Can I cascade two notches to deepen rejection?

Yes — two identical Q=5 notches at 60 Hz in series give deeper-but-narrower combined notch. Cascading is sensitive to f₀ matching; if 0.5% mismatched the result can be worse than single stage. For mains hum, cascading DIFFERENT frequencies (60 + 120 + 180 Hz) to catch harmonics is usually better than stacking identical notches.

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Notch Filter Calculator

Design notch (band-reject) filters to surgically remove a single frequency — most often 50 Hz / 60 Hz mains hum. Three topologies: passive Twin-T, active Twin-T with Q boost, and parallel RLC. Includes mains-hum presets, Bode magnitude + phase, and step response.

Input

Mains-hum preset

Notch frequency f₀

Resistor R

C, R/2, and 2C are computed from f₀ and R. Match components ≤ 1 % for a deep notch.

Result

Notch depth

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Transfer function & key formulas

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Schematic

Bode magnitude — |H(jω)| in dB vs frequency (log)

Bode phase — ∠H(jω) in degrees vs frequency (log)

Step response — output vs time after a unit-step input

About Notch Filters & Twin-T Topology

A notch filter (also called band-reject or band-stop) is the complement of a band-pass — it passes everything except a narrow band around its centre frequency f₀. The classic use case is removing 50 Hz or 60 Hz mains hum from audio without affecting the rest of the spectrum. Other applications: cancelling a known interfering tone, surgically attenuating a problem frequency in a recording, or removing harmonic distortion artefacts.

The Twin-T topology

The Twin-T is the most famous passive notch network — two interleaved T-sections, one low-pass (2 R's + 1 C in the bridge) and one high-pass (2 C's + 1 R in the bridge), with matched components R/2 and 2C in the bridge positions. At f₀ = 1 / (2π·R·C) the two T sections cancel exactly, producing a deep null. Component matching is critical: 1 % mismatch limits notch depth to ~40 dB; 0.1 % matching can give 60 dB or better. Passive Twin-T has fixed Q ≈ 0.25 — wide notch (BW ≈ 4·f₀), which works for mains hum removal but limits utility for sharper applications.

Active Twin-T (Q-boosted)

Adding an op-amp to bootstrap the bottom of the Twin-T network sharpens the notch dramatically. The bootstrap gain K = 2 − 1/(2Q), where K is the non-inverting amplifier's gain (K = 1 + R_b/R_a). For Q = 5, K = 1.9; for Q = 10, K = 1.95. The op-amp adds positive feedback that narrows the notch from the wide 0.25 baseline up to 50+ with careful component selection. Sensitivity to component variation also rises with Q — Q > 20 needs precision metal-film resistors and matched film capacitors.

Parallel RLC notch

A parallel LC tank inserted in the signal path between source and load: at f₀ the parallel tank has infinite impedance and blocks the signal completely. Q is set by the source/load resistance R and the LC characteristic impedance: Q = R · √(C/L). Useful when you have an inductor available (RF work especially) and want a deeper notch than passive Twin-T offers.

Why notches don't kill an entire band

A 2nd-order notch is mathematically the complement of a 2nd-order band-pass: |H_notch|² + |H_BPF|² = 1. Both share the same Q and f₀. A high-Q notch (Q = 10, BW = f₀/10) cuts only ±5 % around f₀ — surgical. A low-Q notch (Q = 0.5, BW = 2f₀) cuts a much wider band — useful when you don't know the exact interferer frequency, or want to suppress an entire octave.

Mains-hum removal in practice

At 60 Hz mains, a Q = 5 notch removes the fundamental but also attenuates 50 Hz – 75 Hz audio. For voice this matters little (fundamental ~85–250 Hz). For music it can affect bass. The trade-off: higher Q (narrower notch) requires more precise f₀ matching — slight drift of mains frequency (±0.1 Hz) might miss the notch's narrow well. Q = 5 with auto-tuning, or cascaded notches at 60 + 120 + 180 Hz (fundamental + harmonics), is the common professional approach.

Frequently Asked Questions

How deep should my notch be?

For mains hum removal, 30–40 dB is typically enough — that brings 60 Hz from "audible buzz" to "below noise floor". Pursuing 60+ dB needs precision components: 0.1% tolerance resistors, matched 1% film capacitors, and careful PCB layout to avoid pickup of the very signal you're filtering. Theoretical Twin-T depth is infinite at f₀ — practical depth is limited by component matching. Each 1% mismatch loses about 20 dB of depth.

Should I use 50 Hz or 60 Hz?

Depends on your country's mains. 60 Hz: USA, Canada, most of South America, parts of Asia (Korea, Philippines, Taiwan), Saudi Arabia. 50 Hz: Europe, Russia, Africa, most of Asia (China, India, Japan-east-of-Itoigawa-Shizuoka, etc.). Japan is uniquely split: 50 Hz east of the Itoigawa-Shizuoka line (Tokyo), 60 Hz west (Osaka). If you're recording an international guest remotely, ask their region. Battery-powered gear avoids the issue entirely.

Why does the notch have a step response that doesn't reach 1?

It does eventually reach 1 — but with ringing for Q > 0.5. The mathematical relationship is: notch step response = 1 − BPF step response. Since BPF starts at 0, rings, and decays to 0, the notch starts at 1, dips and rings around 1, then settles back to 1. The dip depth and ring duration scale with Q. For Q = 5 expect ~17% dip and ~5 cycles of ringing.

Can I cascade two notches to deepen the rejection?

Yes — two identical Q=5 notches at 60 Hz in series give a deeper-but-also-narrower combined notch (depths add in dB at f₀, but the bandwidth of the combined "hole" shrinks). Cascading is sensitive to f₀ matching between the two stages; if they're 0.5% mismatched the result can be worse than a single stage. For mains hum, cascading different frequencies (60 + 120 + 180 Hz) to catch harmonics is usually a better engineering choice than stacking identical notches.

What's the phase doing around the notch?

Phase swings rapidly through ±90° as the magnitude passes through zero at f₀ — a hallmark of any notch. The phase actually flips by 180° as we cross f₀ (going from −90° below to +90° above for the unity-gain notch). For frequencies far from f₀, the phase returns to 0°. This rapid phase swing is a real signal-processing side effect — cascaded notches add up phase swings and can audibly "smear" transients near the notch frequencies.

Can I use a digital biquad notch instead?

Yes, and it's almost always the better choice in software. A biquad implements the same s-domain transfer function via the bilinear transform. Advantages: no component tolerances, no parasitic resistance in caps/inductors, easily adjustable f₀ and Q, can be made adaptive to mains-frequency drift. Use this calculator when you specifically need hardware: front-end of an ADC, headphone amp, vintage gear restoration. Most modern DAWs include parametric EQ with very narrow Q settings that act as notches.

How is this different from the Band-Pass Filter Calculator?

Mathematically: complementary. |H_notch|² + |H_BPF|² = 1 (at any frequency, for matched Q and f₀). The BPF passes only a narrow band; the notch passes everything except that band. They use the same s² + (ω₀/Q)·s + ω₀² denominator; only the numerator differs. The two calculators share most internal math — this one is best for "kill this frequency"; the band-pass for "keep only this frequency".

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