RLC Circuits

Add an inductor to the RC circuit and you get resonance — the ability to select or reject a specific frequency. RLC circuits are the basis of radio tuning, bandpass and notch filters, and any application that needs frequency selectivity. The interplay between the capacitor (storing energy in an electric field) and the inductor (storing energy in a magnetic field) creates oscillatory behaviour that neither component can produce alone.

Resonant Frequency

At a specific frequency, the capacitive reactance and inductive reactance are equal and cancel out. At this resonant frequency, the impedance of a series RLC circuit is purely resistive (minimum), and current is maximum:

f₀ = 1 / (2π√(LC))

Example: L=100µH, C=100pF
f₀ = 1 / (2π × √(0.0001 × 0.0000000001))
f₀ = 1 / (2π × 0.0000316) ≈ 5.03 MHz

Series RLC

Vin ──[R]──[L]──[C]── Vout (across C)

Impedance:

Z = R + j(X_L - X_C)
Z = R + j(2πfL - 1/(2πfC))

|Z| = √(R² + (X_L - X_C)²)

At resonance: X_L = X_C, so Z = R. All other frequencies see additional impedance.

Taking output across C gives a low-pass characteristic below resonance and bandpass behaviour near resonance. Taking output across the series combination of L and C gives a bandpass filter peaked at f₀.

Parallel RLC

     +──[R]──+
     |       |
Vin ─+──[L]──+── Vout
     |       |
     +──[C]──+

At resonance, the parallel LC combination has maximum impedance (the tank circuit), so maximum voltage appears across it — a bandpass response. At other frequencies the impedance drops.

1/Z = 1/R + 1/jX_L + 1/jX_C

Quality Factor (Q)

Q measures how selective the resonance is — how sharp the peak is, how much the circuit discriminates between f₀ and nearby frequencies:

Series:   Q = (1/R) × √(L/C)
Parallel: Q = R × √(C/L)

Bandwidth: BW = f₀/Q

High Q = narrow bandwidth, sharp resonance. Low Q = wide bandwidth, gentle peak. For a radio tuner, high Q means you can separate stations. For a crossover filter in a speaker system, lower Q gives a smoother transition.

Practical Examples

AM Radio Tuner

A variable capacitor in parallel with an inductor forms a tank circuit. Adjust C to tune f₀ to the station's carrier frequency. At that frequency, the tank circuit resonates and the signal is maximum. The inductor is typically coiled around a ferrite rod (the antenna) to pick up the RF signal.

Notch Filter

A series LC in the signal path (not to ground) creates a notch — it blocks the resonant frequency while passing everything else. Used to eliminate a specific frequency: 60Hz hum in audio, or a strong interfering signal in radio work.

          series LC
Vin ──┬──[L]──[C]──┬── Vout
      |             |
     [R]           [R]
      |             |
     GND           GND

At f₀: series LC ≈ short circuit → Vout = 0 (notch)
Away from f₀: series LC ≈ open circuit → Vout ≈ Vin

Damping

The resistance in an RLC circuit determines how the transient response (what happens when you apply or remove voltage) behaves:

• Underdamped (high Q) — oscillates, rings. A square wave input shows a damped oscillation at the edges. Common in poorly designed power supply output filters or speaker crossovers.
• Critically damped — fastest settling without overshoot. The ideal for most control systems.
• Overdamped (low Q) — slow response, no overshoot. Safe but sluggish.