Chương 3 Mach Logic Theo de Cuong Moi

Chng 3: Mch logic

TS. Phm Vn Thnh

([email protected])

1

Ni dung

Cng OR, cng AND, cng NOT, cng NAND, cng NOR

Cng XOR Cc nh lut ca i s Boolean Xy dng hm logic t bng chn l Rt gn biu thc logic Phng php ba Karnaugh (Bi ging in t mn k thut s TS. Trung Kin)

Cc cng logic c bn OR, AND, NOT

Trong phn ny, chng ta s tm hiu v i s Boolean (Boolean Algebra), c dng phn tch v thit k cc mch logic.

i s Boolean l mt cng c ton hc tng i n gin v n m t mi quan h gia cc li ra ca mch logic vi cc li vo di dng cc biu thc i s, hay cn gi l biu thc Boolean.

Cu thnh ca biu thc Boolean l cc ton t c bn NOT, AND, OR, NAND, NOR.

Biu thc Boolean khng phi cng c duy nht phn tch, thit k hay m t mch logic. Mt s cng c khc cng c dng kt hp l bng chn l (truth table), s nguyn l (schematic symbols), gin thi gian (timing diagram).


i s Boolean khc bit vi i s thng thng v cc hng s v bin Boolean ch nhn hai gi tr l 0 v 1.

0 - in th t 0 n 0.8V, 1 - in th t 2V n 5V.

Cc hng s v bin Boolean (Constants and Variables)

Logic 0 Logic 1

False True

Off On

Low High

No Yes

Open switch Closed switch

Cc li vo ca mch logic trong cc biu thc Boolean c gi l cc bin Boolean. Cc ton t to nn biu thc gi l cc ton t logic (logic operations). 3 ton t c bn l NOT, AND, OR


Bng chn l m t li ra ca mch logic theo cc trng thi c th c ca li vo.

Cc li vo c th c nhiu kiu k hiu khc nhau: A, B, C, D; x, y, z, v

Bng chn l (Truth Table)

S trng thi c th c ca li vo tun theo cng thc: s trng thi = 2n. Trong n l s li vo.

Cc li vo Cc li ra

Cc trng thi

c th c ca t

hp cc li vo

Kt qu cc li

ra


Khi vit bng chn l, cn ch tnh ln lt ca cc li vo. V d mch in c 2 li vo th cn m t trng thi A=0, B=0 trc. Tip A=0, B=1. Ri l A=1, B=0, v A=1, B=1.


A B Y

0 0 1

0 1 0

1 0 1

1 1 0

2 li vo



A B C Y

1 0 1

1 1 0

1 1 1

3 li vo

A B C D Y

0 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1

4 li vo

A B C Y

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 0

3 li vo

A B C D Y

0 0 0 0 0

0 0 0 1 0

0 0 1 0 0

0 0 1 1 1

0 1 0 0 1

0 1 0 1 0

0 1 1 0 0

0 1 1 1 1

1 0 0 0 0

1 0 0 1 1

1 0 1 0 1

1 0 1 1 1

1 1 0 0 0

1 1 0 1 1

1 1 1 0 0

1 1 1 1 0

4 li vo


Cng HOC OR gate

A B Y = A +B

0 0 0

0 1 1

1 0 1

1 1 1

Hm logic: Y = A + B

Bng chn l:

Biu tng trong mch in:

U1A

7432N

A

BY


Cng HOC OR gate

c im lu : Li ra cng OR bng 1 nu c t nht mt trong cc li vo bng 1. Li ra cng OR bng 0 khi tt c cc li vo bng 0.

Ch l hm logic, bng chn l, biu tng v cng tng ng u c th pht trin ln cho nhiu li vo. Trong cc trng hp ny vn tun theo c im lu ca cng OR 2 li vo trn. Y = x1 + x2 + x3 + + xn

xny

x1


Cng HOC OR gate

V d v ng dng cng OR trong iu khin cng nghip:

Bo ng s xy ra khi c bo ng v qu nhit hoc qu p sut trong mt qu trnh ha hc. Khi nhit vt qu mc cho php, th VT tng ng s ln hn th tham chiu VTR. Tng ng li ra khi so snh TH s cho ra mc 1. Tng t khi p sut vt qu mc cho php, th VP > th tham chiu VPR, lm cho li ra PH = 1. Hai li ra TH v PH c a vo cng OR. V th, ch cn mt trong hai trng hp trn xy ra, hoc gi l c hai trng hp xy ra th li ra cng OR s bt ln mc cao, lp tc bo ng c bt.


Cng HOC OR gate

V d v ng dng cng OR trong iu khin cng nghip:


Cng HOC OR gate

Cc cng logic thc t vn lun l cc mch in t thun ty, ch l chng c thit k lm vic trong nhng ch c bit.

y

D1

D2

x1

x2R1k

y

+Vcc

5V

x3

0V

x2

0V

x1

5V Q3NPN

Q2NPN

Q1NPN

R51k

+Vcc

5V

Dng diode Dng transistor


Cng V AND gate

A B Y = A +B

0 0 0

0 1 0

1 0 0

1 1 1

Hm logic: Y = A . B

Bng chn l:


yx2

x1


Cng V AND gate

c im lu : Li ra cng AND bng 0 nu c t nht mt trong cc li vo bng 0. Li ra cng AND ch bng 1 khi tt c cc li vo bng 1.

Ch l hm logic, bng chn l, biu tng v cng tng ng u c th pht trin ln cho nhiu li vo. Trong cc trng hp ny vn tun theo c im lu ca cng AND 2 li vo trn. Y = x1 . x2 . x3 . . xn

xn

yx1


Cng V AND gate

Transistor AND gate

Diode AND gate

y

y

Q1PNP

Q2PNP

Q3PNP

x4

0V

x5

5V

x6

5V

+5V

D1

D2

x15V

x2

0V

+5V

R21k

R11k


Cng O NOT gate

A

0 1

1 0

Hm logic:

Bng chn l:



Vi 3 cng logic c bn trn, c th thit k nn bt c cng logic hay mch in t s phc tp no.

Tuy nhin, khi cc yu cu thit k ngy cng pht trin, s lng v chng loi cc IC s ngy cng nhiu, chuyn dng ngy cng cao. Cc linh kin in t ny li c th ni vi nhau thit k thnh cc mch in mong mun.

y = ?

y = ?

A

B

B

A

74LS32

74LS08

74LS32

74LS08

T hp ca cc cng c bn


T hp ca cc cng c bn

A

B

C

D

U5

NOT

U6

AND3

U7

OR2

U8

NOT

U9

AND2


T hp ca cc cng c bn

A

B

C

D

E

U10

OR2

U11

AND2

U12

NOT

U13

OR2 U14

AND2


Cng KHNG HOC NOT-OR = NOR

A B

0 0 1

0 1 0

1 0 0

1 1 0

yQ

x

Hm logic:

Bng chn l:



Cng KHNG V NOT-AND = NAND

A B

0 0 1

0 1 0

1 0 0

1 1 0

Hm logic:

Bng chn l:



BI TP

S5V

y0V

v5V

x0V

z5V

7400-C

7400-B

7400-A

7400-D

74LS20

Q

Vit hm logic v nu chc nng hot ng ca mch in sau.

Vai tr ca chn S l g?


NAND IC h TTL 7400

Trong 7400 c 4 cng NAND 2 li vo


Cc bng mch th

Cc cng logic c bn

Ni dung



Cng Hoc Tuyt i XOR eXclusive OR

A B

0 0 0

0 1 1

1 0 1

1 1 0

Hm logic:

Bng chn l:



74LS86 l IC XOR c 4 cng XOR 2 li vo


c im lu ca cng XOR:

Cng XOR khng phi l mt cng c bn v n c biu din qua t hp cc ton t AND, OR, NOT.

Vi XOR 2 li vo, u ra s bng 1 khi hai li vo khc nhau u ra bng 0 khi hai li vo ging nhau

Vi XOR nhiu li vo, u ra s bng 1 khi c s l li vo 1 u ra bng 0 khi c chn li vo 1


Mt s mch to thnh cng XOR t nhng cng logic c bn

a)b)

y

x

y

x

Yu cu: Chng minh hm logic li ra l hm ca cng XOR


Hm logic:

Bng chn l:

XOR 3 li vo

A B C Y

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1


XOR 3 li vo xy dng t XOR 2 li vo

3-in XOR

74LS136

74LS136

74LS136


Mch to s b 1

S

0V

A40V

A30V

A2

0V

A1

0V

Q4

Q3

Q2

Q1


To bt kim tra chn l

Theo phng php chn l, khi c l cc s 1 trong chui m, bt chn l s bng 1.

Rt may y ta thy nguyn tc hon ton ging tnh cht ca Hoc Tuyt i.

V th, c th dng cc ca Hoc Tuyt i thit k mch to bt chn l ny

D

5V

C

5V

B

5V

A

0V


Hm logic:

Bng chn l:

Ca Khng Hoc Tuyt i XNOR

A B

0 0 1

0 1 0

1 0 0

1 1 1


Ca Khng Hoc Tuyt i XNOR


Mch so snh 2 s nh phn

A4

0V

A3

0V

A2

0V

A1

0V

B4

0V

B3

0V

B1

0V

B20V Q

Khi tt c cc cp bt tng ng ging nhau, 4 li ra ca cc ca Khng Hoc Tuyt i s bng 1. Tng ng li ra ca AND 4 li vo s ch th 1. Tt c cc trng hp cn li, A khc B th li ra s bng 0

Cng XOR

Kt lun tiu m un

Ni dung



Cc nh lut ca i s Boolean

Gi s, cn xy dng mt mch in c hm logic sau:

Nu khng rt gn hm logic, s cn 1 cng OR 3 li vo, 3 cng AND 3 li

vo, 2 cng NOT. Nh vy, phi dng ti 3 IC (nu c).

Nhng nu bit rt gn:

Y = A(B+C)

Lc ny, ch cn 1 cng AND 2 li vo, 1 cng OR 2 li vo.

S lng linh kin gim i ng k, kch thc mch cng gim i rt nhiu,

li ch hn v mt kinh t.

ng c ca vic s dng i s Boolean


Cc nh lut i s Boolean cho hm 1 bin


Cc nh lut i s Boolean cho hm 1 bin

A.A=0A

A.A=AA

A A

A+A=1A

A+A=AA

A A

A.1=A1

AA.0=00

A

A+1=11

AA+0=A

0

A

74LS0874LS08

74LS32 74LS32

74LS0874LS08

74LS3274LS32

A: l 1 bin hoc l 1 biu thc logic bt k


Cc nh lut i s Boolean cho hm nhiu bin

nh lut giao hon (commutative): x + y = y + x (9)

x . y = y . x (10)

nh lut kt hp (associative): x + (y + z) = (x + y) + z = x + y + z (11)

x(yz) = (xy)z = xyz (12)

nh lut phn phi (distributive): x(y + z) = xy + xz (13)

(w + x)(y + z) = wy + wz + xy + xz (14)


Minh ha cho i s Boolean

Php tnh logic

Php tnh OR tng ng h cng tc tip xc mc song song parallel switch contacts



Php tnh logic

Php tnh AND tng ng h cng tc tip xc mc ni tip serial switch contacts



Cc nh lut ca Boolean

Php tnh AND tng ng h cng tc tip xc mc ni tip A = relay contact



Cc nh lut ca Boolean


Cc nh lut i s Boolean cho hm nhiu bin


Mt s nh lut ngoi suy khc

BAACCABA

ABBAA

BABAA

BCACABA

ABAA

AABA

))((

)(

))((

)(Yu cu bin i v chng minh cc nh lut ny


Mt s nh lut ngoi suy khc


Cc nh lut Boolean cho XOR


2 nh lut De Morgan

Hai nh l quan trng trong i s Boolean c ng gp bi nh ton hc ni ting De Morgan. Hai nh l ny cc k quan trng trong vic n gin ha cc hm logic. C th l vic c th chuyn t tng thnh tch, v ngc li, tch thnh tng.

x v y u c th l cc biu thc logic phc tp khc

Ngoi ra, 2 nh l ny cn p dng cho nhiu bin


2 nh lut De Morgan

ngha ca nh lut De Morgan

Bin i php tnh t tng thnh tch

Chuyn t NOR thnh NOT + AND

A.B

B

A

A+BA+B

B

A

74LS04

74LS04

74LS0874LS0474LS32

74LS02

B.ABA


2 nh lut De Morgan

ngha ca nh lut De Morgan

A+BA.BA.B

B

A

B

A

74LS3274LS08

74LS04

74LS04

74LS04

7400

BAB.A

Bin i php tnh t tch thnh tng

Chuyn t NAND thnh NOT + OR


Tnh vn nng ca cng NAND v NOR

Tt c cc biu thc Boolean bao gm hng lot t hp khc

nhau ca cc ton t c bn AND, OR, NOT. V th, bt c biu

thc no cng u c th biu din thng qua cc ton t ny.

Trong cc cng hc trn, 2 cng NAND v NOR c im rt

c bit l c th to nn cc cng AND, OR, NOT. Chnh v th,

ch cn dng duy nht cng NAND, hoc duy nht cng NOR l

c th lp rp nn mch in



NADN gate produce OR

A+B

B

A

NADN gate produce AND

ABB

A

NADN gate produce NOT

A.A = AA

7400

7400

7400

74007400

7400

Tng t chng minh cho cng NOR cng to ra 3 cng c bn OR, AND, NOT



CBABAABCABy

BABABAy

DABCCDABy

))()((

))((

Ch : Hm ca cng XOR

Hm ca XNOR

S dng De Morgan, chng minh: =


V d v s dng De Morgan

Dng De Morgan n gin ha mch in sau:

Cc nh lut Boolean

Kt lun tiu m un

Ni dung



Xy dng hm logic t bng chn l

Trong thit k mch in, cc nh thit k thng bt u bng bng chn l, cng c dng m t chnh xc hot ng ca mch in s. Nhim v ca vic thit k l t bng chn l tm ra mch in ph hp. Mun vy th cn xc nh c hm logic, tc l tm ra mi quan h ton t gia cc li vo.


Cc tch c bn (fundamental products)

Tch c bn l cch n gin nht m t s kt hp ca 2 tn hiu bng mt cng AND 2 li vo.

V d vi 2 tn hiu, s c 4 kh nng c th xy ra:


Cc tch c bn (fundamental products)

Khi li vo trng thi 0, ng tn hiu tng ng c biu din bng gi tr b ca n. Mi tch c bn ny c gi l mt Minterm mi


Cc tng c bn (fundamental summaries)

Khc vi tch c bn, y, khi li vo bng 1 th ng tn hiu tng ng c dng b


Cc tng c bn (fundamental summaries)

Mt s c im ca Minterm v Maxterm

Minterm v Maxterm ca cng mt ch s l b ca nhau:

Cng logic ca cc minterm = 1

Tch logic ca cc maxterm = 0

Hai minterm nhn vi nhau = 0

Cng ca hai maxterm = 1

(Yu cu SV chng minh)


Xy dng hm logic bng phng php Minterm (SOP Sum of Products)

Cho trc bng chn l, tm biu thc Boolean bng cch cng cc tch c bn ca cc u ra c gi tr 1

A B y

0 0 0

0 1 1 AB

1 0 1 AB

1 1 0

BA


Xy dng hm logic bng phng php Minterm

V d khc:



V d khc:

Cu hi: Hm logic

v mch in hp l cha?

BC+AC+AB



V d t phn trc: Chng minh:

x1 x2 x3 y

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1

- Suy hm logic t bng chn l ca v phi - T , s dng i s Boolean v De Morgan khai trin v tri - So snh kt qu hai bn chng minh.


Xy dng hm logic bng phng php Maxterm

Hm logic l tch ca cc tng c bn ca cc li ra c gi tr 0

A B y

0 0 0 A+B

0 1 1

1 0 1

1 1 0 A+B

BA


POS chun

Ch , POS c th c cc dng khc:

Trong biu thc 1, c th c 3 li vo A, B, C. Nhng tha s th nht ch c A v B.

Trong biu thc 2, c 5 li vo A, B, C, D, E. Nhng trong tha s th nht thiu D, E, tha s th 2 thiu A, B, tha s th 3 thiu A, E.

Dng POS chun: l tch cc tng m trong c ht tt c cc bin.


POS chun

V d: Bin i biu thc Boolean sau v dng POS chun

Biu thc ny chng t c 4 li vo A, B, C, D.

Tha s th nht thiu D. V th n c bin i nh sau:

Tha s th 2 thiu A:

Tha s th 3 l dng POS chun. Vy, biu thc POS chun ca c biu thc s l:


Bin i t SOP sang dng POS chun

Cc bc thc hin nh sau:

Xc nh trng thi nh phn tng ng ca cc li vo ca cc tch c bn

Xc nh cc trng thi nh phn cn li khng thuc nhm lit k trn

Vit cc tng c bn ca cc trng thi va xc nh trong bc 2.


Bin i t SOP sang dng POS chun

V d:

Bin i biu thc SOP sau v dng POS chun:

Cc trng thi nh phn tng ng l:

000 + 010 + 011 + 101 + 111

Cc trng thi nh phn khng thuc nhm ny l:

001, 100, 110

Do , dng POS chun ca biu thc trn l:


T SOP thit lp bng chn l

Cho 1 biu thc SOP nh sau:


T POS thit lp bng chn l

Cho 1 biu thc POS nh sau:


So snh 2 phng php SOP v POS

Tm hm logic t phng php minterm SOP

Tm hm logic t phng php minterm POS


Kt lun tiu m un

Ni dung



Rt gn hm logic

Chng ta ch yu s s dng cc nh lut ca i s Boolean rt gn cc

hm logic. phn sau s c thm cng c ba Karnaugh. Ch mt iu l

khng c mt cng c no kim tra xem hm rt gn l ti gin hay

cha. V th, ngi thit k cn phi da vo kinh nghim tm ra p n

cui cng.

Nhn chung, vic rt gn thng qua mt s bc chnh sau:

Bin i biu thc v dng tng ca cc tch c bn SOP (sum of

products). Vic ny c thc hin bng vic nhn cc tha s trong

ngoc v dng 2 nh lut De Morgan.

Nhm cc s hng (cc tch c bn) c th vi nhau. Khi ta c th gim

c cc bin.

Rt gn hm logic

Tr li mt mch in c suy ra t bng chn l bng phng php minterm, vi cu hi l mch ny v hm logic ca n thc s ti gin cha?

Rt gn hm logic

Chng ta th rt gn:

Hm y l dng SOP, v th ch cn tm xem c th c cc s hng no c th nhm li vi nhau

Rt gn hm logic

RT GN MCH IN

Xt mt mch in sau:

rt gn mch, chng ta kh c th nhn cc cng logic on nhn. Thng thng, cn phi vit ra hm logic tng ng ca mch in ny. Sau , ti gin hm tm ra mch in ti gin

Rt gn hm logic

RT GN MCH IN

Rt gn hm logic

Hai mch in hon ton tng ng

y l mt hnh nh khc minh chng tnh cc k thit yu ca vic rt gn hm logic hay mch logic

Rt gn hm logic

Mt bn sinh vin lp rp mt mch in nh trn. Bn cho mch in ny bao nhiu im (/10)?

Rt gn hm logic

Rt gn cc hm logic sau:

Rt gn hm logic

V d: Rt gn hm sau:

p dng De Morgan cho s hng u tin:

Tip tc De Morgan cho 2 tha s:

p dng nh lut phn phi:

Tip tc bin i

Rt gn hm logic

Mch no trong s nhng mch sau tng ng:

Rt gn hm logic

Tm mch in tng ng:

Rt gn hm logic

Kt lun tiu m un

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Chương 3 Mach Logic Theo de Cuong Moi