Derive the resonance frequency of a series RCL circuit.

Setting Xₗ = X_C gives ωL = 1/(ωC), so ω² = 1/(LC) and ω₀ = 1/√(LC) in rad s⁻¹. Dividing by 2π gives f₀ = 1/(2π√(LC)) in Hz.

Describe how the current in a series RCL circuit varies as the supply frequency is swept from well below to well above resonance.

The current rises to a sharp maximum at the resonance frequency and falls away on either side. The peak occurs because the impedance is smallest there.

A series RCL circuit has L = 0.25 H and C = 10 μF. Calculate its resonance frequency.

f₀ = 1/(2π√(LC)) = 1/(2π√(0.25 × 10 × 10⁻⁶)) = 1/(2π × 1.58 × 10⁻³) = 101 Hz. Working back, LC = 1/(2πf₀)² = 1/(635)² = 2.5 × 10⁻⁶ s², which is 0.25 × 10 × 10⁻⁶.

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Option A: Alternating CurrentsOption A

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Option A: Alternating Currents

Resonance in the series RCL circuit and the Q factor

3 things to know
Component 3

Overview

Impedance of a series RCL circuit at resonance, phase angle of a series RCL circuit at resonance and Q factor of a series RCL circuit.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

What condition defines resonance in a series RCL circuit?

Key term

What is the impedance of a series RCL circuit at resonance?

Key term

What is the phase angle of a series RCL circuit at resonance?

Questions from this subtopic

Each one with the model answer underneath.

1
What condition defines resonance in a series RCL circuit?
Model answer
Resonance occurs when the inductive reactance equals the capacitive reactance, Xₗ = X_C. The two reactances then cancel exactly.
2
Derive the resonance frequency of a series RCL circuit.
Model answer
Setting Xₗ = X_C gives ωL = 1/(ωC), so ω² = 1/(LC) and ω₀ = 1/√(LC) in rad s⁻¹. Dividing by 2π gives f₀ = 1/(2π√(LC)) in Hz.
3
What is the impedance of a series RCL circuit at resonance?
Model answer
The impedance falls to its minimum value, equal to the resistance R alone. The reactive terms cancel, so Z = √(R² + 0) = R.
4
What is the phase angle of a series RCL circuit at resonance?
Model answer
The phase angle is zero, so the current is in phase with the supply potential difference. The circuit behaves as a pure resistance.
5
Describe how the current in a series RCL circuit varies as the supply frequency is swept from well below to well above resonance.
Model answer
The current rises to a sharp maximum at the resonance frequency and falls away on either side. The peak occurs because the impedance is smallest there.
6
A series RCL circuit has L = 0.25 H and C = 10 μF. Calculate its resonance frequency.
Model answer
f₀ = 1/(2π√(LC)) = 1/(2π√(0.25 × 10 × 10⁻⁶)) = 1/(2π × 1.58 × 10⁻³) = 101 Hz. Working back, LC = 1/(2πf₀)² = 1/(635)² = 2.5 × 10⁻⁶ s², which is 0.25 × 10 × 10⁻⁶.

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