Digital Electronics Notes.pdf

Unit- III
DIGITAL ELECTRONICS
FUNDAMENTALS
1

Digital and Analog Basic Concepts:
Analog Signals
Amplitude
(peak-to-peak)
Amplitude
(peak)
Period
(T)
Hz
T
1
F 
•Amplitude
Distance above reference line
•Cycle
One complete wave
•Frequency
Cycles per second .Hertz is the unit used for expressing
frequency. Frequency:
2

Example of Analog Signals
 An analog signal can be any time-varying signal.
 Minimum and maximum values can be either positive or
negative.
 They can be periodic (repeating) or non-periodic.
 Sine waves and square waves are two common analog
signals.
 Note that this square wave is not a digital signal because
its minimum value is negative.
0 volts
Sine Wave Square Wave
(not digital)
Random-Periodic
3

Digital and Analog Basic Concepts:
Digital Signals
Amplitude:
For digital signals, this will ALWAYS be 5volts.
Period:
The time it takes for a periodic signal to
repeat. (seconds)
Frequency:
A measure of the number of occurrences
of the signal per second. (Hertz, Hz)
Time High (tH):
The time the signal is at 5 v.
Time Low (tL):
The time the signal is at 0 v.
Duty Cycle:
The ratio of tH to the total period (T).
Rising Edge:
A 0-to-1 transition of the signal.
Falling Edge:
A 1-to-0 transition of the signal.
Amplitude
Hz
T
1
F 
Frequency:
%
100


T
t
DutyCycle H
Period (T)
4

Example of Digital Signals
 Digital signal are commonly referred to as square waves or
clock signals.
 Their minimum value must be 0 volts, and their maximum
value must be 5 volts.
 They can be periodic (repeating) or non-periodic.
 The time the signal is high (tH) can vary anywhere from 1% of
the period to 99% of the period.
0 volts
5 volts
5

Differences of Analog and Digital Signals
Analog Signals
 Continuous both time and
amplitude
 Infinite range of values
 More exact values, but
more difficult to work with
 Storing such a signal
requires large amount of
memory
 Processing requires large
processing power or more
time
 Transmitting requires a
large bandwidth
Digital Signals
 Discrete in time and
quantized in amplitude
 Finite range of values
 Not as exact as analog, but
easier to work with
 Storing such a signal
requires less amount of
memory
 Processing requires low
processing power or less
time
 Transmitting requires a less
bandwidth than analog
6

Number Systems
 A number system defines how a number can be
represented using distinct symbols.
 The number the numeral represents is called its value.
 A number can be represented differently in different
systems
 For example, the two numbers (2A)16 and (52)8 both
refer to the same quantity, (42)10, but their
representations are different
7

Common Number Systems
System Base Symbols
Used by
humans?
Used in
computers?
Decimal 10(D) 0, 1, … 9 Yes No
Binary 2(B) 0, 1 No Yes
Octal 8(O) 0, 1, … 7 No No
Hexa-
decimal
16(H) 0, 1, … 9,
A, B, … F
No No
Four number system
Decimal (10) , Binary (2), Octal (8) and Hexadecimal(16)
8

Decimal numbers
 In the decimal number systems each of the ten digits, 0 through 9,
represents a certain quantity also known as base-10 number systems
 The position of each digit in a decimal number indicates the magnitude of
the quantity represented and can be assigned a weight. The weights for
whole numbers are positive powers of ten that increases from right to left,
beginning with 10º = 1
 For fractional numbers, the weights are negative powers of ten that decrease
from left to right beginning with 10−1.
9

Decimal numbers
10^4 10^3 10^2 10^1 10^0 10^-1 10^-2 10^-3
10000 1000 100 10 1 0.1 0.01 0.001
Base
Right of decimal point
left of the decimal point
10

Decimal numbers
 Example: the value 725.194 is represented in expansion form as follows:
7 * 10^2 + 2 * 10^1 + 5 * 10^0 + 1 * 10^-1 + 9 * 10^-2 + 4 * 10^-3
= 7 * 100 + 2 * 10 + 5 * 1 + 1 * 0.1 + 9 * 0.01 + 4 * 0.001
= 700 + 20 + 5 + 0.1 + 0.09 + 0.004
= 725.194
11

Binary Numbers
• Most modern computer system using binary logic. The
computer represents values(0,1) using two voltage levels
(usually 0V for logic 0 and either +3.3 V or +5V for logic 1).
• The Binary Number System uses base 2 includes only the digits
0 and 1 . The position of a 1 or 0 in a binary number indicates
its weight, or value within the number, just as the position of a
decimal digit determines the value of that digit.
• With 4 digits position we can count from zero to 15. In general,
with n bits we can count up to a number equal to 2ⁿ - 1.
• Largest decimal number = 2𝑛
- 1
12

Binary Numbers
• A binary number is a weighted number. The right-most bit is the
least significant bit (LSB) in a binary whole number and has a
weight of 2º =1. The weights increases from right to left by a
power of two for each bit. The left-most bit is the most
significant bit (MSB); its weight depends on the size of the
binary number.
• The weighted values for each position are :
2^5 2^4 2^3 2^2 2^1 2^0 2^-1 2^-2
32 16 8 4 2 1 0.5 0.25
Base
Right of radix point
left of the radix point
13

Octal Systems
• Computer scientists are often looking for shortcuts to do
things
• One of the ways in which we can represent binary
numbers is to use their octal equivalents instead
• This is especially helpful when we have to do fairly
complicated tasks using numbers
• The octal system is composed of eight digits, Known as
base-8 number systems which are: 0, 1, 2, 3, 4, 5, 6, 7
• To count above 7, begin another column and start over:
10, 11, 12, 13, 14, 15, 16, 17, 20, 21 and so on.
• Counting in octal is similar to counting in decimal, except
that the digits 8 and 9 are not used.
14

Hexadecimal Numbers
• The hexadecimal number system has sixteen digits and is used
primarily as a compact way of displaying or writing binary
numbers because it is very easy to convert between binary and
hexadecimal.
• Long binary numbers are difficult to read and write because it is
easy to drop or transpose a bit. Hexadecimal is widely used in
computer and microprocessor applications.
• The hexadecimal system known as base-16 number systems,
• It is composed of 16 digits and alphabetic characters, which are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• To count above 16, begin another column and start over:
10, 11, 12, 13, 14…. 1D, 1E, 1F, 20, 21 and so on
• The maximum 3-digits hexadecimal number is FFF or decimal
4095 and maximum 4-digit hexadecimal number is FFFF or
decimal 65,535
15

Number Representation in different systems
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
Quantities/Counting (1 of 3)
16

Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
17

Hexa-
decimal
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17 Etc.
18

Number System Conversions
 The possibilities:
Hexadecimal
Decimal Octal
Binary
19

Quick Example
2510 = 110012 = 318 = 1916
Base
20

Decimal to Decimal
Hexadecimal
Decimal Octal
Binary
21

12510 => 5 x 100 = 5
2 x 101 = 20
1 x 102 = 100
125
Base
Weight
Decimal to decimal
22

Fractions
3.14 => 4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14
23

Binary to Decimal
Hexadecimal
Decimal Octal
Binary
24

Binary to Decimal
 Technique
 Multiply each bit by 2n, where n is the “weight” of the bit
 The weight is the position of the bit, starting from 0 on
the right
 Add the results
25

Example
1010112 => 1 x 20 = 1
1 x 21 = 2
0 x 22 = 0
1 x 23 = 8
0 x 24 = 0
1 x 25 = 32
4310
Bit “0”
26

Binary to Decimal
1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1
= 8 + 4 + 0 + 1
(1101)2 = (13)10
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ….
27

Binary to decimal (Fractions)
10.1011 => 1 x 2-4 = 0.0625
1 x 2-3 = 0.125
0 x 2-2 = 0.0
1 x 2-1 = 0.5
0 x 20 = 0.0
1 x 21 = 2.0
2.6875
28

Octal to Decimal
Hexadecimal
Decimal Octal
Binary
29

Octal to Decimal
 Technique
 Multiply each bit by 8n, where n is the “weight” of the bit
the right
 Add the results
30

Example
7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810
31

Octal to Decimal
137 = 1 x 82 + 3 x 81 + 7 x 80
= 1 x 64 + 3 x 8 + 7 x 1
= 64 + 24 + 7
(137)8 = (95)10
 Digits used in Octal number system – 0 to 7
32

Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
33

Hexadecimal to Decimal
 Technique
 Multiply each bit by 16n, where n is the “weight” of the
bit
the right
 Add the results
34

Example
ABC16 =>C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
35

Hex to Decimal
BAD = 11 x 162 + 10 x 161 + 13 x 160
= 11 x 256 + 10 x 16 + 13 x 1
= 2816 + 160 + 13
(BAD)16 = (2989)10
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
36

 Decimal to Base-k [Successive division]
1. Divide the decimal number to be converted by ‘k’
2. The remainder from Step 1 is the least significant digit
3. Divide the quotient of the previous divide by the ‘k’
4. Record the remainder from Step 3 as the next digit
5. Repeat Steps 3 and 4, getting remainders from right to left, until
the quotient becomes zero in Step 3.
6. The last remainder thus obtained will be the Most Significant Digit
37

 Decimal to Base-k [Successive Multiplication]
1. Multiply the fractional decimal number, to be converted, by ‘k’
2. The integer from Step 1 is the highest order digit
3. Multiply the fractional part from Step 1 by the ‘k’
4. Record the integer from Step 3 as the next digit
5. Repeat Steps 3 and 4, getting integers from left to right, until the
fractional part becomes zero in Step 3.
38

Decimal to Binary
Hexadecimal
Decimal Octal
Binary
39

Example
12510 = ?2 2 125
62 1
2
31 0
2
15 1
2
7 1
2
3 1
2
1 1
2
0 1
12510 = 11111012
40

Decimal to Binary
13
6
3
1
0
2
2
2
2
1
0
1
1
(13)10 = (1101)2
MSB
LSB
41

Fractions
 Decimal to binary
3.14579
.14579
x 2
0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
etc.
11.001001...
42

Decimal to Octal
Hexadecimal
Decimal Octal
Binary
43

Example
123410 = ?8
8 1234
154 2
8
19 2
8
2 3
8
0 2
123410 = 23228
44

Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
45

Example
123410 = ?16
123410 = 4D216
16 1234
16 77 2
4 13 = D
16
0 4
46

Decimal to Hex
2989
186
11
0
16
16
16
13
10
11
(2989)10 = (BAD)16
MSP
LSP
47

Binary to Octal
Hexadecimal
Decimal Octal
Binary
48

Binary to Octal
 Technique
 Binary to Octal [Integers]
1. Divide the binary digits into groups of three (starting from the right).
2. Convert each group of three binary digits to one octal digit
 Binary to Octal [Fractions]
1. Divide the binary digits into groups of three (starting from the radix
point).
2. Convert each group of three binary digits to one octal digit
49

Example
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
50

Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
51

Binary to Hexadecimal
 Technique
 Binary to Hexadecimal [Integers]
1. Divide the binary digits into groups of four (starting from the right).
2. Convert each group of four binary digits to one hexadecimal digit
 Binary to Hexadecimal [Fractions]
1. Divide the binary digits into groups of four (starting from the radix
point).
2. Convert each group of four binary digits to one hexadecimal digit
52

Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
53

Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
54

Octal to Hexadecimal
 Technique
 Use binary as an intermediary
55

Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
56

Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary
57

Hexadecimal to Octal
 Technique
 Use binary as an intermediary
58

Example
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
59

Octal to Binary
Hexadecimal
Decimal Octal
Binary
60

Octal to Binary
 Technique
 Convert each octal digit to a 3-bit equivalent binary
representation
61

Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
62

Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
63

Hexadecimal to Binary
 Technique
 Convert each hexadecimal digit to a 4-bit equivalent
binary representation
64

Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
65

Binary Addition
 It is a key for binary subtraction, multiplication, division
 Each position has a value that can be represented by a
character in the set.
 A Carry to the next higher position is generated if the addition
generates a value that cannot be represented by the character
set
66

Adding Two Bits
a b
s = a + b
Decimal Binary
0 0 0 00
0 1 1 01
1 0 1 01
1 1 2 10
68
SUM
CARRY

Adding Three Bits: Include Carry
Input
a b c
s = a + b + c
Decimal value Binary value
0 0 0 0 00
0 0 1 1 01
0 1 0 1 01
0 1 1 2 10
1 0 0 1 01
1 0 1 2 10
1 1 0 2 10
1 1 1 3 11
69
SUM
CARRY

LOGIC GATES
 A logic gate is an elementary building block of a digital circuit.
 Logic gates can be created using transistor technology, that
perform Boolean operations on high (5V) and low(0V) signals.
 There are seven logic gates:
 Basic gates: AND, OR and NOT
 Universal gates: NOR and NAND
 Other gates : XOR and XNOR
70

AND GATE
For AND gate, if both the inputs are 1 the output is 1;
otherwise, the output is 0.
71

OR GATE
OR Gate: For OR gate, if both the inputs are 0 the output is 0;
otherwise, the output is 1.
72

NOT GATE
NOT Gate: A NOT gate accepts one input signal (0 or 1) and
returns the complementary (opposite) signal as output.
73

NAND GATE
NAND Gate: For NAND (“NOT of AND”) gate, if both the
inputs are 1, the output is 0; otherwise, the output is 1.
74

NOR GATE
NOR Gate: The NOR (“NOT of OR”) gate, if both the
inputs are 0, the output is 1; otherwise, the output is 0.
75

XOR GATE
XOR Gate: For XOR gate, if both the inputs are same,
the output is 0; otherwise, the output is 1.
76

XNOR GATE
XNOR Gate: For XNOR gate, if both the inputs are
same, the output is 1; otherwise, the output is 0.
77

PROPERTIES OF BOOLEAN
ALGEBRA
78

BASIC RULES OF BOOLEAN ALGEBRA
79

Proof For A+𝐴 B = A+B
A+𝐴B = A.1 + 𝐴 B
= A(1+B) + 𝐴 B
= A + AB + 𝐴 B
= A + B (A + 𝐴)
80

Proof For (A+B)(A+C) = A+BC
(A+B)(A+C) = A.A + A.C + B.A + B.C
= A + A.C + B.A + BC
= A(1+C)+ BA +BC
= A + BA + BC
= A(1+B) + BC
= A +BC
81

DeMorgan’s Theorems
(i)
(ii)
82

EXAMPLES
1. Implement the Boolean Expression
using basic Gates
83

2. Simplify the boolean expression
and realize using basic gates.
84

3. Write the boolean expression for the given logic diagram
85

HALF ADDER
Half Adder is a combinational arithmetic circuit that adds two
bits and produces sum and carry as output.
Block diagram of half adder is shown below where A and B
are input bits.
86

Truth table of half adder is as shown below where sum is
represented as ‘S’ and carry as ‘C’.
From the truth table, expressions for sum and carry can be written
as follows
S = and C = A.B
87
𝐴 B + A𝐵
=

Sum and carry can be realized using logic gates as follows
88

FULLADDER
Full Adder is a combinational arithmetic circuit that adds three bits
and produces sum and carry as
Output.
Block diagram of full adder is shown below where A, B and Cin
are input bits.
89

Truth table of full adder is as shown below
90

From the truth table, expressions for sum and carry, Cout can be
written and simplified as follows
91

Sum and carry of full adder can be realized as follows
93

MULTIPLEXER
Multiplexing: Multiplexing is a property of combining one or more
signals and transmitting on a single channel. This is achieved by a device
called Multiplexer.
Multiplexer: Multiplexer is a digital switch, also called data selector
which is a combinational circuit with 2ⁿ input lines, 1 output line and n
select lines.
94

2:1 Multiplexer
 A 2:1 multiplexer consists of 2 inputs, 1 output and 1 select line.
 Depending on the select line, the output is connected to either of the
inputs.
 If the select line is 0, the output will be switched to D0 and if select
line is 1, the output will be switched to D1 .
The block diagram of 2:1 mux is shown below
95

Truth table of 2:1 mux is shown below
From the truth table, expression for 2:1 mux output Y = 𝑺. 𝑫𝟏+𝑺.D0
2:1 mux can be implemented using logic gates as follows.
96
Select Data Inputs,
S
Output,
Y
0 D0
1 D1

4:1 Multiplexer
 A 4:1 multiplexer consists of 4 inputs, 1 output and 2 select lines.
 Depending on the select lines, the output is connected to particular
input which is shown in the truth table.
 If the select lines are 0,0 D0 will be connected to the output, if select
lines are 0,1 D1 will be connected to output and so on.
The block diagram of 4:1 mux is shown below
97

Truth table of 4:1 mux is shown below
From the truth table, expression for 4:1 mux output
𝒀 = 𝑺𝟏.𝑺𝟎.𝑫𝟑+ 𝑺𝟏. 𝑺𝟎. 𝑫𝟐 + 𝑺𝟏.𝑺𝟎.𝑫𝟏 + 𝑺𝟏. 𝑺𝟎. 𝑫𝟎
98

4:1 mux can be implemented using logic gates as follows.
99

DECODERS
Decoder is a combinational circuit that has n inputs and 2n outputs.
Depending on the input combinations, particular output will be high for
decoder.
1:2 Decoder
 1:2 decoder has 1 input and 2 outputs.
 If input is 0, D0 will be high(1) and if input is 1, D1 will be set to 1.
The block diagram of 1:2 decoder is shown below
100

The truth table, expression and implementation of decoder is shown in
the figures below.
Truth table:
Output expressions:
𝐷0=𝐴 and 𝐷1=A
Implementation:
101

2:4 Decoder
 2:4 decoder has 2 inputs and 4 outputs.
 If inputs are 0,0 D0 will be set to 1 and if inputs are 0,1, D1 will be set
to 1 and so on as shown in the truth table.
The block diagram of 2:4 decoder is shown below
102

The truth table and implementation of 2:4 decoder is shown in the
figures below.
Truth table:
Output expressions and Implementation :
𝐷0=𝐴1 .𝐴0
𝐷1=𝐴1 .𝐴0
𝐷2=𝐴1.𝐴0 and 𝐷3=𝐴1.𝐴0
103

D FLIP-FLOP
Flip-flop: Flip-flop is a basic building block of sequential circuits that
can store one bit of information either 1 or 0. It has two stable states. The
output of flip-flop changes according to input only when clock is
applied.
D Flip-flop:
 D flip-flop is constructed from gated SR flip-flop with an inverter
added between S and R inputs.
 D flip-flop has one input and two outputs as shown in symbol.
 D input of D flip-flop is connected to S and inverter is used to generate
reset, R.
 Hence, S=D and R = not of D as shown in the block diagram and
circuit diagram.
104

Symbol and circuit of D-flip flop is shown below.
105

Truth table of D-flip flop is shown below
 When D is 1 and clock is high, it sets the flip-flop and the output, Q is
1.
 When D is 0 and clock is high, it resets the flip-flop and the output, Q
is 0.
 When clock is 0 or not given, irrespective of the input D, the output
remains in the previous state.
106

D LATCH
Latch is a basic building block of sequential circuits that can store one
bit of information either 1 or 0. The output of latch changes according to
input only when enable is applied.
D latch construction is same as D flip-flop but in D latch enable is used
in place of clock.
The symbol and circuit of D latch are shown below.
SYMBOL
107

Truth table for D latch is shown below
 When D is 1 and enable is high, it sets the latch and the output, Q is 1.
 When D is 0 and enable is high, it resets the latch and the output, Q is
0.
 When enable is 0 or not given, irrespective of the input D, the output
remains in the previous state.
108

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