What is the primary condition for resonance in an AC circuit?

The primary condition for resonance in any AC circuit containing both inductive and capacitive elements is that the inductive reactance ($X_L$) must be equal in magnitude to the capacitive reactance ($X_C$). Mathematically, this is expressed as $X_L = X_C$. This equality leads to the cancellation of the reactive components of impedance, making the circuit behave purely resistively at the resonant frequency. This condition is crucial for determining the specific frequency at which resonance occurs.

How does a series RLC circuit behave at resonance?

At resonance, a series RLC circuit exhibits minimum impedance, which is equal to the circuit's resistance (Z = R). Consequently, the current flowing through the circuit becomes maximum. The phase angle between the applied voltage and the current becomes zero, meaning the power factor is unity (cos$\phi$ = 1). Individual voltages across the inductor and capacitor can be significantly higher than the source voltage, a phenomenon known as voltage magnification.

How does a parallel RLC circuit behave at resonance?

A parallel RLC circuit at resonance behaves quite differently from a series circuit. It exhibits maximum impedance, which means the total current drawn from the source is minimum. Similar to the series circuit, the phase angle between the source voltage and current is zero, leading to a unity power factor. However, in parallel resonance, the currents circulating within the LC tank circuit (between L and C) can be much larger than the current drawn from the source, a phenomenon called current magnification.

What is the significance of the Quality Factor (Q-factor) in resonant circuits?

The Quality Factor (Q-factor) quantifies the sharpness or selectivity of a resonant circuit. A high Q-factor indicates a very sharp resonance peak, meaning the circuit responds strongly only to a very narrow range of frequencies around the resonant frequency. This is desirable in applications like radio tuners where precise frequency selection is needed. Conversely, a low Q-factor implies a broader resonance, making the circuit less selective but potentially useful for wider bandwidth applications.

Can resonance occur in circuits without a resistor?

Yes, resonance can conceptually occur in ideal LC circuits (without a resistor). In such a scenario, for a series LC circuit, the impedance at resonance would ideally be zero, leading to infinite current. For a parallel LC circuit, the impedance would ideally be infinite, leading to zero current from the source. However, in practical circuits, every component has some inherent resistance, so a purely resistive component (R) is always present, making the impedance finite at resonance.

What is bandwidth in the context of resonant circuits?

Bandwidth (BW) refers to the range of frequencies over which a resonant circuit effectively operates or allows significant power transfer. It is typically defined as the difference between the two half-power frequencies ($f_1$ and $f_2$), where the power dissipated is half of the maximum power at resonance. For a series RLC circuit, these are the frequencies where the current is $1/\sqrt{2}$ times the maximum current. Bandwidth is inversely proportional to the Q-factor ($BW = f_0/Q$), meaning a higher Q-factor results in a narrower bandwidth and greater selectivity.

Resonance in AC Circuits

Physics

NEET UG

Version 1Updated 22 Mar 2026

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Resonance in an AC circuit is a special condition where the inductive reactance ( $X_L$ ) and capacitive reactance ( $X_C$ ) become equal in magnitude, leading to specific and often extreme circuit behaviors. This equality results in the cancellation of their opposing effects, making the circuit purely resistive. The frequency at which this phenomenon occurs is termed the resonant frequency ( $f_0$ ). A…

Quick Summary

Resonance in AC circuits occurs when the inductive reactance ( $X_L$ ) equals the capacitive reactance ( $X_C$ ). This specific frequency is called the resonant frequency ( $f_0 = 1/(2\pi\sqrt{LC})$ ). In a series RLC circuit, resonance leads to minimum impedance (Z=R), maximum current, and unity power factor.

The voltages across L and C can be much larger than the source voltage (voltage magnification). This circuit acts as an 'acceptor' for current at $f_0$ . In contrast, a parallel RLC circuit at resonance exhibits maximum impedance, minimum line current, and unity power factor.

The currents circulating between L and C can be much larger than the source current (current magnification). This circuit acts as a 'rejector' for current at $f_0$ . The Quality Factor (Q-factor) describes the sharpness of resonance, with higher Q indicating a narrower bandwidth and greater selectivity.

Resonance is fundamental to tuning circuits, filters, and oscillators, allowing specific frequencies to be selected or rejected.

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Key Concepts

Resonant Frequency Calculation

The resonant frequency ( $f_0$ ) is the cornerstone of resonance. It's the unique frequency where the energy…

Series Resonance Characteristics

In a series RLC circuit, at resonance, the impedance (Z) reaches its minimum value, which is equal to the…

Quality Factor and Bandwidth Relationship

The Quality Factor (Q) and Bandwidth (BW) are intrinsically linked, describing the selectivity of a resonant…

Resonant Frequency: —

f_0 = \frac{1}{2\pi\sqrt{LC}}

\omega_0 = \frac{1}{\sqrt{LC}}

Series RLC Resonance:

- $X_L = X_C$ - Impedance $Z_{min} = R$ - Current $I_{max} = V/R$ - Phase angle $\phi = 0^\circ$ , Power Factor $\cos\phi = 1$ - Voltage magnification: $V_L = V_C = Q \times V_{source}$ - Q-factor: $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}$

Parallel RLC Resonance:

- $X_L = X_C$ - Impedance $Z_{max}$ (ideally infinite) - Current $I_{min} = V/Z_{max}$ - Phase angle $\phi = 0^\circ$ , Power Factor $\cos\phi = 1$ - Current magnification: $I_L = I_C = Q \times I_{source}$ - Q-factor: $Q = \frac{R}{\omega_0 L} = \omega_0 CR = R\sqrt{\frac{C}{L}}$

Bandwidth: —

BW = f_0/Q

Reactances Equal, Series Minimum Impedance, Parallel Maximum Impedance. (Resonance: $X_L=X_C$ . Series: Min Z, Max I. Parallel: Max Z, Min I).