CVMU digital electronics ppt for students to learn from

Module 5
• Binary Number System
• Logic Gates
• Boolean Algebra
• Half and Full Adders
• Flip- Flops
• Registers and Counters
• A/D and D/A Conversion(Block Diagram Approach only).
PROF. SUNAYANA DOMADIA DIGITAL FUNDAMENTALS (3130704) 2

Introduction
⮚Digital Electronics is that branch of electronics which deals with the digital
signals to perform various tasks and meet various requirements
⮚Digital Electronics is very important in today's life because if digital
circuits compared to analog circuits are that signals represented digitally can be
transmitted without degradation due to noise.
⮚It is based upon the digital design methodologies and consists of digital
circuits, IC’s and logic gates
⮚ It uses only binary digits.

Need for Digital Electronics
⮚Most analog systems were less accurate and were slow in computation and
performance
⮚Digital system have the ability to work faster than analog equivalents
⮚It was much economical than analog methodologies as the performance was
faster.

Applications
⮚Almost all devices we use on a daily basis make use of digital electronics
in some capacity.
⮚Digital electronics simply refers to any kind of circuit that uses digital
signals rather than analogue.
⮚It is constructed using circuits calls logic gates, each of which performs a
different function.
⮚ The circuit will make use of different components that are all standard, but
that are put together in different combinations to achieve the desired result.
It circuit will also include resistors and diodes, which are used to control
current and voltage.

Applications
⮚Many of our household items make use of digital electronics. This could
include laptops, televisions, remote controls and other entertainment systems,
to kitchen appliances like dishwashers and washing machines.
⮚Computers are one of the most complex examples and will make use of
numerous, complex circuits. There may be millions of pathways within the
circuit, depending on how complex the computer and its functions need to be.
⮚Use in Machine learning and AI for mathematical calculations

Block diagram of Computer

Logic Gates
Most basic logical unit of the digital system is gate circuit
Types of gate circuits are as follows
1. AND Gate
2. OR Gate
3. NOT Gate (Inverter)
4. NOR Gate
5. NAND Gate
6. XOR Gate
7. XNOR Gate

AND Gate
• AND Gate has an output which is normally at logic level “0” and only goes
“HIGH” to a logic level “1” when ALL of its inputs are at logic level “1”
A
B
C
Logic Notation
A B C
0 0 0
0 1 0
1 0 0
1 1 1
Truth Table
2-input AND Gate

OR Gate
• OR Gate or Inclusive-OR gate has an output which is normally at logic level
“0” and only goes “HIGH” to a logic level “1” when one or more of its inputs
are at logic level “1”.
A
B
C
Logic Notation
A B C
0 0 0
0 1 1
1 0 1
1 1 1
Truth Table
2-input OR Gate

NOT (Inverter) Gate
• NOT gate has an output which is always opposite to input level.
A C
Logic Notation
A C
0 1
1 0
Truth Table
Inverter Gate

NOR Gate
• NOR Gate is an OR gate followed by an inverter.
A
B
C
Logic Notation
A B C
0 0 1
0 1 0
1 0 0
1 1 0
Truth Table
2-input NOR Gate

NAND Gate
• NAND Gate is an AND gate followed by an inverter.
A
B
C
Logic Notation
A B C
0 0 1
0 1 1
1 0 1
1 1 0
Truth Table
2-input NAND Gate

Exclusive-OR (X-OR) Gate
• X-OR gate that has 1 state when one and only one of its two inputs assumes a
logic 1 state and has 0 state when all of its input are same.
• Also known as anti-coincidence gate or inequality detector.
A
B
C
Logic Notation
A B C
0 0 0
0 1 1
1 0 1
1 1 0
Truth Table
2-input XOR Gate

Exclusive-NOR (X-NOR) Gate
• X-NOR gate that has 1 state when all of its input are same and has 0 state when
one of its input has 0 state and other input is 1 state.
• Also known as equality detector.
A
B
C
Logic Notation
A B C
0 0 1
0 1 0
1 0 0
1 1 1
Truth Table
2-input XNOR Gate

NAND as Universal Gate
NOT using NAND
A A’
A
B
(AB)’ ((AB)’)’ = AB
AND using NAND
A
B
A’
(A’B’)’ = (A+B)
OR using NAND
B’

NOR as Universal Gate
NOT using NOR
A A’
A
B
(A+B)’ ((A+B)’)’ = A+B
OR using NOR
A
B
A’
(A’+B’)’ = AB
AND using NOR
B’

Boolean Algebra is the mathematics we use to analyse digital
gates and circuits. We can use these “Laws of Boolean” to both
reduce and simplify a complex Boolean expression in an attempt
to reduce the number of logic gates required.
Boolean Algebra is therefore a system of mathematics based on
logic that has its own set of rules or laws which are used to
define and reduce Boolean expressions
Boolean Algebra

Boolean Algebra
• ll
``

Boolean Algebra
•

Boolean Algebra
•
1. The complement of a sum of variables is equal to the product of their
individual complements.
2. The complement of a product of variables is equal to the sum of their
individual complements.

A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
A+B+C
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
From truth table, it is clearly visible that L.H.S. = R.H.S. Hence, the complement of a sum of
variables is equal to the product of their individual complements.
L.H.S. R.H.S.

A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
ABC
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
1
1
1
1
1
1
0
From truth table, it is clearly visible that L.H.S. = R.H.S. Hence, the complement of a
product of variables is equal to the sum of their individual complements.
L.H.S. R.H.S.

According to principle of duality "Dual of one expression is obtained by
replacing AND (.)with OR(+) and OR with AND together with replacement of
1 with 0 and 0 with 1.

1. (A + B)'(A' + B')
Reducing Boolean Expression
= (A'B')(A B) = (A'B')(BA) = A'(B'BA) = 0
2. ABC + A'B + ABC' =B
3. A'C' + ABC + AC' = C' + ABC = (C + C')(C' + AB) = AB + C'
4. A'B(D' + C'D) + B(A + A'CD) = B(A'D' + A'C'D + A + A'CD) =
B(A'D' + A + A'D(C + C') = B(A + A'(D' + D)) = B(A + A') = B
5. ABCD + A'BD + ABC'D = ABD + A'BD = BD

Reducing Boolean Expression
•
(Distributive law)
(Distributive law)
(A.A = A)
(1 + A = 1)

Number
Systems

Conversion among Bases
• Possibilities
• Example
Hexadecimal
Decimal Octal
Binary
2510 = 110012 = 318 = 1916 Base

Decimal to Binary
• Technique
• Divide by two, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)
• Second remainder is bit 1 and so on
Decimal Binary

Example (Decimal to Binary)
12510 = ?2 2 125 1
2 62 0
2 31 1
2 15 1
2 7 1
2 3 1
2 1 1
0
12510 = 11111012

Example (Decimal to Binary)
0.687510 = ?2
0.6875 x 2 = 1.3750 1 0.3750
+
0.3750 x 2 = 0.7500 0 0.7500
+
0.7500 x 2 = 1.5000 1 0.5000
+
0.5000 x 2 = 1.0000 1 0.0000
+
0.687510 = 0.10112
integer fraction

Exercise
• (32)10 = ( )2
• (555)10 = ( )2
• (12999)10 = ( 11001011000111 )2
• (157.63)10 = ( )2 = 10011101.10100001010….
• (64.125)10 = ( )2
= 1000002

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
Binary Decimal

Example (Binary to Decimal)
1 0 1 0 1 1
1 x 20
1 x 21
0 x 22
1 x 23
0 x 24
1 x 25 +
+
+
+
+
1
2
0
1010112 =
0
32 +
+
+
+
+
4310
4310
8

Example (Binary to Decimal)
1 1 . 1 1
1 x 20
1 x 21 +
1
2
11.112 =
+
3.7510
3.7510
1 x 2-2
1 x 2-1 +
0.25
0.5 +
+
+

Exercise
• (11011)2 = ( )10= 27
• (101101)2 = ( )10
• (11101111)2 = ( )10
• (110.011)2 = ( )10=6.375
• (1001.0010)2 = ( )10

Decimal to Octal
• Technique
• Divide by eight
• Keep track of the remainder
Decimal Octal

Example (Decimal to Octal)
12510 = ?8 8 125 5
8 15 7
8 1 1
0
12510 = 1758

Example (Decimal to Octal)
0.687510 = ?8
0.6875 x 8 = 5.5000 5 0.5000
+
0.5000 x 8 = 4.0000 4 0.0000
+
0.687510 = 0.548
integer fraction

Exercise
• (32)10 = ( )8= 40
• (555)10 = ( )8
• (12999)10 = ( )8
• (157.63)10 = ( )8= 235.50243656050
• (64.125)10 = ( )8

Octal to Decimal
• Technique
• Add the results
Octal Decimal

Example (Octal to Decimal)
7 2 4
4 x 80
2 x 81
7 x 82 +
+
7248 =
46810
46810
4
16
448 +
+

Example (Octal to Decimal)
4 3 . 2 5
3 x 80
4 x 81 +
3
32
43.258 =
+
35.328110
35.328110
5 x 8-2
2 x 8-1 +
0.0781
0.25 +
+
+

Exercise
• (32)8 = ( )10= 26
• (555)8 = ( )10
• (12333)8 = ( )10
• (157.63)8 = ( )10= 111.796875
• (64.125)8 = ( )10

Decimal to Hexa-Decimal
• Technique
• Divide by 16
• Keep track of the remainder
Decimal
Hexa-
Decimal

Example (Decimal to HexaDecimal)
123410 = ?16 16 1234 2
123410 = 4D216
16 77 13=D
16 4 4
0

Exercise
• (32)10 = ( )16 = 20
• (555)10 = ( )16
• (12999)10 = ( )16
• (157.63)10 = ( )16= 9D.A147AE147AE
• (64.125)10 = ( )16

Hexa-Decimal to Decimal
• Technique
• Add the results
Hexa-
Decimal
Decimal

Example (HexaDecimal to Decimal)
A B C
C x160
B x161
A x162 +
+
ABC16
=
274810
274810
12 x160
11 x161
10 x 162
+
+
12
176
2560 +
+

Exercise
• (FA8)16 = ( )10 = 4008
• (9AC3)16 = ( )10
• (1A74D)16 = ( )10
• (1AC.9A)16 = ( )10= 428.6015625
• (ABC.5AC)16 = ( )10

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

Octal - Binary Table
Octal Binary
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111

Example (Octal to Binary)
7058 = ?2
7 0 5
101
000
111
7058 = 1110001012

Exercise
• (463)8 = ( 100110011)2
• (2056)8 = ( )2
• (2057.64)8 = ( )2
• (6543.04)8 = ( )2
• (7476.47)8 = ( )2

Binary to Octal
• Technique
• Group bits in threes, starting on right
• Convert to octal digits
Binary Octal

Example (Binary to Octal)
10110101112 = ?8
1
011
001
10110101112 = 13278
111
010
3 2 7

Exercise
• (11011)2 = ( 33 )8
• (101101)2 = ( )8
• (11101111)2 = ( )8
• (110.011)2 = ( 6.3 )8
• (1001.0010)2 = ( )8

Hexa-Decimal to Binary
• Technique
• Convert each hexadecimal digit to a 4-bit equivalent binary representation
Hexa-
Decimal
Binary

Hexa-Decimal to Binary
Hexa-
Decimal
Binary Hexa-
Decimal
Binary
0 0000 8 1000
1 0001 9 1001
2 0010 A 1010
3 0011 B 1011
4 0100 C 1100
5 0101 D 1101
6 0110 E 1110
7 0111 F 1111

Example (Hexa-Decimal to Binary)
10AF16 = ?2
1 0 A F
1111
1010
0000
10AF16 = 10000101011112
0001

Exercise
• (FA8)16 = ( 111110101000 )2
• (9AC3)16 = ( )2
• (1A74D)16 = ( )2
• (1AC.9A)16 = ( )2
• (ABC.5AC)16 = ( )2

Binary to Hexa-Decimal
• Technique
• Group bits in fours, starting on right
• Convert to hexadecimal digits
Binary
Hexa-
Decimal

Example (Binary to Hexa-Decimal)
10110101112 = ?16
0010
10110101112 = 2D716
0111
1101
2 D 7

Exercise
• (11011)2 = ( )16= 1B
• (101101)2 = ( )16
• (11101111)2 = ( )16
• (110.011)2 = ( )16
• (1001.0010)2 = ( )16

Octal to Hexa-Decimal
• Technique
• Convert Octal to Binary
• Regroup bits in fours from right
• Convert Binary to Hexa-Decimal
Octal
Hexa-
Decimal

Example (Octal to Hexa-Decimal)
10768 = ?16
1 0 7 6
110
111
000
10768 =23E16
001
1110
0011
0010
E
3
2

Exercise
• (463)8 = ( 133 )16
• (2056)8 = ( )16
• (2057.64)8 = ( )16
• (6543.04)8 = ( )16
• (7476.47)8 = ( )16

Hexa-Decimal to Octal
• Technique
• Convert Hexa-Decimal to Binary
• Regroup bits in three from right
• Convert Binary to Octal
Hexa-
Decimal
Octal

Example (Hexa-Decimal to Octal)
1F0C16 = ?8
1 F 0 C
1100
0000
1111
1F0C16
=
174148
0001
100
001
100
4
1
4
111
7
001
1
000
0

Exercise
• (FA8)16 = ( 7650 )8
• (9AC3)16 = ( )8
• (1A74D)16 = ( )8
• (1AC.9A)16 = ( )8
• (ABC.5AC)16 = ( )8

Half Adder
• A combinational circuit which adds two one-bit binary numbers is called a half-adder.
• The sum column resembles like an output of the XOR gate.
• The carry column resembles like an output of the AND gate.
Inputs Outputs
A B Sum
Carr
y
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
S = A ⊕ B
C = AB
A
B

Limitation of Half-Adder
• In multi-digit addition we have to add two bits along with the carry
of previous digit addition. Such addition requires addition of 3 bits.
This is not possible in half-adders.

Full Adder
• In a full adder, three bits can be added at a time. The third bit is a carry from a
less significant column.
Inputs Outputs
A B Cin S Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
S = A’B’Cin + A’BCin’ + AB’Cin’ + ABCin
= (AB’ + A’B)Cin’ + (AB + A’B’)Cin
= (A ⊕ B)Cin’ + (A ⊕ B)’Cin
= A ⊕ B ⊕ Cin
Cout = A’BCin + AB’Cin + ABCin’ + ABCin
= AB + (A ⊕ B)Cin

Full Adder
Logic diagram for full adder
A
B
Cin
S = A ⊕ B ⊕ Cin
Cout = AB + (A ⊕ B)Cin
Half Adder Half Adder
S = A ⊕ B ⊕ Cin
Cout = AB + (A ⊕ B)Cin

Sequential Switching Circuits
• Sequential switching circuits are circuits whose output levels at any instant of
time are dependent on the levels present at the inputs at that time and on the
state of the circuit, i.e., on the prior input level conditions (i.e. on its past
inputs)
• The past history is provided by feedback from the output back to the input.
• Made up of combinational circuits and memory elements.
• Eg. Counters, shift registers, serial adder, etc.

Flip-flop
• A flip-flop, known formally as bistable multivibrator, has two stable states.
• It can remain in either of the states indefinitely.
• Its state can be changed by applying the proper triggering signal.

SR FLIP FLOP

D Flip flop
D flip-flop operates with only positive clock transitions or
negative clock transitions. Whereas, D latch operates with enable
signal.
That means, the output of D flip-flop is insensitive to the changes
in the input, D except for active transition of the clock signal.
The circuit diagram of D flip-flop is shown in the following
figure.

JK Flip-Flop

T Flip-Flop
T flip-flop is the simplified version of JK flip-flop. It is obtained by connecting the same input ‘T’ to both inputs of
JK flip-flop. It operates with only positive clock transitions or negative clock transitions. The circuit diagram of T
flip-flop is shown in the following figure.

Applications of Flip-Flops
These are the various types of flip-flops being used in digital
electronic circuits and the applications like Counters,
Frequency Dividers, Shift Registers, Storage Registers

Registers
• As a flip-flop (FF) can store only one bit of data, a 0 or a 1, it is referred to as a
single-bit register.
• A register is a set of FFs used to store binary data.
• The storage capacity of a register is the number of bits (1s and 0s) of digital
data it can retain.

Registers
• Loading a register means setting or resetting the individual FFs, i.e. inputting
data into the register so that their states correspond to the bits of data to be
stored.
• Loading may be serial or parallel.
• In serial loading, data is transferred into the register in serial form i.e. one bit
at a time.
• In parallel loading, the data is transferred into the register in parallel form
meaning that all the FFs are triggered into their new states at the same time.

Types of Registers
1. Buffer register
2. Shift register
3. Bidirectional shift register
4. Universal shift register

Buffer Register
Buffer registers are a type of registers used to store a binary word. These can be constructed
using a series of flip-flops as each flip-flop can store a single bit. This means that in order to
store an n-bit binary word one should design an array of n flip-flops. Figure 1 shows a 4 bit
synchronous buffer register formed by cascading four positive edge triggered D flip-flops.
Here the entire input data word X1 X2 X3 X4 is loaded onto the register at a single clock tick

Shift Register
• A number of FFs connected together such that data may be shifted into and shifted out of
them is called a shift register.
• Data may be shifted into or out of the register either in serial form or in parallel form.
• So, there are four basic types of shift registers:
1. serial-in, serial-out
2. serial-in, parallel out
3. parallel-in, serial-out
4. parallel-in, parallel-out
• Data may be rotated left or right. Data may be shifted from left to right or right to left at will,
i.e. in a bidirectional way.
• Also, data may be shifted in serially (in either way) or in parallel and shifted out serially (in
either way) or in parallel.

Data transmission in shift register
Serial-in, serial-out shift-
right, shift register
Serial
data
input
Serial
data
output
Serial-in, serial-out shift-
left, shift register
Serial
data
output
Serial
data
input

Serial-in, parallel-out, shift register
Serial
data
input
Parallel data output
Parallel-in, parallel-out, shift register
Parallel data input
Parallel data output

Parallel-in, serial-out, shift register
Serial
data
output
Parallel data input

Serial-in, Serial-out, Shift register
FF1
D1 Q1
> FF2
D2 Q2
> FF3
D3 Q3
> FF4
D4 Q4
>
Serial Input
CLK
Serial
output

Serial-in, Serial-out, Shift-left, Shift register
FF4
Q4 D4
< FF3
Q3 D3
< FF2
Q2 D2
< FF1
Q1 D1
<
Serial
Output
CLK
Serial
input

Serial-in, Parallel-out, Shift register
FF1
D1 QA
> FF2
D2 QB
> FF3
D3 QC
> FF4
D4 QD
>
Serial Input
CLK
QA QB QC QD

Parallel-in, Serial-out, Shift register

Parallel-in, Parallel-out, Shift register
FF1
D Q
> FF2
D Q
> FF3
D Q
> FF4
D Q
>
A
CLK
B C D
QA QB QC QD

Applications of Shift Registers
Registers are used in digital electronic devices like computers as
Temporary data storage
Data transfer
Data manipulation
As counters.

Counter
“A register that goes through a prescribed sequence of distinct states
upon the application of a sequence of input pulses is called a counter.”
The input pulses could be the clock or some other input that occurs when
the next step in the count should occur.
A counter that follows the binary number sequence is called a binary
counter.
Counters are of two types.
 Asynchronous or ripple counters.
 Synchronous counters.

Asynchronous Counters v/s Synchronous Counters
Asynchronous Counters Synchronous Counters
• In this type of counters FFs are
connected in such a way that the output
of the first FF drives the clock for the
second FF, the output of the second the
clock of the third and so on.
• All the FFs are not clocked
simultaneously.
• Design and implementation is very
simple even for more number of states.
• Main drawback of these counters is
their low speed as the clock is
propagated through a number of FFs
before it reaches the last FF.
• In this type of counters there is no
connection between the output of first
FF and clock input of next FF and so
on.
• All the FFs are clocked simultaneously.
• Design and implementation becomes
tedious and complex as the number of
states increases.
• Since clock is applied to all the FFs
simultaneously the total propagation
delay is equal to the propagation delay
of only one FF. Hence they are faster.

Asynchronous or ripple counters
CLK
QA
QB
0 1 0 1 0
0 1 0

2-bit Ripple Up-Counter using Negative Edge-triggered
Flip-Flop
FF1
J1 Q1
>
K1 Q1’
FF2
J2 Q2
>
K2 Q2’
1 1
Q1 Q2
CLK
CLK
Counter output
Q2 Q1
initially 0 0
0 1
1 0
1 1
0 0
CLK
Q1
Q2
0 1 0 1 0
0 1 0

2-bit Ripple Down-Counter using Negative Edge-triggered
Flip-Flop
CLK
Present
State
Q2 Q1
0 0
1 1
1 0
0 1
0 0
FF1
J1 Q1
>
K1 Q1’
FF2
J2 Q2
>
K2 Q2’
1 1
Q1 Q2
CLK
CLK
Q1
Q2
0 1 0 1 0
0 1 0
Q1
’ 1 0 1 0 1

Synchronous counters
CLK
Present
State
Q1 Q0
0 0
0 1
1 0
1 1
0 0

Classification of counters
Depending on the way in which the counting progresses, the
synchronous or asynchronous counters are classified as follows −
Up counters
Down counters
Up/Down counters

Design 3-bit Binary Counter

Excitation table

Next step is to transfer the flip-flop input functions to Karnaugh maps to derive a
simplified Boolean expressions,

Ring Counter
After
pulses
State
Q1 Q2 Q3 Q4
0 1 0 0 0
1 0 1 0 0
2 0 0 1 0
3 0 0 0 1
4 1 0 0 0
5 0 1 0 0
6 0 0 1 0
7 0 0 0 1

4-bit Johnson Ring Counter

Application of counters
Frequency counters
Digital clock
Time measurement
A to D converter
Frequency divider circuits
Digital triangular wave generator.

A/D and D/A Conversion
Interfacing with the analog world using Analog-to-Digital
Converter (ADC) and Digital-to-Analog Converter (DAC).
Analog-todigital converter (ADC) and digital-to-converter (DAC) are used to interface a computer to the analog world so that
the computer can monitor and control a physical variable

The physical variable is normally a nonelectrical quantity. A transducer is a device that converts the physical
variable to an electrical variable. Some common transducers include thermistors, photocells, photodiodes, flow
meters, pressure transducers, and tachometers. The electrical output of the transducer is an analog current or
voltage that is proportional to the physical variable it is monitoring.
Transducer
Analog-to-Digital Converter (ADC) The transducer’s electrical analog output serves as the analog input to the
ADC. The ADC converts this analog input to a digital output. This digital output consists of a number of bits that
represent the value of the analog input. For example, the ADC might convert the transducer’s 800- to 1500-mV
analog values to binary values ranging from 01010000 (80) to 10010110 (150). Note that the binary output from
the ADC is proportional to the analog input voltages so that each unit of the digital output represents 10mV. The
digital representation of the analog vales is transmitted from the ADC to the digital computer, which stores the
digital value and processes it according to a program of instructions that it is executing.

Digital-to-Analog Converter (DAC) This digital output from the computer is connected to a DAC, which converts it
to a proportional analog voltage or current. For example, the computer might produce a digital output ranging from
0000000 to 11111111, which the DAC converts to a voltage ranging from 0 to 10V.
Actuator The analog signal from the DAC is often connected to some device or circuit that serves as an actuator to
control the physical variable.

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