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Module 6

DIGITAL ELECTRONICS

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

DIGITAL ELECTRONICS
• In analog system, the output can be
continuously controlled by the input & the
output is linearly proportional to the input
...
i
...
they
have only two discrete values and are
called BINARY
...
A logic value of ‘0’ or ‘1’
is often called as BINARY DIGIT or BIT
...
Most common are
binary, decimal, octal & hexadecimal system
...
The binary number system is
also called as Base 2 system or Radix 2 system
...
1011)2
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Convert the given binary number into decimal
equivalent number
1
...
(0
...
5+0+0
...
0625
=(0
...
(10101
...
011)2 =0×2-1+1×2-2+1×2-3
= (0
...
375)10
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Octal Numbers

Octal Number System:
A number system that
uses 8 digit (0-7) is
called
an
octal
number system
...

Example: (723)8, (676)8

Binary
Equivalent
number
4 2 1 (weights)

0
1

000
001

2
3

010
011

4
5
6
7

100
101
110
111

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Hexadecimal
number system

Equivalent binary number
8 4 2 1 (weights)

0
1

0001

2

0010

3

0011

4

0100

5

0101

6

0110

7

0111

8

1000

9

1001

10

1010=A

11

1011=B

12

1100=C

13

Hexadecimal Number
System:
The hexadecimal
number system has
base 16
...

It uses the digits 0-9
& letters A,B,C,D,E,F
as 16 digit symbols
...
(2376)8 = (?)10
= 2×83+3×82+7×81+6×80
= 1024+192+56+6
= (1278)10
2
...
567)8=1×83+2×82+3×81+4×80
+5×8-1+6×8-2+7×8-3
=512+128+24+4+0
...
09375+0
...
7324219)10
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Convert the following hexadecimal number into
decimal number system
1
...
(2B8D
...
875+0
...
88281)10
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Conversion form Decimal number system to
Binary number system:
*The given decimal number (integer) is repeatedly
divided by 2, which is the base number of binary
system till quotient becomes ‘0’ and collect the
remainder from bottom to top
...
The string of integer obtained from
top to bottom gives the equivalent fraction in
binary number system
...

• For fraction part repeatedly multiply by 8,
record carry in integer place & take the
string of integer from top to bottom
...

For traction part repeatedly
multiplied by 16, record
carry in integer place &
take the string of integer
from top to bottom
...

(7423
...


...
245)8= (111100010011
...
Each in
group of binary digit is replaced by its octal equivalent
...


(11101101110
...


...
11111)2 =(3556
...

(347
...


...
28)16= (001101000111
...
Each in group of binary
digit is replaced by its hexadecimal equivalent
...

(110111101
...


0100


...
01)2=(1BD
...

(46) 8 = (?)16
Octal equivalent

4
100

6
110

Hexadecimal
equivalent

0010
2

0110
6

(46) 8 = (26)16
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Conversion from Hexadecimal number system to Octal
number system:
First write down the 4 bit binary equivalent of hexadecimal digit
and then rearranging into group of three bit each
...
9) 16 = (?)8

Hexadecimal equivalent
Octal equivalent

2
A
B
...
1001
001 010 101 011
...
4 4

(2AB
...
44)8

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Decimal number

• BCD Numbers:
• The Binary Coded
Decimal (BCD) is a
combination of four binary
digits that represent
decimal numbers
...
It
has four bits and
represents the decimal
digits 0 to 9
...


BCD number

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
0001 0000
0001 0001
0001 0010
0001 0011
0001 0100
0001 0101

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Represent the (743
...


...
6)10=(011101000011
...
e
...

The carry is taken to the next higher column
...
Write the
sum in that column and carry the carry term to the next
higher significant column
...
65)8 to (671
...


...


...
65)8 to (671
...
01)8
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Addition in Hexadecimal number System: Add the digit in each
column in decimal and convert this sum into hexadecimal number
...

Add ( 7AB
...
71) 16
7
1

9

A
5

B
C

16
23
C=0 S=9 C=1 S=0 C=1 S=7


...



...


6
7

7
1

13
8
C=0 S=D C=0 S=8

( 7AB
...
71) 16 =(0907
...

2
...

For a given number ‘N’ in base-‘r’, we can define two
types of complements
r’s complement
(r-1)’s complement
2’s & 1’s complement for binary numbers
8’s & 7’s complement for octal numbers
10’s & 9’s complement for decimal numbers
16’s & 15’s complement for hexadecimal numbers
1’s complement:
1’s complement form of any binary number can be
obtained by replacing 0’s by 1’s and 1’s by 0’s
...

Step2: Inspect the result obtained in step1 for an end carry
...
(end round
carry) (b) If an end carry doesn’t occur, take 1’s complement of the
number obtained in step1 and place a negative sign in front of it
...

(100100)2
Take 1’s complement of the number = 011011
Add ‘1’ to LSB to get 2’s complement = 011011
+1
011100
2’s complement = (011100)2
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

2’s complement subtraction:
Step1: find 2’s complement of subtrahend
Step2: Add minuend and 2’s complement subtrahend
Step3: (a) If an end carry occurs, discard it
...


Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Example 1
...
(15)10-(31)10
(1111)2-(11111)2
1111—minuend
11111— subtrahend
2’s complement of subtrahend = 00001
Add minuend and 2’s complement of subtrahend,
0
1
1
1
1
+
0
0
0
0
1
No end
1
0
0
0
0
carry
So, take the 2’s complement of the answer and place negative
sign in front of it, i
...
= -(10000)2
(1111)2-(11111)2= -(10000)2
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

BOOLEAN ALGEBRA
George Boole in 1854 invented a new kind of algebra known as
Boolean algebra
...

Boolean algebra is the mathematical frame work on which logic
design based
...

Basic Laws of Boolean algebra:
1
...
i
...
to change 0’s to 1’s and 1’ to 0’s
...
i
...

A+B=B+A
A

B

A+B

B

A

B+A

0

0

0

0

0

0

0

1

1

1

0

1

1

0

1

0

1

1

1

1

1

1

1

1

=

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Property 2: This property of multiplication states that the order
in which the variables are AND’ed makes no difference in the
output
...
e
...
B=B
...
B

B

A

B
...

For three variables i
...
(A OR’ed with B)or’ed with C is same as A
OR’ed with (B OR’ed with C)
i
...
(A+B)+C = A+(B+C)
A

B

C

A+B

B+C

(A+B)+C

A+(B+C)

0

0

0

0

0

0

0

0

0

1

0

1

1

1

0

1

0

1

1

1

1

0

1

1

1

1

1

1

0

0

1

0

1

1

1

0

1

1

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

=

1

Property2: The associative property of multiplication states
that, it makes no difference in what order the variables are
grouped when AND’ing several variables
...
e
...
B)C = A(B
...
B

B
...
B)C

A(B
...
B + A
...
B

7
A
...
B+A
...

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

8

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Duality: The important property to Boolean algebra is called
Duality principle
...

1
...
Change all 1’s to 0’s

3
...
+’s (plus) are replaced by
...
It is extensively useful in simplifying complex
Boolean expression
...


Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

De Morgon’s Second Theorem: It states that compliment of
sum of two variables is equal to product of compliment of two
individual variables
...
A
logic gate is a digital circuit with one or more input signal
and only one output signal
...
A digital circuit is
referred to as logic gate for simple reason i
...
it can be
analyzed on the basis of Boolean algebra
...
They
are OR, AND and NOT gate
...

• The input and output of the binary variables for each
gate can be represented in a tabular column or truth
table
...
OR Gate: The OR gate performs logical additions commonly
known as OR function
...
The operation of OR gate is such that a HIGH(1)
on the output is produced when any of the input is HIGH
...

If A & B are the input variables of an OR gate and c is its output,
then A+B
...

Input
Output
Logical Symbol:
A
A
B

Y= A+B

0

0

0

0

1

1

1

0

1

1

Y

B

1

1

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

• Realization of OR gate using diodes:
• Two input OR gate using "diode-resistor" logic is shown
in figure below
...

D1
X

F

Y
D2

RL

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

• 2
...

The AND gate has two or B
more inputs and a single
Input
output
...
Even if any one of
0
1
the input is LOW, the
output will be LOW
...
B
Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Y
Output
Y=A
...
Where X, Y are inputs and F is the
output
...
Not Gate ( Inverter): The NOT gate performs the basic
logical function called inversion or complementation
...
It has one input and one output
...


A

Y

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

• Realization of NOT gate using Transistors:
• A NOT gate using a transistor is shown in below figure
...

When the input is HIGH, the transistor is in the ON state
and the output is LOW
...

+Vcc
RL
F
A

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

4
...


A
Y
B

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

5
...


A

Y

B

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

6
...
The
circuit is also called as inequality comparator, because it
produces output when two inputs are different
...
XNOR Gate or Exclusive NOR Gate: An XNOR gate is
a gate with two or more inputs and one output
...
i
...
The
output of XNOR gate is High, when all the inputs are
identical; otherwise it is low
...

• NAND gate as Universal gate

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

NOR gate as Universal gate:

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Digital Circuits
Types of Digital Circuits:
Basically digital circuits can be classified into two types
...

In sequential circuits, the output signals are fedback to the input
...
i
...
the output signals
...

In particular, the output of the combinational circuit doesn’t depend upon any
past input or output So that the circuit doesn’t possess any memory
...

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Half Adder: A combinational circuit which performs the
arithmetic addition of two binary digits is called Half
Adder
...

A

Sum
Half Adder
Carry

B

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

Department of Electronics and Communication Engineering,
Manipal Institute of Technology, Manipal, INDIA

• Full Adder: The full adder is a combinational circuit that
performs the arithmetic sum of three input bits
...
Two of the
inputs are variables, denoted by A and B, represent the
two significant bit to be added The third input Cin
represents carry form the previous lower significant
position

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