A choke coil having resistance RΩ and of inductance L Henry is shunted by a capacitor of C farads. The dynamic impedance of the resonant circuit would be:

Explanation:

Dynamic Impedance of a Resonant Circuit

Definition: The dynamic impedance of a resonant circuit refers to the effective impedance offered by the circuit at resonance. At resonance, the inductive reactance and capacitive reactance cancel each other out, leaving only the resistive component to determine the impedance. In this problem, a choke coil with resistance R and inductance L is shunted by a capacitor of capacitance C, forming a resonant circuit.

Resonance Condition:

Resonance occurs when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase. Mathematically:

X_L = X_C, where:

• X_L = ωL is the inductive reactance.
• X_C = 1 / (ωC) is the capacitive reactance.
• ω is the angular frequency, and ω = 2πf.

At resonance, the angular frequency ω can be expressed as:

ω = 1 / √(LC)

Dynamic Impedance Calculation:

The dynamic impedance of the resonant circuit is the impedance offered by the resistive, inductive, and capacitive components at resonance. At resonance, the inductive reactance and capacitive reactance cancel out, leaving only the resistance and its interaction with the reactive components. The dynamic impedance Z_dyn can be calculated using the following relationship:

Z_dyn = L / (R × C)

Here:

• L: Inductance of the choke coil (in henries).
• R: Resistance of the choke coil (in ohms).
• C: Capacitance of the shunted capacitor (in farads).

This formula shows that the dynamic impedance is directly proportional to the inductance and inversely proportional to the product of the resistance and capacitance.

Correct Option Analysis:

The correct option is:

Option 3: L / (R × C)

This option correctly represents the dynamic impedance of the resonant circuit. At resonance, the circuit's behavior is governed by the interplay between the choke coil's inductance, its resistance, and the shunted capacitor's capacitance, as described by the formula Z_dyn = L / (R × C).

Additional Information

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

Option 1: R / (L × C)

This option is incorrect because it does not correctly relate the dynamic impedance to the circuit parameters. The resistance, inductance, and capacitance interact differently, as shown in the correct formula L / (R × C).

Option 2: C / (R × L)

This option is also incorrect. While capacitance, resistance, and inductance play a role in the circuit's behavior, their relationship in the dynamic impedance calculation is not represented by this formula.

Option 4: 1 / (R × L × C)

This option is incorrect because it misrepresents the impedance formula. The dynamic impedance is not the reciprocal of the product of resistance, inductance, and capacitance.

Conclusion:

The dynamic impedance of a resonant circuit is a critical parameter that determines the circuit's effective impedance at resonance. For a choke coil with resistance R and inductance L shunted by a capacitor of capacitance C, the dynamic impedance is given by Z_dyn = L / (R × C). This formula highlights the dependence of the impedance on the circuit's inductance, resistance, and capacitance, with the correct option being Option 3. Understanding resonance and the behavior of reactive components is essential for analyzing and designing resonant circuits effectively.