09.Combinatorial.pdf

8/9/2019 09.Combinatorial.pdf

1/44

LOGIC GATES AND BOOLEAN ALGEBRA

Digital (logic) circuits operate in the binary mode

when each input and output voltage is either a 0or 1.

0 and 1 represent predefined voltage ranges.

For example, for the TTL (Transistor Transistor

Logic), have

Logic Input Range Output Range

0 0-0.8 V 0-0.4 V

1 2-5 V 2.4-5V

Note that the output range is tighterthan the input range

to ensure reliability


2/44

Combinatorial Circuits designed/analyzed usingBoolean Algebra,

In Boolean Algebra, variables can only take onthe logical values 0 (FALSE) or 1 (TRUE)

The Basic Functions in Boolean Algebra are

NOT, OR and AND defined as follows

NOT

OR

AND

B=A (NOT) A B=1 if A=0 or B=0 if A=1

C=A+B C=0 only if A=B=0; else C=1

C=AB C=1 if A=B=1; else C=0

In Boolean Algebra, 1 and 0 are NOT NUMBERS. They merely

represent the logical variables TRUE and FALSE


3/44

Logic Gates

Digital circuits called logic gates can be

constructed from diodes, transistors, and

resistors in such a way that the circuit

output is the result of a basic logicoperation (OR, AND, NOT) performed on

the circuits.

Logic Gates which do more complicated logical

operations are also widely available, e.g. NOR,

NAND, Ex-OR etc..


4/44

Q1Q2

Q3

Q4

A

B

Inputs

Y=AB

Output

+5 V

GND

R1

4 K

R2

1.6 K

R3

1 K

R4

130

D

TTL NAND Gate

Totem Pole or

Active Pull-Up

Output


5/44

Truth Table

A truth table is a tabular representation for

describing how a logic circuits output

depends on the logic levels present at thecircuits inputs


6/44

A

B

X

A B X

0

0

11

0

1

01

0

1

11

Binary Counting

Sequence

No. of input combinations= 2Nfor N- Inputs

Truth Table


7/44

NOT Operation

Called NOT A

or Complement of A

or Inverse of A

AX =

A 0 X=A 1

A 1 X=A 0

= =

= =

Function of

only one

variable


8/44

NOT gate or Inverter

A AX =

A

0 1

1 0

A0

1

1

0

AX =

Truth Table


9/44

Function of Two Binary Variables

X = f(A,B)

Examples: OR, AND

NOR (Not-OR)

NAND (Not-AND)

Ex-OR


10/44

A

BX=A+B

A B X

0

01

1

0

10

1

0

11

1

OR gate

OR operation X = A+B

Truth Table


11/44

A B C X

0

0

0

01

1

11

0

0

1

10

0

11

0

1

0

10

1

01

0

1

1

11

1

11

X=A+B+CABC

Truth Table


12/44

1

A

01

B 0

1C

0

1X

0

timeOR Operation


13/44

AND Operation X = AB

AND gate

AB

X = AB


14/44

A B X

0

0

1

1

0

1

0

1

0

0

0

1

Truth Table


15/44

1

A01

B 0

1

X 0

AND Operation


16/44

NOR and NAND gates

NOR - NOT-OR

Combination of NOT and OR

OR followed by NOT

NAND NOT-AND

Combination of NOT and AND

AND followed by NOT


17/44

X=A+BA

B

NOR gate

A B A+B A+B0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

0

X= A+BA

B

A+B

Truth Table for NOR Gate


18/44

NAND gate

A

BABX =

ABX = AB

A

B


19/44

A B AB AB

0

0

11

0

1

01

0

0

01

1

1

10

Truth Table for NAND Gate


20/44

Ex-OR GateA

B

Y

Y=A B

=AB+AB

Comparator: Output is LOW

if both the inputs are identical;

else output is HIGH

Half-Adder: Y is the sum of

A and B (in binary) ignoring

the carry out that will begenerated by 1+1

Symbol

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

Truth Table


21/44

Describing Logic Circuits Algebraically

A

X= ?B

C

X=AB+C

A

X= ?BC

X=(A+B)C

A

X= ?B

BAX +=

A

X= ?B)BA(X +=


22/44

A

BY

A

B

YAB

AB

AB


23/44

A

D

B

C

X= ?

)DABC(AX +=


24/44

A

B

C

D

E

X = ?

B)C(A +DB)C(A ++

A+B (A+B)C

( )EDB)C(AX ++=


25/44


26/44

Boolean Algebraic Theorems : Useful in simplifyingexpression of logic variables

A variable can have only two values, A = 1 or A= 0

(1)AA =

Also,

)(51AA

(4)AAA

(3)11A

(2)A0A

=+

=+

=+

=+

DualsInterchange + and .

Interchange 0 and 1

)(90A.A

(8)AA.A

(7)00.A

(6)A1.A

=

=

=

=


27/44

( ) (10)ACABCBA +=+ Distributive Law

Dual (11)C)B)(A(ABCA ++=+Proof : RHS=

LHS

BCABCC)A(1

BCCAA

BCCAB)A(1BCCAABA

BCCAABAA

C)B)(A(A

=

+=

++=

++=

+++=

+++=

+++=

++


28/44

Duals

(16)CAABBCCAAB

(15)B)AC)((ACAAB(14)ABAAB

(13)BABAA

(12)AABA

+=++

++=+

=+

+=+

=+

A.(A B) A (17)

A.(A B) A.B (18)

(A B).(A B) A (19)(A B).(A C) AC AB (20)

(A B).(A C).(B C) (A B).(A C) (21)

+ =

+ =

+ + =

+ + = +

+ + + = + +


29/44

(13) A AB A(B 1) AB AB A AB A B(A A) A B+ = + + = + + = + + = +

A1.A)BA(BBAAB(14) ==+=+

(12) A AB A(1+B) A.1 A+ = = =

Some sample proofs

De Morgans Theorem:

(22)................CBA......A.B.C..... +++=

(23)................C.B.A..C.........BA =++


30/44

Ex. Simplify ( )( )

( ) ( )

( )DBCA

D.BC.A

DBCA

DB.CAZ

+=

+=

+++=

++=

Using De Morgans Theorem

Ex. Simplify

( )

B)(CDACB

)CB(CDACB

DCBADCAA)AC(B0

CBADCBADCACBACAA

CBADCBADBACA

CBADCBA)DBAC(ACBADCBA)BDAC(AZ

++=

++=

++++=

++++=

++++=

++++=

++=


31/44


32/44

Using NAND Gates

NOT A

A

B AND

A

B OR

Using NAND Gates

AA.AX ==

ABABAB.ABX ===AB

A

B

BABAB.AX +=+==


33/44

Using NOR Gates

NOT A

A

B

OR

A

B

AND

Using NOR

AAAX =+=

BA +

BABABABAX +=+=+++=

A.BB.ABAX ==+=

A

B


34/44

COMBINATIONAL LOGIC CIRCUITS

orCOMBINATORIAL CIRCUITS

The circuits made up of combination of logic

gates discussed earlier can be classified as

combinational logic circuits because at any

time, the logic level at the output depends onthe combination of logic levels present at the

inputs.

A combinational circuit has no memory, so its

output depends only on the current value of

its inputs.


35/44

Standard Forms For Logical Expression

For Logic circuit simplification and design, wetry to express the logical expression in one of

two standard forms -

Sum-of-Products Form

or Product-of-Sums Form


36/44

Sum-of-Products (SOP)

Examples: (i) (ii)

Note that each expression consists of two or moreAND

terms that areORed together and that -

(i) The same variable used never appears twice in a

product. (Because A.A=A and )

ABC+ABC AB+ABC+CD+D

AA=0

(ii) One complement sign can not cover more than

one variable in a term (e.g. we can not have ABC

as we can then use De Morgans Law to simplify)


37/44


38/44


39/44

Standard SOP Form

A SOP Form in which each product terminvolves all the variables (complemented or

uncomplemented).

Each individual term is referred to as a minterm.

Example Z=ABC+ABC+ABC


40/44

Express in the standard SOP form

Z=A+BC

=A(B+B)(C+C)+(A+A)B C

=(AB+AB)(C+C)+ABC+ABC=ABC+ABC+ABC+ABC+ABC+ ABC

=ABC+ABC+ABC+ABC+ABC


41/44

Product of Sum (POS)

It consists of two or more OR terms (Sums) that

are ANDed together. Each OR term contains oneor more variables in complemented or

uncomplemented form.

e.g. (A+B+C)(A+C)Standard POS : When a function is expressed as

a POS, where each term consists of a sum, the

sum involving all of the variables in either

complemented or uncomplemented form.

e.g. Z=(A+B+C)(A+B+C)


42/44

B C

ABCA +

BCAABX +=

AB

XAB

C

Simplifying Logic Circuits


43/44

X = AB(A + BC)

= AB(A . BC)

= AB[ A . (B + C)]

= AB A(B + C)

= AB(B + C)= ABB + ABC =ABC


44/44

09.Combinatorial.pdf

Documents

Transcript of 09.Combinatorial.pdf