Designing a Combination Circuit with Specific Output Conditions

In the realm of digital electronics and computer science, designing circuits that meet specific logical conditions is a common task. This article focuses on the problem of designing a combination circuit with 4 inputs that outputs 1 when the combination of inputs is a multiple of 3. This task involves a series of logical steps, including the development of a truth table and the design of the circuit based on Boolean logic principles.

Introduction to the Problem

The problem requires us to design a circuit that outputs a 1 only when the combination of 4 inputs is a multiple of 3. This means that out of the 16 possible combinations of 4 bits (which range from 0 to 15), only those which are multiples of 3 (3, 6, 9, 12) should result in an output of 1. The rest should result in an output of 0.

Identifying Multiples of 3

To solve this problem, we first need to identify which numbers between 0 and 15 are multiples of 3:

0 ÷ 3 0 (No multiple of 3) 1 ÷ 3 0.3333 (Not a multiple of 3) 2 ÷ 3 0.6667 (Not a multiple of 3) 3 ÷ 3 1 (Multiple of 3) 4 ÷ 3 1.3333 (Not a multiple of 3) 5 ÷ 3 1.6667 (Not a multiple of 3) 6 ÷ 3 2 (Multiple of 3) 7 ÷ 3 2.3333 (Not a multiple of 3) 8 ÷ 3 2.6667 (Not a multiple of 3) 9 ÷ 3 3 (Multiple of 3) 10 ÷ 3 3.3333 (Not a multiple of 3) 11 ÷ 3 3.6667 (Not a multiple of 3) 12 ÷ 3 4 (Multiple of 3) 13 ÷ 3 4.3333 (Not a multiple of 3) 14 ÷ 3 4.6667 (Not a multiple of 3) 15 ÷ 3 5 (Multiple of 3)

The numbers that are multiples of 3 are 3, 6, 9, and 12. These are the inputs for which the circuit should output a 1.

Designing the Truth Table

Using the identified inputs, we can construct the truth table for our circuit. A truth table is a tabular representation of all possible input combinations along with the corresponding output:

A B C D Output 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 0

From the truth table, we can see that the circuit should output a 1 only for the following input combinations: 011, 100, 101, and 1101 (representing the integers 3, 6, 9, and 12 in base 10 respectively).

Boolean Expression and Circuit Design

Based on the truth table, we can now derive a Boolean expression for the circuit. We identify the minterms that correspond to the outputs of 1 and then sum them up to obtain the Boolean expression:

Output A'B'C' A'BC' AB'C' ABC

Here, A'B'C' corresponds to the binary input 011 (3), A'BC' corresponds to the binary input 100 (4), AB'C' corresponds to the binary input 101 (5), and ABC corresponds to the binary input 1101 (12).

Next, we design the circuit using logic gates that perform the necessary Boolean operations. The circuit can be implemented using a combination of AND, OR, and NOT gates. The exact circuit design will depend on the specific requirements and the available resources, but it can be simplified using Karnaugh maps (K-maps) or logic minimization techniques to reduce the number of gates needed.

Conclusion

In this article, we have explored the process of designing a combination circuit that outputs a 1 when the combination of 4 inputs is a multiple of 3. We started by identifying the multiples of 3 within the range of 0 to 15, then constructed a truth table, and derived a Boolean expression for the circuit. Finally, we discussed how to implement the circuit using logic gates. This exercise highlights the importance of digital design principles and the step-by-step approach in solving complex circuit design problems.

Related Keywords

combination circuit multiple of three digital design