## Digital Electronics Introduction Analogue Signal And Digital Signal

The voltage or current signals passed through some traditional electronic circuits like a rectifier made of p- n junction diode or an amplifier made of the transistor can be varied continuously j withn a definite range. For example. In the CE mode of a j transistor circuit, the input voltage can be varied continuously from j 0 V’ up to 6 Vor 10 V. This kind of signal is called an analog signal and (the corresponding electronic circuit is called analog circuit

On the other hand, In the case of ultra-modern electronic equipment like calculators, computers, etc., there are two discrete states of the Input or output signal low and high In this caw. the correct value of the voltage or current signal is not important at all. For example, If the magnitude of an input or output voltage lies in the range of 0 V to 0.5 V, it can be considered as j low voltage and if it lies between V and 5 V, it can be j considered as high voltage. In this case, the circuit is so designed that the value of the voltage never lies in the range of 0.5 V to 4 V

**Read and Learn More Class 12 Physics Notes**

If an electrical circuit exhibits input and output signals that can be categorised into two separate states, these states can be denoted by two symbols, usually in the form of two digits. This signal is classified as a digital signal, and the resulting circuit is called a digital circuit. A digital circuit is commonly known by this name. The utilisation of the circuit is less complex in contrast to that of an analogue circuit. The accuracy of the input and output signals is rather negligible. The latency between the input application and output acquisition is negligible, and the efficiency of this circuit is remarkably good. Given these factors, digital circuits are widely employed in the current period. In the realm of digital signals, the two discrete states are symbolised by two binary digits: 0 and 1. The numeral 0 represents a value that is considered to be low, whereas the numeral 1 represents a number that is considered to be high. Positive logic is the term used to describe the utilisation of digits.

In the above example, 0 is used as the symbol of voltage the voltage from 4 V to 0 5 V. On the other hand, in the case of less-used negative logic, l Is used to denote a low value and 0 Is used to denote a high value In different types of analog and digital signals, typical analog and digital signals respectively.

Here for analog signal, formation of wave i.e.. the actual form of time-varying voltage is most Important. On the other hand. In the case of digital signals, waveforms are always rectangular, i.e.. The interval of time In which the voltage is in a lower or higher state is Important. Thus the accuracy of the digital signal does not depend on the range or two discrete values of voltage In higher <uul lower state.

## Digital Electronics Number System

**Binary system Definition:** The system of expressing all the real numbers by the two digits 0 and 1 is called the binary system-

By a number, we usually mean a number in the decimal system. 10 digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used in the decimal system. Although in daily life decimal system is widely used. Computer circuits are fabricated by using a new system called binary number system which consists of two digits 0 and 1.

Let us consider a four-digit integral number 1101 in the binary system.

In this case, the places of the digits are given in the following table:

**The value of any number depends on two factors:**

- The magnitude and position of tire digits by which any number is formed and
- The base or radix ofthe number system. The base or radix of a number system refers to the number of digits used in the system.

For example, the bases of the decimal and binary systems are 10 and 2 respectively.

A number formed in a system is expressed by the symbol (number)base. For example, the number (257) formed in the decimal system is written as (257)_{10}. Similarly, the number (1101)_{2} formed in the binary system is written as (1101)_{2}. The two numbers (11)_{10} and (11)_{2} are not the same; the first number is eleven and the second one is three.

**Integers in Decimal and Binary Systems **

** Decimal system:**

Let us consider a four-digit integer 2795 in the decimal system. The number is expressed in words as two thousand seven hundred ninety-five. In this case, the places of the digits are given in the following table

We can determine the value of this number from these places and from the base of the number system in the following way.

Or,

= 2795

The place of a digit indicates the significance of the digit. The digit lying at the extreme left- side of a number has the greatest significance and the digit at the extreme right has the least significance. In this case, the digit 2 has the greatest significance and hence 2 is the most significant digit. The digit 5 has the least significance and hence 5 is the least significant digit

**Binary system:**

Let us consider a four-digit integral number 1101 in the binary system. In this case, the places of the digits are given in the following table:

We can determine the value of this number in the following way:

Or,

= 13 ( decimal value)

**For a better understanding see the following table:**

In the given number, the digit 1 in the place of 8 is the most significant digit and the digit 1 in the place of 1 is the least significant digit

**Fractions In Decimal And Binary Systems**

**Decimal system:**

A fraction in a decimal system is written by a decimal sign (.) and placed some digits right side of this sign. Such as** .417**. In this case, the places of the digits are given in the following table

The method of determining the value of the number is illustrated below with 0. 417 as an example

Or,

= 0.417(decimal value)

**For a better understanding see the following table:**

In the case of a fraction, the digit next to the decimal point is the most significant digit and the digit at the extreme right is the least significant digit. In this case, the digit 4 lying in the one-tenths place is the most significant digit and the digit 7 lying in the one-thousandth place is the least significant digit.

**Binary system:**

Let us consider a fraction of 0.1011 in the binary system.

**In this case, the positions of the digits are:**

The value of the number can be determined with the help of these places

**The base of the number system is shown below:**

Or,

= 0.6875 (Decimal value)

**For a better understanding see the following table:**

In the given fraction, the digit 1 in the place of \(\frac{1}{2}\) is the most significant digit and the digit 1 in the place \(\frac{1}{16}\) of is the least significant digit

**Binary to Decimal Conversion**

The determination of the decimal value of any binary number was discussed in the previous section. Some examples of the conversion of binary to decimal numbers are given below:

1. (10111)_{2}

= (1 × 2^{4}) + (0×2^{3}) + (1 ×2^{2}) + (1 ×2^{1}) + (1×2^{0})

2. (10.111)_{2}

= (1 × 2^{1}) + (0 ×2^{0}) + (1× 2^{-1}) + (l ×2^{-12}) + (1 × 2^{-3})

= 2 + 0 + 0.5 + 0.25 + 0.125 = (2.875)_{10}

3. (0.001)_{2} = (0 × 2^{0}) + (0 ×2^{-1}) + (0 × 2^{-2}) + (1 × 2^{-3})

= 0 + 0 + 0 + 0.125 = (0.125)_{10}

3. (1.001 )_{2} = (1 × 2^{0}) + (0 × 2^{-1}) + (0× 2^{-2}) + (1× 2^{-3})

= 1 +0 + 0 + 0.125 = (1.125)_{10}

**Decimal to Binary Conversion**

For better understanding, in the case of conversion of decimal to binary, it is necessary to discuss integers and fractions separately.

**Conversion of integers:**

Starting from 2°, the ascending powers of 2 are 2^{0}, 2^{1}, 2^{2}, 2^{3}, ………… Multiplying 0 or 1

With these numbers (i..e., 2^{0},2^{1} ……..) and then adding the \ products, any Integer can be expressed.

**For example:**

44 = (2^{5} × 1) + (2^{4} × 0) + (2^{3} × 1) + (2^{2} × 1) + (2^{1} × 0) + (2^{0} × 0)

45 = (2^{5} × 1) + (2^{4} × 0) + (2^{3} × 1) + (2^{2} × 1) + (2^{1} × 0) + (2^{0} × 1)

46 = (2^{5} × 1) + (2^{4} × 0) + (2^{3} × 1) + (2^{2} × 1) + (2^{1} × 1) + (2^{0} × 0)

47 = (2^{5} × 1) + (2^{4} × 0) + (2^{3} × 1) + (2^{2 }× 1) + (2^{1} × 1) + (2^{0} × 1)

If we place those 0s and Is in order of their multiplication with the power of 2 to express the decimal number, the binary form of that decimal number can be easily expressed

So,

(44)_{10} = (101100)_{2}

(45)_{10} = (101100)_{2}

(46)_{10} = (101100)_{2}

(47)_{10} = (101100)_{2}

The process in which decimal numbers are converted into their respective binary numbers as discussed above, is not a correct mathematical process, because these calculations are done orally.

The proper method for conversion is to go on dividing the number and the successive quotients by 2 continuously, writing the remainder in each division till the quotient is zero. Arranging the remainders from bottom to top, i.e., from left side to right side, we get the given number in a binary system. Two examples are given below

**1. Determination of the binary form of (44) _{10}:**

**2. Determination of the binary form of (45) _{10 }:**

(45)_{10} = (101101)_{1}

**Conversion of fraction:**

Starting from 2^{-1}, the descending powers of 2 are 2^{-1}, 2^{-2}, 2^{-3}, ……… All fractions can not be expressed by multiplying 0 or 1 with these numbers (i.e. 2^{-1} 2^{-2}, ……. ) and then adding the products. For example, 0.125 = (2^{-1} × 0) + (2^{-2} × 0) + (2^{-3} × 1), but 0.12 cannot be expressed in this way.

Initially, it is necessary to multiply any decimal fraction by 2. If the product of the fraction is less than 1, write 0. If it is larger than 1, write 1. Place a binary point to the left of either 0 or 1. If the value of the fraction is less than 1, it must be multiplied by 2. If the value of the fraction is more than 1, just the non-integer part should be multiplied by 2. Similarly, if the value of this product is less than 1, assign it a value of 0; otherwise, assign it a value of 1.

The digit 0 or 1 must be placed immediately to the right of the preceding digit 0 or 1. Consequently, the decimal portion of the product value must be repeatedly multiplied by 2. Based on this criterion, input either 0 or 1 repeatedly until the product value reaches 1. However, in the majority of circumstances, the product value is not equivalent to 1. Therefore, it is not possible to represent the decimal fraction as an equivalent binary fraction. The process of converting a fraction from decimal to binary can be best comprehended through the following examples:

**1. Determination of the binary form of 0.5625:**

**2. Determination of the binary form of 0.3:**

In this case, we see that the product can never be equal to 1 and part 1001 repeats again and again. So, 0.3 cannot be expressed in a binary fraction of the exact value. Since the part 1001 repeats itself in the fraction 0.010011001 …, it can be written as (0.0)(1001)_{2}

∴ (0.3)_{10} = (0.0)(1001)_{2}

It should be mentioned here that in case of any calculation, only those significant digits after the point are considered, which are required for the calculation. For example, if you require eight significant digits after the point for a calcula¬ tion, you should write,

(0.3)_{10} = (0.01001100)_{2}

The integral and the fractional parts of any decimal number are converted separately into their corresponding binary values to express the number.

**For examples:**

(44)_{10} = (101100)_{2 }and (0.5625)_{10} = (0.1001)_{2}

∴ (44.5625)_{10} = (101100.1001)_{2}

**Addition, Subtraction, Multiplication, and Division of Binary Numbers: **

The process of addition, subtraction, 1 multiplication, and division are similar for both decimal and binary systems. The only difference between binary and decimal.

The number is, in the decimal system, the numbers of digits are 10 (0, 1, 2, 9) whereas in a binary system, the number of digits 2(0 and 1).

**Binary Addition:**

Rule of Addition

0 + 0 = 0, 0 + 1 = 1 + 0 = 1,1 + 1 = 10

The last equation, 1 + 1 = 10 shows that, in a column, the sum of two binary’ 1 gives 0 in that column with a carry of1 in left

**Example 1:**

- In the first column (from right), the binary sum of 1 and 1 gives 0 in that column with a carry of in the left column (second column from right).
- In the second column (from right), the carry of1 from the first column is added to the sum of 1 and 0, giving 0 in that column with a carry of1 in the third column (from right).
- In the third column, the carry of 1 from the second column is added to the stun of 1 and 1, giving 11 in that column as there is no column in left

**Example 2:**

- In the first column (from right), the binary sum of 1, 1, 0, 1, and 0 gives 1 in that column with a carry of 1 in the left column (second column from right).
- In the second column, the carry of 1 from the first column is added to the sum of 0, 1, 1, and 1, giving 0 in that column with a carry of 10 in the third column from the right.
- HD In the third column, the carry of 10 from the second column is added to the sum of 1 and 1, giving 100 in that column as there is no column in the left.

**Binary Subtraction:**

Rule of subtraction

0 – 0 = 0, 1 – 0 = 1, 1 – 1 = 0, 10 -1 =

The tat three are the same as in decimal. The fourth rule is the only new one. It applies In the borrowing case when the top digit in

A column Is 0 and the bottom digit Is 1. (Remember: In binary, 10 Is pronounced as ‘one-zero’ or ‘two).

The procedure of binary subtraction is shown with an example in the table below : (Alter alignment of the numbers, subtraction proceeds from right to left)

**Red marks indicate borrowing:**

To subtract a large number from a small number,’ we subtract the small number from the large number and put a minus (-). sign before the result. Thus to subtract 11111 from 111001, we ‘ subtract the second number from the first number

As 11001 < 11111

So, 11001 – 11111 = -110

**Binary multiplication and division:**

Rule of multiplication

0 × 0 = 0, 1 × 0 = 0 × 1= 0, 1×1 = 1

The process of binary multiplication is the same as decimal multiplication.

It should be mentioned here that in calculators and computers, multiplication and division are the same as repeated binary addition and subtraction. Hence in application, binary multiplication and division have no importance

## Digital Electronics Number System Numerical Examples

**Example 1. Write the decimal equivalent of (101101) _{2}**

(101101)_{2} = .1 × 2^{5} + 0 × 2^{4} + 1 × 2^{3} + 1 × 2^{2} + 0 × 2^{1} + 1 × 2^{0}

= 32 + 0 + 0 + 4 + 0+1 = (45)_{10}

**Example 2 ****Addition:**

**(11001.101)**_{2}+ (1001.11)_{2 }+ (11.01)_{2}**(10000001)**_{2 }+ (1111)_{2}

**Solution:**

**Example 3. Subtraction:**

**(11001.101)**_{2}– (1001.11)_{2 }**(10000001)**_{2 }– (1111)_{2}

**Solution:**

## Logic Gates

A gate is a special kind of digital circuit which possesses one or more input voltages but only one output voltage. OR, AND and

NOT gates are three basic gates. By joining these gates in different ways we can construct different kinds of circuits. These circuits perform arithmetic calculations and can conclude logical inferences like that performed by a human brain. This kind of mathematics was discovered by George Boole of England in 1854 AD. This form of mathematics is known as Boolean algebra.

Actual mathematical analysis of gates can be done with the help of Boolean algebra. Different gates can perform different algebraic processes and in this way, they can arrive at logical inferences. Thus, these gates are called logic gates or logic circuits

**Boolean algebra:**

To appreciate the importance of binary logic, let us consider, for example,

- Answer to a mathematical problem: is it ‘right’ or ‘wrong’?
- A physical statement: is it ‘true’ or ‘false’?
- Full a switch in a circuit: is it ‘on’ or ‘off’?
- The voltage across a circuit element: is it ‘low’ or ‘high’?
- A particle in a 2-level system: is it ‘up’ or ‘down’?

There are many situations in the physical world where only two such states are available. The method of explaining the truth by finding the answers of some two-valued questions related to a subject was first invented by the Greek philosopher Aristotle. Then some mathematicians understood that it is possible to express the method of finding the truth by the logic step by step by an algebraic method.

Notably, the British mathematician Augustus De Morgan discovered the interconnectedness between logic and mathematics. Subsequently, Bool carried out the remaining tasks. Nearly a century later, in 1938, the American applied mathematician employed this algebra for the first time in the telephone switching circuit.

Typically, the most practical method for representing these two states is by use a pair of numerical symbols, commonly 0 and 1. The decision of whether to employ 0 or 1 is referred to as binary logic. Boolean algebra is used to mathematically analyse a sequence of binary logical systems. The states of the input or output of a digital gate circuit are represented by the binary values of 0 or 1. Furthermore, a gate circuit can serve as a component of a sequence.

The output of a gate can serve as an input for the subsequent gate. Essentially, a digital gate circuit is a system that operates based on binary logic. Boolean algebra is the most suitable mathematical tool for analysing any circuit, regardless of its complexity, that is composed of digital gates.

An OR gate receives two or more input voltages or signals and produces a single output voltage or signal, similar to other types of gates. The gate is named OR gate because when any of the input voltages, whether it is the first, second, third, or any subsequent one, is high, the output voltage will also be high. For instance, if at least one of the input voltages of a two-input OR gate is in a high state, the output voltage of the gate will likewise be in a high state.

**Working principle of OR gate:** From the electrical analogy shown in Fig. 2.2, it becomes clear how an OR gate works. In this circuit, two switches A and B are connected in parallel. Obviously,

- When both switches remain ‘off the bulb does not glow, i.e., the output becomes zero—no output is obtained.
- One of the two switches, A or B, is ‘on’ but the other is ‘off; the bulb glows, i.e., a non-zero output is obtained.
- If both switches are ‘on; the bulb glows and hence an output j is obtained.

So, this circuit works like an OR gate

**OR gate in the electronic circuit:**

An actual electronic two-input OR gate is the simplified form of this circuit. Two input voltages are denoted by A and, B and the output voltage is denoted by Y. Let the two possible states of the two input voltages below (say, 0 V) and high (say, 5V). The resistance R_{L} is permanently connected to the circuit.

**The gate may exist in any of the following four states:**

**Both A and B are low:**In this case, the output voltage remains low. According to Fig. 2.3, if both A and B are low, the two diodes remain non-conducting. As a result, Y also remains low.**When A is low and B is high:**In this case, the output voltage remains high. If A is low, the diode attached to A remains in the non-conducting state. But if B is high, the diode attached to B is forward-biased, and hence Y remains high.**When A is high and B is low:**In this case, the output voltage remains high. According to if B is low, the diode connected with B remains non-conducting. But if A is high, the diode connected with A is forward-biased, and hence Y remains high.**Both A and B are high:**In this case, the output voltage remains high. According to if both A and B are high, the diodes connected with them are forward biased, and hence Y remains high.

**OR gate Truth table:**

A table can be prepared by taking the possible inputs and outputs of a gate. This table is called the truth table of that gate.

**The truth table of a two-input OR gate is given here:**

As one of the inputs or the output of a gate can remain only in the two states low or high, this state can be easily expressed by binary digits. Expressing the lower state by 0 and the higher state by 1, the above truth table is prepared. By observing the truth table minutely, we understand that, if the state of any one of the two inputs is 1, the output state becomes 1. So, the state of Y becomes 1, if the state of either A or B or both A and B is 1.

In another way, we can regard an OR gate as an **‘any-or-all’** gate, i.e., the output state will be 1 if the states of any one or all inputs are I.

** OR gate Positive and negative logic:**

In the case of digital signal, if lower and higher states are represented by 0 and 1 respectively, it is called positive logic. On the other hand, If lower and higher states are represented by 1 and 0 respectively, it is called negative logic. Naturally, both positive and negative logic cannot be used simultaneously.

**OR gate Symbol:** The symbol of an OR gate is Digital circuits can be drawn using this symbol

**Boolean algebra related to OR gate:**

In Boolean algebra, the ‘OR’ operation is denoted by the symbol ‘+’ The Boolean algebraic equation related to the OR gate, shown in Fig. 2.5 is,

Y = A+B

This equation is read as: ” Y equals A OR B”.

When A = 0 = B, then Y = 0 + 0 = 0 .

When A = 0 and B = 1 , then Y = 0 + 1 = 1 .

When A = 1 and B = 0 , then Y = 1 + 0 = 1 .

When A = 1 = B, then Y = 1 + 1 = 1.

The equation 1 + 1 = 1 may appear to be absurd. But in Boolean algebra, the ‘+’ sign does not indicate addition. So, the equation 1 + 1 = 1 is read as “1 or 1 equals 1” For an OR gate circuit containing three inputs, the truth table, symbol, and Boolean algebraic equation are given below.

**AND Gate**

In an AND gate, there are two or more input voltages or signals, and like any other gate, there is one output voltage or signal.

This gate is called AND gate because, If all the input voltages, i.e., the first the second, and the third und input voltages are high, only then will the output voltage be high. For example, if Both the input voltages of a two-input AND gate are high, then only its output voltage will be high.

**Working principle of AND gate:**

From the electrical analog) it becomes clear how an AND gate works.

- In this circuit, two switches A and B are connected in series. Obviously,
- When both the switches remain ‘off; the bulb does not glow, i.e., output becomes zero — no output is obtained.
- One of the two switches, A or B, is ‘on’ but the other is ‘off,’ even then the bulb does not glow, i.e., output becomes zero no output is obtained.
- Sial only when both the switches arc ‘on,’ the bulb glows, i.e., the output is obtained.

So, this circuit works like an AND gate

**AND gate In the electronic circuit:**

An actual electronic j two input AND gate Is The simplified form of j earthing. This circuit. The two input voltages are denoted by A and B, and the output voltage by Y. Let the two possible states of the two Input voltages below (say, 0 V) and high (say, 5 V). The resistance Rj and the battery V_{CC} (= 5 V); are permanently connected with the circuit.

**The gate may exist in any of the following four states:**

**Both A and II arc low:**In this case, the output voltage remains low. According to if both A and R are low, due lo V_{CC}the two diodes are forward biased, i.e., two diodes are Conducting. As a result, the voltages of A, B, and Y are the same, I.e., Y Is also low.**If A low and B it high:**In this case, the output voltage is low. According to if R Is high, the diode connected with R is reverse biased, l.e„ this diode Is Non-conducting. Rut If A Is low, due to the diode connected with A Is forward biased, I.e., this diode is Conducting. As a result, the voltages of A and Y are the same, i.e., Y is low.**If A is high and B is low:**- In this case, the output Is low The reason for this Is similar to that described In 2 above.
**Both A and B are high:**In this case, the output Is high. According to if both A and B are high, the two diodes are reverse biased, I.e., the diodes arc in the non-conducting state. As a result, no current flows through R_{L,}and hence due to V_{CC}, the output Y Is high.

By comparing, we can see that an Or gate easily becomes an AND gate if we alter the ends of the diodes and achieve a proper voltage (Vcc) at one end of RL instead of earthing

**AND gate Truth table:**

Observing the truth table carefully, we understand that. If the states of both the Inputs be l. the state of the output will be I. So, the state of V’ will be 1 If states of both A and R arc I. lit another way, we can regard an AND gate as an all or none gate i.e the out state will be 1 if the states of all input be 1 otherwise output state will be 0.

**AND gate Symbol:** The symbol of an AND gate Is shown in the Digital circuits that can be drawn using this symbol.

**Boolean algebra rotated to AND gate:**

In Boolean algebra, ‘AND’ operation Is denoted by the symbol the Boolean algebraic equation related to AND gate, shown in

Y= A.B = AB

This equation is read as: ” Y equals A AND B”.

When A = 0 = B, then 7= 0.0 = 0

When A = 0 and B = 1 , then 7= 0 .1 = 0

When A = 1 and 5 = 0, then 7=1.0 = 0

When A = 1 = 5, then 7=11 = 1.

A three-input AND gate circuit, its truth table, symbol and

**Boolean algebraic equations are given below:**

**AND gate Digital Signal:** The waveforms of two digital signals A and B are

**For OR gate – A + B= Y:**In time intervals EF and GH, both signals A and 5 are in a lower state i.e., 0. So, for these time intervals, output 7 of the OR gate will be 0. For all other intervals, A or 5 or both A and 5 are in a higher state, i.e., 1. Hence, output state 7 will be 1. This output waveform

**For AND gate – AB = Y:**In this case, signals A and B are in a higher state i.e., 1 for the time intervals CD, FG, and IJ So, for these time intervals, the output of the AND gate will be in a higher state i.e., 1. For all other intervals, A or 5 or both A and 5 are in a lower state i.e., 0. Thus output 7 will be in a lower state ) i.e., 0. This output waveform.

**NOT Gate**

In a NOT gate, there is one input and one output voltage or signal. This gate is called NOT gate because the states of the output voltage and input voltage can never be the same. So, if the input voltage in a NOT gate is low, the output voltage will be high and vice versa. This gate is also known as an inverter.

**Working principle Of NOT gate:**

From the electrical analogy it becomes clear how a NOT gate works. In this circuit, a switch A and a bulb are connected in parallel.

Clearly,

- When the switch remains off, the bulb glows, i.e., an output is obtained.
- When the switch remains on, the bulb does not glow, i.e., output becomes zero — no output is obtained. So, this circuit works like a NOT gate

**NOT gate in the electronic circuit:**

An actual electronic NOT gate. The simplified form of this gate. Note that, diodes cannot form a NOT gate; at least one transistor is necessary.

Here, the input and the output voltages are denoted by A and 7 respectively. Let the two possible states of the input and output

Voltages are low (say, 0 V) and high (say, 5 V). Two resistances R_{B} and R_{C} and the battery Vcc (= 5V) are connected permanently to the circuit.

**The gate can exist in any of the two states given below:**

**A is low:**In this case, the output voltage is high. According to, Fig. 2.14, if A is low, the values of Rg and Rc are so chosen that the transistor is in the cut-off region, i.e., almost no current passes through the transistor. As a result, Y remains high due to Vcc.**A high:**In this case, the output voltage is low. According to Fig. 2.14, if A is high, due to Rg and Rc, the transistor is in the saturation region, i.e., maximum current flows through Rc. As a result, the point P is in a low state, i.e., Y is low

**NOT gate Truth table:**

**NOT gate Symbol:**

The symbol of a NOT gate. Digital circuits can be drawn by using this symbol.

**Boolean algebra related to NOT gate:**

In Boolean algebra, the **‘NOT’** operation Is denoted by giving a ’ sign above A. The Boolean algebraic equation related to NOT gate, shown

Y = \(\bar{A}\)

This equation is read as: “ Y equals NOT A ”

When A = 0 , then Y = \(\bar{0}\) = 1.

When A = 1 , then Y = \(\bar{1}\) = 0.

**NOT output of a digital Input:**

According to each lower state I.e., 0 of the input signal A (in time intervals DF and GH), the output Y remains in the higher state the other hand, for higher state of A i.e., 1 (in time Intervals CD, FG, and HJ), the output Y remains in lower state i.e., 0

- OR, AND, and NOT gates are called basic logic gates because, any other logic gate is the combination of these three basic gates, In any manner.
- A logic gate cannot be formed by using any one of the basic gates repeatedly. For example, combining a large number of OR gates, no AND gate or NOT gate can be constructed.
- The NOR gate and the NAND gate, discussed in the next section have some special advantages. A logic gate of any kind can be constmcted by a combination of any number of NOR gates or NAND gates only. Thus, NOR and NAND gates are called universal logic gates, although none of them are basic gates.
- To construct different logic gates by using more than one basic gate, DC Morgan’s theorem can hr applied.
**Dc Morgan’s theorem:**- \(\overline{A+B}=\bar{A} \cdot \bar{B}\)
- \(\overline{A \cdot B}=\bar{A}+\bar{B}\)

**NOR Gate**

Joining the Input of a NOT gate with the output of an OR gate, a NOT-OR, i.e., a NOR gate is constructed So, the voltage or signal that comes at the output of an OR gate, goes to die input of a NOT gate. NOT gate Inverts this voltage and sends it to the output.

**NOR Gate Truth table:**

In the case of a NOR gate, only if the states of all of the Inputs are 0, the output state becomes 1.

**NOR gate Symbol:**

Two symbols of a NOR gate are. Digital circuits can be drawn by using any one. The second symbol is more commonly used

**Boolean algebra related to NOR gate:**

The Boolean algebraic equation related to the NOR gate

This equation is read as Y equals A NOR B”

When.A = 0 r: B, then V = \(\overline{0+0}\) = 1.

When A = 0 and B = 1 , then , Y = \(\) = 0.

When A = 1 and B = 0 . then Y = \(\overline{1+0}\) = 0.

When A – 1 and B = 1 . then Y = \(\overline{1+1}\) = 0.

**Design of basic gates using one or more NOR gates:**

**1. From NOR gate to OR gate:**

**2. From NOR gate to AND gate:**

**3. From NOR gate to NOT gate:**

**NAND Gate**

Joining the input of a NOT gate with the output of an AND gate, a NOT-AND, i.e., a NAND gate is constructed. So, the voltage or signal that comes at the output of an AND gate, goes to the input of a NOT gate. NOT gate inverts this voltage and sends it to the output.

**NAND Gate truth table:**

In case of a NAND gate, if the state of any of the inputs be 0, the output state becomes 1.

**NAND Gate Symbol:**

Two symbols of a NANI) gate arc shown In Rig. 2.22. Digital circuits can ho drawn using any one. The second symbol Is more commonly used

**Boolean algebra related to NAND gate:**

The Boolean algebraic equation related to the NAND gate,

This equation is read as: ” Y equals A NAND It”.

When A = 0 = B, then Y = \(\overline{0.0}\) = 1 .

When A = 0 and B = 1 , then Y = \(\overline{0.1}\) = 1

.

When A = 1 and B = 0 , then Y = \(\overline{1.0}\) = 1.

When A = 1 and B = 1 , then Y = \(\overline{1.1}\) = 0

**Design of basic gates using one or more NAND gates:**

**1. From NAND gate to OR gate:**

**2. From NAND gate to AND gate:**

**3. From NAND gate to NOT gate:**

**NOR and NAND outputs of two digital inputs:**

The waveforms of two digital signals A and B are as an example.

**For NOR gate \(\overline{A+B}\) = Y:**In time intervals EF and GH, both the signals A and B are in a lower state i.e., 0. So, for these two-time intervals, the output Y of the NOR gate will be in a higher state i.e., 1. For all other intervals, A or B or both A and B are in a higher state i.e., 1. Hence, the output state Y will be 0. This output waveform.**For NAND gate \(\overline{A \cdot B}\) = Y:**Here, for the time intervals CD, FG, and IJ, both the signals A and B are in a higher state i.e., 1. So, for these intervals, output Y of the NAND gate will be in a lower state i.e.„ 0. On the other hand, for all other intervals, A or B or both A and B are in a lower state i.e., 0. Hence output state Y will be 1. This output waveform

**Some Relations of Boolean Algebra**

Verification of the following relations of logic signal A, by putting A = 0 and A = 1 in each of these relations

**Some other useful theorems:**

A + AB = A

A + \(\bar{A}\) = A + B

A(A + B) = A

A + (\(\bar{A}\)) = A + B

A(A + B) – AB

**Three Laws of Boolean Algebra**

**Commutative law:** A + B = B+A; AB = BA

**Associative law:** A + (B+ C = (A+B) + C

A . (B. C) = (A . B). C

**Distributive law:** A.(B+C) = A. B + A.C

**Determination of Boolean Algebraic Relation and Design of Logic Circuit from Truth Table**

To understand the process of determination of Boolean algebraic relation, first, we shall discuss a truth table as an example is given below:

Here, we consider the rows in which Y = 1. So, the second and third rows of the table are taken into consideration. The first and fourth rows are ignored.

In second row, for A = 1, write A, and for B = 0, write B. The Combined relation of these two will be AB. Again in the third row, for A = 0, write \(\bar{A}\), and for B = 1, write B.

The combined relation of these two will be \(\bar{A}\) B. As Y = 1 for A\(\bar{B}\) OR \(\bar{A}\)B, so, the Boolean algebraic relation will be Y = A\(\bar{B}\) + \(\bar{A}\)B.

The logic circuit associated with this Boolean relation is. Here, by using the NOT gate, input A is converted to \(\bar{A}\). In the same way, input B is converted to \(\bar{B}\).

Now, A and B are applied as inputs to an AND gate. Similarly, \(\bar{A}\) and B are applied as inputs to another AND gate The outputs of these two AND gates are A\(\bar{B}\) and \(\bar{A}\) B respectively. These outputs are applied to an OH gate as inputs. Then, the final output will be V = \(A \vec{B}+\overline{A B}\)

In the Practice field, this gate is known as an Exclusive- OR or XOR. For example, two-way switching in which two switches are used to control a bulb or a fan in a house.

In this system, a bulb or a fan is connected in such a way that if any one of these two switches is in an ‘on’ state, the current will flow through the bulb or the fan. But if both of these two switches are in the ‘on’ or ‘off’ state, then no current will flow i.e., the die bulb or the fan will be in the ‘off state. This type of circuit system is an XOR gate.

**Example 1: ****Truth table of AND gate:**

The given truth table shows that Y = 1 in only die fourth row of the table. Hence the rest of the rows need not be taken into consideration.

In the fourth row, for A = 1 , write A, and for B = 1, write B. As A = 1 AND 5=1, so, final Boolean algebraic relation is Y= AB

From above, it is clear that this is well known Boolean algebraic relation of AND gate. Hence the corresponding logic circuit is an AND gate.

**Example 2: ****The truth table of OR gate:**

The given truth table shows that Y = 0 in only the first row of the table. Hence excluding the first row, we see that for the second row, the relation is AB; for the third row, the relation is A\(\bar{B}\) and the relation for the fourth row is AB.

So, for Y = 1, the relation will be A \(\bar{B}\) OR \(\bar{A}\) B OR AB. Hence the Boolean algebraic relation for the given truth table is Y = \(A \bar{B}+\bar{A} B+A B\) + AB. The corresponding logic circuit is shown below.

**By simplifying the relation:**

Y = \(A \bar{B}+\bar{A} B+A B\)

= \(A \bar{B}+(\bar{A}+A) B=A \bar{B}+B\)

= \(A \bar{B}+(A+1) B \quad\{ A+1=1 \mid\)

= \(A \bar{B}+A B+B=A(\bar{B}+B)+B\) =

= A+B \(A+B \quad \bar{B}+B=1]\)

This is the well-known Boolean relation of the OR gate. Hence the corresponding logic circuit is an OR gate. So, the circuit is equivalent to a two-input OR gate.

## Digital Electronics & Logic Gates Integrated Circuit Or IC

Until recently, it was customary to construct electronic circuits with passive components such as resistors, capacitors, and inductors and active components such as diodes and transistors connected by conducting wires. In practice, however, such circuits have two major disadvantages.

**1. Connection problem:**

The connection between the various components has to be done necessarily by soldering the wire to the element. Naturally, there are chances of solving leading to the entire circuit being rendered useless. This defect is also very difficult to remove because itis practically impossible to locate the defective soldering joints.

**2. Size of the circuit:**

Most often complex electronic circuits comprising of a large number of elements are necessary. This causes an unusual increase in the size and production cost of the electronic instrument.

In later years, an integrated circuit(IC), was invented by physicist** JG Kilby** played a remarkable role in removing these disadvantages in manufacturing electronic circuits, As IC is just a small bit of silicon** crystal** or **chip.**

The relative advantages and disadvantages of an IC , over discrete electronic circuits are brief or chip.

Tim relative advantage and disadvantages of an IC over (discrete l electronic circuits arc briefly stated below

**IC Advantages:**

- No soldering Is necessary, The entire connection Is built up inside the IC. Hence It Is very reliable,
- In possible to build up a large number of electronic components within a single chip, 10 used In
**INTEL PENTIUM**microprocessor consists of more than ten lakh electronic components, - Due to Its small size, the cost Is very low, Hence replacement Is better than repairing a defective element
- Due to small size and Jow costing, sufficiently complex circuits are possible to construct using IC which also increases the efficiency of the circuits.

**IC Disadvantages:**

- It is impossible to fabricate transformers or any other kind of inductor onto the integrated circuits. (~in Power rating of IC is sufficiently low; besides this, it cannot withstand high fluctuation of voltage or temperature.
- If hulk production is not done instead of small-scale production, it is not commercially viable. Moreover, advanced technology is essential for the production of perfect ICs.
- Still, it can be safely remarked that the advantages of IC far outweigh the disadvantages which are being brought under effective control with the help of superior technology.

**IC Classification:**

Based on their working principle and uses, ICs are classified into two types.

**Linear or analog IC:**Here the relation between input and output is linear. Also, the input and output voltages and curpioTJoyi rents change continually within a certain range. Linear IC is used in amplifiers, oscillators, and especially in operational amplifiers (OPAMP).**Digital IC:**In this type, input and output voltages can have only two states either high or low. No continuous change occurs in this voltage or current. Digital IC is used in simple digital circuits, microprocessors, etc.

## Digital Electronics & Logic Gates Synopsis

1. If the input and the output signals of an electronic circuit have two discrete states, they can be denoted by two digits. These kinds of signals are called digital signals and the circuits are called digital circuits.

2. The system of expressing all the real numbers by the two digits 0 and 1 only, is called a binary system.

3. With the help of Boolean algebra, mathematical analysis of gate circuits can be done and the different gates can perform different algebraic processes. In this way, they can arrive at logical Inferences. These gates are known as logic gales or logic circuits.

The fundamental logic gates are OR, AND, and NOT gates.

4. The Boolean algebraic equation of a 2-Input OR gate is,

Y = A+B

5. The Boolean algebraic equation of a 2-input AND gate is,

Y = A.B

6. The Boolean algebraic equation of a NOT gate is,

Y = \(\bar{A}\)

7. The Boolean algebraic equation of a 2-input NOR gate is

Y = \(\overline{A+B}\)

8. The Boolean algebraic equation of a 2-input NAND gate is

Y = \(\overline{A \cdot B}\)

9. NOR and NAND gates are called universal logic gates.

10. De Morgan’s theorem related to Boolean algebraic relations:

- Y = \(\overline{A+B}=\bar{A} \cdot \bar{B}\)
- Y = \(\overline{A B}=\bar{A}+\bar{B}\)

11. Each IC is a very small single block or chip of silicon crystal in which several electronic circuit components (e.g. transistors, diodes, resistors, etc.)

12. Are electrically interconnected and their interconnections form a complete electronic circuit Hence, an IC performs a complete electronic function

## Digital Electronics & Logic Gates Very Short Question And Answers

**Question 1. Of the two binary numbers, 10010111 and 10011001, which one is the greater?**

**Answer:** 100111001

**Question 2. Of the two numbers (10) _{10} and (11)_{2}, which one is the greater**

**Answer:**(10)

_{10}

**Question 3. What is radix in decimal and binary systems respectively?**

**Answer:** 10, 2

**Question 4. (33)10 = (____________) _{2} . =**

**Answer:**100001

**Question 5. Is it possible to convert an analog circuit into a digital circuit**

**Answer:** Yes

**Question 6. Is it possible to convert a digital circuit into an analog circuit?**

**Answer:** Yes

**Question 7. If A and B are the two inputs of an AND gate, what will be the value of the output?**

**Answer:** A.B

**Question 8. What role will an AND gate play if negative logic is used instead of positive logic **

**Answer:** OR gate

**Question 9. When in Input signal 1 Is applied to a NOT gate. What will be its output?**

**Answer:** 0

**Question 10. If the output of a NOR gate Is fed to the Input of a NOT gate then this combination will be called what**

**Answer:** 1

## Digital Electronics & Logic Gates Assertive Type

- Statement 1 is true, statement 2 Is true; statement 2 Is a correct explanation for statement 1.
- Statement 1 is true, and statement 2 Is true; statement 2 is not a correct explanation for statement 1.
- Statement 1 Is true, and statement 2 Is false.
- Statement 1 Is false, and statement 2 Is true.

**Question 1. **

**S****tatement 1:** The conversion of binary number 1101 to decimal number can be written as 2^{4}+ 2^{3} + 0+ 2^{1} = 26

**Statement 2:** In a binary system, the base is 2, and the digits used are 0 and 1

**Answer:** 4. Statement 1 Is false, statement 2 Is true.

**Question 2.**

** Statement 1:** The conversion of binary fraction 0 101 to decimal fraction can IK* written as 2^{-1} + 0 + 2^{-3} = 0.625

**Statement 2:** In a binary system, the base is 2, and the digits used are 0 and 1.

**Answer:** 1. Statement 1 is true, statement 2 Is true; statement 2 Is a correct explanation for statement 1.

**Question 3. **

**Statement 1:** OH gate is a basic logic gate

**Statement 2:** Any logic gate can be made by using more than one OK gate in an appropriate combination

**Answer:** 3. Statement 1 Is true, and statement 2 Is false.

**Question 4. **

**S****tatement 1:** NOR gate is a basic logit: gate.

**Statement 2:** Any logic gale tan he made by using more than one NOR gate In appropriate combination.

**Answer:** 4. Statement 1 Is false, statement 2 Is true.

**Question 5. **

**Statement 1:** Boolean algebraic equation oI NOT gate Is

Y = \(\bar{A}\)

**Statement 2:** NOT gate is used to Invert the state of a digital signal between Its two states.

**Answer:** 1. Statement 1 is true, statement 2 Is true; statement 2 Is a correct explanation for statement 1.

**Question 6. **

**Statement 1:** According to Boolean algebra,2

**Statement 2:** The output of an AND gate becomes ‘on’ only if all the Inputs remain In the ‘on’ state.

**Answer:** 2. Statement 1 is true, and statement 2 Is true; statement 2 is not a correct explanation for statement 1.

**Question 7. **

**Statement 1:** The decimal value of the binary number 111 is 7 therefore (0.11) = (7/2³)_{10}

**Statement 2:** Decimal fraction 0.1 1 1 can be written as (111/10³)

**Answer:** 1. Statement 1 is true, statement 2 Is true; statement 2 Is a correct explanation for statement 1.

## Digital Electronics & Logic Gates Match The Columns

**Question 1. Some decimal numbers and their corresponding binary values are given column 1 and column 2 respectively**

**Answer:**

**Answer:** 1-D, 2-B, 3-A, 4-C,

**Question 2. Match the logic gates in column 1 with truth tables in column 2**

**Answer:** 1-B, 2-C, 3-D, 4-A,

**Question 3. Some logic gates and their corresponding Boolean algebraic equations are given in column 1 and column 2 respectively**

**Answer:** 1-C, 2-A, 3-D, 4-B,