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

Practical Applications of Cutoff Frequency

Audio Engineering

Used in Crossover Networks to direct low frequencies to woofers and high frequencies to tweeters.

RF & Telecommunications

Essential for Antenna Design and limiting noise in wireless signal transmission.

Power Supplies

Filters out unwanted AC Ripple to provide clean DC voltage for sensitive electronics.

Data Acquisition

Prevents Aliasing in Analog-to-Digital Converters (ADC) by filtering high-frequency noise.

Design Calculator: Finding Component Values

If you have a target cutoff frequency (f_c) and a known resistor (R), use this formula to find the required capacitor (C):

C = 1 / (2πRf_c)

Design Example: To create a Low-Pass Filter with a 10 kHz cutoff using a 1 kΩ resistor:

• C = 1 / (2π × 1000 × 10,000)
• C = 15.9 nF

Passive vs. Active Cutoff Filters

While RC and RL are passive filters, many modern circuits use active filters involving Operational Amplifiers (Op-Amps).

Feature	Passive Filter (RC/RL)	Active Filter (Op-Amp)
Power Requirement	None	Requires External Power
Signal Gain	Lossy (Insertion Loss)	Can Amplify Signal
Input Impedance	Lower	Very High (Ideal for buffering)

Frequently Asked Questions

What is the significance of the -3dB point?

The -3dB point is the cutoff frequency where the output power is halved. In terms of voltage, the output is exactly 70.7% of the input voltage. Engineers use this standard to define the effective bandwidth of a system.

How does tolerance affect cutoff frequency?

Real-world components have tolerances (e.g., ±5% for resistors). This causes the actual cutoff frequency to vary slightly from the calculated theoretical value. For precise filters, 1% metal film resistors and 5% film capacitors are recommended.

Try Interactive Online Simulator

1. 1. Interactive RLC Simulator (Cut-off, Impedance, Phase, etc.) (first order)
2. 2. Interactive Sallen-Key filter Simulation (2nd order LPF)

Further Reading

1. 1. Q Factor in Band-Pass Filter
2. 2. What is - 3dB Frequency Response? Applications ...
3. 3. Resistive Cutoff Frequency vs Resonant Frequency 
4. 4. Corner Cutoff Frequency, Bandwidth and Resonance Frequency 
5. 5. Lower & Upper Cut-off Frequencies

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