All questions of Combinational Logics for Electrical Engineering (EE) Exam

D

Understanding Half Adder
A half adder is a digital circuit that computes the sum of two binary digits. It has two main outputs:
- Sum (S): Represents the addition of A and B.
- Carry (C): Represents the carry from the addition.
The truth table for a half adder is as follows:
| A | B | Sum (S) | Carry (C) |
|---|---|---------|-----------|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Designing Half Adder Using NAND Gates
To realize a half adder using 2-input NAND gates, we need to derive the Sum and Carry expressions based on the truth table.
- Sum (S) can be expressed as: A XOR B
- Carry (C) can be expressed as: A AND B
Using NAND gates, we can derive these functions as follows:
Sum (S) Logic
1. A NAND B gives us the complement of A AND B.
2. To achieve A XOR B using NAND gates, we can manipulate the outputs of NAND gates:
- S = (A NAND (A NAND B)) NAND (B NAND (A NAND B))
Carry (C) Logic
1. To find Carry (C = A AND B):
- C = (A NAND B) NAND (A NAND B)
Counting the NAND Gates
- For Sum (S): 4 NAND gates are needed.
- For Carry (C): 1 NAND gate is needed.
Thus, the total number of 2-input NAND gates required to realize a half adder is:
- Total = 4 (for S) + 1 (for C) = 5 NAND gates.
Conclusion
Therefore, the correct answer is option 'A': 5 NAND gates are required to realize a half adder circuit.

Decoder in Digital Electronics:

Decoders are essential components in digital electronics that convert n inputs into 2^n outputs. In other words, a decoder takes a binary-coded input and activates one of its outputs based on the input combination.

Explanation:

- Number of Outputs: The number of outputs in a decoder is determined by the number of inputs it has. For n inputs, a decoder will have 2^n outputs. This is because each input line can have two possible states (0 or 1), leading to 2^n possible combinations.

- Binary to Decimal Conversion: Decoders are often used to convert binary information to decimal. For example, a 2-input decoder can be used to decode binary numbers 00, 01, 10, and 11 to corresponding decimal numbers 0, 1, 2, and 3.

- Addressing Multiple Devices: Decoders are also used in address decoding in memory and input/output devices. By using a decoder, a microprocessor can select a specific device or memory location based on the address provided.

- Applications: Decoders are widely used in various digital systems such as multiplexers, demultiplexers, memory systems, arithmetic logic units, etc. They play a crucial role in data processing and control operations.

Conclusion:

In summary, a decoder converts n inputs to 2^n outputs, making it a versatile and fundamental component in digital electronics. Understanding the working principles of decoders is essential for designing and implementing complex digital systems.

Full Adder and its Inputs

A full adder is a combinational logic circuit that performs the addition of three bits. The three input bits of a full adder are A, B, and C (where C is the carry bit from the previous addition). The circuit adds the three bits and produces two outputs: Sum and Carry.

Sum Output of a Full Adder

The Sum output of a full adder is the result of adding the three input bits. The sum is calculated using the XOR (exclusive OR) gate. The XOR gate produces a 1 output when the inputs are different, and a 0 output when the inputs are the same. Therefore, the sum output of a full adder is given by:

Sum = A XOR B XOR C

Explanation:

- When A and B are different, the XOR gate produces a 1 output.
- If C is 1, then the XOR gate produces a 0 output because the carry bit is already included in the previous addition.
- If C is 0, then the XOR gate produces a 1 output because there is no carry bit to include in the addition.
- When A and B are the same, the XOR gate produces a 0 output.
- If C is 1, then the XOR gate produces a 1 output because the carry bit needs to be included in the addition.
- If C is 0, then the XOR gate produces a 0 output because there is no carry bit to include in the addition.

Therefore, the sum output of a full adder is given by A XOR B XOR C.

Conclusion

In conclusion, the sum output of a full adder is calculated using the XOR gate. The XOR gate produces a 1 output when the inputs are different, and a 0 output when the inputs are the same. Therefore, the sum output of a full adder is given by A XOR B XOR C.

**Octal-to-Binary Encoder**

An octal-to-binary encoder is a digital circuit that converts an octal input signal into a binary output signal. It takes in three inputs (A2, A1, A0) representing the octal number and produces four outputs (Y3, Y2, Y1, Y0) representing the binary number.

**Logic Diagram**

The logic diagram of an octal-to-binary encoder can be implemented using OR gates, as shown below:

```
_______
A2 --------| |
A1 --------| OR |------ Y3
A0 --------|_______|

_______
A2 --------| |
A1 --------| OR |------ Y2
A0 --------|_______|

_______
A2 --------| |
A1 --------| OR |------ Y1
A0 --------|_______|

_______
A2 --------| |
A1 --------| OR |------ Y0
A0 --------|_______|
```

**Explanation**

- The octal input signal has three bits (A2, A1, A0), representing the numbers 0 to 7.
- The binary output signal has four bits (Y3, Y2, Y1, Y0), representing the numbers 0 to 15.
- Each output bit is produced by a separate OR gate.
- The inputs A2, A1, and A0 are connected to each OR gate as shown in the logic diagram.
- The output of each OR gate will be high (1) if any of its inputs are high (1).
- Therefore, the OR gates perform a logical OR operation on the input bits and produce the corresponding output bits.

**Number of OR Gates**

To implement an octal-to-binary encoder, we need one OR gate for each output bit. Since there are four output bits (Y3, Y2, Y1, Y0), we need a total of four OR gates.

Hence, the correct answer is option A) 3.

Half Adder:
A half adder is a logical circuit that adds two single-bit binary numbers A and B. It has two inputs, A and B, and two outputs, the sum (S) and the carry (C).

Inputs:
In a half adder, the inputs A and B represent the two bits that need to be added.

Carry:
The carry output of a half adder is a logic gate that determines if there is a carry generated when adding A and B together. It indicates whether there is a need to carry over to the next bit when adding multiple bits.

Logic Gates:
Logic gates are the fundamental building blocks of digital circuits. They perform logical operations on one or more binary inputs to produce a single binary output.

AND Gate:
An AND gate is a logic gate that outputs a 1 if and only if all of its inputs are 1. Otherwise, it outputs a 0. The truth table for an AND gate is as follows:

```
A | B | Output
--------------
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1
```

Explanation:
In a half adder, the carry output is determined by the AND operation between the inputs A and B. The carry output is 1 only when both inputs A and B are 1, indicating that a carry is generated.

Let's consider the inputs A and B:

```
A | B
-----
0 | 0
0 | 1
1 | 0
1 | 1
```

When A and B are both 0, the carry output is 0. When A is 0 and B is 1, or A is 1 and B is 0, the carry output is also 0. However, when A and B are both 1, the carry output is 1.

Therefore, the carry output of a half adder is given by A AND B, which corresponds to option 'A'.

The number of 1's in the K-map depends on the values of inputs A and B. Without specific values for A and B, it is not possible to determine the number of 1's in the K-map.

Half Adder and its Inputs
A half adder is a digital circuit that is used to add two binary digits together. It has two inputs, A and B, and two outputs, sum and carry. The sum output gives the result of adding the two inputs, while the carry output indicates if there is a carry from the least significant bit to the most significant bit.

Logic Gates in Half Adder
The half adder uses two logic gates, an XOR gate and an AND gate, to perform the addition. The XOR gate gives a 1 output when one and only one of its inputs is 1. The AND gate gives a 1 output only when both of its inputs are 1.

Sum of Half Adder
To find the sum of the two binary digits using a half adder, we use the XOR gate. The sum output is the result of the XOR gate.

Sum Output
The sum output is given by the Boolean expression A XOR B. This means that if A and B are different, the sum output will be 1. If A and B are the same, the sum output will be 0.

Conclusion
Therefore, the sum output of a half adder is given by A XOR B. This is because we want the sum output to be 1 only when one of the inputs is 1, but not both. The XOR gate is the logic gate that satisfies this condition.

Explanation:

In digital electronics, a counter is a sequential circuit that counts the number of occurrences of an event. A counter can be implemented using flip-flops, which are basic building blocks of sequential circuits.

Modulus Counter:

A modulus counter is a type of counter that counts up to a specific modulus value before resetting back to zero. In this case, we are looking for a counter that can count up to six before resetting.

Implementing a Mod-6 Counter:

To implement a mod-6 counter using flip-flops, we need to use three flip-flops. Each flip-flop will represent a binary digit, and the combination of these flip-flops will represent the count.

Binary Representation:

The modulus value can be represented in binary as 110, which means we need three bits to represent the count from 0 to 5.

Using D Flip-Flops:

D flip-flops are commonly used to implement counters. Each D flip-flop has two inputs - D (data) and CLK (clock). The output of one flip-flop is connected to the CLK input of the next flip-flop, creating a ripple effect.

Implementing a Mod-6 Counter using D Flip-Flops:

Here is how we can implement a mod-6 counter using three D flip-flops:

- Connect the Q output of the first flip-flop to the D input of the second flip-flop.
- Connect the Q output of the second flip-flop to the D input of the third flip-flop.
- Connect the Q output of the third flip-flop to the D input of the first flip-flop.

Working:

The counter starts at 000. On the rising edge of the clock signal, the first flip-flop increments its count by 1. When the count reaches 2 (010 in binary), the second flip-flop increments its count by 1. When the count reaches 4 (100 in binary), the third flip-flop increments its count by 1.

Reset:

When the count reaches 6 (110 in binary), the counter resets back to 000 and starts counting again. This creates a mod-6 counter.

Conclusion:

Therefore, a mod-6 counter can be implemented using three flip-flops.

Understanding Full Adder Circuits
A full adder is a crucial component in digital electronics, used for binary addition. Let's break down its characteristics:
Inputs of a Full Adder
- A full adder requires three inputs:
- Two inputs represent the bits to be added (let's call them A and B).
- One input is the carry-in (C_in) from a previous less significant bit addition.
Outputs of a Full Adder
- The full adder produces two outputs:
- The sum output (S) results from the addition of the input bits and the carry-in.
- The carry-out (C_out) indicates if there is a carry that needs to be added to the next more significant bit.
Functionality of a Full Adder
- The full adder operates based on the following logic:
- The sum (S) can be calculated as: S = A XOR B XOR C_in.
- The carry-out (C_out) can be calculated as: C_out = (A AND B) OR (C_in AND (A XOR B)).
Conclusion
In summary, a full adder indeed has three inputs and two outputs. It plays a vital role in arithmetic operations within digital circuits, especially in constructing arithmetic logic units (ALUs) and adding circuits. Option 'A' is correct as it accurately reflects the input-output configuration of a full adder.

Introduction:
A circuit for comparing two n-bit numbers is used to determine the relationship between the two numbers. It compares each corresponding bit of the two numbers and generates an output based on the result of the comparison. The truth table of this circuit represents all possible input combinations and their corresponding outputs.

Explanation:
To determine the number of entries in the truth table for a circuit comparing two n-bit numbers, we need to consider the total number of possible input combinations.

Number of Bits:
Let's assume that we have two n-bit numbers, A and B. Each number has n bits, where the most significant bit (MSB) is bit n-1 and the least significant bit (LSB) is bit 0.

Number of Input Combinations:
For each bit, there are two possible values, 0 or 1. Therefore, the total number of possible input combinations is given by 2^n for both numbers. As there are two numbers being compared, the total number of input combinations for the truth table can be calculated as (2^n) x (2^n).

Number of Entries in Truth Table:
The truth table consists of all possible input combinations and their corresponding outputs. Each input combination represents a row in the truth table, and the number of rows is equal to the number of entries in the truth table.

Since there are (2^n) x (2^n) possible input combinations, the truth table for the circuit comparing two n-bit numbers will have (2^n) x (2^n) entries.

Answer:
Therefore, the correct answer is option 'C', which states that the circuit for comparing two n-bit numbers has (2^n) x (2^n) entries in the truth table.

Summary:
- A circuit for comparing two n-bit numbers is used to determine the relationship between the two numbers.
- The truth table of this circuit represents all possible input combinations and their corresponding outputs.
- The number of entries in the truth table is equal to the total number of possible input combinations.
- For two n-bit numbers, the total number of input combinations is given by (2^n) x (2^n).
- Hence, the circuit for comparing two n-bit numbers has (2^n) x (2^n) entries in the truth table.

Obtaining Carry in Addition
When performing addition, the carry is obtained when the sum of two digits in a particular position exceeds the base of the number system being used. For example, when adding two binary numbers, if the sum of two bits in the same position is 2 (10 in binary), a carry of 1 is generated and added to the next higher position.

Example:
1101
+ 1011
-------
10000
In this example, when adding the rightmost bits (1 + 1), the sum is 10, which results in a carry of 1 to the next higher position.

Carry in Subtraction
In subtraction, the concept of borrowing is used instead of carrying. When subtracting a larger digit from a smaller one, borrowing is required to perform the operation.

Example:
1011
- 0110
--------
001
In this example, when subtracting the second digit from the first digit, borrowing is needed to perform the subtraction operation.

Conclusion
In the operation of addition, the carry is obtained when the sum of two digits exceeds the base of the number system. In contrast, borrowing is used in subtraction when subtracting a larger digit from a smaller one. Therefore, carry is obtained in the addition operation.

Explanation:
To understand why the final output of three T flip-flops in a cascade is 12.5 MHz, let's first understand what a T flip-flop does and how it works.

T Flip-Flop:
A T flip-flop is a type of flip-flop that toggles its output based on the input signal. The input to a T flip-flop is called the "toggle" input, and it can either be a logic high (1) or a logic low (0). When the toggle input is high (1), the output of the flip-flop toggles between its current state and its complemented state. When the toggle input is low (0), the output remains unchanged.

Cascade Connection:
When multiple flip-flops are connected in cascade, the output of one flip-flop becomes the input to the next flip-flop. In this case, we have three T flip-flops connected in cascade.

Frequency Division:
Each T flip-flop in the cascade divides the input frequency by two. This means that the output frequency of the first flip-flop is half of the input frequency, the output frequency of the second flip-flop is half of the output frequency of the first flip-flop, and so on.

Calculating the Output Frequency:
Given that the input frequency is 100 MHz, let's calculate the output frequency of the cascade.

- The output frequency of the first flip-flop is half of the input frequency, which is 100 MHz / 2 = 50 MHz.
- The output frequency of the second flip-flop is half of the output frequency of the first flip-flop, which is 50 MHz / 2 = 25 MHz.
- The output frequency of the third flip-flop is half of the output frequency of the second flip-flop, which is 25 MHz / 2 = 12.5 MHz.

Therefore, the final output of the three T flip-flops in a cascade is 12.5 MHz.

Conclusion:
The final output of three T flip-flops in a cascade, when the input is a 100 MHz signal, is 12.5 MHz. This is because each flip-flop divides the frequency by two, resulting in a progressive reduction of the frequency as it passes through each flip-flop.

Encoder:

Explanation:
- An encoder is a device that converts a decimal input number into binary.
- It is commonly used in digital systems to convert analog signals into digital signals.
- The encoder takes a decimal input number and converts it into a binary output based on the specified encoding scheme.
- This process involves assigning a unique binary code to each decimal input number.
- The binary output from the encoder can then be processed by other digital devices such as ALU or memory.
- In the context of digital systems, an encoder is essential for converting data between different formats and ensuring compatibility between devices.
- Therefore, in the given options, an encoder is the device that specifically performs the conversion of a decimal input number to binary.

Sequential Circuit

A sequential circuit is a digital circuit that has memory. It means the output of the circuit depends not only on the present input but also on the past input. The sequential circuits can be classified into two categories: Synchronous and Asynchronous circuits.

Synchronous circuits are the circuits that use clocks to synchronize the input and output signals. The input signals are sampled at the rising or falling edge of the clock, and the output signals are updated accordingly. Flip-flops, counters, and shift registers are examples of synchronous circuits.

Asynchronous circuits are the circuits that do not use clocks to synchronize the input and output signals. The output signals are updated whenever the input signals change. Asynchronous circuits are slower and more complex than synchronous circuits.

The Answer

The correct answer to the given question is option 'D', i.e., Multiplexer. A multiplexer is a combinational circuit that selects one input signal out of multiple input signals based on the control signals. A multiplexer does not have any memory or feedback, so it is not a sequential circuit.

Conclusion

In conclusion, a sequential circuit is a digital circuit that has memory, and the output depends on the present and past input signals. Synchronous and asynchronous circuits are two categories of sequential circuits. Flip-flops, counters, and shift registers are examples of synchronous circuits, while multiplexers are not sequential circuits.

Full Adder Configuration

To understand the number of gates required for the configuration of a full adder, let's first discuss what a full adder is and its logical representation.

A full adder is a combinational circuit that takes in three inputs: A, B, and Cin (carry-in) and produces two outputs: Sum (S) and Cout (carry-out). It is used to add two binary numbers along with a carry input from the previous stage.

The logical representation of a full adder can be expressed using basic logic gates such as AND, OR, and EXOR gates.

Logical Representation of Full Adder

The logical representation of a full adder is as follows:

1. Sum (S) output:
- It is the output of the EXOR gate.
- It is calculated by taking the EXOR of A, B, and Cin.

2. Carry-out (Cout) output:
- It is the output of the OR gate.
- It is calculated by taking the OR of the AND gates' outputs.

Calculating the Number of Gates

Now, let's calculate the number of AND, OR, and EXOR gates required for the configuration of a full adder.

1. AND gates:
- We require 2 AND gates to calculate the carry-out (Cout) output.
- Each AND gate takes 2 inputs, so a total of 2 AND gates are needed.

2. OR gates:
- We require 1 OR gate to calculate the carry-out (Cout) output.
- The OR gate takes 2 inputs.

3. EXOR gates:
- We require 2 EXOR gates to calculate the sum (S) output.
- Each EXOR gate takes 3 inputs, so a total of 2 EXOR gates are needed.

Hence, the number of AND, OR, and EXOR gates required for the configuration of a full adder is as follows:

- AND gates: 2
- OR gates: 1
- EXOR gates: 2

Therefore, the correct answer is option 'B' (2, 1, 2).

There are 10 different 3-digit numbers.

Understanding a 6-Bit Counter
A 6-bit counter is a digital device that can store and count values using 6 binary digits (bits). Each bit can represent two states: 0 or 1.
Calculating the Maximum Count
- The maximum number of unique values a counter can represent is calculated using the formula:
2^n, where n is the number of bits.
- For a 6-bit counter:
2^6 = 64 unique values.
Counting Sequence
- A counter typically starts counting from 0, which means it will count from 0 to 63.
- Therefore, the values it can represent are:
0, 1, 2, ..., 63.
Determining the Value of n
- Since the counter starts at 0 and goes up to the maximum of 63, the highest value n that the counter can reach is:
n = 63.
Conclusion
- The correct answer to the question, "The value of n is _____" is:
Option D: 63.
This highlights the capability of a 6-bit counter to represent a range of values, emphasizing its utility in digital circuits and applications.

Priority Encoding:
Priority encoding is a technique used in digital circuits to assign a unique code to the highest priority active input. It is commonly used in applications where multiple inputs need to be prioritized and encoded into a binary code.

Working Principle:
A priority encoder consists of multiple inputs and an output. The inputs are typically labeled as D0, D1, D2, etc., and the output provides a binary code corresponding to the highest priority active input. When multiple inputs are active simultaneously, the priority encoder assigns a code to the input with the highest priority.

Answer Explanation:
When two inputs are active on a priority encoder, the output will be coded based on the higher value input. This means that the input with the higher priority will be encoded on the output.

To understand this, let's consider a simple example of a 4-input priority encoder. The inputs are labeled as D0, D1, D2, and D3 (D0 being the highest priority). If both D2 and D3 are active, the output will be coded based on D3 because it has a higher priority than D2.

The priority encoder follows a priority order, where each input has a higher priority than the inputs below it. When multiple inputs are active, the priority encoder will prioritize the input with the highest priority and encode it on the output.

Example:
Let's consider a priority encoder with inputs D0, D1, and D2. The priority order is D0 > D1 > D2. If D1 and D2 are active simultaneously, the output will be coded based on D1 because it has a higher priority than D2. Similarly, if D0 and D2 are active simultaneously, the output will be coded based on D0.

In summary, when two inputs are active on a priority encoder, the input with the higher priority will be coded on the output. This ensures that the highest priority input is given precedence in the encoding process.

The size of the encoder required for a 4-bit flash-type ADC is 16:4.

Explanation:
Flash type ADC is a type of analog-to-digital converter that uses a combination of comparators to convert an analog input voltage into a digital output. In this type of ADC, the number of comparators used is equal to 2 raised to the power of the number of bits.

In a 4-bit flash ADC, we have 4 bits of resolution, which means we need 2^4 = 16 comparators. Each comparator compares the input voltage to a reference voltage and produces a high or low output depending on the comparison result.

To convert these 16 comparator outputs into a 4-bit digital output, we need an encoder. An encoder is a combinational logic circuit that converts the parallel output of the comparators into a serial digital output. The encoder maps the comparator outputs to the corresponding digital code.

In this case, we need to map the outputs of 16 comparators to a 4-bit digital output. Since we have 16 comparators, we need 16 input lines for the encoder. The number of output lines is equal to the number of bits in the digital output, which is 4 in this case.

So, the size of the encoder required for a 4-bit flash type ADC is 16:4, meaning 16 input lines and 4 output lines.

To summarize:
- Flash type ADC uses a combination of comparators to convert an analog input into a digital output.
- The number of comparators used in a flash ADC is equal to 2 raised to the power of the number of bits.
- In a 4-bit flash ADC, we need 16 comparators.
- An encoder is used to convert the parallel outputs of the comparators into a serial digital output.
- The size of the encoder required for a 4-bit flash ADC is 16:4, meaning 16 input lines and 4 output lines.