Resonance in AC Circuits

Why it matters

Resonance in AC circuits is crucial for designing efficient communication systems, filters, and oscillators. It provides frequency selectivity through energy exchange between inductance and capacitance; losses remain and resonance does not guarantee maximum efficiency, which is essential in applications like radio and television broadcasting.

Key ideas

• Resonance occurs in an AC circuit when the inductive reactance equals the capacitive reactance, resulting in a purely resistive impedance at a particular frequency known as the resonant frequency.
• Resonant Frequency (f₀): The frequency at which resonance occurs. At this frequency, the circuit can store and transfer energy between the inductor and capacitor with minimal loss.
• Quality Factor (Q): A dimensionless parameter that describes the sharpness of the resonance peak. A higher Q indicates a narrower and higher peak, meaning the circuit is more selective about the frequencies it resonates with.
• Bandwidth (BW): The range of frequencies over which the circuit can effectively operate. It is inversely proportional to the quality factor.

The following formulas and example apply to a series RLC circuit, with BW measured between the half-power frequencies. At resonance its impedance is minimum and equals R. Parallel resonant circuits have different impedance and bandwidth expressions.

Formulas

• Resonant Frequency: f₀ = 1 / (2π√(LC))
  • f₀: Resonant frequency (Hz)
  • L: Inductance (H)
  • C: Capacitance (F)
• Quality Factor: Q = f₀ / BW
  • Q: Quality factor (dimensionless)
  • BW: Bandwidth (Hz)
• Bandwidth: BW = R / (2πL)
  • R: Resistance (Ω)

Worked example

Given: L = 50 mH, C = 20 µF, R = 10 Ω

1. Calculate the resonant frequency using f₀ = 1 / (2π√(LC)).
   • f₀ = 1 / (2π√(50×10⁻³ H × 20×10⁻⁶ F))
   • f₀ ≈ 159.15 Hz
2. Calculate the bandwidth using BW = R / (2πL).
   • BW = 10 Ω / (2π × 50×10⁻³ H)
   • BW ≈ 31.83 Hz
3. Calculate the quality factor using Q = f₀ / BW.
   • Q = 159.15 Hz / 31.83 Hz
   • Q ≈ 5.0

Final Answer: Resonant Frequency = 159.15 Hz, Bandwidth = 31.83 Hz, Quality Factor = 5.0

Common mistakes

• Confusing the units of inductance and capacitance, leading to incorrect calculations.
• Forgetting that resonance occurs when the inductive and capacitive reactances are equal.
• Misinterpreting the quality factor as having units, when it is actually dimensionless.

For GATE EC

Questions often involve calculating the resonant frequency, quality factor, or bandwidth of a given circuit. Practice problems that require understanding the relationship between these parameters and their impact on circuit performance.

Quick check

1. What is the condition for resonance in an AC circuit?
2. How does the quality factor affect the bandwidth of a circuit?
3. What happens to the impedance of a circuit at resonance?

Answers: 1. Inductive reactance equals capacitive reactance. 2. Higher quality factor means narrower bandwidth. 3. Impedance is purely resistive.