Question

The impedance of a circuit consists of \(3 \Omega\) resistance and \(4 \Omega\) reactance. The power factor of the circuit is. (a) \(0.4\) (b) \(0.6\) (c) \(0.8\) (d) \(1.0\)

Short answer

The power factor of the circuit can be calculated using the formula Power Factor = \(cos(\theta)\) = \(\frac{R}{Z}\). First, calculate the impedance Z with the given resistance R and reactance X: \(Z = \sqrt{R^2 + X^2} = \sqrt{(3 \Omega)^2 + (4 \Omega)^2} = 5 \Omega\). Then, find the power factor: Power Factor = \(\frac{3 \Omega}{5 \Omega} = 0.6\). The correct answer is (b) 0.6.

Step-by-step solution

Calculate the impedance

To calculate the impedance, use the formula: \(Z = \sqrt{R^2 + X^2}\) Plug in the given resistance and reactance values: \(Z = \sqrt{(3 \Omega)^2 + (4 \Omega)^2}\)

Simplify the expression

Evaluate the squares and sum inside the square root: \(Z = \sqrt{9 + 16}\) Add the numbers inside the square root: \(Z = \sqrt{25}\) Take the square root of 25: \(Z = 5 \Omega\)

Calculate the power factor

Now that we have the impedance, we can find the power factor using the formula: Power Factor = \(cos(\theta)\) = \(\frac{R}{Z}\) Plug in the given resistance and calculated impedance values: Power Factor = \(\frac{3 \Omega}{5 \Omega}\)

Simplify the expression

Divide 3 by 5 to get the power factor: Power Factor = \(0.6\) The power factor of the circuit is 0.6, so the correct answer is (b) 0.6.

Key concepts

Impedance Calculation

Impedance is a fundamental concept in alternating current (AC) circuits. It combines resistance and reactance into a single measure to characterize how a circuit element opposes the flow of an alternating current. When we calculate impedance, we use the formula:

• Impedance, \( Z = \sqrt{R^2 + X^2} \)

Here, \( R \) denotes resistance, which restricts current flow, and \( X \) denotes reactance, which opposes the change in current.

In the given exercise, the resistance is \( 3 \Omega \) and the reactance is \( 4 \Omega \). Plugging these values into the impedance formula, it becomes:

• \( Z = \sqrt{(3 \Omega)^2 + (4 \Omega)^2} \)
• \( Z = \sqrt{9 + 16} = \sqrt{25} = 5 \Omega \)

This shows that the impedance of the circuit is \( 5 \Omega \). This value indicates how strongly the circuit resists the AC current.

Resistance and Reactance

Resistance and reactance are two components of impedance that play crucial roles in analyzing electrical circuits. Let's explore each:

• Resistance (R): This is the measure of how much an object opposes the passage of current through it. It is dependent on the material and dimensions of the conductor, and is constant regardless of the frequency of the AC signal. Measured in ohms (\( \Omega \)), resistance in the exercise is \( 3 \Omega \).


• Reactance (X): Unlike resistance, reactance changes with the frequency of the current. It represents the opposition to changes in current flow caused by inductance and capacitance in the circuit. In the exercise, reactance is \( 4 \Omega \). There are two types:
• Inductive Reactance, due to inductors, which increases with frequency.
• Capacitive Reactance, due to capacitors, which decreases with frequency.

Together, resistance and reactance form the vector components needed for complete impedance calculations, crucial for understanding the circuit's behavior.

Electrical Circuits

Electrical circuits are paths or networks through which electrical current flows. These circuits can be as simple as a battery and light bulb, or as complex as those in computers.

There are two basic types of electrical circuits: series and parallel.

• Series Circuits: All components are connected end-to-end, making the current flow through each component in sequence. If one component fails, the circuit is broken and current stops flowing.
• Parallel Circuits: Components are connected across common points or junctions, and the current can flow through multiple paths. If one path fails, current can continue to flow through other paths.

In the context of the exercise, the AC circuit's impedance results from both the resistance and reactance components arranged in a specific configuration, affecting how efficiently the circuit can deliver power. Understanding how impedance, resistance, and reactance behave in an electrical circuit is vital for solving circuit problems and optimizing performance.