Cutoff Frequency (including RC, RL, and RLC Circuit)

Cutoff Frequency

1. Definition

The cutoff frequency (f_c) is the frequency at which the output of a circuit drops to a specific fraction of its maximum value:

• Voltage or current drops to 70.7% of the maximum (1/√2 ≈ 0.707)
• In decibels: this corresponds to the −3 dB point

2. Cutoff Frequency Formulas

a) RC Filter

• RC Low-Pass: fc = 1 / (2πRC)
• RC High-Pass: fc = 1 / (2πRC)

b) RL Filter

• RL Low-Pass: fc = R / (2πL)
• RL High-Pass: fc = R / (2πL)

c) Series RLC Circuit

For a series RLC circuit, the resonant frequency is:

f0 = 1 / (2π√(LC))

The −3 dB cutoff frequencies (lower and upper) are:

fL = f0 - R/(4πL)
fH = f0 + R/(4πL)

Bandwidth:

BW = fH - fL = R / (2πL)

3. Step-by-Step RC Low-Pass Example

Given:

• R = 1 kΩ = 1000 Ω
• C = 0.1 μF = 0.1 × 10⁻⁶ F

Cutoff frequency:

fc = 1 / (2πRC)
fc = 1 / (2π × 1000 × 0.1 × 10⁻⁶) ≈ 1591 Hz

Interpretation: Frequencies below 1.591 kHz pass mostly unaffected; above 1.591 kHz, the output attenuates.

4. Summary Table

Filter Type	Cutoff Frequency
RC Low-Pass	fc = 1 / (2πRC)
RC High-Pass	fc = 1 / (2πRC)
RL Low-Pass	fc = R / (2πL)
RL High-Pass	fc = R / (2πL)
Series RLC	f0 = 1 / (2π√(LC)), Bandwidth = R / 2πL