Current In Branch CD Analysis Of A 50V Constant Voltage Circuit

by BRAINLY IN FTUNILA 64 views
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In the realm of electrical circuits, understanding the flow of current through different branches is paramount. This analysis delves into a specific scenario involving a circuit where a constant voltage of 50V is maintained between points A and B. Our primary objective is to determine the current flowing through branch CD of the circuit. To achieve this, we will meticulously dissect the circuit, applying fundamental principles of circuit analysis.

Circuit Overview

The circuit under consideration comprises several resistors interconnected in a network. The key components and their respective resistances are as follows:

  • Resistor 1: 1╬й
  • Resistor 2: 2╬й
  • Resistor 3: 2╬й
  • Resistor 4: 4╬й

A constant voltage source of 50V is connected across points A and B, establishing a potential difference that drives the current through the circuit. Our focus is on determining the current flowing specifically through the branch CD, which consists of Resistor 4 (4╬й).

Methodology

To determine the current in branch CD, we will employ a systematic approach rooted in fundamental circuit analysis principles:

  1. Circuit Simplification: The initial step involves simplifying the circuit by combining resistors in series and parallel configurations. This simplification reduces the complexity of the circuit, making it easier to analyze.
  2. Total Resistance Calculation: Once the circuit is simplified, we calculate the total resistance offered by the entire circuit to the flow of current. This total resistance is crucial for determining the overall current flowing through the circuit.
  3. Total Current Calculation: Using Ohm's Law (V = IR), we calculate the total current flowing through the circuit. This current represents the total charge flow driven by the 50V voltage source.
  4. Current Division Analysis: To determine the current flowing through branch CD, we apply the principle of current division. This principle states that in a parallel circuit, the total current divides among the branches in inverse proportion to their resistances.
  5. Branch CD Current Calculation: Finally, we apply the current division formula to calculate the current flowing specifically through branch CD (Resistor 4).

Step-by-Step Analysis

Let's embark on a step-by-step analysis to determine the current in branch CD:

1. Circuit Simplification

The circuit can be simplified by recognizing that Resistors 2 and 3 (both 2╬й) are connected in parallel. The equivalent resistance of these parallel resistors can be calculated as:

1/R_eq = 1/R_2 + 1/R_3 = 1/2╬й + 1/2╬й = 1/1╬й
R_eq = 1╬й

Therefore, the equivalent resistance of the parallel combination of Resistors 2 and 3 is 1╬й.

2. Total Resistance Calculation

Now, the simplified circuit consists of Resistor 1 (1╬й), the equivalent resistance of Resistors 2 and 3 (1╬й), and Resistor 4 (4╬й). These three resistances are connected in series. The total resistance of the circuit is the sum of these individual resistances:

R_total = R_1 + R_eq + R_4 = 1╬й + 1╬й + 4╬й = 6╬й

The total resistance of the circuit is 6╬й.

3. Total Current Calculation

Using Ohm's Law (V = IR), we can calculate the total current flowing through the circuit:

I_total = V / R_total = 50V / 6╬й = 8.33A

The total current flowing through the circuit is approximately 8.33A.

4. Current Division Analysis

To determine the current flowing through branch CD, we need to consider the current division between the branch containing Resistor 1 and the equivalent resistance of Resistors 2 and 3, and the branch containing Resistor 4. The current divides in inverse proportion to the resistances of these branches.

Let I_123 be the current flowing through the branch containing Resistor 1 and the equivalent resistance of Resistors 2 and 3, and let I_4 be the current flowing through branch CD (Resistor 4). According to the current division principle:

I_4 = I_total * (R_1 + R_eq) / (R_1 + R_eq + R_4)

5. Branch CD Current Calculation

Plugging in the values, we get:

I_4 = 8.33A * (1╬й + 1╬й) / (1╬й + 1╬й + 4╬й) = 8.33A * 2╬й / 6╬й = 2.78A

Therefore, the current flowing through branch CD (Resistor 4) is approximately 2.78A.

Conclusion

Through a meticulous step-by-step analysis, we have successfully determined the current flowing through branch CD of the circuit. By applying fundamental principles of circuit simplification, total resistance calculation, total current calculation, and current division, we arrived at a precise value of approximately 2.78A. This analysis highlights the importance of understanding circuit principles for effectively analyzing and predicting the behavior of electrical circuits.

This comprehensive analysis provides a clear understanding of how to approach circuit analysis problems, particularly those involving current determination in specific branches. The principles and techniques employed here can be extended to analyze more complex circuits and electrical systems.

Further Exploration

To further enhance your understanding of circuit analysis, consider exploring these additional topics:

  • Kirchhoff's Laws: These fundamental laws provide a systematic approach to analyzing complex circuits with multiple loops and nodes.
  • Th├йvenin's Theorem and Norton's Theorem: These theorems offer techniques for simplifying circuits by reducing them to equivalent circuits with a single voltage source or current source and a single resistor.
  • Superposition Theorem: This theorem allows you to analyze circuits with multiple independent sources by considering the contribution of each source individually.
  • AC Circuit Analysis: Explore the analysis of circuits with alternating current (AC) sources, which introduces concepts like impedance and reactance.

By delving into these topics, you can significantly expand your knowledge of circuit analysis and become proficient in solving a wide range of electrical circuit problems.

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  • Current in branch CD: This keyword directly addresses the specific problem being solved in the analysis.
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