In which of the following circuit can resonance occur?

Explanation:

Resonance in Electrical Circuits

Definition: Resonance in electrical circuits occurs when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase, causing them to cancel each other out. This results in the circuit impedance being purely resistive at a specific frequency, known as the resonant frequency. At resonance, the circuit can store and transfer energy between the inductor and capacitor with maximum efficiency, leading to a significant increase in current or voltage.

Working Principle: Resonance in an R-L-C circuit (where R stands for resistance, L for inductance, and C for capacitance) is achieved when the inductive reactance (X_L) and capacitive reactance (X_C) are equal. The condition for resonance can be mathematically expressed as:

X_L = X_C

Since inductive reactance (X_L) is given by:

X_L = 2πfL

And capacitive reactance (X_C) is given by:

X_C = 1 / (2πfC)

Where f is the frequency of the AC supply, L is the inductance, and C is the capacitance. At resonance:

2πfL = 1 / (2πfC)

Solving for the resonant frequency (f₀):

f₀ = 1 / (2π√LC)

At this frequency, the total impedance of the circuit is at its minimum if it is a series R-L-C circuit or at its maximum if it is a parallel R-L-C circuit, and the circuit exhibits resonant behavior.

Correct Option Analysis:

The correct option is:

Option 2: R-L-C parallel

This option correctly identifies a circuit where resonance can occur. In an R-L-C parallel circuit, the inductive and capacitive reactances can cancel each other out at a specific frequency, leading to a condition of resonance. This results in a high impedance at the resonant frequency, and the circuit can exhibit significant voltage magnification.

In a parallel R-L-C circuit, the total admittance (Y) is given by the sum of the individual admittances:

Y = Y_R + Y_L + Y_C

Where:

Y_R = 1 / R

Y_L = 1 / (jωL)

Y_C = jωC

At resonance, the inductive susceptance (B_L) and capacitive susceptance (B_C) cancel each other out:

B_L = -B_C

This results in the total susceptance being zero and the total admittance being purely resistive, leading to resonance.

Important Information

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

Option 1: R-L series

Resonance cannot occur in an R-L series circuit because it lacks a capacitive component. Resonance requires both inductance and capacitance to create a condition where inductive and capacitive reactances can cancel each other out. Without a capacitor, this balance cannot be achieved.

Option 3: R-C series

Similarly, resonance cannot occur in an R-C series circuit because it lacks an inductive component. Resonance requires both inductance and capacitance to achieve the condition where inductive and capacitive reactances cancel each other out. Without an inductor, this balance cannot be achieved.

Option 4: R-L parallel

Resonance cannot occur in an R-L parallel circuit either because it lacks a capacitive component. As with the series circuits, both inductance and capacitance are required to achieve the resonance condition. The absence of a capacitor means that the necessary balance between inductive and capacitive reactances cannot be established.

Conclusion:

Understanding resonance in electrical circuits is crucial for designing circuits that can efficiently store and transfer energy at specific frequencies. In the context of the options provided, the R-L-C parallel circuit is the correct choice where resonance can occur. This is because it includes both inductive and capacitive components, which are essential for achieving the condition of resonance. By analyzing the other options, it becomes evident that the absence of either the inductive or capacitive component makes resonance impossible in those circuits.