Lcr Circuit

The RMS (Root Mean Square) value of a voltage is related to the peak value by the formula: For a series circuit, the applied voltage is the vector sum of the peak voltages across each component. The total peak voltage is calculated as: Given that: - , - , - , Substituting the values: Now, we calculate the RMS value: Thus, the RMS value of the applied voltage is approximately 70.7 V.

The correct option is (C): 70.7 V

The power factor of an R-L circuit is given by the formula: Where:
is the resistance,
is the inductive reactance.
Given that the power factor is and ,
we substitute these values into the formula: Squaring both sides: Cross-multiplying: Thus, the value of resistance is .

Given:
The graph of impedance (Z) vs. angular frequency (ω) for a series LCR circuit shows resonance clearly at .
At resonance, impedance is purely resistive and minimum.

Step 1: Understanding the Quality Factor (Q)

The Quality factor of a series LCR circuit at resonance is defined as:

where:
= Resonant angular frequency,
= Inductance,
= Capacitance,
= Resistance at resonance.

Step 2: Determine Quality Factor from the given graph:

Quality factor can also be graphically determined using:

From the given graph:
- Resonance occurs at .
- At resonance, minimum impedance, .
- Bandwidth ( ) is the frequency range at which the impedance is .

Observing carefully from the given graph, impedance reaches at two points:

• Lower frequency
• Higher frequency

Thus, bandwidth is:

Step 3: Calculate Quality Factor (Q):

Final Conclusion:
The quality factor of the series LCR circuit is 1.

Step 1: Resonance condition in an LCR circuit:

At resonance, the inductive reactance ( ) and capacitive reactance ( ) become equal:

Step 2: Implication for circuit impedance:

At resonance, the net reactance is zero. Hence, the impedance ( ) of the LCR circuit at resonance is purely resistive, equal to :

Step 3: Phase difference at resonance:

In a purely resistive circuit, voltage and current are in phase. Thus, the phase difference ( ) between voltage and current at resonance is:

Final Conclusion:
At resonance, the voltage and current in a series LCR circuit are in phase (phase difference = 0).

In a series LCR circuit, the circuit behaves capacitively when the capacitive reactance is greater than the inductive reactance . This occurs when the frequency reaches the point where exceeds , which is shown at point in the graph. Thus, the circuit becomes capacitive at frequency point .

The resonance frequency of an LCR circuit is given by:
.

As the capacitance increases, the resonance frequency decreases, because is inversely proportional to the square root of . This means that as increases, decreases, resulting in a graph with a downward slope as increases.

The -factor of an LCR circuit is an important parameter for communication systems as it determines the sharpness of resonance. The quality factor is given by the formula:
.

For higher -factors (which are desirable for better tuning), we require a lower resistance and larger inductance and capacitance values.
After calculating the -factor for each combination, the combination in option (D) yields the highest -factor, which is ideal for tuning in communication circuits.