Electronics and Communication Engineering : Digital circuits, THE GATE ACADEMY

DIGITAL CIRCUITS
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Syllabus Digital Circuits
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Syllabus for Digital Circuits
Boolean algebra, minimization of Boolean functions; logic gates; digital IC families (DTL, TTL,
ECL, MOS, CMOS). Combinatorial circuits: arithmetic circuits, code converters, multiplexers,
decoders, PROMs and PLAs. Sequential circuits: latches and flip-flops, counters and shift-
registers. Sample and hold circuits, ADCs, DACs. Semiconductor memories.
Microprocessor(8085): architecture, programming, memory and I/O interfacing.
Analysis of GATE Papers
(Digital Circuits)
Year ECE EE IN
2013 6.00 5.00 5.00
2012 6.00 5.00 5.00
2011 9.00 5.00 10.00
2010 9.00 8.00 8.00
Over All
Percentage
7.5% 5.75% 7%

Contents Digital Circuits
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CC OO NN TT EE NN TT SS
Chapter Page No.
#1. Number systems & Code Conversions 1 –35
 Base or Radix of a Number System 1 – 2
 System Conversions 2 – 6
 Coding Techniques 6 – 9
 Error Detecting Codes 9 – 16
 Number System Arithmetic 17 – 24
 Signed Binary Numbers 25 – 27
 Assignment 1 28 – 30
 Answer Keys 32
 Explanations 32 – 35
#2. Boolean Algebra & Karnaugh Maps 36 – 62
 Boolean Algebra 36
 The Basic Boolean Postwater 36 – 42
 Karnaugh Maps (k-maps) 42 – 45
 Comparators 45 – 46
 Decoder 46 – 50
 Answer Keys 56
#3. Logic Gates 63 – 90
 Logic systems 63 – 66
 Relation of basic Gates using NAND & NOR gates 66 – 69
 Code Converters 69 – 79
 Answer Keys 87
#4. Logic Gate Families 91 – 126
 Classification of Logic Families 91
 Caracteristics of Digital IC’s 91 – 95
 Resistor Transistor Logic 95
 Transistor Logic 96
 Direct Coupled Transistor Logic Gates 96 – 97

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 Emitter Coupled Logic Circuit 97 – 98
 MOSFET Gates 99 – 103
 Operating Regions of MOS Transistor 104
 CMOS Inverter 104 – 107
 Important Points 107 – 113
 Advantages & Disadvatages of Major Logic Families 113 – 115
 Answer Keys 123
#5. Combinational Digital Circuits 127 – 167
 Introduction 127
 Combinational Digital Circuits 127 – 133
 Multiplexers 133 – 141
 Flip-Flops 141 – 146
 Registers and Shift Registers 146 – 148
 Counters 148 – 149
 Assignment 1 150 -157
 Answer Keys 161
#6. AD /DA Convertor 168 – 185
 Introduction 168
 D/A Resolution 168 – 170
 ADC Resolution 170 – 172
 Answer Keys 180
#7. Semiconductor Memory 186 – 192
 Types of Memories 186
 Memory Devices Parameters or Chatacteristics 187 – 189
 Assignment 1 190
 Answer Keys 191
#8. Introduction to Microprocessors 193 – 225
 Basics 193 – 195
 8085 Microprocessers 196
 Signal Description of 8085 196 – 200

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 Classification Based on Operation 200 – 204
 Classification of Instructions As Per Thier Length 204 – 205
 Addressing Modes 205 – 206
 Memory Mapped I/O Technique 206 – 208
 Interfacing 208 – 209
 Answer Keys 219
Module Test 226 – 246
 Test Questions 226 – 240
 Answer Keys 241
 Explanations 241 -246
Reference Book 247

Chapter 1 Digital Circuits
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CHAPTER 1
Number Systems & Code Conversions
Important Points
The concept of counting is as old as the evolution of man on this earth. The number systems are
used to quantify the magnitude of something. One way of quantifying the magnitude of
something is by proportional values. This is called analog representation. The other way of
representation of any quantity is numerical (Digital). There are many number systems present.
The most frequently used number systems in the applications of Digital Computers are Binary
Number System, Octal Number System, Decimal Number System and Hexadecimal Number
System.
Base or Radix (r) of a Number System
The Base or Radix of a number system is defined as the number of different symbols (Digits or
Characters) used in that number system.
The radix of Binary number system = 2 i .e. it uses two different symbols 0 and 1 to write the
number sequence.
The radix of Octal number system = 8 i.e. it uses eight different symbols 0, 1, 2, 3, 4, 5, 6 and 7 to
write the number sequence.
The radix of Decimal number system = 10 i.e. it uses ten different symbols 0, 1, 2, 3, 4, 5, 6, 7, 8
and 9 to write the number sequence.
The radix of Hexadecimal number system = 16 i.e. it uses sixteen different symbols 0, 1, 2, 3, 4,
5, 6, 7, 8, 9,A, B, C, D, E and F to write the number sequence.
The radix of Ternary number system = 3 i.e. it uses three different symbols 0, 1 and 2 to write
the number sequence.
To distinguish one number system from the other, the radix of the number system is used as
suffix to that number.
Eg: 102 Binary Numbers; 108 Octal Numbers;
1010 Decimal Number; 1016 Hexadecimal Number;
Characteristics of any number system are
1. Base or radix is equal to the number of digits in the system,
2. The largest value of digit is one (1) less than the radix, and
3. Each digit is multiplied by the base raised to the appropriate power depending upon the
digit position.
 The maximum value of digit in any number system is given by (Ω-1), where Ω is radix
Example: maximum value of digit in decimal number system = (10 – 1) = 9.

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Positional Number Systems
In a positional number systems there is a finite set of symbols called digits. Each digits having
some positional weight. Below table shows some positional number system and their possible
symbols
Number system Base Possible symbols
Binary 2 0, 1`
Ternary 3 0, 1, 2
Quaternary 4 0, 1, 2, 3
Quinary 5 0, 1, 2, 3, 4
Octal 8 0, 1, 2, 3, 4, 5, 6, 7
Decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Duodecimal 12 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B
Hexadecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
 Binary, Octal, Decimal and Hexadecimal number systems are called positional number
systems.
 Any positional number system can be expressed as sum of products of place value and the
digit value.
Eg: 75610 =
156.248 = 1
 The place values or weights of different digits in a mixed decimal number are as follows:
decimal point
 The place values or weights of different digits in a mixed binary number are as follows:
binary point
 The place values or weights of different digits in a mixed octal number are as follows:
octal point
 The place values or weights of different digits in a mixed Hexadecimal number are as
follows:
hexadecimal point
System Conversion
Decimal to Binary conversion
(a) Integer number: Divide the given decimal integer number repeatedly by 2 and collect the
remainders. This must continue until the integer quotient becomes zero.

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Eg: 3710
Operation Quotient Remainder
37/2 18 +1
18/2 9 +0
9/2 4 +1
4/2 2 +0
2/2 1 +0
1/2 0 +1
Note: The conversion from decimal integer to any base-r system is similar to the above example
except that division is done by r instead of 2.
(b) Fractional Number: The conversion of a decimal fraction to a binary is as follows:
Eg: 0.6875510 = X2
First, 0.6875 is multiplied by 2 to give an integer and a fraction. The new fraction is multiplied
by 2 to give a new integer and a new fraction. This process is continued until the fraction
becomes 0 or until the numbers of digits have sufficient accuracy.
Eg: Integer value
1
0
1
1
(
Note: To convert a decimal fraction to a number expressed in base r, a similar procedure is used.
Multiplication is done by r instead of 2 and the coefficients found from the integers range in
value from to (Ω-1).
 The conversion of decimal number with both integer and fraction parts are done separately
and then combining the answers together.
Eg: (41.6875)10 = X2
4110 = 1010012 0.687510 = 0.10112
Since, (41.6875)10 = 101001.10112.
Eg: Convert the Decimal number to its octal equivalent: 15310 = X8
Integer Quotient Remainder
153/8 +1
19/8 +3
2/8 +2
1 0 0 1 0 1
Fig 1

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Eg: (0.513)10 = X8
(153)10 ( …… 8
Eg: Convert 25310 to hexadecimal
253/16 = 15 + (13 = D)
15/16 = 0 + (15 =F)
.
Eg: Convert the Binary number 1011012 to decimal.
101101 =
= 32 + 8 + 4 + 1 = 45
(101101)2 = 4510.
Eg: Convert the Octal number 2578 to decimal.
2578 =
= 128 + 40+7 = 17510.
Eg: Convert the Hexadecimal number 1AF.23 to Decimal.
1AF.2316 =
Important Points
1. A binary will all ‘n’ digits of ‘ ’ has the value
2. A binary with unity followed by ‘n’ zero has the value it is an n + 1 digit number
e.g.
(a) Convert binary 11111111 to its decimal value
Solution: All eight bits are unity. Hence value is = 255
(b) Express as binary
Solution: is written as unity followed by zero 10000000000
Same rule apply for other number code
Eg. Express in octal system
Solution ( (
= ( (
Solution (
Binary to Decimal Conversion (Short Cut Method)
Binary to Decimal Binary → octal → Decimal
Eg. Convert 101110 into decimal
Solution (⏟ ⏟ ( (

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Note: For converting Binary to octal make group of 3 bit starting from left most bit
Binary to Decimal Conversion (Equation Method)
Where a and →the last sum term
Eg. ( to decimal
So ( (
Note: we can use calculator (scientific) but there is a limit of digit as input in calculator. We can
use transitional way of multiplying each digit with (where n is the position of digit in
binary number) and adding in the last but for large binary digit its again a tedious task
Eg. ( to decimal
So ( (
Octal to Decimal Conversion (Equation Method)
Above equation can be used for octal to decimal conversion with small modification
Eg. convert (3767)8 to decimal
( (
Note: In general recursive equation to convert an integer in any base to base 10 (Decimal) is
b a
Where b → base of the integer.
Binary Fraction to Decimal
Since conversion of fractions from decimal to other bases requires multiplication. It is not
surprising that going from other bases to decimal required a division process
3
3
7
+
24
6
+
248 2032
7
31 254 2039
8 8 8
+
1 1 1 0 1 0 1 1 1 1 0
2 6 14 28 58 116 234 470 942 1886
1 3 7 14 29 58 117 235 471 943 1886
+ + + + ++ + + + +
1
1
1
+
2
0
+
6 12
1
3 6 13
2 2 2
+
1

Electronics and Communication Engineering : Digital circuits, THE GATE ACADEMY

Electronics and Communication Engineering : Digital circuits, THE GATE ACADEMY

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