan"s therem
f $ (x1x2’)’ = x1’ + x2 7 sa*e as
48 soltionETR 271 *i+it"l Circuits ,"ll 2014 71
Synthesis o9 o+ic ;etor!
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= Psin+ oole"n "l+ebr"% reuce this 9unction
f (x1, x2) = x1’x2’ +
x1’x2 + x1x2 f (x1, x2) = x1x2 + x1’x2’ +
x1’x2 +x1’x2 f (x1, x2) = x2(x1 + x1’) + x1’ (x2
+ x2’) f (x1, x2) = x2(1) + x1’ (1)
f (x1, x2) = x2 + x1’
f x2 x1
ETR 271 *i+it"l Circuits ,"ll 2014 72
C"nonic"l 5rouctDo9DSums
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= 3;* e"ch minterm ith the /"lue o9 f’ 9or th"t
minterm:
1
:
2
f 6:1%
:20 00 1
1 01 1
11
01
f $ (x1x2’)’ = x1’ + x2 7 sa*e as
48 soltion f $ (x1x2’)’ = *2’ = 92
ETR 271 *i+it"l Circuits ,"ll 2014 7#
Synthesis o9 o+ic ;etor!
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= 5r"ctice
0ite the Cannical %rduct-Of-&ums
:
1
:
20 00 0
0 10 1
01
00
:
#01
01
1 01 0
1 11 1
01
01
11
10
f 6:1% :2% :#
ETR 271 *i+it"l Circuits ,"ll 2014 74
Synthesis o9 o+ic ;etor!f6
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f = 9 • 92 • 93 • 9: = 8 (9, 92, 93, 9: ) = 89(, 2, 3, :)
:
1
:
20 00 00 10 1
0100
:
#0101
1 01 01 11 1
0101
1110
f 6:1% :2% :#
f = (x1 + x2 + x3) (x1 + x2’ + x3) (x1 + x2’ + x3’)
(x1’ + x2’ + x3’) %edce to minimum cst
8odct7of74*sETR 271 *i+it"l Circuits ,"ll 2014 7&
inimum C!st 5-S: : f6
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f = 9
• 92
• 93
• 9:
:
1
:
20 00 00 10 1
0100
:
#0101
1 01 01 11 1
0101
1110
f 6:1% :2% :#
f = (x1 + x2 + x3) (x1 + x2’ + x3) (x1 + x2’ + x3’)
(x1’ + x2’ + x3’)f = (x1 + x3) (x2’ + x3’)
ETR 271 *i+it"l Circuits ,"ll 2014 7'
S-5 "n 5-S= *r" the minimum cost re"liL"tion o9 the
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= *r" the minimum cost re"liL"tion o9 theS-5 "n 5-S 9orB
:
1
:
20 0
0 00 10 1
0
100
:
#0
101
1 01 01 11 1
0101
1110
f 6:1% :2% :#
ETR 271 *i+it"l Circuits ,"ll 2014 77
*u"lity
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78
y
The u"l o9 " oole"n e:pression c"n beritten by
− Repl"cin+ 3;* ith -R% "n -R ith 3;*
− Repl"cin+ 0 ith 1% "n 1 ith 0
− e"/in+ liter"ls unch"n+e− 6 < ⋅
See the oole"n l"s "n theorems%pre/iously iscusse% 9or e:"mples o9 oole"n
e:pressions "n their u"ls$
ETR 271 *i+it"l Circuits ,"ll 2014
*eMor+"ns "
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79
+
Graphical representation of DeMoran!s "a#
x
y
x
y
x
y
(X·Y)' X' + Y'
x
y
x
y
x
y(X+Y)' X'·Y'
ETR 271 *i+it"l Circuits ,"ll 2014
S-5 "n 5-SB E:"mples,or e"ch o9 the 9olloin+ oole"n e:pressions%ienti9y hether it is in S-5 or 5-S 9ormB
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80
ienti9y hether it is in S-5 or 5-S 9ormB
1$ ,63%%C < 63$63C$6C
2$ ,63%%C < 3$$C $C 3$C
3$$C
#$ ,63%%C < 3 $C $C 3$$C
4$ ,63%%C < 63C$6C$63C$6
&$ ,63%%C < 3$$C 3$6C 63C$ETR 271 *i+it"l Circuits ,"ll 2014
S-5 U 5-S
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Sum o9 proucts eu"tions ritten 9or
hen the output is " G1 D minterms5rouct o9 sums eu"tions ritten 9or
hen the output is " G0 – m":terms
These "re complements o9 e"ch other>hich 9orm is use m"y epen on
hich h"s the +re"test u"ntity o9 G1s
or G0s in the output$ ener"lly thesm"ller number +i/es the better solution
81ETR 271 *i+it"l Circuits ,"ll 2014
Simpli9yin+ oole"nE:pressions
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8$
E:pressions
oole"n "l+ebr" c"n be use in se/er"l"ys to simpli9y " oole"n e:pressionB
− Combine terms
− Elimin"te reun"nt or consensus terms
− Elimin"te reun"nt liter"ls
− 3 reun"nt terms to be combineith or "llo the elimin"tion o9 other
terms
ETR 271 *i+it"l Circuits ,"ll 2014
Eui/"lency o9 oole"nE:pressions
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83
E:pressions To oole"n e:pressions "re eui/"lent i8 both
e:pressions e/"lu"te to the s"me /"lue 9or "llcombin"tions o9 the /"ri"bles in the e:pressions
The eui/"lency c"n be pro/en usin+
− 3 Truth t"ble− oole"n "l+ebr" theorems to m"nipul"te one
e:pression until it is ientic"l to the other$
− oole"n "l+ebr" theorems to reuce bothe:pressions inepenently to the s"me
e:pression$
ETR 271 *i+it"l Circuits ,"ll 2014
Import"nce o9 oole"n 3l+ebr"
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8%
oole"n "l+ebr" is use to simpli9y oole"ne:pressions
Simpler e:pressions le"s to simpler lo+iccircuits
− Reuces cost
− Reuces "re" reuirements− Reuces poer consumption
The ob.ecti/e o9 the i+it"l circuit esi+ner is toesi+n "n re"liLe optim"l i+it"l circuits
− Thus% oole"n "l+ebr" is "n import"nt tool tothe i+it"l circuit esi+ner$
ETR 271 *i+it"l Circuits ,"ll 2014
5roblem ith oole"n 3l+ebr"
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85
In +ener"l% there is no e"sy "y toetermine hen " oole"n e:pression h"sbeen simpliVe to " minimum number o9terms or " minimum number o9 liter"ls$
"rn"u+h M"ps ill pro/ie " bettermech"nism 9or the simpliVc"tion o9oole"n e:pressions$
ETR 271 *i+it"l Circuits ,"ll 2014
o+ic Circuits9 i f b b i h
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86
3 9unction f c"n be represente by either "
minterm e:p"nsion or " m":term e:p"nsion$ oth 9orms o9 the 9unction c"n be re"liLe usin+
lo+ic +"tes th"t implement the b"sic lo+icoper"tions$
Minterm E:p"nsion 6St"n"r S-5− Consists o9 the sum 6-R o9 prouct 63;* terms$
− Re"liLe usin+ "n 3;*D-R circuit$
M":term E:p"nsion 6St"n"r 5-S− Consists o9 the prouct 63;* o9 sum 6-R terms$
− Re"liLe usin+ "n -RD3;* circuit$
ETR 271 *i+it"l Circuits ,"ll 2014
Incompletely SpeciVe,unctions
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87
,unctions 3 9unction f is completely speciVe hen its
output is eVne 6i$e$ either 0 or 1 9or "ll
combin"tions o9 its inputs$ Hoe/er% i9 the output o9 " 9unction f is not
eVne 9or "ll combin"tions o9 its inputs%then it is s"i to be incompletely speciVe$
− Those combin"tions o9 the inputs 9or hichthe output o9 9unction f is not eVne "re
re9erre to "s Gont c"re outputs$
ETR 271 *i+it"l Circuits ,"ll 2014
Incompletely SpeciVe,unctions
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88
The truth t"ble representin+ "n incompletelyspeciVe 9unction inclues "n G: 6or " G ine"ch ro corresponin+ to "n inputcombin"tion 9or hich the output is noteVne$A B C F
0 0 0 0
0 0 1 &
0 1 0 1
0 1 1 &
1 0 0 11 0 1 0
1 1 0 &
1 1 1 1
“don't care” for !" = 001
“don't care” for !" = 011
“don't care” for !" = 110
ETR 271 *i+it"l Circuits ,"ll 2014
Incompletely SpeciVe,unctions
" ; <
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89
" ; <
0 0 0 0
0 0 1
0 1 0 1
0 1 1 1 0 0 1
1 0 1 0
1 1 0
1 1 1 1
'he (inter( e)pansion is*
'he (a)ter( e)pansion is*
,-/ 2 Σ(,$%7 + Σ,136
,-/ 2 Π M,05 . Π D,136
“don't care” #$nter#%
“don't care” #axter#%
3 Gont c"re c"n be either " 0 or 1$ Select "
/"lue 9or e"ch Gont c"re th"t ill help simpli9y the
9unction$ETR 271 *i+it"l Circuits ,"ll 2 014
Incompletely SpeciVe,unctions
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90
" ; < 0 0 0 0
0 0 1 1
0 1 0 10 1 1 2
1 0 0 1
1 0 1 0
1 1 0 #
1 1 1 1
-ssu(e &1 2 0 &$ 2 0 &3 2 0*
-ssu(e &1 2 1 &$ 2 1 &3 2 1*
,-/ 2 -!/! + -/!! + -/
,-/ 2 ,-+/+⋅,-!+/+!,-/ 2 --!+-/+-!+-!/+
//+/!+-!+/+!,-/ 2 / + -! + -!
-ssu(e &1 2 0 &$ 2 1 &3 2 1*
,-/ 2 / + -!
ETR 271 *i+it"l Circuits ,"ll 2014
Con/ersion o9 Reuirements toEu"tionsSteps 9or esi+nin+ sin+le output combin"tion"l circuit
*eVne sitchin+ 9unction th"t speciVes esire beh"/ior
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+ p
*eVne simpliVe "l+ebr"ic e:pression 9or the 9unction
Re"liLe 9unction usin+ "/"il"ble lo+ic elements
E:"mple To st"rt the c"r you must h"/e the tr"nsmission in ;eutr"l or
5"r!% h"/e your 9oot on the br"!e or the p"r!in+ br"!e en+"+e"n turn the !ey to i+nition$*eVne , 6P%N%>%%%W< St"rt C"r
P < Tr"nsmission in ;eutr"l
N< Tr"nsmission in 5"r!
>< ,oot r"!e En+"+e
< 5"r!in+ r"!e En+"+e
< ey Inserte 6note the Gors int inic"te this% it "s G"ssume
W < I+nition 5osition Selecte
)1
Con/ersion o9 Sentences toEu"tions, 6N%>%%%W < 6PN⋅ 6>⋅ ⋅W
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tr"nsmission "n br"!e "n !ey
"n i+nition
6p"r! or neutr"l 69oot or h"n
;oteB coul be " Gont c"re i9 you c"nt put
the sitch in the i+nition position ithout " !ey
bein+ inserte$ Hoe/er Gont c"re positionsc"n be " problem uner some circumst"nces th"tperh"ps "re not Gnorm"l 6h"t i9 the i+nitionloc! oesnt or! properly$ ou h"/e to bec"re9ul re+"rin+ Gont c"re eVnitions$Sometimes you nee to be on the Gs"9e sie$
)2ETR 271 *i+it"l Circuits ,"ll 2014
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93
Kuestions?
ETR 271 *i+it"l Circuits ,"ll 2014
SimpliVc"tion o9 o+ic,unctions
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9%
o+ic 9unctions c"n +ener"lly be simpliVeusin+ oole"n "l+ebr"$ Hoe/er% to problems "riseB
– It is iXcult to "pply to oole"n "l+ebr"
l"s "n theorems in " system"tic "y$ – It is iXcult to etermine hen "
minimum solution h"s been "chie/e$
Psin+ " "rn"u+h m"p is +ener"lly 9"ster
"n e"sier th"n usin+ oole"n "l+ebr"$
ETR 271 *i+it"l Circuits ,"ll 2014
SimpliVc"tion usin+ oole"n3l+ebr"i/enB ,-/ 2 Σ(,0 1 $ 5 6 7
,in i i 4O i
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95
,inB (ini(u( 4O e)pression
Combinin+ terms in one "yB
Combinin+ terms in " i8erent "yB
ETR 271 *i+it"l Circuits ,"ll 2014
"rn"u+h M"ps
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96
i!e " truth t"ble% " "rn"u+h m"p
speciVes the /"lue o9 " 9unction 9or "llcombin"tions o9 the input /"ri"bles$
ETR 271 *i+it"l Circuits ,"ll 2014
ToDN"ri"ble Dm"p
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97
0
1
0 1
m0
m$
m 3m 1
/
-ow>
" ;*inte
*
0 0 0 m0
1 0 1 m1
2 1 0 m2
# 1 1 m#
ETR 271 *i+it"l Circuits ,"ll 2014
ToDN"ri"ble Dm"pB E:"mple
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98
0 2
1
Minter( e)pansion* ,-/ 2 Σ (,0 1 2 -!/! + -!/ 2 -!
Ma)ter( e)pansion* ,-/ 2 Π Μ,$ 3 2 ,-!+/⋅,-!+/! 2 -!-!+-!/!+ /-!+//! 2 -!
n#er$c aebra$c
ow>
" ;
0 0 0 1
1 0 1 1
2 1 0 0
# 1 1 0
ETR 271 *i+it"l Circuits ,"ll 2014
"rn"u+h M"ps
3 " A
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3 "
0 00 11 01 1
1100
B0
1
0 1# $
$#
A
%r!up minterms
= &nly inclu'e cells ere f = #
= %r!up must *e n
cells (#,,,-, etc)3 < 0% <0@1 3
' $ A ))ETR 271 *i+it"l Circuits ,"ll 2014
ThreeDN"ri"ble Dm"pow " ; < *inte
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100
> *0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
# 0 1 1 m#
4 1 0 0 m4
& 1 0 1 m&
' 1 1 0 m'
7 1 1 1 m7
m 0 m%
m5
m1
/
-
m 3 m7
m6m $
0 0
0 1
1 1
1 0
0 1
r"y Coe
Import"nt to set up the Dm"p so th"t "."cent cells ch"n+ETR 271 *i+it"l Circuits ,"ll 2014
ThreeD/"ri"ble Dm"pBE:"mple
ow>
" ; <
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101
# 7
2 '
0 4
1 &
Minterm e:p"nsionB ,63%%C < S m62% #% 4% '
M":term e:p"nsionB ,63%%C < 5 M60% 1% &% 7
0 0 0 0 01 0 0 1 0
2 0 1 0 1
# 0 1 1 14 1 0 0 1
& 1 0 1 0
' 1 1 0 1
7 1 1 1 0
ETR 271 *i+it"l Circuits ,"ll 2014
MinimiL"tion usin+ Dm"ps Dm"ps c"n be use to eri/e the
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10$
−
Minimum Sum o9 5roucts 6S-5 e:pression− Minimum 5rouct o9 Sums 65-S e:pression
5roceureB− Enter 9unction"l /"lues in the Dm"p
− Ienti9y "."cent cells ith s"me lo+ic"l /"lue 3."cent cells i8er in only one bit
− Pse "."cency to minimiLe lo+ic 9unction HoriLont"l "n Nertic"l "."cency Dm"p r"ps 9rom top to bottom "n le9t to ri+ht
ETR 271 *i+it"l Circuits ,"ll 2014
MinimiL"tion usin+ Dm"ps o+ic"l 3."cency is use to
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103
− Reuce the number o9 liter"ls in " term− Reuce the number o9 terms in " oole"n
e:pression$
The "."cent cells−
,orm " rect"n+le− Must be " poer o9 2 6e$+$ 1% 2% 4% (% Y
The +re"ter the number o9 "."cent cells th"tc"n be +roupe to+ether 6i$e$ the l"r+er the
rect"n+le% the more the 9unction c"n bereuce$
ETR 271 *i+it"l Circuits ,"ll 2014
Dm"ps – o+ic"l 3."cency
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10%
*ray code
ETR 271 *i+it"l Circuits ,"ll 2014
MinimiL"tionB E:"mple Z1
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105
MinimiLe the 9olloin+ lo+ic 9unction usin+ ""rn"u+h m"pB
,-/ 2 Σ (,$ 6 7
4pecif the euialent (a)ter( e)pansion.
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MinimiL"tionB E:"mple Z#
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107
Pse " "rn"u+h m"p to etermine the
1$ minimum S-5 e:pression
2$ minimum 5-S e:pression
,or the 9olloin+ lo+ic 9unctionB
,-/ 2 Σ (,0 1 5 7
4pecif the euialent (a)ter( e)pansion.
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MinimiL"tionB E:"mple Z&
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109
,or the 9olloin+ truth t"bleB# A B C F
0 0 0 0 0
1 0 0 1 1
$ 0 1 0 0
3 0 1 1 1
% 1 0 0 1
5 1 0 1 0
6 1 1 0 07 1 1 1 1
ETR 271 *i+it"l Circuits ,"ll 2014
E:"mple Z&
Speci9y theB
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110
Speci9y theB
1$ minterm e:p"nsion
2$ m":term e:p"nsion
Pse " Dm"p to etermine theB
1$ minimum S-5 e:pression
2$ minimum 5-S e:pression
ETR 271 *i+it"l Circuits ,"ll 2014
MinimiL"tionB E:"mple Z'
,or the 9olloin+ truth t"bleB
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111
,or the 9olloin+ truth t"bleB
# A B C F
0 0 0 0 0
1 0 0 1 1
$ 0 1 0 1
3 0 1 1 1
% 1 0 0 0
5 1 0 1 1
6 1 1 0 07 1 1 1 0
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Minim"l ,orms
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113
C"n " lo+ic 9unction h"/e more th"n oneminimum S-5 e:pression?
C"n " lo+ic 9unction h"/e more th"n oneminimum 5-S e:pression?
ETR 271 *i+it"l Circuits ,"ll 2014
Dm"ps – To minim"l 9orms,-/ 2 Σ (,01$567 2 Π M,3%
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114
ETR 271 *i+it"l Circuits ,"ll 2014
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11
Kuestions?
ETR 271 *i+it"l Circuits ,"ll 2014
• 'he slies inclue herein so(e #ere taen fro( the (aterials acco(panin Digital Fundamentals : 'ho(as lo; Fundamentals of Logic Design, 6 th Edition : <oth an =innean Dr. rai "orie Depart(ent of >lectrical an o(puter >nineerin at Geore Mason?niersit.
eference%
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?niersit.
ETR 271 *i+it"l Circuits ,"ll 2014 11'
C"nonic"l SumD-9D5roucts= 3;* e"ch minterm ith the /"lue o9 f 9or th"tminterm
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:
1
:
2
f 6:1%
:20 00 1
1 01 1
11
01
f (x1, x2) = x1’x2’ +
x1’x2 + x1x2
f = * • 1 + *1 • 1 + *2
• + *3• 1
f = * + *1 + *3
Cannical sum-f-!rductsfrmat ETR 271 *i+it"l Circuits ,"ll 2014 117
In SumDo9D5roucts 6S-5 9orm% b"sic
Combin"tion"l o+ic Circuits
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ETR 271 *i+it"l Circuits ,"ll2014
combin"tion"l circuits c"n be irectlyimplemente ith 3;*D-R combin"tions i9the necess"ry complement terms "re"/"il"ble$
JK J
K
A
B AB
Product terms
Sum-of-products
Product term
C
D
CD
AB + CD + + JK . . .
Combin"tion"l o+ic Circuits
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ETR 271 *i+it"l Circuits ,"ll2014
3n e:"mple o9 "n S-5 implement"tion isshon$ The S-5 e:pression is "n 3;*D-Rcombin"tion o9 the input /"ri"bles "n the"ppropri"te complements$
S-5
DE
ABC
A
B
C
E
D
X = ABC+ DE
>hen the output o9 " S-5 9orm is in/erte%
Combin"tion"l o+ic Circuits
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ETR 271 *i+it"l Circuits ,"ll2014
the circuit is c"lle "n 3;*D-RDIn/ert circuit$ The 3-I conV+ur"tion lens itsel9 to prouctDo9Dsums 65-S implement"tion$
3n e:"mple o9 "n 3-I implement"tion is
shon$ The output e:pression c"n bech"n+e to " 5-S e:pression by "pplyin+*eMor+"ns theorem tice$
5-SDE
ABC AB
C
E
D
X = ABC+ DE X = ABC+ DE
X = 6 A + B + C6D + E
X = 6 ABC6DE
3-I
*eMor+"n
Implementin+ " S-5 e:pression is one byVrst 9ormin+ the 3;* termsA then the terms
Implementin+ Combin"tion"lo+ic
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ETR 271 *i+it"l Circuits ,"ll2014
"re -R to+ether$Sho the circuit th"t ill implement theoole"n e:pression X < ABC + ABD + BDE.63ssume th"t the /"ri"bles "n theircomplements "re "/"il"ble$
C
AB
EDB
AB
D
St"rt by 9ormin+ the terms usin+ three #Dinput
3;* +"tes$ Then combine the three terms usin+ " #Dinput -R+"te$
X < ABC + ABD +BDE