Lecture 3 ETR 271 Fall 2014 Boolean Logic

8/9/2019 Lecture 3 ETR 271 Fall 2014 Boolean Logic

http://slidepdf.com/reader/full/lecture-3-etr-271-fall-2014-boolean-logic 1/120

September 2014 Johnny Cornett

ETR 271 – MicrocomputerElectronics I



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

ETR 271 *i+it"l Circuits ,"ll 2014 2



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




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

ETR 271 *i+it"l Circuits ,"ll 2014 &



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

ETR 271 *i+it"l Circuits ,"ll 2014 '



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




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% :# < ?

ETR 271 *i+it"l Circuits ,"ll 2014 (



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



ETR 271 *i+it"l Circuits ,"ll2014

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.




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




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




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




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$




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.)




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 &




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.




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




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.



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)




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

ETR 271 *i+it"l Circuits ,"ll 2014 2#



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

ETR 271 *i+it"l Circuits ,"ll 2014

24



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$

ETR 271 *i+it"l Circuits ,"ll 2014 2&

3 lo+ic +"te th"t prouces " HIH outputhen one or more inputs "re HIH$

S l t T



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$

ETR 271 *i+it"l Circuits ,"ll 20142'

K i



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


KuiL

K i



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



#$ 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



'$ 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

ETR 271 *i+it"l Circuits ,"ll 2014 #0

KuiL

K i



($ 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


KuiL

K i



)$ 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


KuiL

o+ic Symbols



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

ETR 271 *i+it"l Circuits ,"ll 2014 ##

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






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



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?

ETR 271 *i+it"l Circuits ,"ll 2014#7

3



ETR 271 *i+it"l Circuits ,"ll 2014#(

oole"n 3l+ebr"

oole"n 3ition




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




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




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




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 "




*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"



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


oole"n 3l+ebr"



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"



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'


oole"n 3l+ebr"



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


47

Nenn *i"+r"ms



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


4(

Terminolo+y



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


4)

5receence@3ppe"r"nce



@ 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


&0

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


&1




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




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 &#




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





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’




= 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 &'




y +

= 5r"ctice

= *r" the minimum circuit% usin+ 3;*% -R% ;-T+"tesO

x1 x2

11 1 1

1

11

f (x1, x2)





= 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



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


Sum o9 5roucts 6S-5



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


Sum o9 5roucts 6S-5



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$


Minterms 6S-5



= More system"tic metho – Minterms

ETR 271 *i+it"l Circuits ,"ll 2014 '2

Minterm



'

#

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$




5rouct o9 Sums 65-S



'&

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$


M":terms 65-S



= Minterms "re here f < 1% M":terms "re here f <

0

ETR 271 *i+it"l Circuits ,"ll 2014 ''

M":term



'

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$





= 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 '(




= 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 ')



C"nonic"l 5rouctDo9DSums



= 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




= 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


C"nonic"l 5rouctDo9DSums



= 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#




= 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% :#


Synthesis o9 o+ic ;etor!f6



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



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



= *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% :#


*u"lity



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$


*eMor+"ns "



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'


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



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



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



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


Eui/"lency o9 oole"nE:pressions



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$


Import"nce o9 oole"n 3l+ebr"



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$


5roblem ith oole"n 3l+ebr"



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$


o+ic Circuits9 i f b b i h



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$


Incompletely SpeciVe,unctions



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$





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





" ; <



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




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 / + -!


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



+ 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



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



93

Kuestions?


SimpliVc"tion o9 o+ic,unctions



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"$


SimpliVc"tion usin+ oole"n3l+ebr"i/enB ,-/ 2 Σ(,0 1 $ 5 6 7

,in i i 4O i



95

,inB (ini(u( 4O e)pression

Combinin+ terms in one "yB

Combinin+ terms in " i8erent "yB


"rn"u+h M"ps



96

i!e " truth t"ble% " "rn"u+h m"p

speciVes the /"lue o9 " 9unction 9or "llcombin"tions o9 the input /"ri"bles$


ToDN"ri"ble Dm"p



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#


ToDN"ri"ble Dm"pB E:"mple



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


"rn"u+h M"ps

3 " A



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



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>

" ; <



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


MinimiL"tion usin+ Dm"ps Dm"ps c"n be use to eri/e the



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


MinimiL"tion usin+ Dm"ps o+ic"l 3."cency is use to



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$


Dm"ps – o+ic"l 3."cency



10%

*ray code


MinimiL"tionB E:"mple Z1



105

MinimiLe the 9olloin+ lo+ic 9unction usin+ ""rn"u+h m"pB

,-/ 2 Σ (,$ 6 7

4pecif the euialent (a)ter( e)pansion.




MinimiL"tionB E:"mple Z#



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.




MinimiL"tionB E:"mple Z&



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


E:"mple Z&

Speci9y theB



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


MinimiL"tionB E:"mple Z'

,or the 9olloin+ truth t"bleB



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




Minim"l ,orms



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?


Dm"ps – To minim"l 9orms,-/ 2 Σ (,01$567 2 Π M,3%



114




11

Kuestions?


• '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%



?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



:

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




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 . . .





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%





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




"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

Lecture 3 ETR 271 Fall 2014 Boolean Logic

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Transcript of Lecture 3 ETR 271 Fall 2014 Boolean Logic