physics investigatory project for class 12 logic gates

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S.No . Contents Page-No I Introducti on II BOOLEAN ALGEBRA III Basic Gates IV OR Gate v AND Gate VI

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S.No.

Contents

Page-No

I Introduction

II BOOLEAN ALGEBRA

III Basic Gates

IV OR Gate

v AND Gate

VI NOT Gate

VII NOR Gate

VII NAND Gate

vIII EX-OR Gate

IX EX-NOR Gate

X experiment

In the modern world of electronics, the term Digital

is generally associated with a computer because the term Digital is derived from the way computers perform operation, by counting digits. For many years, the application of digital electronics was only in the computer system. But now-a-days, digital electronics is used in many other applications

SignalSignal can be defined as a physical quantity, which contains some information. It is a function of one or more than one independent variables. Signals are of two types.

Analog Signal Digital Signal

Analog Signal

KEY POINT

*Following are some of the examples in which Digital electronics is heavily used.

Industrial process control

Military system Television Communication

system Medical equipment Radar Navigation

INTRODUCTION

An analog signal is defined as the signal having continuous values. Analog signal can have infinite number of different values. In real world scenario, most of the things observed in nature are analog. Examples of the analog signals are following.

Temperature Pressure Distance Sound Voltage Current & power

Graphical representation of Analog Signal (Temperature)

The circuits that process the analog signals are called as analog circuits or system. Examples of the analog system are following.

Filter Amplifiers Television receiver Motor speed controller

Disadvantage of Analog Systems

Less accuracy Less versatility More noise effect More distortion More effect of weather

Digital SignalA digital signal is defined as the signal which has only a finite number of distinct values. Digital signals are not continuous signals. In the digital electronic calculator, the input is given with the help of switches. This input is converted into electrical signal which have two discrete values or levels.

One of these may be called low level and another is called high level. The signal will always be one of the two levels.

This type of signal is called digital signal. Examples of the digital signal are following.

Binary Signal Octal Signal Hexadecimal Signal

Graphical representation of the Digital Signal (Binary)

The circuits that process the digital signals are called digital systems or digital circuits. Examples of the digital systems are following.

Registers Flip-flop Counters Microprocessors

Number System & Conversion

"A set of values used to represent different quantities is known as Number System". For example, a number system can be used to represent the number of students in a class or number of viewers watching a certain TV program etc. The digital computer represents all kinds of data and information in binary numbers. It includes audio, graphics, video, text and numbers. The total number of digits used in a number system is called its base or radix. The base is written after the number as subscript such as 51210.

Some important number systems are as follows.

Decimal number system Binary number system Octal number system Hexadecimal number system

The decimal number system is used in general. However, the computers

ADVANTAGE AND COMPARISON

Advantage of Digital Systems

More accuracy More versatility Less distortion Easy communicate Possible storage of information

Comparison of Analog and Digital Signal

S.N.

Analog Signal

Digital Signal

1 Analog signal has infinite values.

Digital signal has a finite number of values.

2 Analog signal has a continuous nature.

Digital signal has a discrete nature.

3

Analog signal is generated by transducers and signal generators.

Digital signal is generated by A to D converter.

4Example of analog signal − sine wave, triangular waves.

Example of digital signal − binary signal.

use binary number system. The octal and hexadecimal number systems are used in the computer.

Decimal number SystemThe Decimal Number System consists of ten digits from 0 to 9. These digits can be used to represent any numeric value. The base of decimal number system is 10. It is the most widely used number system. The value represented by individual digit depends on weight and position of the digit.

Each number in this system consists of digits which are located at different positions. The position of first digit towards left side of the decimal point is 0. The position of second digit towards left side of the decimal point is 1. Similarly, the position of first digit towards right side of decimal point is -1. The position of second digit towards right side of decimal point is -2 and so on.

The value of the number is determined by multiplying the digits with the weight of their position and adding the results. This method is known as expansion method. The rightmost digit of number has the lowest weight. This digit is called Least Significant Digit (LSD). The leftmost digit of a number has the highest weight. This digit is called Most Significant Digit (MSD). The digit 7 in the number 724 is most significant digit and 4 is the least significant digit.

 Example:

The weights and positions of each digit of the number 453 are as follows:

Position 2 1 0Weights 10

2101

100

Face value 4 5 3

 The above table indicates that:

The value of digit 4     =          4x102   =          400

The value of digit 4     =          5x10    =          50       

The value of digit 3     =          3x10    =          3

The actual number can be found by adding the values obtained by the digits as follows:

400 + 50 + 3    =45310

Example:

The weights and positions of each digit of the number 139.78 are as follows.

Position 2 1 0   -1 -2Weights 10

2101

100 .10-1 10-2

Face Value 1 3 9   7 8

The above table indicates that:

The value of digit 1     =          1x102   =          100

The value of digit 3     =          3x101   =          30       

The value of digit 9     =          9x100   =          9

The value of digit 7     =          7x10-1  =          0.7      

The value of digit 8     =          8x10-2   =         0.08

The actual number can be found by adding the values obtained by the digits as follows:

100 + 30 + 9 + 0.7 + 0.8          =          139.78

Binary Number SystemDigital computer represents all kinds of data and information in the binary system. Binary Number System consists of two digits 0 and 1. Its base is 2.

Each digit or bit in binary number system can be 0 or 1. A combination of binary numbers may be used to represent different quantities like 1001.

The positional value of each digit in binary number is twice the place value or face value of the digit of its right side. The weight of each position is a power of 2.The place value of the digits according to position and weight is as follows:

Position 3 2 1 0Weights

23

22

21

20

 

Example: Convert 101112 decimal number

Position 2 1 0 -1 -2Weights 10

2101

100 10-1 10-2

Face Value 1 3 9 7 8

 101112             =          1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20

                        =          1 x 16 + 0 + 1 x 4 + 1 x 2 + 1 x 1

                        =          16 + 0 + 4 2 + 1

                        =          2310

Example: Convert 101.1012

Position 2 1 0   -1 -2 -3Face Value 1 0 1 .1 0 1

Weight 24

21

20   2-1 2-2 2-3

 101.1012                    =          1 x 22 + 0x21 + 1 x 20 + 1x 2-1 + 0 x 2-2 + 1 x 2-3

                                 =          1 x 4 + 0 + 1 x 1 + ½ + 0 + 1/8

                                 =          4 + 0 + 1 + 0.5 + 0.125

                                 =          5.62510

Decimal to Binary conversionThe following technique, called the (Division – Remainder Technique) is easiest & simple method used to convert decimal to binary numbers.

Step – 1

Divide the decimal number to be converted by the value of the new base. In this case divide it by 2.

Step – 2

Record the remainder from Step – 1 as the rightmost digit.

Step – 3

Divide the quotient of the previous by the new base.

Step – 4

Record the remainder from Step – 3 as the next digit (to the left) of the new base number.

Step – 5

Bottom to top sequence of remainder will be the required converted number. Repeat Step – 3 & Step – 4, recording remainders from right to left, until the quotient becomes less than the digit of new base so that it cannot be divided.

All these steps are performed in the following picture for converting decimal numbers to binary. Isn't that simple & easy.

Binary system complements

As the binary system has base r = 2. So the two types of complements for the binary system are 2's complement and 1's complement.

1's complementThe 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows.

2's complementThe 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.

2's complement = 1's complement + 1

Example of 2's Complement is as follows.

Boolean algebraBoolean algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in 1854.

Rule in Boolean AlgebraFollowing are the important rules used in Boolean algebra.

Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.

Complement of a variable is represented by an over bar (-). Thus, complement of variable B is represented as . Thus if B = 0 then = 1 and B = 1 then = 0.

O Ring of the variables is represented by a plus (+) sign between them. For example O Ring of A, B, C is represented as A + B + C.

Logical AND of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC.

Boolean LawsThere are six types of Boolean Laws.

Commutative law

Any binary operation which satisfies the following expression is referred to as commutative operation.

Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

TOPICES-

-Rules in Boolean algebra.- Boolean Algebra law.

1. Commutative law

1.1- Associative law

1.2- Distributive law

1.3- AND law

1.4-OR law

2. INVERSION law

3. Theorems & K-map

4. Boolean

Associative law

This law states that the order in which the logic operations are performed is irrelevant as their effect is the same.

Distributive law

Distributive law states the following condition.

AND law

These laws use the AND operation. Therefore they are called as AND laws.

OR law

These laws use the OR operation. Therefore they are called as OR laws.

INVERSION law

This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself.

Important Boolean TheoremsFollowing are few important boolean Theorems.

Boolean function/theorems Description

Boolean Functions Boolean Functions and Expressions, K-Map and NAND Gates realization

De Morgan's Theorems De Morgan's Theorem 1 and Theorem 2

Boolean function:-Boolean algebra deals with binary variables and logic operation. A Boolean Function is described by an algebraic expression called Boolean expression which consists of binary variables, the constants 0 and 1, and the logic operation symbols. Consider the following example.

Here the left side of the equation represents the output Y. So we can state equation no. 1

Truth Table Formation

A truth table represents a table having all combinations of inputs and their corresponding result. It is possible to convert the switching equation into a truth table. For example, consider the following switching equation. The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth table for this equation is shown in Table (a). The number of rows in the

truth table is 2n where n is the number of input variables (n=3 for the given equation). Hence there are 23 = 8 possible input combination of inputs.

Truth Table Formation

A truth table represents a table having all combinations of inputs and their corresponding result. It is possible to convert the switching equation into a truth table. For example, consider the following switching equation.

The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth table for this equation is shown in Table (a). The number of rows in the truth table is 2n where n is the number of input variables (n=3 for the given equation). Hence there are 23 = 8 possible input combination of inputs.

Methods to Simplify a Boolean FunctionThe methods used for simplifying a Boolean function are as follows:

Karnaugh-map or K-map, and NAND gate method.

Karnaugh-map or K-map

The Boolean theorems and the De-Morgan's theorems are useful in manipulating the logic expression. We can realize the logical expression using gates. The number of logic gates required for the realization of a logical expression should be reduced to a minimum possible value by K-map method. This method can be done in two different ways, as discussed below.

Sum of Products (SOP) Form

It is in the form of sum of three terms AB, AC, BC with each individual term is a product of two variables. Say A.B or A.C etc. Therefore such expressions are known as expression in SOP form. The sum and products in SOP form are not the actual additions or multiplications. In fact they are the OR and AND functions. In SOP form, 0 represents a bar and 1 represents an unbar. SOP form is represented by . Given below is an example of SOP.

Product of Sums (POS) Form

It is in the form of product of three terms (A+B), (B+C), or (A+C) with each term is in the form of a sum of two variables. Such expressions are said to be in the product of sums (POS) form. In POS form, 0 represents an unbar and 1 represents a bar. POS form is represented by .

]

Given below is an example of POS.

Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a certain logic. Based on this, logic gates are named as AND gate, OR gate, NOT gate etc.

DE MORGAN'S THEOREMS De Morgan has suggested two theorems which are extremely useful in Boolean Algebra. The two theorems are discussed below.

Theorem 1

The left hand side (LHS) of this theorem represents a NAND gate with inputs A and B, whereas the right hand side (RHS) of the theorem represents an OR gate with inverted inputs. This OR gate is called as Bubbled OR.

Table showing verification of the De Morgan's first theorem:

Theorem 2

The LHS of this theorem represents a NOR gate with inputs A and B, whereas the RHS represents an AND gate with inverted inputs. This AND gate is called as Bubbled AND.

Table showing verification of the De Morgan's second theorem:

Basic GatesAND GateA circuit which performs an AND operation is shown in figure. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

OR Gate

A circuit which performs an OR operation is shown in figure. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

NOT Gate

NOT gate is also known as Inverter. It has one input A and one output Y.

Logic diagram

Truth Table

NAND Gate: -

A NOT-AND operation is known as NAND operation. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

NOR Gate

A NOT-OR operation is known as NOR operation. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

XOR Gate

XOR or Ex-OR gate is a special type of gate. It can be used in the half adder, full adder and subtract or. The exclusive-OR gate is abbreviated as EX-OR gate or sometime as X-OR gate. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

XNOR GateXNOR gate is a special type of gate. It can be used in the half adder, full adder and subtract or. The exclusive-NOR gate is abbreviated as EX-NOR gate or sometime as X-NOR gate. It has n input (n >= 2) and one output.

Logic diagram

Truth Table

AIM To find truth table & Boolean expression from given Diagram of given Boolean expression.

GIVEN -

Boolean expression: Y = [(A.B)+B’].A A B B A.B (A.B)

+B Y= [(A.B+B)].A

0 0 1 0 1 0

0 1 0 0 0 0

1 0 1 0 1 1

1 1 0 1 1 1TRUTH TABLE:

OUTPUT:

Fig-1

I/P TP 1 TP 2 0/P+ H H H+ L H

I/P IP1 IP2 O/P- L H L- H L

I/P IP1 IP2 O/P+ L H H

- H H

I/P IP1 IP2 O/P- L L L+ L L