Digital Electronics

COMBINATIONAL LOGIC
 Combinational Circuits

■ Output is function of input only

i.e. no feedback

Combinational
n inputs • • m outputs
• Circuits •
• •


When input changes, output may change (after a delay)
 Combinational Circuits

■ A combinational circuit operates in three steps ,

a) It accepts n-different inputs.
b) The combination of gates operates on the inputs .
c) “m” different outputs are produced as per requirement .

■ Example of Combinational Circuit:

1) Adder , subtractors 2) Encoder, decoder
3) Comparator 4) Multiplexers, Demultiplexers,
5) Code converters
 Combinational Circuits
■ Analysis
?
A

– Given a circuit, find out its function

B
F1
C
A
B

– Function may be expressed as:

C
A
B

?
■ Boolean function
A
F2
C

B
C

■ Truth table
■ Design
– Given a desired function, determine its circuit
– Function may be expressed as:
■ Boolean function
■ Truth table
?
 Analysis Procedure
■ Boolean Expression Approach
A
B
F1
C
ABC
A A+B+C
B AB'C'+A'BC'+A'B'C
C
A (A’+B’)(A’+C’)(B’+C’)
B

A
F2
C AB+AC+BC

B
C
F1=AB'C'+A'BC'+A'B'C+ABC
F2=AB+AC+BC
 Analysis Procedure
■ Truth Table Approach A B C F1 F2
A= 0
0
0 0 0 0 0
B= 0 0
F1
C= 0
A= 0 0
B= 0 0
C= 0
0 1
A= 0
B= 0
0
0
A= 0
F2
C= 0 0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

A= 0
0
0 0 0 0 0
B= 0 1
C= 1
F1 0 0 1 1 0
A= 0 1
B= 0 1
C= 1 1
0
A= 0
B= 0
0
A= 0 0
F2
C= 1
0
B= 0
C= 1
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0
B= 0 0 0 0 0 1 1 0
F1
C= 0 0 1 0 1 0
A= 0 0
B= 0 0
C= 0
0 1
A= 0
B= 0
0
A= 0 0
F2
C= 0
0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0
B= 0 0 0 0 0 1 1 0
F1
C= 0 0 1 0 1 0
A= 0 0 0 1 1 0 1
B= 0 0
C= 0
0 1
A= 0
B= 0
0
A= 0 0
F2
C= 0
0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0
0
0 0 1 1 0
B= 0 0
C= 0
F1 0 1 0 1 0
A= 0 0
0 1 1 0 1
B= 0 0 1 0 0 1 0
C= 0
0 1
A= 0
B= 0
0
A= 0 0
F2
C= 0
0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0
0
0 0 1 1 0
B= 0 0
C= 0
F1 0 1 0 1 0
A= 0 0
0 1 1 0 1
B= 0 0 1 0 0 1 0
C= 0
0 1 1 0 1 0 1
A= 0
B= 0
0
A= 0 0
F2
C= 0
0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0 0 0 1 1 0
0
B= 0 0
F1 0 1 0 1 0
C= 0
0 1 1 0 1
A= 0 0
B= 0 0 1 0 0 1 0
C= 0
1 1 0 1 0 1
0
A= 0 1 1 0
B= 0 0 1
0
A= 0 0
F2
C= 0
0
B= 0
C= 0
 Analysis Procedure

■ Truth Table Approach A B C F1 F2

0 0 0 0 0
A= 0 0 0 1 1 0
0
B
=0
0
F1 0 1 0 1 0
C
=0
A= 0
0 1 1 0 1
0
B= 0 0 1 0 0 1 0
C= 0
1 1 0 1 0 1
A= 0 0
B= 0
1 1 0 0 1
0 1 1 1 1
1
A= 0 0 F2
C
=0
0
B= 0 B
C= 0 B
0 1 0 1
0 0 1 0
A 1 0 1 0
A 0 1 1 1
C
C
F1=AB'C'+A'BC'+A'B'C+ABC F2=AB+AC+BC
 Basic Adders

■ Adders are important in

– computers
– other types of digital systems in which numerical
data are processed

■ We must know about adders.

Two type of Adder.
1) Half adder
2) Full adder
 The Half-Adder

■ Basic rule for binary addition.

■ The operations are performed by a logic ckt called a half-

adder.
 Binary Adder
■ Half Adder
x S
– Adds 1-bit plus 1-bit HA
y C
– Produces Sum and Carry
x
+ y
x y ───
C S
C S
0 0 0 0
0 1 0 1
x S=AB+A
1 0 0 1
B
1 1 1 0
y
C=AB
 Binary Adder
■ Full Adder using half adder

x S
y HA HA

z C

x
S

y
C

z
 Half Subtractor

 Logic diagram for a half subtractor.

 The half-subtractor is a combinational circuit which is used to
perform subtraction of two bits. It has two inputs, X and Y
and two outputs D (difference) and B (borrow).
 Truth table:
 The truth table for the half subtractor is given below

X
D=XY+XY X Y D B
Y 0 0 0 0
B=XY
0 1 1 1
1 0 1 0
1 1 0 0
 Half Subtractor using NANDgate

A B

•The expression for

difference output is,
D=AB+AB
D=AB+ BA

•Taking the
double
B=AB inversion

D=AB+AB
 Half Subtractor using NANDgate

 Applying the De Morgan ‘s theorem,

D=AB . AB

 Similarly the other output B is given by,

B = AB = AB
 Full subtractor
 The full-subtractor is a combinational circuit which is used to
perform subtraction of three bits.

 It has three inputs, X and Y and x y z D B

0 0 0 0 0
 Z and two outputs D (difference) 0 0 1 1 1
 and B (borrow).
0 1 0 1 1

0 1 1 0 1
 D=X-Y-Z (don't bother about sign)
1 0 0 1 0
B = 1 If X<(Y+Z)
1 0 1 0 0

1 1 0 0 0
 Truth table 1 1 1 1 1
 The truth table for the fullsubtractor
 is given below.
Full subtractor

=X Y Z

=XY +YZ +XZ

Binary code to Gray code converter
Decimal Binary I/p Gray O/p

B3 B2 B1 B0 G3 G2 G1 G1

0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 0 0 0 1 1
3 0 0 1 1 0 0 1 0
4 0 1 0 0 0 1 1 0
5 0 1 0 1 0 1 1 1
6 0 1 1 0 0 1 0 1
7 0 1 1 1 0 1 0 0
8 1 0 0 0 1 1 0 0
9 1 0 0 1 1 1 0 1
Gray code to Binary code converter
Decimal Gray code I/P Binary O/p
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
3 0 0 1 1 0 0 1 0
2 0 0 1 0 0 0 1 1
6 0 1 1 0 0 1 0 0
7 0 1 1 1 0 1 0 1
4 0 1 0 0 0 1 1 1
12 1 1 0 0 1 0 0 0
13 1 1 0 1 1 0 0 1
15 1 1 1 1 1 0 1 0
14 1 1 1 0 1 0 1 1
Excess-3 to Binary code converter
Excess-3 I/p BCD O/P
E3 E2 E1 E0 D3 D2 D1 D0
0 0 1 1 0 0 0 0
0 1 0 0 0 0 0 1
0 1 0 1 0 0 1 0
0 1 1 0 0 0 1 1
0 1 1 1 0 1 0 0
1 0 0 0 0 1 0 1
1 0 0 1 0 1 1 0
1 0 1 0 0 1 1 1
1 0 1 1 1 0 0 0
1 1 0 0 1 0 0 1
 Binary to BCD code converter
Decimal Binary I/p BCD O/P

B3 B2 B1 B0 D4 D3 D2 D1 D0
0 0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 1
2 0 0 1 0 0 0 0 1 0
4 0 1 0 0 0 0 1 0 0
5 0 1 0 1 0 0 1 0 1
6 0 1 1 0 0 0 1 1 0
7 0 1 1 1 0 0 1 1 1
10 1 0 1 0 1 1 0 0 0
11 1 0 1 1 1 0 0 0 1
Binary to Excess-3 converter
Binary I/p Excess-3 O/P
B3 B2 B1 B0 E3 E2 E1 E0
0 0 0 0 0 0 1 1
0 0 0 1 0 1 0 0
0 0 1 0 0 1 0 1
0 0 1 1 0 1 1 0
0 1 0 0 0 1 1 1
0 1 0 1 1 0 0 0
0 1 1 0 1 0 0 1
0 1 1 1 1 0 1 0
1 0 0 0 1 0 1 1
1 0 0 1 1 1 0 0
THANKS…