If R is the resistance of an R-L series circuit and reactive inductance is X_L , then the power factor of a series R-L circuit is given by _______. 

Explanation:

Power Factor of Series R-L Circuit

Definition: In an R-L series circuit, the power factor is a measure of how effectively electrical power is converted into useful work. It represents the cosine of the angle (\(\theta\)) between the voltage and current vectors in the circuit. The power factor is crucial in AC circuits as it determines the efficiency of power usage.

The power factor of a series R-L circuit is given by:

\(\rm \cos \theta = \frac{R}{Z}\)

Where:

• \(R\): Resistance of the circuit
• \(Z\): Impedance of the circuit

Impedance (\(Z\)): In an R-L series circuit, the impedance is the vector sum of the resistance (\(R\)) and the reactive inductance (\(X_L\)). It is given by:

\(\rm Z = \sqrt{R^2 + X_L^2}\)

Here:

• \(X_L\): Reactive inductance, which is calculated as \(X_L = \omega L\), where \(\omega\) is the angular frequency and \(L\) is the inductance.

Detailed Explanation:

In an R-L series circuit, the total impedance (\(Z\)) is a combination of the resistive and inductive components. The resistance (\(R\)) represents the real part of the impedance and is associated with energy dissipation as heat. The inductive reactance (\(X_L\)) represents the imaginary part of the impedance and is associated with energy storage in the magnetic field of the inductor.

The phase angle (\(\theta\)) between the voltage and current in the circuit is determined by the relationship between the resistance (\(R\)) and the reactive inductance (\(X_L\)):

\(\rm \tan \theta = \frac{X_L}{R}\)

From this, the cosine of the phase angle (\(\cos \theta\)) is calculated as:

\(\rm \cos \theta = \frac{R}{Z}\)

This formula indicates that the power factor depends on the ratio of the resistance to the total impedance of the circuit. A higher resistance relative to the impedance results in a better power factor, whereas a higher reactive inductance results in a lower power factor.

Correct Option Analysis:

The correct option is:

Option 3: \(\rm \cos \theta = \frac{R}{Z}\)

This option correctly defines the power factor of a series R-L circuit. The power factor is the ratio of the resistance (\(R\)) to the total impedance (\(Z\)) of the circuit. It is a measure of how efficiently the electrical power is being converted into useful work.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm \cos \theta = \frac{X_L}{Z}\)

This option is incorrect. The ratio of the reactive inductance (\(X_L\)) to the impedance (\(Z\)) does not represent the power factor. Instead, it represents the sine of the phase angle (\(\sin \theta\)), which is associated with the reactive component of the circuit.

Option 2: \(\rm \cos \theta = \frac{Z}{R}\)

This option is incorrect. The ratio of impedance (\(Z\)) to resistance (\(R\)) does not define the power factor. Instead, it is inversely related to the power factor and does not have a direct physical interpretation in this context.

Option 4: \(\rm \cos \theta = \frac{R}{X_L}\)

This option is incorrect. The ratio of resistance (\(R\)) to reactive inductance (\(X_L\)) is related to the tangent of the phase angle (\(\tan \theta\)), not the cosine of the phase angle (\(\cos \theta\)). It is not a measure of the power factor.

Conclusion:

In summary, the power factor of a series R-L circuit is given by \(\rm \cos \theta = \frac{R}{Z}\), which accurately represents the ratio of the resistance to the total impedance. This formula is essential for understanding the efficiency of power usage in AC circuits. Correctly identifying the power factor formula is crucial for analyzing and optimizing the performance of R-L circuits, especially in applications where power efficiency is critical.